Discourse on a New Method Reinvigorating the Marriage of History and Philosophy of Science EDITED BY
Mary Domsl(i and Michael Dicl(son WITH A CONCLUDING ESSAY BY
Michael Friedman
OPEN COURT Chicago and La Salle, Illinois
[lJ Introduction Discourse on a New Method) or a Manifesto for a Synthetic Approach to History and Philosophy of Science MARY DOMSIZI and MICHAEL DICIZSON
Questions about the marital status of history of science and philosophy of science are older than the flurry of new departments, centers, and programs in the history and philosophy of science that emerged in the midtwentieth century. 1 But quite naturally their creation, and the ongoing joint meetings of the History of Science Society (HSS) and Philosophy of Science Association (PSA), have propelled these questions to the foreground on several occasions. One such event took place at the University of Minnesota in 1969, at a conference dedicated to the rationale of the union of history of science and philosophy of science. 2 The conference papers were published under the title Historical and Philosophical Perspectives of Science (1970), as Volume 5 of the Minnesota Studies in the Philosophy of Science (edited by Roger H. Stuewer). A few years later, Ron Giere wrote a provocative review of the book, entitled "History and Philosophy of Science: Intimate Relationship or Marriage of Convenience?" (Giere 1973).3 Giere considers various arguments for the necessary union of these disciplines-arguments put forward by an esteen1ed group that includes Herbert Feigl, Paul Feyerabend, Mary Hesse, Ernan McMullin, and Arnold Thackray-and, on the whole, he finds their appeal to history as a remedy for the failings of the logical empiricist movement in the philosophy of science unconvincing. Furthermore, Giere does not simply find fault with the arguments presented in the Minnesota Studies volume; he suggests that, although the pairing of history and philosophy of science had already become fashionable, contemporary philosophers of science needn't worry themselves with appeal to the historian's 'internalist' studies of scientific development or scientific practice. Attention to what he terms "real science"-the science that can be learned in textbooks-will serve their philosophical purposes. As such, he argues that the union of these disciplines "lacks a strong 1
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conceptual rationale," and thus the n10st (or best) we can say is that the history of science and the philosophy of science share a common interest in science (Giere 1973, 296). Unsurprisingly, Giere's forceful remarks have been met with equally forceful replies from historically-minded philosophers. Ernan McMullin offered his direct response to Giere in a 1974 PSA paper entitled "History and Philosophy of Science: A Marriage of Convenience?,,4 McMullin argued that, despite Giere's fervent suggestions to the contrary, there are in fact particular issues in the philosophy of science that cannot be adequately addressed without remaining sensitive to the history of science, and especially to the historical development of scientific practices and scientific theories. The principal issues he cites concern the assessment of theories, the nature of scientific growth, and the ontology of theoretical entities. He aims to show that if "an adequate treatment even of the n10st 'logical' of these three questions, viz., that of theory-assessment, cannot avoid reference to the history of science," then we may safely conclude that the "n1arriage of the history and philosophy of science is not just one of convenience" (McMullin 1974, 586). Even without a thorough assessment of McMullin's argument, our current state of practice in the philosophy of science indicates that McMullin and company have won the debate. For there is no question that philosophers of science have in the past few decades made it commonplace to bring the history of science and the history of the philosophy of science to bear on their philosophical discussions, and they have done so quite effectively. For instance, historical examples have figured prominently in some of the more compelling arguments concerning the nature of scientific growth. Larry Laudan's (1981) anti-realist appeal to the 'pessin1istic n1etainduction' draws its force from revolutionary periods in the history of science' and John Worrall's (1989) 'structural realist' reply earns credibility from what he understands as Poincare's understanding of how scientific theories represent the world. 5 So much the worse for Giere's arguments it seems, and so much the better for historically-minded philosophers (and philosophically-minded historians) who can appeal to the continued and successful application of the history of science in the philosophy science as justification that these disciplines are intimately related. However, the debate cannot be settled on these grounds alone. For when Giere claimed that the history of science has no significant bearing on the philosophy of science, he was not claiming that philosophers should ignore the science and scientific practices of the past. The argument was hardly this simple-minded. Giere was instead, and more pointedly, arguing that the sort of history that historians ofscience tal(e as paradigmatic to their discipline-a fine-grained and often complex history, focused, more or less, on the specific events, context, and influences surrounding the develop-
Introduction
3
ment and acceptance or rejection of a concept or theory-is not the sort of history from which philosophers can prima facie draw usefullessons. 6 Much philosophical use of history (e.g., as exercised by Laudan) does not involve history of that sort. Surely Giere would have had exactly this point made to him by Edward Grant, Richard ('Sam') Westfall, and his other historian colleagues at Indiana University at the time. For Giere, then, the fact that our contemporary philosophical discourse is peppered with appeals to and examples from the past hardly counts as genuine evidence that the history of science and the philosophy of science are involved in an intimate relationship. Any such evidence for this relationship will come from a more reflective consideration of the conceptual ties between these disciplines. In particular, Giere urges that historicallyminded philosophers "not be content merely to practice their art but . . . make repeated efforts to explain and argue the rationale for their approach" (Giere 1973, 291), and he solicits from historian-philosophers an account of how "philosophical conclusions may be supported by historical facts." "Until this is done," he writes, "the historical approach to philosophy of science is without a conceptually coherent program" (1973, 292). Our aim in this 'manifesto' is in part to provide the sort of historiographical and methodological account that we think Giere rightly urges. The account we offer talces its inspiration from the historiography that has been masterfully applied by Michael Friedman over the past few decadesa historiography which we have dubbed a "new method" in the title of this book. Though Friedman is not the only historian-philosopher to apply the methodology we describe and encourage, we believe that attention to his course of work grants us important insights into how best to understand the complexity and intimacy of the relationship between the history of science and the history of philosophy. Our intention in this volume is to explicate and pursue these insights (though certainly not uncritically), and thereby provide a means by which to reinvigorate the very marriage of the history and philosophy of science that Giere and others have continued to bring into dispute.
"Remarl(s on the History of Science and the History of Philosophy"7 Some twenty years after Giere's review was published, Friedman (1993) made what perhaps seemed at the time a modest plea to contemporary philosophers of science. Taking for granted that Thomas IZuhn's Structure of Scientific Revolutions (1962) had convincingly shown that philosophers of science ought to take seriously the historical development of science, Friedman suggests that historians of philosophy ought to take seriously the
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historical role that science has played in philosophy. They should, that is, consider the historical interaction between the history of science and the history of philosophy. Relying on IZuhn's historiography of science, and adopting a more or less Kuhnian characterization of scientific revolutions, Friedn1an argues, in particular, that an important task for the historian of philosophy is to situate the emergence and evolution of philosophical ideas in their revolutionary scientific context. Friedman does not have in mind what he rightly supposed to be, by then, a commonplace in the history of philosophy, namely, the recognition that broad themes from the history of science-and especially the largescale transformations in philosophical concerns that often accompany scientific revolutions-have some part to play in our understanding of the history of philosophy. In this vein, Friedman mentions the philosophical concerns of figures such as Descartes, whose work can be understood in the context of the shift from Aristotelian-Scholastic natural philosophy to the mechanical philosophy. Other examples spring to mind: the role of the systematization of geometry in the development of various ancient Greek accounts of knowledge; the role of the developn1ent of computer science in the development of various theories of mind; and the role of the shift from classical to quantum theory (together with the perceived 'acausality' and attendant 'irrationality' of the latter) in the development of various twentieth century 'irrationalist' philosophies. Recognition of such broad parallels between the history of science and the history of philosophy (and corresponding discussion of the direction of causality, if any) is widespread. Without discounting the importance of such broad-brushed considerations, Friedman has in mind a finer-grained discussion of the connections between science and philosophy, one that takes account of the specific content of, and especially the central problems driving, the science of the day. More specifically, he emphasizes that many philosophical enterprises are motivated by the attempt to understand (and in some cases provide) the detailed foundations of scientific theories. In his 1993 paper, Friedman explores the possible fruits of adopting this historiographical perspective in the context of two examples. The first concerns the change from an Aristotelian-Scholastic physical dynamics to a mechanical, ultimately Newtonian, dynan1ics-a change motivated by the work of figures such as Galileo, Descartes, and, of course, Newton himself. That shift, argues Friedman, raised new foundational problems in science that in turn gave rise to philosophical questions concerning the status and possibility of our knowledge of the natural world. Crucial among the problems emerging from Newton's system of the world was our ability to distinguish states of true motion from states of true rest and, perhaps even more important, states of inertial motion from states of non-inertial (accelerated) motion. Specifically, it is unclear how the new dynamics of the
Introduction
5
Principia Mathematica has any empirical content, in part because the theory is Galilean-invariant, so that there is nothing given (such as an Earth at absolute rest, as there was in the older system) relative to which, for example, inertiallTIotion is defined. 8 But then how does Newton's law of inertia (a law about inertial bodies) have any empirical content? (The second law does not help here, unless one already knows all of the forces affecting the systelTI, a point that would much later push Poincare towards his conventionalism about geometry.) Friedman suggests that the dialogue that ensued between science and philosophy in the wake of the Principia's success and acceptance centered on "this fundamental problem lying at the basis of the new physical dynamics" (Friedman 1993,41). And this suggestion motivates reconsideration of how to best understand the course of modern thought. For if we take seriously the interaction between science and philosophy and, in turn, approach the evolution of the modern philosophical tradition as successive attempts to come to terms with the fundamental problem of relativity of motion that lay at the basis of the new physical dynamics, then, on Friedman's account, we are brought to a conception of the evolution of modern philosophy that is more illuminating than the conventional picture of a succession of largely futile attempts to solve the problems of skepticism about the external world. (43) This re-evaluation of n10dern thought, centered on the revolutionary character of the history of science, sheds new light on the progress of epistemology and metaphysics. Specifically, Friedman claims that contrary to the opinion of some contemporary historical writers, the philosophers of the modern tradition are not best understood as attempting to stand outside the new science so as to show, from some mysterious point outside of science itself, that our scientific knowledge somehow 'mirrors' an independently existing reality. (48) Rather, as Friedman argues, they are best understood as trying to come to terms with the foundations of the new science. IZant, for Friedman, is paradigmatic here, because he proposes that metaphysics should articulate the a priori conditions of the possibility of experience. Starting from the so-called synthetic a priori forms of knowledge expressed in Euclidean geoll1etry and Newtonian dynamics, lZant sought not to justify this knowledge but to investigate those very conditions of human cognition that make such synthetic a priori knowledge possible. By grounding the project and goal of metaphysics in mathematics
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and natural science, IZant at the same time established "a reorganization and reinterpretation of metaphysics itself" (49), and thus instigated his own 'revolution', not in science, but in philosophy. On Friedman's account, then, Kant's great service to philosophy was to bring metaphysics down from its lofty place 'above' human knowledge precisely by taking pure mathematics and pure natural science as the starting point of his metaphysical investigations. In his second example, Friedman points out that a similar concern with our most successful claims to scientific and mathematical knowledge emerges in the early twentieth century as philosophers faced another revolution in the natural sciences: the special and general theories of relativity grounded on non-Euclidean geometry. Remaining sensitive to their historical situation-and, in particular, to the ongoing scientific revolutions of the early twentieth century-Friedman contends that we can understand logical positivists as attempting philosophically to come to terms with the profound conceptual revolutions that initiated twentieth-century science. These thinkers should be seen not as attempting to justify twentieth-century science from son1e sterile and futile external vantage point but rather as once again refashioning the basic concepts and principles of philosophy so as to accommodate and comprehend the new scientific developments. That is, their aim is not to justify twentieth-century science from some supposed "higher" standpoint but rather to provide a rational reconstruction of that science and to find thereby a new, non-metaphysical task for philosophy. (49) With the supplanting of Newtonian dynamics and its framework of Euclidean geometry-and thus with the possibility of scientific knowledge as understood by IZant no longer on center stage-the logical positivists sought to trace the development of scientific ideas and scientific theories by embedding the development of scientific theories in a logical language in order to exploit the rational interconnections among our scientific claims. However, although the protagonists had, out of historical necessity, abandoned certain aspects of IZantianism in its original form, the spirit of IZant's project for metaphysics was still alive during the early twentieth century. In particular, Friedman argues that the logical positivists continued to "advocate a modified lZantian position according to which there is a fundamental distinction between the spatiotemporal framework of physical dynamics and the empirical laws formulated within this framework" (50). This fundamental distinction is not simply a IZantian distinction between the a priori and the a posteriori; faced with revolutionary changes in math-
Introduction
7
ematics and science, it was transformed into a distinction between the relativized a priori and the a posteriori, between historically dynamic matheInatical and physical laws on the one hand, and the understanding of the world that depends constitutively on then1 on the other hand. Recognizing this IZantian thread running through early twentieth century philosophy of science, Friedman concludes that although the logical positivists' preoccupation with the a priori did indeed thereby preclude them from using the history of science as a philosophical tool, this did not prevent them from recognizing the profound philosophical significance of conceptual revolutions in science. On the contrary, their effort to articulate a coherent conception of the relativized a priori must, I think, count as the most rigorous attempt we have yet seen philosophically to come to terms with precisely such conceptual revolutions. Of course, as we have also seen, this heroic attempt of the logical positivists was in the end a failure. Yet I do not myself think that we will ever progress beyond this point until we possess a fuller appreciation of the historical evolution of our own philosophical predicament. And this means, as I have tried to emphasize throughout, that we must attend more closely to the history of science' the history of philosophy, and to the essential interaction between them. (54) In these concluding remarks, Friedman proposes a project not only for historians of philosophy, but also for philosophers in general. Note, in particular, the word 'essential' in the final sentence. Friedman's implicit claim is that the historical interaction between philosophy and science is not a quirk of history, but an essential fact about the two disciplines. It is of the nature of science and philosophy to interact. Without placing too much weight on this point, or seeking to follow through on this idea in detail (which in any case would require a long discussion of the nature of science and philosophy), we must nonetheless emphasize the distinction between the claim that philosophy has, as a matter of historical fact, interacted with science in a variety of ways (as exemplified in Friedman's two examples), and the claim that philosophy by nature does, and ought, to proceed in that manner. Friedman quite clearly endorses the former claim, and appears to endorse the latter as well. Indeed, the two claims are not entirely independent. Intellectual disciplines do not exist in an historical vacuum. The standards, problems, goals, and methods of a discipline provide the context that malces its practices intelligible, and they are in part inherited from its own past. In the case of philosophy, insofar as this context arises from substantial historical interaction with science, understanding that historical interaction is required for
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understanding our present practices. Hence, while the two claims that we have thus far attributed to Friedman are indeed distinct, the first does give reason to believe the second. Nonetheless, one might say that Friedman has taken the role of science in the history of philosophy to an unhealthy extreme. The argument runs thus: Remaining so firmly wedded to a scientific perspective on, say, the modern period provides us at best only half the picture. For example, skepticism was in fact in bloom and helped shape epistemology from Descartes to IZant and beyond. Though initially persuasive, notice that this sort of argument is grounded on a worrisome either-or: either we examine how the history of science has influenced the history of philosophy, or we examine how skepticism influenced the history of philosophy. We certainly do not deny that Friedn1an has adopted the history and philosophy of science as the focal point of his investigations. But, as his recent work has illustrated, adopting this strategy does not force us to thereby ignore other factors that contributed to the development of novel philosophical insights. For Friedman's project is not simply one of illuminating how the history of philosophy has helped shape the history of the philosophy of science, and vice versa. Rather, by turning our focus toward the history and philosophy of science, he has shed new light on the standard intellectual concerns surrounding skepticism, metaphysics, and theology that shaped (and continue to shape) the historical progress of philosophy.9 Thus, what some might take as the overly radical nature of Friedman's work we talce, more sympathetically, as a strategy for drawing attention to an interaction between philosophy and science that has, until very recently, attracted inadequate scholarly attention, as well as a fruitful strategy for giving old questions new flavor.
A Tradition of Historical Philosophy and Philosophical History There is more to Friedman's story than indicated from the above sketch of his 1993 article. From a plea to talce seriously the interaction between the history of science and the history of philosophy, Friedman arrives not only (apparently) at the conclusion that philosophy and science ought to interact in a certain way, but also (clearly) at a IZantian, or better, neo-IZantian, conception of the nature of that interaction, which rests on the relativized a priori. IO If one finds the science-influenced story that Friedman tells hard to digest, then his IZant-influenced story about the history of science and the history of philosophy may be poison for the palate. But we need to take care, as always, not to throw the baby out with the (IZantian) bath-
Introduction
9
water, and ask whether embracing the sort of historiography of philosophy that Friedman promotes in fact demands that we at the same time embrace a IZantian or neo- IZantian understanding of the growth of scientific knowledge and the methodology of philosophy. From our perspective, the two are not inextricably bound. In fact, there is a long tradition of pursuing the sort of historically informed philosophy and philosophically informed history that has become a hallmark of Friedman's work. Seeing Friedman's own work in this context will help us to understand, first, how specifically he differs from his predecessors, and also how one might pursue a program broadly sympathetic to Friedman's approach without necessarily ending up a neo-IZantian.l 1 On the side of history, Friedman stands in a tradition of philosophical historians that includes Richard Westfall, who approached the historical development of ideas with an eye to the philosophical issues that loomed over seventeenth-century science. Westfall's Force in NewtonJs Physics (1971) provides an example. In his masterful explication of Newton's formulation of the second law of motion, Westfall puts less emphasis on the attempt to formalize the law (F= rna) and more emphasis on the philosophical context that allowed Newton to ask the right questions in the first place. It was Newton's proposed resolution to "the metaphysical question of the ontological status of force" that, according to Westfall, brought him to the notion of force that lay at the heart of his mature physical dynamics. In a similar fashion, Friedman has brought us to a deeper appreciation for the philosophical questions that enabled Einstein to develop relativity theory. On Friedman's account, Einstein's attention to the philosophical questions surrounding the newly discovered non-Euclidean geometries, as illustrated by his close reading of Helmholtz and Poincare, partly allowed him to develop a theory of relativity that stands on the foundation of a non-Euclidean geometry (cf. Friedman 1999, 2001). On the side of philosophy, Friedman attests to his own position in a tradition of historical philosophers, most notably, Thomas IZuhn. Quite clearly, their respective philosophies of science take revolutionary changes in the history of science as their touchstone. Where IZuhn (and followers) see, in a scientific revolution, an opportunity to draw in1portant generic philosophical lessons about general philosophical issues, such as the nature of scientific knowledge, Friedman sees in them an opportunity to understand the specific problems that drove the development of philosophy at the time of the revolution and beyond. Friedman also draws significant motivation from another historicallyminded philosopher, Ernst Cassirer. Cassirer famously attempted to preserve a lZantian understanding of science and knowledge in the wake of the discovery of non-Euclidean geometry and Einstein's theory of relativity. Relying heavily on IZant's notion of "the regulative use of reason," Cassirer
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presented a (Marburg neo- I
Introduction
11
justice, once again, to both sides of lZant's original philosophical synthesis. And he alone is thus in a position to mediate the increasing intellectual tension between the now notorious "two cultures"between the naturwissenschalftliche and geisteS1vissenschaftliche intellectual orientations. (Friedman 2000,151-52) IZeeping with the spirit of Cassirer's project, Friedman has comn1itted himself to treading a path between history and philosophy, a path that, as demonstrated through his own work, can bring us to a deeper appreciation of the dialogue between history and philosophy that continues to shape our understanding of science. Or perhaps being more faithful to his general project, we should say that his work lies less on a road between two disciplines as it does at the intersection of the history and philosophy of science and mathematics. The synthesis of the knowledge, concepts, terminology, and methods emerging from these disciplines has fueled an influential corpus that has blossomed over recent decades. Now, of course, we face non-trivial questions: What precisely does this synthesis entail? Why do we call it a 'synthesis' in the first place? Given his lZantian motivations, one might take Friedman's synthesis of knowledge sin1plyas a 'going beyond' or amplification, so that his goal is to bring philosophy to bear on history and history to bear on philosophy in order that we might 'see beyond' what either can offer alone. Surely there is this somewhat IZantian tinge to Friedman's general project. But taking an even broader view, the synthesis Friedman promotes is premised not merely on an amplification but on a unity-a unity that holds between the intersecting disciplines and a unity that we, as scholars, share with the history of philosophy and the history of science.
Synthesis and 'Synthetic History' As we just said, the "new method" for examining the history of philosophy, to which we refer in the title of this volume and which is illustrated, in various ways, by the essays contained in this book, may appropriately be called 'synthetic'. Now, among historians generally, the phrase 'synthetic history' has come to mean something like a history that offers 'the big picture', incorporating lessons from various more detailed histories, themselves narrowly circumscribed in subject n1atter, time period, geographical area, and the like. We have something quite a bit more specific in mind, and in fact the sort of history of philosophy that we describe here (and that we take to be exemplified by essays in this volume) is not synthetic in this sense. To get a handle on the sort of synthesis that we have in mind, it is useful to recall the philosophical origins of the term 'synthesis', and the
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related term 'analysis' as found in Aristotle's two treatises on 'analytics'. These initiated a long tradition of philosophical and scientific usage of these terms .13 A cursory review of this tradition might easily lead one to conclude that 'analysis' has come to mean nothing more than 'taking apart' or, more philosophically, 'reducing to the fundamental parts', as illustrated, for example, by Francis Bacon (1620) when he writes, in the Preface to The Great Instauration: Now what the sciences stand in need of is a form of induction which shall analyze experience and take it to pieces, and by a due process of exclusion and rejection lead to an inevitable conclusion. (Bacon [1620] 1999, 78) The emphasis in such uses of the term 'analysis' is on the idea that the object of analysis is naturally conceived in terms of fundamental parts, so that the conception in those terms, achieved by analysis, amounts to a foundational understanding of it. Our contemporary (philosophical) use of the term may often appear to be along similar lines. For example, an 'analytically true' proposition is true in virtue of the meanings of its constituent terms, so that, for example, understanding the proposition as composed of its constituent terms is the same as understanding the proposition. However, neither Aristotle nor indeed contemporary philosophy should be understood in quite this way. Note, for example, that understanding a proposition as containing (conceptually) certain terms does not constitute understanding the (n1eaning of the, or truth of the) proposition, for at the least, one must also understand how those terms compose the proposition. ('All philosophers are n1usicians' differs from 'all musicians are philosophers.') Aristotle is useful on tl1is point, because in his usage, the term 'analysis' did not carry, necessarily, the connotation of 'breaking into fundamental parts', but involved, rather, the idea of 'loosening' (the everyday meaning of the Greek term). Aristotle occasionally uses the term in this everyday sense, as when, for example, he tells us that 'in most birds, the gut is thin, and simple when loosened out [analyzed]' (History of Animals 11.17 509aI7). When it comes to propositions-for example, the propositions that science seeks to understand-Aristotle uses the term to refer to the process offinding an argument whose conclusion is the proposition in question. We shall see in a moment how this process is a kind of 'loosening'. As Byrne (1997) points out, Aristotle was likely influenced by the emerging 'method of analysis' (perhaps in the Platonic Academy) in this more philosophical use of the term. In its mature forn1, famously (but cryptically) described by Pappus several centuries later, the method of analysis seems to have involved taking the thing to be shown (or con-
Introduction
13
structed) for granted, and working 'backwards' from it to something known (or known to be constructible), "and the proof," as Pappus says, '''vill be the reverse of the analysis" (Mathematical Collections, Book VII). Suppose, for example, that we are faced with the problem of constructing a square equal in area to a given rectangle (Euclid 11.14). Using the n1ethod of analysis, we assume that we have constructed the square, and we seek a diagram that somehow unites the square and the rectangle in such a way that, from the rectangle, one can construct the diagram, and thence the square. Aristotle adopts a similar model in his conception of science. Science seeks to understand propositions relating types, for example, 'All As are Bs.' Scientific understanding, for Aristotle, amounts to finding an argument whose conclusion is such a proposition. But not just any argull1ent vvill do. Among the several conditions Aristotle places on the prell1ises, perhaps the most important is that they express the 'cause' of the truth of the conclusion, which means, in the context of our simple example, that they contain a middle term that unites the two terms in the proposition, so that the argument we seek has the form 'All As are Cs; All Cs are Bs; Therefore, All As are Bs.' In other words, all As are Bs because they are all Cs. (For exan1ple: All planetary orbits are ellipses because they are the closed paths of bodies governed by gravitational forces.) In Aristotle's conception, analysis has 'loosened up' the proposition, 'stretching apart' the extren1e terms A and B, showing how 'between' them there lies a middle term, C, which is responsible for the truth of the proposition 'All As are Bs.' Crucially, the analysis lies not in the observation that the proposition in question is composed of two terms, nor even in any observation about how these terms compose the proposition, but in the observation that the truth expressed by the proposition arises from a particular relationship between those terms. That relationship may be expressed as a kind of'synthesis' of the extreme terms effected by the middle term, C; and analysis is thus the discovery of this 'synthetic' relationship between A and B. We take this conception of analysis and synthesis as the model for the sort of history of philosophy and history of science that we have in mind. Let us see how it plays out in a brief example, discussed by Friedman at the beginning of his contribution to this volume. It is well known that Newton advocated the view that space is some sort of 'real thing'. He had clear scientific reasons for doing so, not the least of which was the apparent failure of Descartes's view of space (which, roughly, identified spatial extension with matter) to yield a coherent account of motion, and a coherent account of true motions, in particular. This negative diagnosis of Descartes's identity of space with corporeal extension is spelled out in Newton's De Gravitatione,14 where he also claims that Descartes's system of nature is such that "not even God himself could
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define the past position of any moving body accurately and geometrically" (Newton 2004, 20). In both De Gravitatione and later in the General Scholium of the Principia,15 Newton responds to the Cartesian program by adducing several positive arguments for the existence of 'absolute space', including the famous 'bucket' thought experiment, which purports to show that the real physical effects of acceleration are best explained in terms of an absolute space in terms of which all acceleration is defined. Newton characterizes this absolute space as the 'sensorium of God'. As Friedman points out, the doctrine is intimately connected with God's omnipresence, and God's ability to create Inatter at any point in space as an immediate act of the will, in a way that is closely analogous to the ability of a soul (which is present throughout the body) to move its body immediately, as an act of the will. Newton took a position that was thus close to that of the Cambridge Platonist Henry More, whom Newton knew and whose work he read. 16 In his Enchiridion Metaphysicum (1671) and elsewhere, More proposed a form of Platonism, in reaction against Cartesian dualism, arguing, among other things, that the properties of space mirror those of God. In particular, space is extended (More believed that all substances, material and immaterial alike, are extended things), indivisible, infinite, immobile, and penetrable (while matter is divisible and impenetrable). For More, as for Newton, God's omnipresence and God's ability to create matter at any point in space as an immediate act of the will is the n1eans by which the laws of nature are 'upheld'. According to the point of view adopted here, while it may be initially helpful to parse Newton's view about space into its scientific, philosophical, and theological aspects, ultimately those aspects must be understood as unified, for surely Newton did not talce himself to be doing, at one moment, philosophy, and at another, 'physics', and at yet another, theology, nor to be doing these three 'separate' things somehow 'at once'.l7 Here is not the place to suggest specific 'middle terms' that reveal the unity, for Newton, of these activities. Our point is simply to set up the problem in such a way that a solution could be recognized. In particular, a solution would show how, and why, Newton himselfwas able to bring all of these different sorts of considerations to bear in the context of a single discussion. It would render Newton's struggles concerning the nature of space intelligible by revealing the manner in which Newton's philosophical, scientific, and theological concerns were, in his own way of thinking, united, and ultimately addressed by his account of space. And notice that even identifying those concerns itself requires historical work, tracing them to More, Descartes, Galileo, and of course many others. We return, now, to the general account of synthetic history, in order to clarify a few points. First, analysis and synthesis are best conceived not as opposites but as two views of, or two ways of examining, a single phe-
Introduction
15
nomenon (such as, Newton's account of space): Analysis reveals the cause of the phenomenon, and synthesis demonstrates, or expresses, or explains, the unity of the various components that constitute the phenomenon. We might just as well, then, have called the conception of the history of philosophy that we have in mind here 'analytic'. Indeed, in some ways, this term would have been more appropriate (if also more misleading), because the fundamental task of such history is to find and explore the causes of specific unities between philosophy and science. Second, as we said above, analysis is not merely a matter of breaking something down into its constituent parts. Instead, analysis seeks the cause of the unity between the parts, the cause of their constituting one thing. In terms of the synthetic history that we are attempting to describe here, we presume the initial distinction between, for example, scientific and philosophical concerns or issues as (in this case, non-exhaustive) constituent parts of something broader, and seek to describe them in a way that makes clear not only their unity, but also the cause of their unity. Third, synthesis and analysis, while fundamentally about the unified phenomenon, may be (and often are) undertaken with a focus on just one of the parts. For example, as a scientific (as opposed to purely mathematical) activity, the proof that closed paths of bodies under the influence of a gravitational potential are elliptical ultimately serves the purpose of understanding why the orbits of the planets are elliptical (and thus understanding the cause of the 'unity' between 'orbits of the planets' and 'ellipses'); however, the proof may very well be undertaken without an explicit concern for the planetary orbits. That is, the notion of a planetary orbit per se does not enter the analysis (though it may well, and historically did, motivate the analysis). Similarly, one may engage in the sort of history of science that we have in n1ind "\vithout explicitly 'doing philosophy'. Hence, for example, we noted above that in his account of the development of the second law of motion, Westfall focuses on the philosophical context that allowed Newton to develop the requisite concepts. We would say that Westfall is doing history of science, not history of philosophy, but he is doing it synthetically insofar as his account helps to reveal the essential unity between philosophical and scientific concerns. Similar remarks hold, ll1utatis mutandis, for the history of philosophy, and indeed for philosophy itself.
Unity and the Growth of IZnowledge I.et us return to Friedman's own work, work that we take to be broadly characterized by the synthetic conception of history of philosophy that we have sketched above. We have suggested that Friedman's approach to history and philosophy need not commit us to the IZantian conclusions he
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draws about the interaction between the history and philosophy of science and mathematics, conclusions centered on his notion of the relativized a priori. I8 Synthetic history, as we understand it, may be approached from any of a wide variety of frameworks, all of which may be capable of providing insight into the unity of specific intellectual projects that we find emphasized in Friedman's general project. What these frameworks will have in common is their capacity to shed light on that unity. We find, therefore, that Friedn1an's work brings to light a notion of unity that transcends any Kantian commitments and places us in a dialogue with our intellectual past as well as with the very disciplines we adopt as the objects of our investigations. In The Dynamics of Reason (2001), where the notion of the dynanlic and relativized a priori is presented in its most mature form, Friedman gives a reflective account of the inspiration that lies behind his neo-Kantian reading of science. Drawing on the successes and the shortcomings ofideas forwarded by Iunt, Carnap, IZuhn, and Quine, Friedman defines more clearly his notion of the relativized a priori as that indelible feature of scientific knowledge that has withstood revolutionary change in the physical sciences and, in so doing, he self-consciously effects a synthesis of many philosophical and scientific ideas of the past. Moreover, we can see in Friedman's work an illustration of one of the most important rewards of this synthetic approach to the history of philosophy. As suggested in the prefaces and introductions of his most recent books, Friedman's ultimate goal is to put history to work for us by drawing on the resources of the past, specifically by acknowledging both the positive and negative consequences of philosophies gone by. Only in so doing can we can draw important lessons about our current situation in the philosophy of science and sustain a hope for progress. (Recall our discussion, above, of how history renders our current intellectual practices intelligible.) In some sense, we might be tempted to say that Friedman is pursuing a natural selection of sorts by embracing those concepts, ideas, and methods that can serve us best in our attempt to delimit the present and future tasks of philosophy in regard to knowledge, and scientific knowledge in particular. Analogies tend to break down very quickly, and this case is no exception. For though there is a process of selection at the heart of Friedman's project, a Darwinian notion of "survival of the fittest" (which is central to 'Whiggish' forms of history) will not suit our purposes. What Friedman seeks is not simply an appropriation of concepts, methods, and ideas from the past but an extension, exploration, modification and above all, a synthesis of those concepts, methods and ideas that present the greatest potential for an account of developments in both the history of science and the history of philosophy.
Introduction
17
In this respect, Friedn1an's notion of historically informed philosophy reaches beyond the somewhat naive claim that our best philosophy must account for historical episodes. Too often we see this path taken in contemporary scholarship. A philosophical doctrine is put forward and legitimized on the basis of the history of science; history serves to justify philosophy simply in virtue of the scattered intersecting points they share in the course of the history of ideas. However, as Friedman's work illustrates, synthetically pursued history does more than justify or legitilnate; at its best, it informs contemporary philosophy, offering us a mirror that allows us see where we now stand and what methods and ideas we should pursue in order to in1prove our current philosophical condition. Recognizing the dialectic between philosophy, science, history, and mathelnatics, our task is to locate those issues that bind these disciplines together so that we can make the Inost informed decisions about our own n1ethods and concepts as we attempt to move forward. And in Friedman's hands especially, vve've seen that this idea has inspired progress, for by granting history, science, and mathematics a voice in the pursuit of philosophical questions, he has produced a corpus of considerable range and depth. Besides inspiring a particular methodology characterized by synthesis, Friedman has thus inspired a particular perspective on our current state of practice. He has, with elegance and sincerity, taken a seemingly humble stand, bringing us down from our own lofty place 'above' the past. History is not, for him, a tool, or a string of repeated failures that tells us what not to do. Rather, history is our inheritance, an ancestor of sorts, and we can effectively steer its continuing course only by en1bracing our place in the dialectic between history and philosophy. As seen in Friedman's own work, beyond sensitivity to a dialogue between history and philosophy, he is fully aware of the dialogue we, as contemporary scholars, n1ust continue to share with the science and philosophy that came before us. And thus, the dialectic central to Friedman's work extends beyond an examination of how the past has contributed to our present; placing us in this dialogue, we now have from him inherited a responsibility to examine how the past can best inform our philosophical future.
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Mary Domski and Michael Dickson NOTES
1. Among others, the departments, centers, or programs at Boston University, Cambridge University, Indiana University, the University of Minnesota, the University of Notre Dame, the University of Sydney, Pittsburgh University, and the University of Toronto were created in the mid-nventieth century. 2. The conference was sponsored by the United States National Committee for the International Union of History and Philosophy of Science, itself founded in 1956. 3. Giere's attention to the relationship between history and philosophy of science in 1973 very likely went beyond purely academic interest. As a then member of the Department of History and Philosophy of Science at Indiana University, the question of this relationship was presumably a real issue for him, with day-to-day consequences. 4. There have, of course, been numerous other occasions at which the questions raised by Giere's review article have been addressed. For instance, in 1992 Michael Ruse organized a PSA symposium entitled "Do the History of Science and the Philosophy of Science have anything to Say to Each Other?" which included papers by Robert Richards, Rachel Lauden, and Marga Vicedo. The complete texts can be found in PSA: Proceedings ofthe Biennial Meeting ofthe Philosophy ofScience Association (1992), Volume Two: Symposia and Invited papers, 467-96. A recent installment of this ongoing discussion was a conference with the now familiar title "Do Historians and Philosophers of Science have Anything to Say to Each Other?" held at Duke University on March 23-24, 2007. 5. Laudan's "A Confutation of Convergent Realism" and Worrall's "Structural Realism: The Best of Both Worlds?" are reprinted in D. Papineau (ed.) (1996). 6. More specifically, he contends that the history of science, as practiced by historians of science, has no significant bearing on our understanding of what he takes to be the main problem areas for philosophy of science: the structure of theoretical knowledge, the validation of knowledge claims, or the strategy and tactics of research (Giere 1973, 292-93). 7. This is the title of Friedman (1993), which we discuss below. 8. This, of course, was not a problem for Newton himself, who proposed that the center of the solar system is at rest relative to absolute space. We discuss the relevance of Newton's proposal of an absolute space for his own program of natural philosophy below. 9. The broader scope of Friedman's approach to the history of philosophy and history of science is brought to light in his own contribution to this volume. 10. The relativized a priori emerges as a centerpiece of Friedman's historiography as early as his first book, Foundations of Space-Time Theories (1983). 11. Certainly, the tradition of historically informed philosophy and philosophically informed history we refer to includes notable contributors other than those mentioned explicitly. We could have very well focused on others. With limitations of space and time being what they are, we've restricted our attention to cases we find especially illustrative of Friedman's place in this tradition, with no intention of belittling the credit owed to others.
Introduction
19
12. See Friedman (2001), 65-68 for Friedman's own explicit appeal to Cassirer's "logical idealism." 13. For an extended discussion of the use of the term 'analysis' in Aristotle, see Byrne (1997). Our discussion in the few paragraphs below follows his chapter 2, though vve disagree with Byrne's view that in the modern period 'analysis' came to Inean nothing more than 'reducing to parts'. 14. The full title of De Gravitatione is De Gravitatione et aequipondio fluidorum. The manuscript first appeared in English translation, with a Latin transcription, in A.R. Hall and M.B. Hall (1962). A new translation by Christian Johnson, with the assistance of Andrew Janiak, appears in Janiak (2004). 15. There is some dispute about the when De Gravitatione was written, whether in the 1660s or in the early 1680s, in1mediately prior to the publication of the first (1687) edition Principia. Whatever the case, there's no dispute that the manuscript was penned prior to the General Scholium, which was added to the second (1713) edition Principia. 16. Hall (1948) describes evidence that Newton vvas reading and studying More possibly as early as 1661. In any case, it is clear that the two knew one another, and that Newton admired More. Newton apparently read other Catnbridge Platonists as well, with sufficient interest to take copious notes in at least the case of Ralph Cudworth; see Sailor (1988). For Newton's broader Cambridge Platonist commitments, see the classic McGuire and Rattansi (1966). 17. On tl1is issue, we (and Friedman) are sensitive to the point, emphasized by several historians of science, that our contemporary disciplinary divisions were foreign to Newton and to early n10dern natural philosophers in general. See, for instance, Cunningham (1991). 18. For an alternative, though complementary, approach to the generally IZantian features of Friedman's work, see Mark Wilson's insightful paper, which is included as the penultimate contribution to this collection.
REFERENCES Bacon, F. ([1620J 1999). Selected Philosophical Works. Edited with an Introduction by Rose-Mary Sargent. Indianapolis: Hackett Publishing Company. Byrne, P.R. (1997). Analysis and Science in Aristotle. Albany: SUNY Press. Cunningham, A. (1991) "How the Principia Got Its Name; Or, Taking Natural Philosophy Seriously." History ofScience 29: 377-92. Friedman, M. (1983). Foundations of Space-Time Theories: Relativistic Physics and Philosophy of Science. Princeton: Princeton University Press. - - - . (1993). "Remarks on the History of Science and the History of Philosophy." In Horwich (ed.), 37-54. - - - . (1999). Reconsidering Logical Positivism. Cambridge: Cambridge U niversity Press. - - - . (2000). A Parting ofthe Ways: Carnap) Cassire1) and Heidegger. Chicago: Open Court.
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- - - . (2001). Dynamics of Reason. The 1999 I(ant Lectures at Stanford University. Stanford: CSLI Publications. Giere, R. (1973). "History and Philosophy of Science: Intimate Relationship or Marriage of Convenience?" British Journal for the Philosophy of Science 24: 282-97. Hall, A.R. (1948). "Sir Isaac Newton's Notebook." Cambridge HistoricalJournal. 9:239-50. Hall, A.R. and M.B. Hall (1962). Unpublished Scientific papers of Isaac Newton; a selection from the Portsmouth collection in the University Library) Cambridge. Cambridge: Cambridge University Press. Horwich, P. (ed.) (1993). World Changes: Thomas ICuhn and the Nature ofScience. Cambridge, Mass.: The MIT Press. Laudan, L. (1981). "A Confutation of Convergent Realism." In Papineau (ed.), 107-38. McGuire, J.E. and P.M. Rattansi (1966). "Newton and the 'Pipes of Pan'." Notes and Records ofthe Royal Society ofLondon. 21 (2): 108-43. McMullin, E. (1974). "History and Philosophy of Science: A Marriage of Convenience?" PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association. Edited by R.S. Cohen et aI., 585-601. Newton, I. ([1687] 1999). Mathematical Principles ofNatural Philosophy (3rd edition). Translated by LB. Cohen and A. Whitman. Berkeley: University of California Press. - - - . (2004). Philosophical Writings. Edited by A. Janiak. Cambridge: Cambridge University Press. Papineau, D. (ed.)(1996). The Philosophy of Science (Oxford Readings in Philosophy). Oxford: Oxford University Press. Sailor, D.B. (1988). "Newton's Debt to Cudworth." Journal ofthe History ofIdeas. 49:511-18 Stuewer, R. H. (ed.) (1970). Historical and Philosophical Perspectives ofScience. VOl. S of the Minnesota Studies in the Philosophy of Science. Minneapolis: University of Minnesota Press. Westfall, R.S. (1971). Force in NewtonJs Physics: The Science of Dynamics in the Seventeenth Century. London: Macdonald and Co.; New York: American Elsevier. Worrall, J. (1989). "Structural Realism: The Best of Both Worlds?" In Papineau (ed.), 139-65.
PART I
The Newtonian Era
[2J The Axiomatic Tradition in Seventeenth-Century Mechanics DOMENICO BERTOLONI MELII
Over the last few years several scholars have devoted increasing attention to the changing roles of contrived experiments in the seventeenth century, an age when they gained a significant if problematic role as a component of our knowledge about nature. Special attention has been devoted to the transition from generalized everyday experiences, at times associated with an Aristotelian or peripatetic tradition, to the contrived experiments performed at special places and times by the new natural philosophers. Historians have studied how experiments pertinent to several disciplines were conceived, performed, privately recorded, witnessed, and reported in print. 2 There is no question that for some scholars, such as the French Minim Marin Mersenne, the nobleman Robert Boyle, or the Paris academician Edme Mariotte, for example, experiments were the key source of knowledge about nature and they made no secret of it. Other scholars in the mathen1atical disciplines, however, did not wish to rely on contrived experiments at all in the formal presentation of their sciences. Rather, they sought either principles to which the mind naturally consents, such as symmetry, or principles based on generalized experiences describing the normal course of nature. Mental operations or thought experiments would be more appropriate terms to describe how they proceeded. This search for new principles is a significant feature of seventeenthcentury investigations and formulations of the mathematical disciplines that has attracted less attention than experiments. One root of this approach can be found in classical Greece and especially in Archimedes, who was among the first to try to formulate a science dealing with nature-mechanics or the science of the balance-in a mathematical fashion resembling Euclid's Elements. 3 My essay focuses on mechanics both because this was Archimedes's subject in On the Equilibrium ofPlanes, and 23
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also because mechanics provides ample material to reflect on the way mathematicians like Galileo and Christiaan Huygens maneuvered in the transition from the science of equilibrium to the new science of motion. Recently Michael Mahoney has identified the engineering tradition as a source of rules or principles in mechanics, namely: 1. 2. 3. 4. 5.
You cannot build a perpetual-motion machine. You cannot get more out of a machine than you put into it. What holds an object at rest is just about enough to get it n10ving. Things, whether solid or liquid, do not go up by themselves. When you press on water or some other liquid, it pushes out equally in all directions.
According to Mahoney, starting in the 1580s with the Dutch mathematician and engineer Simon Stevin and with Galileo, engineers aspiring to become natural philosophers began transforming such maxims into formal mathematical principles of mechanics. 4 In this essay I explore this Archimedean tradition and the attempts to reformulate classical mathematical disciplines, notably the doctrine of the equilibrium of the balance, or formulate new ones, such as the science of motion or of the collision of bodies, relying on generalized experience and principles to which the mind consents, rather than contrived experiments. While Mahoney is certainly right in identifying a significant source in the engineering tradition, mathematicians appealed also to abstract principles, such as symmetry, and progressively reformulated and expanded all principles regardless of their provenance in ways bearing only at best a vague relation to engineering. S Moreover, often mathematicians were only too keen to suppress their engineering background to address a more philosophical audience; therefore, engineering connections were more likely to be suppressed than emphasized. Several mathematicians adopted various axioms and postulates in their works on mechanics and motion. Since it would be impossible to provide an exhaustive account, I will discuss only a few of the protagonists of this approach besides Stevin and Galileo, namely the latter's successor and follower, Evangelista Torricelli, who formulated a new principle of the science of motion, and Huygens, who provided an axion1atic forn1ulation of the science of collision among bodies. It is Galileo, however, that is going to attract most of the attention and whose project and concerns I examine in greater detail. Starting from Descartes and his laws of motion, several philosophers and mathematicians, such as Leibniz, appealed to God, and to broader theological and philosophical reasons, in order to establish the foundations of several sciences, especially with regard to conservation principles. At times, the boundaries between theologically and not-theologically based princi-
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pIes vvere quite blurred, with principles introduced on theological grounds being later posed on non-theological ones. Descartes, for example, justified his laws of motion having recourse to God's immutability, but Huygens used the first two laws, clain1ing that bodies left to themselves tend to move with a rectilinear uniform motion, dropping God out of the picture. Those theological and philosophical considerations add factors about God and his relations to the world going beyond the scope of my contribution. 6 It goes without saying that specific rhetorical techniques of mentioning witnesses of contrived experiments, such as those mentioned by Simon Schaffer, Steven Shapin or Peter Dear, do not apply here. But one may still wonder what counted to late 16th- and 17th century mathematicians as an acceptable axiom or principle, how they were presented in print and justified, and how the perception of what was acceptable changed with time. 7
1. Archimedes and Axiomatic Foundations A key exemplar from Antiquity is Archimedes's work in mechanics, especially his treatise On the equilibrium ofplanes, which established the doctrine of the balance and determined centers of gravity of different geometrical figures. His other work on mechanics, On floating bodies, has a more problematic axiomatic structure, so much so that the meaning of its only postulate is unclear. 8 As we are going to see below, in his 1612 work on bodies in water Galileo tried to provide foundations for hydrostatics based on the balance, thus underscoring its primacy over other areas of mechanics. In On the equilibrium ofplanes Archimedes relied on an axiomatic style derived from mathematics in a work about nature, starting from seven postulates. The first two, reproduced below, give us a sense of his enterprise: 9 1. We postulate that equal weights at equal distances are in equilibrium, and that equal weights at unequal distances are not in equi1ibrium' but incline towards the weight which is at the greater distance. 2. If, when weights at certain distances are in equilibrium, something be added to one of the weights, they are not in equilibrium, but incline towards that weight to which something has been added. Archimedes did not take as a postulate the statement that "magnitudes are in equilibriun1 at distances reciprocally proportional to their weights," but rather he tried to prove it as a theorem. That statement lacked the required
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criteria to be naturally accepted by our mind. If some generalized experiences may have guided him in his choice of some postulates, others must have appeared less suitable and contrived experiments were certainly not a basis on which to proceed; even such relatively common experiences as those provided by a statera, or balance with unequal arms, did not look sufficient to him. I am unsure whether we have reflected with sufficient care on which statements could be chosen as postulates at different locations and times. I often paused when reading Peter Dear's claim: "Recent research has shown that Galileo aimed at developing scientific knowledge, whether of moving bodies or of the motion of the earth, according to the . Aristotelian (or Archin1edean) deductive formal structure of the n1ixed mathematical sciences" (Dear 1995, 125-26). My sense is that Archimedes would have required stricter criteria for what can be accepted as a postulate and would have used common experience more sparingly than Aristotle. But this is a claim needing a more extensive elaboration than can be given here. The actual proof of equilibrium provided by Archimedes when the distances are inversely as the weights was seen as problematic or at least improvable by several commentators, such as Galileo, and has attracted attention to the present day.lO In propositions 6 and 7 of On the equilibrium ofplanes, Archimedes tried to provide somewhat cumbersome proofs of the equilibrium of balances with unequal arms and unequal weights in two cases, first with commensurable and then with incommensurable magnitudes. Nonetheless, Archimedes's work constituted an example of how to formulate a mathematical science dealing with nature that was especially influential in the late sixteenth and seventeenth centuries.
2. Stevin and Galileo: From the Balance to Falling Bodies Much as Archimedes in his study of the equilibrium of the balance, Stevin and Galileo did not wish to rely on contrived experiments in their formal presentation of the science of mechanics, hydrostatics, the science of the resistance of materials, and the science of motion. Stevin provided axiomatic formulations of his theory. In addition, both in his celebrated study of the inclined plane and of the equilibrium of water in a container, he relied on a principle such as the denial of perpetual motion (see Illustration 1 ).1 1 Galileo gave the balance pride of place in his work on mechanics. The balance was both the device that had been treated and formalized by Archimedes, and the basis for understanding other machines and problems, such as the other simple machines, the siphon, and the loaded bean1.
The Axiomatic Tradition in Seventeenth-Century Mechanics
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T
s
v
o
G
ILLUSTRATION 1: Stevin and the Inclined Plane. According to Stevin the chain of spheres on the inclined plane will not move on its own accord, lest we have perpetual motion. The lower part SON ... GV can be removed with symmetry consideration. The remaining portion STV shows that the weights are in equilibrium when they are proportional to the lengths of the inclined planes.
In some cases, such as the science of resistance of materials, Galileo believed he could extend the doctrine of the balance to new domains by seeking to show visually how a beam could be conceptualized as a balance (see Illustration 2). Similarly, in the case of the siphon he tried to provide foundations based on a balance with unequal arms by relying on the analogy
ILLUSTRATION 2: Galileo's beam. Galileo argued that a loaded beam could be seen as a balance or lever. One arm is eB, the fulcrum is B, and the other arm is AB, namely the cross section of the beam attached to the wall. The "weights" in this case would be represented by the resistance offered by the beam's fibers.
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between the speed of the two weights and of water in the two branches of the siphon (see Illustration 3).1 2
ILLUSTRATION 3: Galileo's siphon. Here Galileo argued that if water in the larger arm is pressed down from GH to QO, water in the smaller arm will rise from L to AB. The speed of water in the two arms is proportional to GQ and LA and is inversely as the cross sections. Therefore the siphon works like a balance where the weights are as the water surfaces in the two arms and the speeds are as the distances from the fulcrum.
Although Galileo's reformulation of the doctrine of the balance does not tell us much that is new about the contents of mechanics, in my interpretation it is the key to understanding his formal axiomatic presentation strategy in other areas, especially the science of motion. Here I am going to outline his reformulation and its implications. Both in his Padua lecture notes, Le mecaniche, and later in the Discorsi, Galileo presented a new proof of the key condition for the equilibrium of a balance, namely, that the weights are inversely as the distances of their suspension points from the fulcrum. Nobody doubted the truth of the proposition; the trouble was how to prove it starting from indubitable assumptions. If Galileo had sought to ground his science on experience, the claim that the balance is in equilibrium if the weights are inversely as the distances of their suspension points from the fulcrum would have looked like a perfect candidate. Such balances had been commonly used for centuries on every market square and nobody doubted the principle on which they operated. However, Galileo wished to prove that principle by means of a series of operations that he thought would be accepted as legitimate by the mind or our intuition (see Illustration 4). Much as for Archimedes, the starting point was a variant of the perfectly symmetrical case of equal weights hanging from equal distances. Galileo attempted to simplifY Archimedes's proofs, but followed a rather similar approach. The main idea consists in attaching a homogeneous bar, such as a cylinder or a prism, parallel to a horizontal balance;
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then by a series of operations that do not appear to our mind to alter the equilibrium conditions, like cutting the prism at some point and suspending it in a different fashion, one is left with a configuration whereby the weights are inversely to their respective distances from the center. Much as in Archimedes, intuition and reason are invoked rather than actual experiments. 13
A
ILLUSTRATION 4: Picture of Galileo's balance. The prism AB is attached at A and B to the balance with fulcrum C. Cutting the prism in D and attaching the two parts to the balance in E leaves the equilibrium conditions unchanged. Next Gableo suspends the two parts from their middle points L at G and M at F, again without altering the equilibrium conditions. The distances CG and CF are found with some calculations to be inversely as the weights of AD and DB of the two portions of the prism.
From the perspective adopted here, a distinction between private manuscripts and publication in Galileo is insufficient. Within Galileo's published works we have to distinguish between informal and formal presentation. I have no doubt that privately Galileo performed important experiments either heuristically or to confirm results he had found by calculation. 14 In print Galileo referred to experiments in the Dialogo, for example, but in the second and third days of the Discorsi his main concern was with establishing an axiomatic science of motion on the example of Archimedes. In De motu antiquiora Galileo tried to establish foundations for the science of motion from the balance, but he soon realized that he needed a different starting point and therefore attempted to find suitable foundations elsewhere. Galileo performed several heuristic experiments privately, and discussed some of them in print in a variety of texts, but my main concern here is with the formal axiomatic presentation of his sciences. Whereas Galileo's experimental forays have been frequently commented upon, his foundational efforts have attracted less attention, yet they constitute a major episode in the history of science. Galileo's letters to Paolo Sarpi in 1604 and Luca Valerio in 1609 show him desperately seeking principles to which the mind naturally consents, in order to construct an
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axiomatic science based on definitions and propositions. It is somewhat ironic to notice how hard Galileo was struggling to find axioms that looked natural. Indeed, this is often a feature of axioms, to be contrived in the extreme in order to look perfectly natural and straightforward, much like some experiments. Both in Le mecaniche and in a letter to Sarpi, Galileo used the word "undoubted" or "indubitable" to qualify an axiom or postulate. In Le mecaniche he stated: "We can take as an undoubted axiom this conclusion: heavy bodies, once we have removed all external and adventitious impediments, can be moved on the plane of the horizon by a whatsoever smallest force." Sin1ilarly in the letter to Sarpi he stated that thinking over the problem of motion, he realized he lacked a "totally indubitable principle to be put as an axiom." His "indubitable" principle turned out to be the erroneous proportionality between speeds and distances, but this does not affect my point.l 5 The rhetorical strategy and language used to present postulates requires the same amount of care and scrutiny we have devoted to experimental narratives. Galileo did not succeed fully in providing definitions and axioms in a form that would appear as an intuitive and convincing account of how nature operates. Rather, he defined naturally accelerated motion as that motion where the speed increases proportionally to the time, adding a postulate stating: "The degrees of speed acquired by the same moveable over different inclinations of planes are equal whenever the heights of those planes are equal. "16 There is no question that, despite his pretences to the contrary, Galileo's formulation was highly artificial and contrived. Even a cursory survey of the manuscript evidence and correspondence shows that he struggled for decades to present his construction in a "natural" fashion. On the basis of his definition and postulate Galileo built a mathematical theory that in principle may have remained just that, a mathematical theory devoid of any physical significance. It is curious that despite this pretence, Galileo spent some time trying to defend his axiom by claiming both that it is naturally accepted by the mind, or the "lume naturale" ("good sense" is Dral<e's translation),l? and that an experiment or "esperienza" confirms it; pendulums, for example, reach the same height from which they have been released. Galileo is aware that a fall along a rectilinear inclined plane is different from a fall along a circle arc as in an oscillating pendulum. In spite of this and other difficulties, Galileo argues that "l'intelletto resti capace," or "the mind understands" (following Drake) that the bob returns to the same height. "Hence let us take this for the present as a postulate, of which the absolute truth will be later established for us by our seeing that other conclusions, built on this hypothesis, do indeed correspond with and exactly conform to experience."18 This provisional
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acceptance of a postulate that is later corroborated by the conclusions underscores the lack of certainty of the postulate and highlights the weakness of Galileo's foundations. From the definition Galileo proved that the spaces traversed by a body falling with a naturally accelerated Illotion are proportional to the square of the times. This statement is shown to be equivalent to the odd-nun1ber rule, vvhereby the spaces traversed in successive time intervals are as 1, 3, 5, 7, etc. At this point he introduced the celebrated experiment with the inclined plane showing with a great degree of accuracy that real bodies actually follow this rule. The experiment involves an inclined plane about twelve braccia long, arranged so as to have a rather small inclination, raised on one side only one or two braccia. Galileo measured time with a water clock. This contrived experiment is presented as having no foundational role at all. Its role is to show that the mathematical construction he has provided actually describes the behavior of real bodies in the world. Galileo's mathematical theory would remain valid even if nature behaved differently. Galileo perceived the reference to experiments justifying his axioms as a weakness and after publication he continued to reflect on the matter. Eventually, stimulated by his pupil Vincenzo Viviani, he produced a proof of his axiom that was first published in the posthun10us second edition of the Discorsi in 1656.1 9 Galileo's works were read in strikingly different fashion. Among his disciples, Torricelli was probably the one who was most concerned with the problem of axiomatic foundations. In the 1644 De motu he put forward the postulate or principle whereby two combined bodies do not move unless their common center of gravity descends. There is no question that he presented his postulate as a proposition accepted by the n1ind based on common intuition and general observations. His rather general examples were not contrived experiments, but rather instances of a general statement, such as systems of weights on inclined planes and connected via a pulley, for example. In his reformulation Torricelli was quite successful, since his postulate was both powerful and convincing. Mersenne, by contrast, had no interest at all in axiomatization and frequently focused instead on the empirical adequacy of Galileo's claims, as with the distance covered by a body in free fall in a given time or with spheres rolling down inclined planes. In both instances he worried about numerical results from specific propositions rather than logical deductive structures. 20 Next we are going to examine a previous attempt at justifying the law of fall that was later discarded by Galileo. In the debates on falling bodies in the 1640s several scholars were to adopt a sin1ilar approach.
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3. Falling Bodies and Unit Invariance An instructive case occurred with the debates on the rule followed by heavy bodies in free fall. In a letter to Benedetto Castelli, Gianbattista Baliani reported that Galileo had defended the odd-number rule byarguing that it was the only one invariant on the choice of the distance. 21 The statement is somewhat cryptic, but I believe it can be interpreted as follows. Units of distance are arbitrary; indeed in Galileo's Italy they varied almost from town to town. Galileo seenled to argue that if nature follows a rule according to which heavy bodies fall, that rule should not depend on arbitrary factors as the units used at Florence or Rome, or indeed any units at all, but must be independent of them. Units are local and conventional, whereas nature's operations should be universal and independent of human conventions. Following the odd-number rule, if a body were to fall a given distance in an arbitrary time, in the second tin1e interval, equal to the first, the body would fall three times that distance, five in the third, and so on. This proposition remains true for any choice of the initial distance fallen. For example, if one were to choose an initial distance four times greater as the initial unit, the correspondent time would be doubled. In the second time interval, equal to the first, the body would again fall three times the initial distance, since 5+7 (=12) is three times 1+3 (=4); the same would happen in successive intervals, since 9+11 (=20) is five times 1+3 (=4). Several other rules lack this invariance property. For example, if a body fell in successive equal time intervals by distances as the natural numbers' 1, 2, 3, 4, etc., then if one were to chose a different unit of distance, the proportion based on the natural numbers-rule would not be preserved. Galileo's choice is not unique, but the problem of determining which rules are invariant under transformation of units is beyond the scope 'of this paper. It is not clear what the origin of this invariance rule is, but it is tempting to identify the engineering tradition as a possible source. Bridges or towers stand or fall regardless of whether they are measured in Roman or Florentine units and one can see why a sin1ilar reason could be applied to falling bodies. Galileo in the end did not make much of this argument because he deemed it only probable, but a closely related one was used by several scholars of the following generation. Unlike Galileo, who according to Baliani relied on space, those other scholars took time as the independent variable. Despite this change, their reasoning resembled Galileo's in a major respect, because their condition reads like a constraint imposed by our mind on the proportion between space and time of fall. Once again, it is not a contrived experiment that serves as a foundation for the mathematical theory, but rather a condition imposed by our mind on the possible or reasonable form experimental results can take. This strategy shows
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some similarity to imposing a symmetry principle, but it also appears somevvhat less direct and more sophisticated in that it relies on the notion of unit lueasure. The seventeenth-century scholars who adopted this form of reasoning were luatheluaticians Torricelli, Huygens, and Jacques Le Tenneur, as well as the physician Theodore Deschamps. In his correspondence with Mersenne in the late 1640s, for example, Huygens ridiculed alternative rules based on the sequence of natural numbers 1, 2, 3, 4, etc. or on a geoluetric relation, 1, 2, 4, 8, etc., arguing that they violated the invariance of unit measures. Such alternative rules had actually been proposed by the Bishop Jean Caramuel, the Jesuits Honore Fabri and Pierre Le Cazre, and the Genua nobleman Gianbattista Baliani. 22 While empirically it may have been difficult to refute those rules, the request that they satisfy unit invariance ruled them out in the eyes of the scholars mentioned above. Later in the century, interest in this problem shifted from invariance to physical causes and Galileo's odd-number rule was considered as a viable approximation. Probably this is the reason why such debates attracted no further attention.
4. Huygens and Impact In the Latin and French editions of Principia philosophiae of 1644 and 1647, Descartes put forward seven rules of impact for hard bodies. The rules relied on the third law of motion, stating the conservation of quantity of motion, or magnitude of a body times its speed, with no regard to direction. Descartes presented his laws of motion with the help of theological considerations about God's immutability, whereas he claimed that the seven impact rules were self-evident. He did not follow the Archimedean axiomatic approach and did not produce a deductive structure; rather, he argued that his book should be read as a novel. Overall, Descartes favored heuristic approaches and disliked axiomatic presentations' so much so that even his Geometrie contains no axioms. Yet it is perfectly possible to seek alternative formulations both of his mathematics and rules of collision. 23 Huygens was one of the first readers to realize that Descartes's rules were problen1atic and to tackle the issue in a new fashion. The problem looked in son1e respects similar to the equilibrium of the balance with equal arms and equal weights, in that one could start from the symmetrical case of equal bodies colliding with equal and opposite speeds. Much like the case of the balance, symmetry considerations require that the two bodies behave in the same way after the impact, but this condition is insufficient to deal with more general cases. The outcome of his efforts was the
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treatise De motu corporum ex percussione, which remained unpublished in his lifetime and saw the light only posthumously in 1703. It consists of five hypotheses and thirteen propositions. It was not entirely uncommon to find the term "hypothesis" meaning "presupposition" or "postulate," as done by Huygens here. The first hypothesis states: 24 "Any body once moved continues to move, if nothing prevents it, at the same constant speed and along a straight line." This statement corresponds to Descartes's first two laws of nature, but it is devoid of the theological justifications one finds in Descartes. In fact, Huygens provided no justification, suggesting that the matter appeared unproblematic in his eyes. The second hypothesis poses a restriction on the type of bodies investigated, namely hard bodies. For this special case Huygens's hypothesis claimed that when two equal bodies collide with equal speeds, they rebound with the same speeds reversed. It is the third hypothesis that is of special interest here, because Huygens had recourse to the principle of relativity of motion. Galileo had discussed a similar principle in the Dialogo, defending it with a series of observations on the behavior of bodies on a n10ving ship, but he was far from being the first to do so. In the opening of day four of the Discorsi Galileo relied again on relativity of motion in order to argue that horizontal projection does not affect the uniforn1ly accelerated motion of falling bodies. Huygens formulated relativity of n10tion as a hypothesis or postulate and applied it in a quantitative fashion. In this way he was able to move from a perfectly symmetric case to one where one body at rest is hit by a supervening equal body. With the example of unit invariance for falling bodies fresh in our mind, we notice here a similar approach whereby an invariance condition enables the formulation of propositions about nature. The case of unit measures, however, was entirely devoid of empirical presuppositions: nobody performed experiments in Florence and Rome to test whether the different units of length en1ployed there had any impact on the outcome. Empirical considerations, however, probably entered the principle of relativity of motion in a mediated way. Huygens did not feel the need to justify it empirically, however, as if human intuition by the mid-seventeenth century authorized his move. His account of impacts on a moving barge reads not like a description of an empirical test, but more like the mental operations on weights described by Archimedes and Galileo in their discussions of the equilibrium of the balance: 25 The motion of bodies and their equal and unequal speeds are to be understood respectively, in relation to other bodies which are considered as at rest, even though perhaps both the former and the latter are involved in another common motion. And accordingly, when two bod-
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ies collide with one another, even if both together are further subject to another uniform motion, they will move each other with respect to a body that is carried by the same comn1on motion no differently than if this motion extraneous to all were absent. This abstract statement is instantiated in a more intuitive fashion with the example of the moving barge. Huygens seems to have recourse to a mixture between intuition and experiences that would have been common enough to readers fan1iliar with travels on boat along canals, a common means of transport in the Netherlands and elsewhere: 26 Thus, if someone conveyed on a boat that is moving with a uniform motion were to cause equal balls to strike one another at equal speeds vvith respect to hin1self and the parts of the boat, we say that both should rebound also at equal speeds with respect to the same passenger, just as would clearly happen if he were to cause the same balls to collide at equal speeds in a boat at rest or while standing on the ground. The list of hypotheses does not cover all Huygens's assumptions and axioms. Buried in the text of Proposition 8 there is a staten1ent of one of his favorite axion1s. The proposition states that in the collision between two bodies with speeds inversely as their magnitudes, the bodies will rebound with the same speeds with which they approached each other. In the proof Huygens converted horizontal to vertical motion, a rather straightforward move if one considers the collision between pendulum bobs. In this context Huygens affirmed: 27 "in mechanics it is a most certain axiom that the common center of gravity of bodies cannot be raised by a motion that arises from their weight." This axiom derives in all probability from a combination of Torricelli's principle and the denial of perpetual motion and, much like its ancestors, is not justified by having recourse to experiments. Huygens probably assumed that his readers would grant it based on common intuition and a wide range of experiences. In 1673 Edme Mariotte, Huygens's colleague at the Paris Academie, published a treatise on the collision of bodies where he put forward a theory closely resembling Huygens's. Unlike the Dutch scholar, however, Mariotte chose as the foundation for his theory what he called "principles of experience," namely propositions based on collision experiments. Whereas Huygens had struggled to build his science almost a priori in Archimedean fashion, Mariotte's treatise is a collection of statements of what happens when bodies with a given n1agnitude and speed collide. 28
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5. A Newtonian Coda The case of Isaac Newton is considerably complex and I am therefore going to discuss here briefly only his three axioms or laws of motion in Principia mathematica. It is significant that Newton qualified the three laws as "axiomata sive leges": they indeed do serve as axioms to his system. His first law, the law of inertia, subsumes Descartes's first two laws of motion. 29 The second law states that the change in the quantity of motion is proportional to the force impressed, whether applied continuously or in impulses. Historians have extensively discussed this law, despite the fact that neither Newton nor his contemporaries made much of it. 30 The third law states that the mutual actions of two bodies are always equal and in opposite directions. A corollary to this law states the conservation of quantity of motion in one direction, a result previously established by Huygens. Huygens and Newton presented their equivalent propositions in different ways. In the brief essay on collision published in 1669 in the Journal des Sfavans, Huygens gave general rules that could be seen either as presuppositions or results of his investigations. The fourth rule echoes proposition VI of De motu corporum ex percussione, stating that quantity of motion (in the Cartesian sense) of two bodies can be diminished or augmented as a result of collision. In rule 5 in the 1669 essay, however, he added that quantity of motion in the same direction is conserved. 31 Newton felt the need to justify his third law in various ways, first by providing examples, then in the scholiun1 by discussing thought experiments, and even reporting the outcome of real experiments, including numerical data. 32 In the scholium he considered both collisions among bodies and attractions. For collisions he sought to test the law by means of pendulun1 experiments showing that quantity of motion, taking direction into account, is not changed by the impact among bodies, whether hard or soft (see Illustration 5). He concluded the experimental report with the words: 33 "In this manner the third law of motion-insofar as it relates to impacts and reflections-is proved by this theory [ofimpact], which plainly agrees with experiments." In the case of attractions, Newton could not experiment with gravity; therefore he justified the third law first by means of an experiment with magnetic bodies placed on floating supports in a bowl of water, showing that action equals reaction because the bodies stay still where they come together, rather than moving in one direction. This case highlights his eagerness to rely on experiments if at all possible. In the case of gravity he had recourse to a thought experiment consisting in cutting the earth in two unequal parts and showing that they do not move as a result of their mutual attractions. 34 In general, despite the case of linear momentum, Newton did not rely on conservation principles and did not mal(e much of
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ILLUSTRATION 5: Newton's pendulum experiments. Two pendulums CA and DB collide and rebound. By carefully subtracting the effects of air resistance (ifRV is the effect of air resistance in a full to and fro oscillation, one quarter of it is 51' or st), Newton showed that action equals reaction for all bodies, whether hard or soft.
them in his system. For example, he still believed that Cartesian quantity of motion (without taking direction into consideration) was a significant measure of motion in the universe and rejected the conservation of living force (mass times the square of the speed) in all its manifestations.
6. Conclusion Seventeenth-century scholars tackled the problem of presenting and justifYing new knowledge about nature in different ways. Some, like Mersenne, Mariotte, and Boyle, attributed a key foundational role to experiment. Despite considerable differences in their approaches, they all believed experiment to be crucial in their investigations as well as in their formal presentation. Others, such as Stevin, Galileo, Torricelli, and Huygens, saw experiment as inherently problematic in this role and sought to find secure foundations elsewhere. I have no doubt that all of them, especially Galileo and Huygens, were remarkable and creative experimentalists. However, they shared a common concern for establishing knowledge about nature in an axiomatic fashion, on the example of mathematics and in a tradition going back to Archimedes. The key idea seems to be that some propositions can appear natural to the mind, yet they entail a number ofless natural-looking consequences. These consequences can help to establish a science or portions of it, or at least they can rule out a number of competing alternatives, as in the case of unit invariance for falling bodies. Thus, premises to which the mind naturally consents, often arranged in a contrived fashion, can have surprising empirical implications. Some axioms look quite convincing, such as the principle of symmetry for the case of the balance with equal arms and equal weights and for equal bodies colliding with equal and opposite speeds, but they do not allow one
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to go very far. Torricelli's principle provides some flexibility in the formulation of a science of n10tion. Other principles, such as that later named by Newton as the principle of inertia, became increasingly more acceptable in the second half of the seventeenth century as a result of a conceptual reorientation rather than specific contrived experiments. In this specific case the acceptance of the principle has a historical dimension that becomes especially significant in the course of the seventeenth century. The case of the unit invariance for falling bodies is peculiar in some respects, since it was not formulated as an axiom, but rather as a criterion to rule out competing rules for falling bodies. It does share with other principles the feature of being a requirement of reason limiting the ranges of possible behavior of falling bodies. The tradition of formulating principles or axioms we expect nature to observe shows us mathematicians in action seeking propositions about nature not, or not only, through experimental activities, but by thinking about a reasonable course nature is expected to follow and by imposing conditions and restrictions on the forn1 of the n1athematical relations describing nature's course. This tradition is significant in the formal presentation of a science and is an important component, alongside the muchmore frequently celebrated experimental n1ethod, in the history of science and philosophy in the seventeenth century. The contents of my essay resonate with some of the themes of this volume, such as the importance of a priori or non-empirical principles in science' and the temporal and historical dimension of those principles. Of course, I am not advocating a teleological reading of I
NOTES 1. I wish to thank all those who offered comments on previous versions of this essay at the University of South Carolina, Columbia, and the University of California, San Diego. Mike Mahoney was kind enough to offer helpful observations on an earlier draft of this essay. A special thanks to the editors for their thoughtful and constructive comments and to Jordi Cat for a fruitful conversation.
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2. Dear (1995); Daston (1991); Newman (2004), ch. 5. 3. Another exaluple is Euclid's Optica (Euclid, 1985). See Lindberg (1976), 12-13. As Tal Golan pointed out, Euclid's Elements too can be seen as a matheInatical science dealing with nature, notably physical space. 4. Mahoney (1998), 707-8 and (1985), 4-5 of the online version. Those Inaxims, however, were not generally accepted. Guidobaldo dal Monte, for example, did not accept no. 3; see Drake and Drabkin (1969), 300, 316, 318. 5. My vievvs do not differ from Mahoney's on this point. 6. Steinle (2008). Funkenstein (1986), ch. 3. 7. Schaffer and Shapin (1985). Shapin (1994). Dear (1995). Some topics relevant to my discussion were introduced in Holton (1973, 1988). 8. Dijksterhuis (1987), 373-77;. Duhem (1991), 247-49. 9. Dijksterhuis (1987), 286-87. I have slightly altered the translation. 10. For a brief survey see Dijksterhuis (1987),290-98. II. Stevin (1955),179 and 401. Stevin was familiar with Archimedes's works in the edition of Federico Commandino. 12. Galileo (1890-1909), 4: 77-78 and 8: 156-57. 13. Galileo (1890-1909), 8:152-54. 14. On the leaning tower experiment, for example, see Camerota (2004), 61-62. 15. Camerota (2004), 91, 138. Galileo (1890-1909), 2:180; 10: 115-16, Galileo to Sarpi, Venice, 16 October 1604; 10: 248-49, Valerio to Galileo, Rome, 18 July 1609. Wisan (1978), especially 5, 44. 16. Galileo (1974), 162. Galileo's alleged claim at 169 that: "sensoryexperiences ... are the foundations of all resulting structure" is a serious mistranslation by Drake. 17. Galileo (1974), 162. 18. Galileo (1974), 164. 19. Galileo (1974), 171-75. 20. For Mersenne's changing attitudes to Galileo see Dear (1988). 21. Galileo (1890-1910), 13: 348-49, Castelli to Baliani, 20 November 1627. 22. Bertoloni Meli (2004), 182. 23. Descartes (1983), 59-69. 24. Huygens (1888-1950), 16: 30. Here and in later quotations I rely on the translation by Michael Mahoney at http://www.princeton.edu/---hos/mike! texts/huygens/impact/huyimpct.html 25. I-Iuygens (1888-1950), 16: 32. 26. Huygens (1888-1950),16: 30. Michael Mahoney drew my attention to Huygens's drawing of a barge on a canal, 16: 122. 27. Huygens (1888-1950),16: 56. Duhem (1991), 261-64 and 349-56. 28. Mariotte (1673). 29. But see Gabbey (1980), 272-97. 30. Westfall (1971) devoted a whole book to the history of the second law in the seventeenth century. Newton (1999), 416-17. 31. Huygens (1888-1950), 16: 48-49 and 180. Huygens's essay in the Journal is reproduced in 16: 179-81. It is worth noticing that Huygens did not make an axiom of the conservation of the sum of the products of the bodies'
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magnitudes times the square of their speeds (for hard bodies), but rather he took it to be a result of his investigations in proposition XI. Huygens (1888-1950), 16: 73 and 180, where the same proposition appears as rule 6. 32. In the Principia Newton generally relied on a combination of high-power mathematics and empirical data, whether from experin1ents or astronomical observations. See Smith (2002). For optics see Shapiro (1993), 1-40. 33. Newton (1999), 417, 420-21; quotation from 427. 34. Newton (1999),427-29. Friedman (2001),39 n.46.
REFERENCES Bertoloni Meli, D. (2004). "The Role of Numerical Tables in Galileo and Mersenne." Perspectives on Science 12:164-90. Butts, R.E. and J.C. Pitt (eds.) (1978). New Perspectives on Galileo. Dordrecht: Reidel. Camerota, M.(2004). Galileo Galilei e la cultura scientifica nelFeta della controriforma. Roma: Salerno editrice. Daston, L.J. (1991). "Baconian Facts, Academic Civility, and the Prehistory of Objectivity." Annals ofScholarship 8:337-63. Daston, L.J. and M. Stolleis (eds.) (2008). Natural Law and Laws of Nature in Early Modern Europe. Surrey, UIZ: Ashgate. Dear, P. (1988). Mersenne and the Learning of the Schools. Ithaca: Cornell University Press. - - - . (1995). Discipline and Experience: The Mathematical Way in the Scientific Revolution. Chicago: University of Chicago Press. Descartes, R. (1983). Principles of Philosophy. Translated, with explanatory notes, by V.R. Miller and R.P. Miller. Dordrecht: Reidel. Dijksterhuis, E.J. (1987). Archimedes. With a New Bibliographic Essay by Wilbur R. I(norr. Princeton: Princeton University Press. Duhem, P. (1991). The Origins ofStatics: The Sources ofPhysical Theory. Translated from the French by G.F. Leneaux, V.N. Vagliente, G.H. Wagener. Dordrecht: IZluwer. Drake, S. and I.E. Drabkin (eds.) (1969). Mechanics in Sixteenth-Century Italy: Selections from Tartaglia) Benedetti) Guido Ubaldo) & Galileo. Madison: University of Wisconsin Press. Euclid (1895). Optica) opticorum recensio theonis) catoptrica) cum scholiis antiquis. Edited by J.L. Heiberg. In vol. 7 of Euclid, Opera Omnia (1883-1916), 9 vols. Leipzig: Teubner. Friedman, M. (2001). Dynamics of Reason. The 1999 I(ant Lectures at Stanford University. Stanford: CSLI Publications. Funkenstein, A. (1986). Theology and the Scientific Imagination from the Middle Ages to the Seventeenth Century. Princeton: Princeton University Press. Gabbey, A. (1980). "Force and Inertia in the Seventeenth Century: Descartes and Newton." In S. Gaukroger (ed.), 230-320. Galilei, Galileo (1890-1909). Opere, 20 vols in 21. Firenze: Giunti Barbera.
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- - - . (1974). Two New Sciences) Including Centers of Gravity and Force of Percussion. Translated, with introduction and notes, by S.Drake. Madison: University of Wisconsin Press. Garber, D. and M. Ayers (eds.) (1998). The Cambridge History of 17th-Century Philosophy. Cambridge: Cambridge University Press. Gaukroger, S. (ed.) (1980). Descartes: Philosophy) Mathematics and Physics. Sussex: Harvester Press. Holton, G. (1973, rev.1988). Thematic Origins of Scientific Thought: I(epler to Einstein. Cambridge: Harvard University Press. Huygens, C. (1888-1950). Oeuvres completes. 22 vols in 23. La Haye: Martinus Nijhoff. - - . (1703). De motu corporum ex percussione. In Huygens (1888-1950), 16: 30-91. Translated by M. Mahoney. (www.princeton.edu/-hos/mike/texts/ huygens/in1pact/huyimpct.html) . Lindberg, D.C. (1976). Theories of Vision from Al-I(indi to I(epler. Chicago: University of Chicago Press. Mahoney, M. (1985). "Diagrams and Dynamics: Mathematical Reflections on Edgerton's Thesis." In J.W. Shirley and F.D. Hoeniger (eds.), 168-220 (www.princeton.eduj-mike/articles/diagdyn/diagdyn.html). - - - . (1998). "The Mathematical Realm of Nature." In Garber and Ayers (eds.), 702-55 .(http://www.princeton.edu/-mike/articlesjmathnat/mathnatfr .html). Mariotte, E. (1673). Traitte de la percussion ou chocq des corps. Paris: E. Michallet. Newman, W.R. (2004). Promethean Ambitions. Alchemy and the Quest to Perfect Nature. Chicago: University of Chicago Press. Newton, I. (1999). The Principia. Mathematical Principles of Natural Philosophy. Translated by LB. Cohen and A.Whitman. Berkeley: University of California Press. Schaffer, S. and S. Shapin (1985). Leviathan and the Air Pump. Princeton: Princeton University Press. Shapin, S. (1994). A Social History ofTruth. Chicago: University of Chicago Press. Shapiro, A. (1993). Fits) Passions) and Paroxysms. Cambridge: Cambridge University Press. Shirley, J.W. and F.D.Hoeniger (eds.) (1985). Science and the Arts in the Renaissance. Cranbury, NJ: Associated University Presses. Smith, G. (2002). "The Methodology of the Principia." In LB. Cohen and G. Smith (eds.), The Cambridge Companion to Newton. Cambridge: Cambridge University Press, 138-73. Steinle, F. (2008). "From Principles to Regularities: Tracing 'Laws of Nature' in Early Modern France and England." In Daston and Stolleis (eds.), 215-31. Stevin, S. (1955). The Principal Works of Simon Stevin) General Introduction) Mechanics. Edited by E.J. Dijksterhuis, vol. 1 of 5, (1955-1966). Amsterdam: C. V. Swets & Zeitlinger. Westfall, R.S. (1971). Force in NewtonYs Physics: The Science of Dynamics in the Seventeenth Century. London: MacDonald, New York: American Elzevier. Wisan, W.L. (1978). "Galileo's Scientific Method: A Reexamination." In Butts and Pitt (eds.), 1-57.
[3] The Reduction to the Pristine State in Robert Boyle's Corpuscular Philosophy WILLIAM R. NEWMAN
The editors of the present volume have conveniently produced a name for Michael Friedman's wide-ranging intellectual endeavor. They call his method "synthetic," by which they refer to his emphasis on the essential interaction between philosophy and science, or rather the underlying unity between these seemingly distinct fields. In the present paper, I hope to show just how important this synthetic n1ethod is to understanding one of the great historical movements at the basis of modern science, namely, the shift to atomic or corpuscular theories of matter that characterized the mechanical philosophy of the seventeenth century. Although few historians would explicitly deny that a close relation existed between philosophy and natural science in the early modern period, the modern disciplines of "history of science" and "history of philosophy" make far too little effort to understand one another's domains, even when the disciplines now identified respectively as natural science and philosophy may well have formed one field before the creation of the academic disciplines dedicated to their retrospective study. In my view it is precisely the absence of an approach like Friedman's that has led both historians and philosophers to overlook one of the major reasons for the move to corpuscular matter theory in the early modern period. Without an awareness of the interaction between scholastic natural philosophy and empirical practice in this period one is simply adrift in a sea of unrelated facts that lend themselves all too easily to the facile misinterpretations populating past and current surveys of the Scientific Revolution. Let me make my fundamental thesis brief. Natural philosophers, physicians, theologians, and mathematicians who received a university education during the sixteenth and seventeenth centuries were generally-and often heavily-exposed to a range of scholastic positions arguing for the nonretrievability of ingredients from a genuine mixture. In a word, the
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most widespread interpretations ofAristotelian matter theory in this period stated that it was not possible to re-isolate the initial constituents of a homogeneous substance once those constituents had combined to form a mixture, and such homogeneous mixts were widely thought to include materials as commonplace as metals, flesh, wood, milk, and wine. 1 During the Late Middle Ages and the early modern period, this theory came increasingly into conflict with a host of empirical examples supplied above all by the discipline of alchemy or "chymistry," a field where corpuscular theories of matter had been circulating in the Latin West since the thirteenth century. It was no accident that Robert Boyle, the famous popularizer of the mechanical philosophy and debunker of substantial forms, was himself a chymist. He was, in fact, giving further articulation and modifications to the views of alchemists as expressed over a period of several hundred years. To make matters short, it was the field of chymistry that supplied Boyle's primary ammunition against early modern scholastic matter theory as taught in the universities. For chymistry provided a way out of the impasse resulting from a strict interpretation of substance and mixture first promulgated by Thomas Aquinas and later adopted by other scholastic schools that had forbidden the persistence and retrievability of ingredients within a mixture. Before proceeding further, it will be useful now to consider the fallacy of those historians who have not followed something like the synthetic method. The approach of most historians who have written on the Scientific Revolution as a whole has been either to fall into an Enlightenment tradition of dismissing alchemy as an outright delusion or to marginalize it by placing it within the category of "the occult," where it might be viewed as occupying a transitional position between scholasticism and the Scientific Revolution, but not as having had any central role in formulating "the new science." The first approach reached its apex in the heyday of progressivist histories of science, nan1ely between the 1940s and 1960s. One sees it clearly in the derision with which Herbert Butterfield, in his celebrated Origins of Modern Science of 1949, would disn1iss historians of alchemy as being "tinctured with the same type of lunacy they set out to describe"(Butterfield 1951, 98). Similarly Rupert Hall, in his The Scientific Revolution: 1500-1800 of 1962, denied alchemy any status as a forerunner to chemistry, and went so far as to describe it as "the greatest obstacle to the development of rational chemistry"(Hall 1962, 310).2 Marie Boas [Hall], in a similar vein, referred to alchemy as a "mystic science" opposed to the rational developments of the seventeenth century in her much cited "Establishment of the Mechanical Philosophy" of 1952.(Boas [Hall] 1952, 428) Even E.J. Dijksterhuis, whose Mechanization of the World Picture (1950, and in English, 1961) remains, in other respects, a model
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to be emulated, could only see alchemy as "a mysterious trifling with ilnpure substances, guided by mystical conceptions and hazy analogies, in vvhich credulity played a considerable part.,,3 Clearly authors like Butterfield, Hall, Boas [Hall], and Dijksterhuis saw little or no place for alchemy in any reformulation of matter theory. The same view, in more subtle terms, may be found in the recent work of writers on the Scientific Revolution such as Steven Shapin and Peter Dear. The failure to recognize a role for chymistry in the development of the mechanical philosophy has not altered in most recent comprehensive treatlnents of the subject, despite the emphatic claims to originality made for new historiographical approaches such as the sociology of scientific knowledge. 4 The well known Leviathan and the Air Pump (1985) of Steven Shapin and Simon Schaffer explicitly refers to "alchemical secretists" as a foil to the experimental science of Boyle and the Royal Society. 5 This is singularly ironic, given that Boyle himself engaged in a largely secret quest for the philosophers' stone that extended over 40 years, from his earliest laboratory training at the hands of the American chymist George Starkey up until Boyle's death in 1691. Shapin's recent survey, The Scientific Revolution (1996), n1erely reinforces his earlier approach. Alchemy mal(es a brief appearance here an10ng the "pseudosciences" (Shapin's term), whose interaction with the "proper sciences" such as chemistry was "intensely problematic." Shapin may be relating what he views as broad seventeenth-century categories, but if so, he is badly mistaken. In fact, the imposition of a n1eaningful distinction between alchemy and chemistry is highly anachronistic for most of the seventeenth century, and especially for Boyle. 6 Peter Dear, another recent entrant into the survey-genre, takes an approach similar to Shapin's (and to the earlier survey-writers) in his Revolutionizing the Sciences (2001). Like Shapin, Dear views alchemy as essentially secretist and bound up with the spiritual perfection of the would-be adept, unlike the emerging science of chemistry, which he anachronistically distinguishes from seventeenth-century alchemy.7 Neither Shapin nor Dear reveals any knowledge of the role that alchemy played in the growth of corpuscular theory and the formulation of Boyle's mechanical philosophy. Having stated my fundamental thesis and outlined the general view of my opponents, let me now proceed to the details of my claim. The degree to which early modern scholasticism was committed to the position that ingredients could not be retrieved from a genuine mixture has been largely overlooked by historians of science. By a "genuine mixture," of course, I refer to the Aristotelian concept of mixis-an absolutely hon10geneous combination of ingredients. In order to understand the n1eaning of Aristotelian mixis) the contemporary reader must make a conscious effort to forget the terminology of modern chemistry, which refers to
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mechanical juxtapositions of particles as "mixtures" and distinguishes such uncombined ingredients from those that have entered into a "chemical compound" joined by "chemical bonds." The language employed by chemists today reverses the terminology of Aristotle, for whom "mixture" meant a homogeneous combining of ingredients and "compound" or "composition" meant a mere juxtaposition of uncombined parts. Aristotle had claimed in Book I, Chapter 10 (328al0-12) of his Degeneratione et corruptione that genuine mixis occurred only when the ingredients of mixture acted upon one another to produce a state of absolute hOlllogeneity. Otherwise, he asserted, a sufficiently keen-sighted person, such as the classical hero Lynceus, would be able to see the heterogeneous particles that made up what had seemed to be a genuinely uniform substance. Aristotle's predecessor Empedocles had of course espoused precisely the sort of theory that Aristotle was here debunking. Empedocles had maintained a century before Aristotle that the four elements were composed at the micro-level of immutable particles, which lay side-by-side to form compounds (what chemists today would call "mixtures"). Aristotle argued that such corpuscles could only form an apparent mixture, like wheat and barley in a jar: he dubbed such illusory mixture synthesis-literally "settingtogether." Aristotle himself did not believe that the ingredients of a genuine mixture were incapable of retrieval. At De generatione et corruptione I 10 327b27-29 he argues the contrary, and his ancient followers, especially John Philoponus, spoke of separating mixtures by means of oiled sponges, river lettuce, and the like. 8 We are not speaking of the ancient commentators here, however, but rather the scholastics of the Middle Ages and their early modern heirs. The Jesuits, to name one early modern current, had adopted Thomas Aquinas as their master in theology, at the urging of Roberto Bellarmino in the 1590s (Matton 1998, 383). Hence it is no surprise to find that the great Jesuit De generatione et corruptione commentaries, such as those of Franciscus Toletus and the Coimbrans, assume an explicitly Thomistic position on the subject of mixture. Even before the Jesuits appeared on the scene, the Thomistic view had become, as Anneliese Maier has shown in her study of mixture, the majority view among scholastics (Maier 1952, 89). Like all scholastic Aristotelians, Thomas viewed matter as consisting of the four elements, fire, air, water, and earth. These in turn contained four "primary qualities"-hot and dry in fire, wet and hot in air, cold and wet in water, and dry and cold in earth. Although the pairs of these qualities, along with an undifferentiated "prime matter" (materia prima), constituted the fundamental stage of material analysis, the primary qualities were not immutable, for the hot could pass away and be replaced by cold, just as the wet could pass away and be replaced by dry. This opened the door to the possibility of elemental transmutation: if, for example, the hot
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and dry in a sample of fire were replaced by cold and wet, that portion of fire would be transmuted into ,vater. 9 But the situation was still more complicated than this, for Thomas's hylomorphism insisted that Aristotelian 'fnixis, the one type of mixture that led to a genuinely homogeneous product' could only occur if a new substantial form, called the "form of the mixture" (forma mixti), was imposed on the four elements.l° This process occurred in a well-defined series of steps. First the four primary qualities of the elements produced, as a result of their mutual action and passion, a single medial quality preserving something of the extremes; this medial quality then provided the disposition necessary for the induction of the new substantial form, the form of the mixture. Yet in such a case, Thon1as insisted, the imposition of the new form of the mixture meant that the four antecedent elements would be destroyed-the generation of the one entailed the corruption of the other. All that remained of the fire, air, water, and earth would be the primary qualities-the hot, cold, wet, and dry-that had been paired within the elements before their destruction, and which were somehow responsible for the dispositive medial quality that prepared the way for the form of the mixture. Even here it is not clear that the four qualities that remained were the original ones underlying the elements or rather similar ones that had been nevvly generated, for in general Thomas insisted that the primary qualities were accidents of the substantial form. If the substantial form itself had been newly introduced to the ingredients, then how could its accidents be the same ones that had been present before in the preexistent elements (which had now been destroyed)? As for the elements themselves, they were now present within the mixture only in virtute or virtualiter -"virtually"-as a result of the said primary qualities.l 1 To employ a distinction made in many later scholastic treatments of mixture (though not in that of Thomas), one could not get the original ingredients back out again in numero, since they had been destroyed by the very act of mixing. If one could perhaps retrieve fire, air, water, and earth that were the same as the original elements in specie, there was no guarantee that they would return in the same relative quantities in which they had entered the mixture. Mter all, the original fire, air, water, and earth had been destroyed by the process of mixture, and there was no reason to think that the primary qualities would reassemble into exactly the same pairings in proportions identical to those that they originally possessed. Hence the empirical correlation between input and output had been severed-mixture was effectively a black box linking substances with no shared material identity. A straightforward example of this viewpoint may be seen in a work ascribed-probably falsely-to the late-thirteenth century follower of Thomas, Aegidius Romanus. In his De generatione commentary, pseudo-Aegidius deals with the issue of burning wood in
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order to determine whether combustion leads to a resolution of the mixt into its elements. Pseudo-Aegidius provides an unequivocal denial of this claim. Rather than being pure elements, the smoke, fire, ash, and fluid released by burning are "certain imperfect mixts," which did not pre-exist in the unburnt wood, but were manufactured by the process of combustion. On the other hand, the liquid that visibly exudes from the ends of the logs during their combustion may actually have pre-existed there, but that is not relevant to the question, since Pseudo-Aegidius obviously considers this material to have been an alien substance lodged in the log rather than a component of the mixture making up the wood. The seen1ing elements observed during and after con1bustion of the wood are not the real elements but "mixts similar to the elements themselves," since the process of mixture has elin1inated the possibility of the original elements' return. The reason that these mixts look like the elements is that the elemental virtues remaining when the substantial form of the wood was made exercise their powers on the matter during its combustion to produce them. 12 The problem of the recapture of ingredients from a mixt became even n10re intractable when one proceeded beyond consideration of a mixture made from the simple elements to the mixture of substances that were themselves made from the elements, such as wine and water. Aristotle himselfhad briefly considered the case of mixing wine with water at Degeneratione et corruptione I 10 328a26-31. A genuine mixture of the ingredients could occur when the two components were roughly balanced, so that one was not merely transmuted into the other, but a homogeneous medial state was produced. Since Aristotle had asserted that the ingredients of a mixture could in fact be separated from one another some lines before (I 10 327b27-29), the wine-water example could then serve as a useful illustration for discussing such reacquisition.l 3 Thus we encounter the wine-water n1ixture in the Jesuit Coimbrans' commentary on De generatione et corruptione) where they refer to a statement, made by anonymous alchemists, that rushes stripped of their cortex can separate water previously mixed with wine, and to Pliny's similar claim that wine can pass through a vessel n1ade of ivy-wood, leaving the water behind. 14 But on the strength ofthe Thomistic position) how could one maintain the reality of the substantial form of the mixture while also accounting for this reacquisition of the ingredients? Clearly the numerical reacquisition of the water would have to be rejected, but again taking a cue from Thomas, the Coimbrans exclude the possibility that the separation enacted by the rushes or ivywood could dissolve the form of the mixture and produce a water that was even specifically the same as the original water. IS Hence they provide the following solution: if the liquid that separates from the wine is real water, then it cannot have undergone genuine mixture with the wine in the first place. Either the wine was too weak to exercise its power on the water, or
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there was insufficient time to do so before the separation took place. On the other hand, if the wine was sufficiently strong and there was enough tin1e for mixture to take place, but a separation was n1ade all the same, then "that which is extracted from the wine is not water, but a liquor of the wine itself, similar to water.,,16 In other words, the Coimbrans cannot admit that the separation of wine and water by rushes or ivy-wood is a genuine separation of ingredients even if the phenomena seem to indicate otherwise. The head-on approach of the Coimbrans was not the only one available to early modern scholastics, however. Another prominent Jesuit, Franciscus Toletus, took a different tack when addressing the problen1 of regaining mixed ingredients. In his con1n1entary to De generatione et corruptione, Toletus accepts the fact that one can recapture the ingredients of a mixture, but only in a very special sense. First he cites the testimony of the ancient commentator Philoponus, who argued that the ingredients can be separated and that "they return to their pristine degree, as when water is extracted froll1 infused wine by means ofart.,,17 To Philoponus's view Toletus then contrasts that of "many Latins," meaning above all the Thomistic position: "But those who say that the elements do not remain [in a mixture] except according to their virtues say that they can be separated, because the elements that were contained in the mixture in virtute can again be generated out of the mixture.,,18 Toletus has clearly shifted the discussion away fron1 the problem of separating water from wine, and has placed it on the very different issue of recapturing the four elements. According to the majority Latin view, he adds, the elements can indeed be "separated"-but only by being regenerated (and thus the new elements will of course be nun1erically different from the old ones). A few folios later, Toletus affirms this position as his own in the following words: "I say that the elements of the mixt are produced de novo during [its] resolution. "19 This statement was relatively unproblematic, since Aristotle himself had argued that the elements can undergo a "circular" generation, whereby water, for example, becomes air and then the air becomes water. 20 But does it follow that the wine can be separated from its mixture with water by a similar regeneration? It is very hard to see how Toletus could have responded affirmatively, since he accepted the Thomistic view that all the forms of the ingredients were destroyed during the process of mixture, with only the "virtues" of the elements remaining. 21 Once the higher form of the wine had disappeared, how could the wine itself be regained from the mixture? What would guide the four primary qualities to reassemble into the same proportional quantities that defined the original mixture of four elements? Despite his finessing of the issue, Toletus's position was not in the end markedly different from that of the Coimbrans.
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The Reduction to the Pristine State We now have a reasonable vantage point from which to see where things stood with scholastic matter theory around the beginning of the seventeenth century. There was a widespread consensus-especially but not exclusively among Thomistically oriented authors-that one could not regain the original numerical ingredients from a mixture, and in certain instances even recapture in specie presented problems. Let me stress again that this striking fact is not en1phasized in historical studies of scholasticism' including the laudable works of Anneliese Maier. Equally underappreciated is the fact that alchemists were already arguing against the Thomistic position and related perspectives from the thirteenth century onwards. Already at the end of the thirteenth century one finds the nascent form of an argument that would eventually prove devastating to the view that ingredients could not be recaptured. I will refer to the en1pirical phenomenon upon which this new argument was founded as the "reduction to the pristine state," a straightforward rendering of the Latin expression reductio in pristinum statum. What this usually refers to is the reacquisition of a substance after it has been combined with another. In such cases the combination with another substance generally results in the ten1porary disappearance of the first substance, and its reduction means its return to its initial uncombined state. The earliest known use of the reduction to the pristine state against the view that mixture must involve the loss of the ingredients' substantial forms occurs in a little-known Theorica et practica ascribed to a Franciscan lecturer at Assisi named Paul of Taranto. 22 Paul's brief is to defend the Arabic alchemical theory that the metals are all composed of sulfur and mercury in varying quantities, states of purity, color, volatility, and n1elting point. It is essential for hin1 that the sulfur-n1ercury theory be correct, since this theory provided the basis for his alchemical practice. A base metal was generally improved by removing some of its impure sulfur and replacing this with purified mercury. The added mercury was supposed to bond with the pre-existent mercury in the base metal, and to convert the metal into gold. Clearly this practical procedure assumed that mercury and sulfur were present within metals in a fully formed state. Hence Paul's opposition to the theory that mixture effectively destroys the forms of the ingredients being mixed followed from a fundamental tenet regarding the theory and practice of alchemy. Paul explicitly argues that the Thomistic view would make it impossible for mercury and sulfur to exist within fully formed metals, since the two principles must necessarily possess their own substantial forms in order to persist. According to the Thomistic position, such persistence of robust intermediate principles within a mixed body having its own substantial form would be excluded from the outset.
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An extraordinary chemical insight allows Paul to argue that the existence of intermediate principles can be proven by means of laboratory operations. His Theorica et practica is so little known that it will be requisite to quote the relevant passage: This is expressly proven by certain experiments of this art (experimentias huius artis) , for all metals and minerals are incinerated and calcined in their own ways, as if by the resolution of their substance they are reduced to the nature of an earth. But then they are resolved by techniques of art into a water, then into air through vapor and smoke, and presently through the resolution of their smoke they are reduced to the nature of water; then they are solidified by cooking into a powder or earth, and finally, having been fused by a strong fire, they return to their own original nature of whatever mineral body or metal. But if there were a complete resolution to the simple elements and not to certain mineral or metallic principles which are nearer than the first simpIe bodies, the metal or such and such a body would no more return fron1 them upon [its exposure] to fire than any thing else made up of the simple elements, and gold would no more return from gold than would stone or wood [return from gold], especially since fire is a comn10n agent, behaving alike towards all and each. But since these [metals and minerals] return just the same as before, it is manifest that they were only resolved to certain components of theirs and not to the simpIe elements or to the prime matter, as those foresaid [philosophers] mistakenly assert. 23
As one can see, Paul n1aintains that the metals could not be calcined at high temperature, their calces dissolved in solvents and subsequently sublin1ed, and then finally reduced back to the original metal, if the metals had been corrupted all the way up to the four elements or to the Aristotelian prime matter. Although relying on a much earlier technology ofweak acids and bases rather than the powerful parting waters available to seventeenthcentury corpuscularians and metallurgists, Paul was able to make much the same argument that figures such as Daniel Sennert and Robert Boyle would employ in the 1600s. The initial analytical agent, identified by Paul as the fire of calcination, is a "common agent" that acts the same on all materials subjected to it. Even if n1aterials react differently to it-since metal melts and wood burns-the action of the heating itself is the same in all cases. If the metals were substances that underwent a profound dissolution into their elelnents (present only in virtute in the Thomistic view) upon analysis, there is no compelling reason that the final reducing fire should then recombine those same elements to form the initial analysand, instead of wood or stone. The fact that the metals could be regained
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unchanged was sufficient evidence for Paul, then, that they had only been dissolved into intermediate principles that still accounted for the specificity of metals. Although Paul of Taranto was not a professed atomist, there is a corpuscularian emphasis throughout his work. It is the bonding and dissociation of tiny particles of mercury and sulfur, each retaining its own nature, that account for the transn1utation of metals, and more superficial metallic phenomena, such as rusting or dissolution of metals in solvents, are the result of interactions between fully formed metallic particles. This same direction, would re-emerge in the seventeenth century in the work of selfstyled "atomists" such as the Wittenberg medical professor Daniel Sennert or the much better known French reviver of Epicurus, Pierre Gassendi. Sennert and Gassendi were both serious readers of chymical texts, but Sennert pushed the reduction to the pristine state much further than Gassendi did. Indeed, Sennert developed an experiment in which silver was dissolved in nitric acid, filtered, and then reduced by means of potassium carbonate. The fact that the dissolved silver-invisible within the acid-left behind no residue demonstrated that the metal in solution must have been divided into very small particles if not actually destroyed. The subsequent reduction of the silver to its metallic state, on the other hand, ensured that it must not have been annihilated by the acid, but must have been present in the solution all along despite its temporary disappearance. Sennert meant this demonstration as an attack on the general scholastic consensus that mixture must involve the destruction of the forms of the ingredients intermediate between the four elements or four qualities and the form of the mixt, just as Paul of Taranto had employed his similar case some four centuries earlier. 24 To the follower of Thomas Aquinas, the initial dissolution of the silver in the acid might well have seemed to signify the production of a new n1ixt. Both the corrosive action of the solvent and the absence of residue left behind upon filtering would have suggested that dissolving silver in acid was something different from shaking wheat and barley in a jar. But the generation of the new form of the mixture would have to presuppose the corruption of all the preceding substantial forms, so how could one then corrupt the form of the acid-silver mixture to regain the pre-existent silver? After all, the silver had been destroyed during the process of mixture, even if its four elements or elementary qualities remained "virtually," as the Thomistic theory dictated. Assuming that the elements or elen1entary qualities could be reconstituted from their virtual state, why should the elements or qualities of the silver recombine in the same proportions that they had possessed in the original ingredients, rather than forming some new substance? "Reversible" chemical reactions provided the Thomist with a major difficulty-the problem ofrecombination. 25 In order to give a more
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concrete example, let us imagine for the sake of argument that a quantity lof silver and acid came together to produce a genuine Aristotelian n1ixture, land that some salt of tartar was then thrown into the solution with the result that the silver re-emerged. Even if the silver, acid, and base each had its own characteristic proportion of fire, air, water, and earth (or lacking this, a fixed proportion of the qualities hot, cold, wet, and dry), the elementswould be free in the solution to recombine as a whole, for the forms of the silver, aquafortis, and salt of tartar would no longer exist to direct the ensuing recombination. Since the silver and the aquafortis had both lost their substantial forms and become a single mixt under the new form of the mixture, why should their combined elements reassemble in the flask into new silver when salt of tartar was dropped in? Why should aardvarks or artichokes not emerge instead of a bright, heavy metal? In short, there is no reason to suppose that the dissociated four elements or elementary qualities would spontaneously recombine in the proportions necessary to form a n1etal instead of any other sublunary substance. 26 Sennert's reduction to the pristine state, performed with powerful and rapid-acting mineral acids rather than with the vinegar and sal ammoniac available to the medieval technology of Paul of Taranto, was able to make a strong impression upon his learned audience. The young Robert Boyle tacitly appropriated Sennert's demonstration of the existence of enduring atoms or corpuscles in his own unpublished essay "Of the Atomicall Philosophy" con1posed in the early 1650s. That work contains two descriptions of the reduction to the pristine state borrowed almost verbatim from Sennert's 1636 Hypomnemata physica. Similarly, Boyle's highly celebrated work, The Sceptical Chymist, displays a Sennertian reduction prominently at the beginning of "the First Part," where Boyle uses it to prove that corpuscles of gold and mercury can be subjected to powerful reagents without losing their nature, despite their assumption of various "disguises" before their reduction. As Boyle points out in Part I, gold, mercury, and other substances may themselves perhaps be divisible into still smaller particles, called minima naturalia, but under normal circumstances, they form semipermanent aggregates that Boyle calls prima mixta. It is precisely the reduction to the pristine state that provides Boyle with his most direct experimental evidence that such prima mixta exist beneath the threshold of sense. If Boyle had stopped at this point, his corpuscular theory would have had little to distinguish it from Sennert's, or for that matter Paul of Taranto's. But Boyle, of course, was a mechanical philosopher, not just a corpuscularian. And here, too, surprising connections with alchemy emerge. For Boyle relied on Sennert's reductions to the pristine state not merely to show that semi-permanent corpuscles existed in nature, but as part of a comprehensive strategy to reveal that most qualitative change was at basis mechanical. In
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addition to using Sennert's experiment to argue for the existence of enduring corpuscles in mixtures, he extends the reduction to the pristine state to form an integral part of his program of reducing "particular qualities" to their mechanical origins. An example of the mechanical removal and replacement of secondary qualities appears in The Sceptical Chymist, where Boyle, in the guise of the interlocutor Carneades, alters the sensible qualities of lead without removing or adding any material components: And to let you see, Eleutherius) that 'tis sometimes at least, upon the Texture of the small parts of a body, and not alwaies upon the presence, or recesse, or increase, or Decrement of any of its Principles, that it may lose some such Qualities, and acquire some such others, as are thought very strongly inherent to the bodies they Reside in; I will add to what may from my past discourse be refer'd to this purpose, this Notable Example from my Own experience; That Lead may without any additament' and only by various applications of the Fire, lose its colour, and acquire sometimes a gray, sometimes a yellowish, sometimes a red, sometin1es an amethistine colour; and after having past through these, and perhaps divers others, again recover its leaden colour, and be made a bright body.27 Here we see the "small corpuscles" or aggregate particles bearing the essential properties of lead, being subjected to color-changes by fire, which Boyle assun1es on Baconian grounds to be a mechanical agency.28 As in the reduction to the pristine state of gold dissolved in aqua regia, the initial lead bearing its original properties is recovered at the end of the experiment. This recovery ensures that no substantial change has occurred-the lead has persisted beneath the various disguises that it has assumed. In the text of The Sceptical Chymist Boyle follows this with two further experiments: the lead can be "n1ade as brittle as glass" by heating alone, and then returned to its former n1alleability; similarly, heating lead can transform it into a transparent substance, which can again be returned to its former opacity by altering "the manner and method of exposing it to the fire." In each case the lead is returned to its pristine state for the same demonstrative purpose: Boyle wants to show that the qualities imposed by heating are, as he would put it in his 1670 "History of Particular Qualities," "extra-essential. ,,29 In other words, such qualities can be induced or removed without changing the essence of the lead itself. In order to make the claim that these mechanically induced properties were not essential to but emergent from matter, however, Boyle obviously required a n1aterial substrate from which they could emerge. And since the point of his argument was that the material substrate remained essentially unchanged, he required semi-permanent corpuscles whose rearrangement, shifting, and
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other local change accounted for the changes in color. He could not invoke the prin1ary qualities of the prima naturalia here, for he had no experimental verification that such minimal corpuscles existed. Hence the empirical force of Boyle's claim that mechanical change at the macro-level induces mechanical change at the micro-level, resulting in a change of sensible quality, relied on the demonstrated existence of semi-permanent aggregate corpuscles that persisted through the induction and removal of extra-essential qualities. The appeal of the reduction to the pristine state here lies in its role as a guarantor that no substantial change has occurred in the corpuscles undergoing extra-essential modification by mechanical means. Boyle assumes, just as Sennert had done, that these aggregate corpuscles do not lose their essential nature at the beginning of a process, only to regain it unchanged at the end. If Boyle's composite corpuscles are then1selves experiencing no essential modification, then the phenomenal change that they undergo must result from an alteration in their mutual texture, brought on by association, dissociation, or transposition. The essence particular to the aggregate corpuscle remains unchanged while the qualities stemming from its texture undergo alteration if the texture changes. In this fashion, the reduction to the pristine state served Boyle not only as an empirical warrant of the reality of semipermanent corpuscles beneath the level of sense, but also as a guarantor of the mechanical nature of material change wrought by processes such as heating, cooling, rubbing, grinding, and a host of other operations.
Conclusion In this brief paper, I have tried to place the arguments of seventeenth-century corpuscularians like Sennert and Boyle within the context supplied by their rejection of Thomistic Aristotelianism. As I have argued, Sennert and Boyle were heirs to a scholastic (but not Thomist) corpuscularian matter theory that extends back to thirteenth century alchemy. Does the continuity that this research has unearthed, with its surprising revelation that Boyle's mechanical philosophy had an unlikely source in alchemy-even in its medieval incarnation-in1ply that the Scientific Revolution is itself a misnomer or a mirage? Is this another Shapinesque argument about a revolution that did not occur?30 An answer to this question quickly emerges if we consider that the theories and practices employed by Paul of Taranto and Sennert took their aim against a long-lived and massively popular reading of Aristotle based on the denial of retrievable corpuscles that endured during mixture. The view that genuine mixture could only result when the ingredients of the n1ixt were destroyed was already established
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by Thomas Aquinas in the late thirteenth century, but it reached its widest audience in the printed handbooks and commentaries of the generation directly before Boyle, particularly (though by no means exclusively) in those composed and disseminated by the Jesuits. The theory of perfect mixture and the concomitant denial of its reversibility were iconic features of a conventional scholasticism whose decisive overthrow was genuinely epoch-making. But the equally scholastic corpuscular alchemy inherited and revised by Boyle had always contained the seeds of hylonlorphism's undoing, since its structural explanations suffered an uneasy cohabitation with the peripatetic substantial form. Nonetheless, when Boyle chose to bring this awkwardness to the fore, and to highlight the advantages of a purely structural type of explanation, he was joining a new battle even if his weapons were not all of his own making. Boyle's ceaseless war on hylomorphism and his reduction of the sensible world to mechanical causes have just as much right to the name of revolution as any political event that results in the deposition of an old and established dynasty, satisfied in its ways and arrogant in its desire to limit discussion to its kingly imperatives while the world moves on. Even if the new regime established by Boyle and his peers soon gave way to Newtonian dynamics, the scholastic "dictatorship of substantial forn1" had come to an abrupt and decisive end at the hands of the mechanical philosophers. Like a conquering horde bent on executing its prisoners rather than interrogating them, the mechanical philosophers allowed only a distorted picture of the previous order to remain. One casualty of this philosophical "ethnic cleansing" was alchemy, whose intermediate role between hylomorphism and mechanism made it an irresistible target both of borrowing and of subsequent repudiation. By casting a stronger beam on premodern chymistry, we therefore acquire a better sense of both old and new, and begin to see a variety of debts and disjunctures previously undisclosed. Nor should we forget a lesson imparted by one of Michael Friedman's earliest papers-that despite the revolutionary character of seventeenth-century n1echanism, the micro-world, in the end, is just as inhospitable to mechanical philosophers as it is to Aristotelians. 31
NOTES 1. The term "mixt," though somewhat archaic, is useful for distinguishing a substance that has undergone mixture from the process of mixture itself. In this paper I will use "mixture" in the common fashion both to indicate the process of mixing and the product of mixing, but for the sake of clarification I will also occasionally employ "mixt" for the product.
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2. The first edition of Hall's The Scientific Revolution 1500-1800 appeared in 1954. 3. Dijksterhuis (1961), 440. Dijksterhuis should have known better-he was a\vare of various medieval concepts of mixture as well as alchemical corpuscular theory (200-209), suspected the influence of Daniel Sennert on Boyle (282, 436, 439), and recognized Boyle's ongoing involvement with alchemy (440). Nonetheless, in addition to his wrongheaded view of Boyle's chrysopoetic inclinations, he could still utter the following dismissive and ill-considered words about medieval alchemy (160): The \vhole subject, which originally had fitted in sufficiently with general philosophical and scientific thought to be entitled to a certain amount of understanding and appreciation, thus descended more and more to the rank of a rather unedifying record of the degeneration of science. In order to study it seriously, one must decidedly either be interested in the pathology of thought or, with e.G. Jung, be able to relate the subject to the areas of present-day depth-psychology. For the inadequacy of Jungian psychology as a tool for analyzing alchemy, see Newman (1996) and Principe and Newman (2001). 4. An exception to this tendency may be found in Henry (1997), 42-72. Henry does consider the corpuscular views of such authors in the alchemical tradition as Geber and Sennert, albeit in passing. 5. This has been commented upon at some length by Principe (1998), 107-11. 6. Shapin (1996), 6. For the unsustainability of the alchemy-chemistry distinction in Boyle, see Newman and Principe (1998), 58. The fact that Shapin is merely putting "old wine in new bottles" has also received notice from Margaret J. Osler. See the introduction to Osler (2000), 3-22, especially 19-20 and note 52. For Boyle's career-long involvement in traditional chymistry, see Newman and Principe (2002), and Principe (1998). 7. Dear (2001), 27, 53. Pace Dear, the view of the chymist Andreas Libavius as a champion of open knowledge and as an opponent of alchemy has been explicitly debunked in Newman (1999). 8. Aristotle, De generatione et corruptione at I 10 327b27-29: "it is clear that the ingredients of a mixture first come together after having been separate and can be separated again" (in the translation of E.S. Forster). For Philoponus, see de Haas (1999), 26, n. 22. 9. Aristode points out that this process has a cyclical character: if the dry in fire passes away and is replaced by wet, the fire will become air; if the hot in air is replaced by cold, the air will become water; if the wet in water is replaced by dry, the water will become earth; and if the cold in earth is replaced by hot, the earth will become fire. See Aristotle, Degeneratione et corruptione II 3-4 330a30-332a2, especially II 4 331 b2-4. 10. Maier (1952), 31-35 et passim. A much inferior study to Maier's, though still useful on certain points, is Pfeifer (1879). Thon1as's discussion of mixture may be found in Thomas (1976). As Maier points out, the corresponding section of Thomas's De generatione et corruptione commentary is interpolated. See Maier (1952),31-32.
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11. Maier, An der Grenze, 33-35. Thomas's position on mixture fit very nicely with his view that every substance could have only one substantial form (the socalled "unity of forms" theory). Nonetheless, the "unity of forms" theory did not follow necessarily from Thomas's theory of mixture, since many scholastic authors who upheld the Thomistic position on nuxture believed that one substantial form could be subordinated to another, even in a single substance. Those authors who maintained a plurality of substantial forms in a given substance often invoked the human body and soul as a case of such subordination. Although the soul was the substantial fornl of man per se, the body had its own subordinate form, which accounted for its ability to resist decomposition into d1e elements for some time after death. See Zavalloni (1951),303-81. 12. Pseudo-Aegidius Romanus (1604), 490: To the fifth it must be said that those things into which green wood is resolved are not the four elements but rather certain imperfect mixts, and that these did not pre-exist in the wood in actu, except perhaps for that humor which exits through the end [or a log], for since the virtues of the elements are present in mixts, it often happens that dissolution is made into mixts similar to the elements themselves, and this is what Aristotle says-that the elements are segregated from a mixt-although it is also not at all absurd that now one, now several, elements be generated from the mixt, which were not there before in actu. Ad quintam dicendum est illa in quae lignum viride resolvitur, non esse quatuor elementa, sed mixta quaedam imperfecta, eaque non praeexistisse actu in ligno, nisi fortasse illum humorem qui per extremitatem egreditur, nam cum in mistis sint virtutes elementorum, fit ut saepenumero dissolutio fiat in mixta similia ipsis elementis; & hoc est quod dicit Aristoteles, elementa segregari ex mixto, quanquam etiam nullum sit absurdum ex mixto generari, modo unum, modo plura elementa, quae tamen antea ibi non erant actu. The same position is upheld by the Coimbrans, for which see COlumbricenses (1606), book 1, chapter 10, question 3, article 3, 373-74. 13. A clear treatment of the wine-water issue in Honore Fabri may be found in des Chene (2001), 363-79. 14. Conimbricenses (1606), book 1, chapter 10, question 1, article 1, 361. The reference to rushes used by alchemists to separate a wine-water mixture derives from Thomas Aquinas's commentary on Peter Lombard's Sentences. See Thomas, vol. 4, distinctio XI, expositio textus, 489: "Alchimistae dicunt quod per immissionem junci decorticati potest [mixtio] separari." See Pliny, book 16, chapter 63, 488 (Latin), 489 (English). The same claim is made by Cato (1935), chapter Ill, 100-103. 15. The Coimbrans' inspiration here is again book 4, distinction 11 of Thomas's Sentence-commentary, to which they explicitly refer. Thomas says there (Thomas 1947, vol. 4, 489-90): "And it must be said that a substance is removed from the wine, because of the similarity of [the substance] with the rush, which is also porous; [the substance] was [already] there, but yet the water that was added [to the wine] is not separated, because it has already been made wine (Et dicendum quod a vino aufertur substantia quae ibi est, propter similitudinem junci quod etiam porosum est; non tamen illa aqua quae fuit apposita, separatur, quia jam facta
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est vinum)." It may seem odd at first face that Thomas did not consider the regeneration of water from its reassembled primary qualities here. It is likely, however, that he vievved the attempts at separation made by alchemists as "artificial" and hence insufficiently powerful to alter the wine's substantial form fundamentally. This is his position on the production of "artificial waters" (such as rosewater) by alchemical distillation. Although such fluids may SeelTI to be real water, the process of distillation in a flask cannot really resolve them to their elements, unlike the cycle of evaporation and condensation in nature. Hence, in Thomas's vievv, rosewater lacks the substantial form of water. Similarly, the substantia that is separated from vvine by alchemists is not water. The Coimbrans took this substantia to be "a liquor of the vvine itself similar to water." See the following footnote for their analysis and reference to Thomas. See Newman (2004), 94-95 for Thomas's discussion of "alchelnical waters." 16. Conimbricenses (1606), book 1, chapter 10, question 1, article 2, 363: Wherefore, because they say tl1at water can be separated from wine, it must be said that if that which is separated is true water, the separation can occur because the wine is so weak that it cannot convert the water into itself, or else both [the wine and water] are so related mutually that a third thing does not result fron1 them. Or because the wine, even if it could convert the water into itself did not have the time to do so. But where none of these circumstances are present, it must be asserted that that which is removed is not water, but a liquor of the wine itself similar to water, not yet, to be sure, thoroughly decocted, which can even be extracted from pure wine, as blessed Thomas has noted in the literal exposition in book 4, distinction 2 [of the Sentences]. Quapropter quod aiunt posse aquam a vino excerni, dicendum si id quod separatur, revera aqua sit, earn excretionem posse fieri, cum vinun1 ita est imbecillum, ut nequeat convertere in se aquam; vel & ambo ita se invicem habent, ut ex ipsis non resultet tertium. Vel cum vinum, etsi possit in se convertere aquam; nondum tamen ei per tempus licuit earn in se convertere. Ubi vero sese istiusmodi circunstantiae non interponunt, asserendum est id, quod a vino eximitur non esse aquam, sed liquorem ipsius vini aquae similem, nondum videlicet exquisite decoctum, qui etiam ex puro vino elici potest; ut annotavit D. Thomas in 4. Dist. II. in expositione litera[e]. 17. Toletus (1602), fo1. 54r: "Philoponus explains that miscibles can be sepa-rated, which sometimes are separated and [then] return to [their] original degree, as when water is extracted by art from wine that has been poured in" (Explicat Philop. quod possunt separari miscibilia, quae aliquando separantur, & redeunt ad pristinum gradum, ut cum arte extrahitur aqua, vino infuso)." For a current study of Philoponus on mixture, see de Haas (1999). 18. Toletus (1602), fo1. 54r: "At vera qui dicunt non manere elementa nisi secundum virtutes, dicunt posse separari, quia ex mixto rursus elen1enta generari possunt, quae virtute continebantur in mixto." 19. Toletus (1602), fo1. 59r: "Ad septimum dico, quod elementa mixti resolurione de novo producuntur, ut alibi dicemus, & ex parte iam diximus." See also his comments on the preceding folio (58v): "I say also that they are separable because the elenlents can again be generated from the mixt, just as the mixt is generated
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from the elements" (Dico etiam, quod separabilia sunt, eo quod ex nuxto possunt elementa generari rursus, sicut ex elementis generatum est mixtum)." 20. See Toletus's discussion of this "circular generation" at his 1602, fo1. 93v. 21. Toletus, De generatione et corruptione, fo1. 60v: Fron1 all these things I gather that the elements do not remain according to their own forms: and therefore they do not remain in the mixture in actu) but they remain according to their matter and tempered virtues, and the form of the nnxture succeeds in the place of these forms of the elements, embracing these virtues eminently, and thus they remain in virtute and in potentia. Ex omnibus his colligo elementa non manere secundum formas proprias: ac propterea non manere actu in mixto, manere auten1 secundum materias, & virtutes temperatas, & loco illarum formarum elementorum succedere formam nuxti, virtutes illas eminenter complectentem, atque ita manere virtute, & potentia. 22. Paul of Taranto is a very dark star in the history of scholasticism. See Newman (1985a) and Newn1an (1985b). 23. Paul of Taranto, Theorica et practica, in Newman (1986), vol. 3,40-41: "Hoc auten1 expresse probatur per quasdam experientias huius artis, nam metalla et nnneralia omnia suis incinerantur et calcinantur modis, quasi per resolutionem eorum substantie reducantur ad terre naturam. Deinde autem per artis ingenia resolvuntur in aquam, deinde in aerem per vaporem et fun1um, moxque per ipsius resolutionem funn reducuntur in aque naturam, et deinde per decoctionem solidantur in pulverem sive teram, ultimo vere per ignem validum fusa redeunt ad propriam naturam priorem cuiuscunque mineralis corporis sive metalli. Si igitur plena esset resolutio ad elementa simplicia et non ad principia quedam nnneralia seu metallica que sunt citra prima simplicia corpora, iam ex eis ad ignem non n1agis rediret metallum vel tale sive tale corpus, quam aliud de compositis ex elementis simplicibus, et non magis ex auro rediret aurum quam lapis vel lignum, presertim cum ignis commune agens sit, ad omnia et equaliter se habens ad singula. Quoniam autem redeunt hec eadem sicut prius, manifestum est ea ad quedan1 sua componentia tantum resoluta fuisse et non ad elementa vel ad primam materiam ut mentiuntur prefati." 24. Sennert's argument is directed not only against the Thonnsts and Scotists, but also ag~inst certain Averroist thinlcers, lilce Jacobus Zabarella, who argued for the permanence of the elements in a mixture but denied the persistence of higher forms. In this short paper it is not been possible, of course, to present more than a bare- bones and selective outline of the different scholastic schools of thought on nnxture. In addition to Maier (1952)) the reader can also consult Newman (2006). 25. I use the term "reversible" here to mean any series of simple inorganic reactions that can be reversed easily to regain the initial ingredients. This does not correspond to the sense in which modern chemists use the expression, namely to mean a reaction that never comes to completion in a closed system. In short, my use of "reversible reaction" has nothing to do with the modern notion of chenncal equilibrium, but rather with the alchemists' rebuttal of the strict Aristotelian concept of "perfect mixture," according to which (at least in the eyes of the major scholastic schools of thought) there was no possibility of reversing the process that we now refer to as a chenncal reaction.
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26. A variant of this problem is described in Pyle (1995),308-309. In fact, the return of the immediate silver after the destruction of its form would also have violated an important scholastic maxim, that "a return from privation to a habit is not given according to nature" (a privatione in habitum non potest fieri regressus secundum naturam ). A nice discussion of this principle, heavily dependent on Toletus, is found in Stahl (1645), 406-409. I treat the significance of this rule at length in Newman (2006). 27. Boyle (1999-2000), vol. 2, 342. Boyle's description of lead was probably inspired by Gassendi. See the following passage from Gassendi's Syntagma philosophicum, in Gassendi (1964), vol. 1, 380A: In which place what the chymists IG10W is marvelous-[namely] that lead, though it is a thing so dense and opaque, where it is urged with a very vehen1ent fire while it is blown upon, changes, is woven, is fashioned, into a certain appearance almost like that of a hyacinth, which is very transparent; not only does rarity contribute to this perspicuity, but also a certain disposition of its parts. I omit that the lead [placed back] in the fire again reclaims its own appearance, by an inverse disposition of parts. Quo loco mirabile est, quod Chymici norunt, plumbum, cum res adeo densa, & opaca, sit, ubi urgetur igne vehementissimo, intereaque insufflatur, abire, contexi, conformari in quandam quasi hyacinthi speciem, quae sit perspicua admodum; tantum ad perspicuitatem valet non raritas modo, verun1 etian1 partium certa dispositio. Omitto autem, ut igne rursus suam plumbi speciem repetat, inversa rursus dispositione partium. 28. Heating was a purely mechanical operation to Boyle, on grounds that he inherited from Francis Bacon. See Boyle (1999-2000), vol. 8, 343-44. 29. Boyle (1999-2000), vol. 6, 280. The "History of Particular Qualities," was originally published as part of Boyle's Cosmical Qualities in 1670 30. I have in mind the opening words of Shapin (1996), 1: "There was no such thing as the Scientific Revolution, and this is a book about it." 31. I refer to the "familiarity condition" imposed by Boyle's mechanical philosophy, according to which an explanation acquires its force by virtue of explaining the phenomena in terms of more familiar phenon1ena. As Michael Friedman points out, this criterion does not apply to quarks or many other entities. of the modern physicist's world. See Friedman (1974).
REFERENCES [Pseudo-] Aegidius Romanus (1604). Commentationes physicae et metaphysicae. Ursel: Jonas Rhosius. Boas [Hall], M.(1952). "The Establishment of the Mechanical Philosophy." Osiris 10: 412-541. Boyle, R. (1999-2000). The lVorks ofRobert Boyle. Edited by M.Hunter and E.B. Davis, 14 vols. (London: Pickering and Chatto). All references to Boyle are taken from this edition.
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Butterfield, H. (1949). The Origins of Modern Science 1300-1800. New York: Macmillan. Cato (1935). On Agriculture. Can1bridge, MA: Harvard University Press. Conimbricenses (1606). Commentarii collegii conimbricensis societatis iesu in duos libris de generatione et corruptione Aristotelis Stagiritae. Lyon: Horatius Cardon. Dear, P. (2001). Revolutionizing the Sciences: European I(nowledge and its Ambitions) 1500-1700. Princeton: Princeton University Press. des Chene, D. (2001). "Wine and Water: Honore Fabri on Mixtures." In Liithy, Murdoch, and Newman (eds.), 363-79. Dijksterhuis, E.J. (1961). The Mechanization of the World Picture. Oxford: Oxford University Press. Translated from the 1950 De Mechanisering van het Wereldbeeld. Friedman, M. (1974). "Explanation and Scientific Understanding." Journal of Philosophy 71: 5-19. Galison, P. and E. Thompson (eds.) (1999). The Architecture of Science. Cambridge, MA: MIT Press. Gassendi, P. (1658). Petri Gassendi Dinensis . . . opera omnia. Stuttgart-BadCannstatt: Friedrich Fromann. Facsimile reprint of Lyon (1658). Greiner, F. (ed.) (1998). Aspects de la tradition alchimique au XVlle siecle. Paris: S.E.H.A. Haas, F.A.J. de (1999). "Mixture in Philoponus. An encounter with a Third I
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Nevvman, W.R. and A.Grafton (eds.) (2001). Secrets of Nature: Astrology and Alchemy in Early Modern Europe. Cambridge, MA: MIT Press. Ne,vman, W.R., and L.M. Principe (1998). "Alchemy vs. Chemistry: The Etymological Origins of a Historiographic Mistake." Early Science and Medicine 3: 32-65. Nevvman, W.R., and L.M. Principe (2002). Alchemy Tried in the Fire: Starkey) Boyle) and the Fate of'Helmontian Chyl1/j,istry. Chicago: University of Chicago Press. Osler, M.J. (ed.) (2000). Rethinking the Scientific Revolution. Cambridge: Cambridge University Press. Pfeifer, X. (1879). Die Controverse iiber das Beharren der Elemente in den Verbindungen von Aristoteles bis zur Gegenwart) Programm zum Schlusse des Studienjahrs 1878/79. Dillingen: . t. \dalbert IZold. Pliny (1968). Natural History. Cambridge, MA: Harvard University Press. Principe, L.M. (1998). The Aspiring Adept: Robert Boyle and his Alchemical Quest. Princeton: Princeton University Press. Principe, L. M. and W.R. Newman (2001). "Some Problems with the Historiography of Alchemy." In Newman and Grafton (eds.), 385-431. Pyle, A.(1995). Atomism and its Critics. Bristol: Thoemmes Press. Shapin, S. (1996). The Scientific Revolution. Chicago: University of Chicago Press. StaW, D. (1645). Axiomata philosophica sub titulis xx. comprehensa. Cambridge: Roger Daniel. Thijsen, J.M.M.H. and H.A.G. Brakhuis (eds.) (1991). The Commentary Tradition on Aristotle)s ceDe generatione et corruptione)). Turnhout: Brepols. Thomas Aquinas (1947) S. Thomae Aquinatis ... scriptum super sententiis magistri Petri Lombardi. Edited by Maria Fabianus Moos, 4 vols. Paris: P. Letheielleux. - - - . (1976). De mixtione elementorum in Sancti Thomae de Aquino opera omnia. Rome: Editori de San Tommaso, vol. 43, 127-30. Toletus, F. (1602). D. FrancisciToleti societatis iesu) nunc s.r.e. cardinalis ampliss. commentaria una cum quaestionibus) in duos libros Aristotelis de generatione et corruptione. Venice: Iuntae. Zavalloni, R., O.F.M. (1951). Richard de Mediavilla et la controverse sur la pluralite des formes. Textes inedits et etude critique. [Philosophes medievaux. Tome ILJ Louvain: Editions de l'institut superieur de philosophie.
[4] Newton as Historically-Minded Philosopher MARY DOMSI(ll
1. Introduction Newton began his career as an enthusiastic student of Cartesian philosophy and mathematics, as the papers from his early studies at Cambridge make clear. However, the papers from the late 1660s onwards reveal a change of heart towards Descartes's mechanical system of nature. For instance, in the unpublished De Gravitatione, penned son1etime before publication of the Principia mathematica (1687),2 Newton questions Descartes's relationist conception of space and time, and in the Principia, Newton proposes his alternative "absolutist" account of space and time in opposition to Descartes. 3 Newton also lays bare the faults of Descartes's vortex theory of planetary n10tion in the General Scholium that was added to the second (1713) edition Principia as he argues for the superi0rity of his new system of the world. More generally, Newton's proposal of an immaterial and universal force of gravity indicates his rejection of the Cartesian reduction of nature to extended matter in motion, and the very title of Newton's masterpiece-The Mathematical Principles of Natural Philosophy-signals his intent to replace the mechanical view of nature, and along with it, the system of natural philosophy forwarded in Descartes's Principles ofPhilosophy (1644). 4 In the realm of mathematics, the scene is no different. As a young student, Newton pored over the novel system of geometry forwarded by Descartes in the Geometrie (1637); but from the early 1670s onwards, Newton censures the Cartesian approach to algebra and geometry that he mastered in his early studies. This anti-Cartesian attitude in mathematics characterized the remainder of Newton's life and works, as evidenced by the following report from Henry Pemberton, the editor of the third (1727) edition of the Principia:
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I have often heard hin1 censure the handling of geometrical subjects by algebraic calculations; ... and he frequently praised Slusius, Barrow and Huygens for not being influenced by the false taste, which then began to prevail. He used to commend the laudable attempt of Hugo de Omerique to restore the ancient analysis, and very much esteemed Apollonius' book De sectione rationis for giving us a clearer notion of that analysis that we had before ... Sir Isaac Newton has several times particularly recon1mended to me Huygens's style and n1anner. He thought him the most elegant of any mathematical writer of Il10dern times, and the n10st just imitator of the ancients. Of their taste and form of demonstration Sir Isaac always professed himself a great admirer: I have heard hin1 even censure himself for not following them more closely than he did; and speak with great regret of his mistake at the beginning of his mathematical studies, in applying himself to the works afDes Cartes and other algebraic writers before he had considered the elements of Euclide with that attention, which so excellent a writer deserves. (Pemberton 1728, Preface) A mistake to begin his mathematical career with Descartes? This is certainly a curious remark, since it's generally agreed that Newton would not have struck upon the startling advances in mathematics that mark his miracle decade of the 1660s had he not started his mathematical career with close study of Cartesian analytic geometry.5 To be sure, study of Cartesian mathematics could not alone place Newton on the path to his fluxional calculus. Given the inner limitations of Descartes's canon of mathematics-a canon that precluded treatment of the infinite and infinitesimal quantities in geon1etry-Newton needed to appeal to other, non-Cartesian mathematical techniques in order to advance mathematics as far as he did. 6 However, perhaps even more curious than calling his early study of Descartes a mistake, Newton does not cite such inner limitations as the reasons for his regret. The issue for Newton was not one of mathematical power. Consistent with Pemberton's account, the basic thrust of Newton's criticism against Cartesian analytic geometry is that this new style of mathematics violates the standards of mathematical practice embraced by Euclid, Archimedes, and other ancient mathematicians. For those of us who worry about how to situate Newton in the philosophical context of the modern period, Newton's arguments against Descartes's approach to geometry offer us a puzzle and a challenge. For from the early 1670s onwards, as Newton makes his distaste for Descartes's geometrical style clear, he invokes an argument strategy that is perhaps disappointing. Rather than offer a systematically laid out philosophy of mathematics that can serve as an alternative to Descartes's rationalist system, we instead get commentaries that suggest Newton is not worried about pin-
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pointing the criteria that guarantee the certainty of mathematical knowledge; he takes the certainty yielded by ancient mathematics as his given. Based on this strategy for argument-one that may appear to be no more than a strong yet invalid argun1ent from authority-we may be tempted to dravv the unsavory conclusion that Newton was simply not as adept at practicing philosophy as he was at practicing natural science or mathematics. For clearly, with no explicitly stated first principles or criteria for true mathematical knowledge, Newton's arguments against the Cartesian approach to n1athematics fail to live up to the standards of "doing philosophy" that characterize Descartes's manner of argumentation-and also, . in large degree, characterize the way in which twenty-first century philosophy is practiced. I concede this point: Newton's style of philosophy does not meet the standards of Descartes's style of philosophy. However, given that Newton rejected many of the tenets central to Cartesian natural philosophy in his later career, this should hardly come as a surprise. For as his career drew on, Newton was not attempting to emulate Descartes's practice of natural philosophy; he was attempting to replace it. Ifwe thus embrace the "nonsystematic" manner in which Newton presents his philosophy of mathen1atics as part and parcel of his general distaste for Cartesian natural philosophy, we circumvent the unsavory conclusion that the absence of first principles or an epistemological foundation for mathematical knowledge in Newton's arguments signals philosophical naivete, or worse, philosophical failure. Instead, by accepting Newton's appeal to ancient authority as central to how he practices philosophy, we gain insight into an alternative, anti-Cartesian manner of doing philosophy that eschews the primacy of "clear and distinct" ideas before the mind and instead embraces history as the foundation for argument. 7 Before examining Newton's arguments more carefully, I'll begin with a brief overview of those aspects of Descartes's n1athen1atics that Newton would come to find so worrisome in the later part of his mathematical career.
2. Cartesian Mathen1atics Taking off from the analytic trends in mathematics that took shape in the late sixteenth century, Descartes offers aground-breaking system of geometry in the Geometrie, a work included with the anonymously published Discourse on Method (1637).8 A primary achievement of this short work is Descartes's novel interpretation of the relationship between curves constructible in the geometrical plane and the variables, equations, and operations of the abstract framework of algebra. 9 In order to solidify the
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relationship between these two fields of mathematics, Descartes clarifies what he takes to be the ambiguities surrounding the ancient classification of curves, and in doing so, he redefines the domain of geometry. 10 In particular, Descartes establishes two criteria for "geometrical" curves, i.e., curves that can be employed in the solution to a geometrical problem and therefore belong in the proper domain of geometry. 11 As we'll see below, according to Descartes's criteria, the status of a curve as "geometrical" ultimately hinges on whether the curve is geometrically intelligible, i.e., on whether the curve's construction can be clearly and distinctly perceived. In the first instance, he asserts that a curve is "geometrical" if and only if the curve can be constructed by "continuous and successive motions" (Descartes 1925, 43). Taking into account the various construction techniques invoked in the Geometrie, Henk Bas has helpfully generalized this criterion for geometrical curves as follows: according to Descartes, a curve is admissible into geometry's domain if and only if the curve is constructible by generalized straightedge and compass (cf. Bas 2001; Bos 1981,298). To clarify "Thy such constructions are intertwined with geometrical intelligibility, we can turn to one of Descartes's examples of a construction that proceeds by "continuous and successive motions." In the first example of Book 2, he presents an infinity of rulers (BC, CD, DE, and so on) that are connected to an instrument YZ (Figure 1; Descartes 1925,45-47). Initially, YX is laid upon YZ, and the points B, C, D, E, F, and H all coincide at point A. IZeeping YZ stationary, YX is moved so that the angle XYZ increases. As a result of this motion, the ruler BC pushes the ruler CD, which pushes the ruler DE, and so on, and the "continuous and successive motions" of these rulers thereby generates the curves AB, AD, AF, and AH. Though these curves can be ranked in their complexity-with AD more complex than a circle, AF n10re complex than AD, and AH the most complex of all the curves-Descartes asserts that he sees no reason why the description of [AD] cannot be conceived as clearly and distinctly as that of the circle, or at least as that of the conic sections; or why that of[AF, AH], or any other that can be thus described, cannot be as clearly conceived as the first; and therefore I see no reason why they should not be used in the same way in the solution of geometric problems. (Descartes 1925, 47) By requiring that "continuous and successive motions" be used in the construction of a curve, Descartes secures the intelligibility of the curves generated. For given the steps laid out above, we "clearly and distinctly" perceive the motion of the rulers whereby the curves are constructed: one ruler pushes the next which pushes the next and so on. The human mind
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FIGURE 1.
can follow these sorts of motions, and can thus clearly and distinctly conceive the resultant curves. However, according to Descartes, the human mind cannot clearly conceive the simultaneous motions of straight and curved lines that vary in velocity; the relationship between these velocities can only be known approximately.12 As a consequence, curves constructed by "two separate movements whose relation does not admit of exact determination," such as the Archimedean spiral and the quadratrix, are not geometrically intelligible and are therefore inadmissible in the domain of geometry.l3 Borrowing his terms from the ancients, Descartes calls this class of curves "mechanical." Though Descartes begins his discussion with an account of the proper modes of construction, the heart of his system lies in his account of how geometry and algebra are related. Namely, in Cartesian analytic geometry, a geometrical curve has a corresponding algebraic expression from which we can gather information about the curve. In particular, we can appeal to a curve's algebraic representation, rather than its construction in the plane, in order to rate its simplicity. As he states, ... I think the best way to group together all [geometrical] curves and then classify them in order, is by recognizing the fact that all points of those curves which we may call 'geometric', that is, those which admit of precise and exact measurement, must bear a definite relation to all points of a straight line, and ... this relation must be expressed by means ofa single equation. (Descartes 1925, 48) Descartes distinguishes between those curves that can be represented by a closed (finite) polynomial equation and those that cannot. The former are those we now call "algebraic," and the latter, which are represented by an indefinite equation, are those we term "transcendental.,,14 Descartes's stated goal is to pr9ve that the class of "algebraic" curves is coextensive with the class of "geometrical" curves, i.e., that those curves that can be represented by a closed polynomial equation are the same curves that can be constructed by generalized straightedge and compass. Though he did
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not prove that "geometrical" and "algebraic" curves are coextensive,15 Descartes's system of 'analytic' geon1etry is nonetheless grounded on an assumed intimacy between geometry and algebra such that the possibility of representing a curve by means of a finite polynomial equation indicates whether the curve is in fact geometrically intelligible. For Descartes's unproven yet assumed claim is that if a curve can be represented by a finite polynomial equation, then the curve is constructible by generalized straightedge and compass and thereby has a rightful place in geometry's domain. Even in the absence of a proof for this claim, the ground-breaking status of what Descartes proposed in the Geometrie shouldn't be underestimated. By capitalizing on the relationship between algebra and geometry, Descartes opened the way for the analytic treatment of curves and made room for new techniques by which, for instance, to find the tangent to a curve and to determine the area of a surface subtended to a curve-techniques not recognized by Descartes himself but developed and embraced by Newton in his early mathematical studies. To be sure, the innovativeness of Descartes's approach to algebra and geometry was not lost on Newton, as made clear by the mathematical papers Newton penned in the early 1660s. However, the commentaries of the 1670s and beyond make it equally clear that, whatever Newton's level of expertise in Cartesian geometrical analysis, he was entirely unwilling to subscribe to the relationship between algebra and geometry that Descartes took as foundational to his analytic form of mathematics.
3. Newton's Appeal to Ancient Mathematical Wisdom The earliest signs of Newton's distaste for the underpinnings of Cartesian analytic geometry emerge in the early 1670s, a period in which Newton evidently developed a greater appreciation for the wisdom of the ancients.l 6 In his early Lectures on Algebra (1673-83)17, for instance, he remarks, Multiplications, divisions and computations of that sort have recently been introduced into geometry, but the step is ill-considered and contrary to the original intentions of this science: for anyone who examines the constructions of problems by the straight line and circle devised by the first geometers will readily perceive that geometry was contrived as a means of escaping the tediousness of calculation by the ready drawing of lines. Consequently these two sciences ought not be confused. The Ancients so assiduously distinguished them from the other that they never introduced arithmetical terms into geometry;
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while recent people, by confusing both, have lost the simplicity in which all elegance in geometry consists. (MP, 5.429)18 This remark is clearly aimed at Descartes, who, at the very outset of the Geometrie, introduces arithmetical terms into geometry "for the sake of greater clearness" (Descartes 1925, 2-5). On Descartes's reading, the ancients' inability to put arithmetical terms to use in geometry made it impossible for them to "see clearly the relation between the two subjects" and "caused them much obscurity and embarrassment, in their attempts at .explanation" (Descartes 1925, 21). Newton, on the other hand, takes the ancient distinction between algebra and geometry to be philosophically motivated; on his reading, they were reluctant-not unable-to introduce arithmetical terms into geometry, because the two sciences were not to be confused. On this score, then, Descartes has both misunderstood and violated the ancient practice of mathematics, giving Newton grounds for rejecting the Cartesian approach to geometry and algebra. To be clear, Newton doesn't question that algebra can service the geometer, or vice versa; he most certainly understands the usefulness of appropriating techniques from different fields and does so himself with great ease. 19 But integrate mathematical techniques as he may, we see from the above passage that Newton questions the collapse of the distinction between geometry and algebra, because such a collapse tarnishes the simplicity of geometrical solutions. 2o In general, Newton insists that algebra and geometry should retain their independence, and as such, neither discipline should be granted a prescriptive role over the other. For instance, Newton argues that the classification of geometrical curves should be grounded on the descriptions of these curves in the plane, not-as Descartes has it-on the dimensions of their corresponding algebraic equations (MP, 5.423-25). In this regard, Newton writes, In constructions which are of equal geometrical rating the simpler ones are always to be preferred. This law overrides all exception. On the simplicity' indeed of a construction the algebraic representation has no bearing. Here the descriptions of curves alone come into the reckoning. This consideration alone swayed the geometers who joined the circle with the straight line. According as these descriptions are easy or difficult the construction is rendered easy or difficult. It is hence alien to the nature of the subject to prescribe laws for constructions on any other basis. (MP, 5.426-27, emphasis added)21 When it comes to the construction of geometrical curves in the plane, our attention ought, according to Newton, remain focused on construction
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procedures. Algebra should have no normative force over geometrical matters, and thus the question of whether a curve can be represented by a finite equation does not come to bear on the status of a curve as geometrically intelligible. Though we may classify "algebraic" curves as those having a corresponding closed polynomial equation, no criterion based on this algebraic representation should determine the proper solutions or the proper scope of geometry. On what basis, then, are "ve to determine which curves belong in geometry's domain? As Newton suggests above, constructions playa key role in ancient geometry, and serve as the basis for determining the simplicity of a curve. However, even constructions-or the limitations of construction techniques, to be more precise-do not have the final word when it comes to determining the sorts of objects that the geometer can treat. The ultimate arbiter of geometry's domain is the mathematician; put simply, if the geometer can solve a problem using a particular curve, then we can rightly deem the curve to be geometrically intelligible. And importantly, the question of whether a curve can in fact be used in the solution to a problem is not, for Newton, intertwined with the construction of the curve by generalized straightedge and compass as it is for Descartes. For instance, in the unpublished tract Geometria (ca. 1692), Newton claims that when the geometer encounters a figure not constructible by straightedge and compass, such as a solid, the geometer describes the construction in reference to "any manual operation which shall seem simplest." He then continues, But the geometer does not order that operation in composition-he merely hints it as a possibility, or proposes it hypothetically and as a species of theorem, or deduces it from the assumption of what is required, or finally assumes it granted in the circun1stances of the problem. (MP, 7.302-303) With no demand that construction procedures be made explicit in the solution to a problen1, the curves the geometer can treat and the curves that can be mechanically constructed are not two sides of the same coin. On this point, Newton again calls upon the "wisdom of the ancients," and the wisdom of Archimedes in particular. Urging us to follow the ancients in order that we "keep geometry unspoiled and at the same time practice it whole," Newton describes ancient geometry as follows: [Geometry] demands that operations for the construction of easier problems be in the power of all its disciples and consequently commands at will that they be performed; it thinks it absurd, however, to postulate that novices should control the more difficult operations, and
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accordingly does not dictate these for all but n1erely takes them within its scope and obliquely propounds them for the use of those who are more proficient and have acquired in themselves a mastery over them. It solves all problems, some in a practical manner, the others speculatively, some by laying down precepts and the rest by instruction without precept. Geometrical and mechanical construction differ solely in this) that geometry commands the former and hence has it in its powe1) while it has no power over the other but merely speculates upon it. (MP, 7.303, emphasis added) On Iny reading of these passages, Newton allows the geometer to hint at ithe possible modes of construction, present construction techniques indirectly, or simply assume a construction as given when attempting to solve a lproblem. 22 As such, whether the geometer has laid out reliable rules for constructing a curve-or whether the geometer has laid out any rules at all-she can still solve problems using the curve in question, and the curve therefore earns a rightful place in the domain of geometry. Thus, Newton declares, "Geometrical constructions . . . are ones which are achieved by means of straight lines and circles and any figures whatever in the plane"; he then adds, "for, of course, figures given in the circumstances of a question are in the geometer's power" (MP, 7.303, emphasis added). 23 In contrast then to Descartes's treatment of "geometrical" curves, constructability by means of idealized instruments does not detern1ine which objects n1ay be accepted into geometry. In the absence of well-defined rules for construction, Newton's geometer is free to speculate about the possible rules for the construction of a complex curve in the plane, and the possibility-the real possibility-of generating the curve by the rules proposed is a matter only for the practical mechanic. These claims are clearly connected to Newton's general concern with utility and "practicality"-or perhaps practicability, to be more accurate; for here we have a case where what the n1echanic can do in practice with instruments does not impinge on what a geometer-an ancient geometer-does in practice with ideal mathematical objects. 24 Clearly, there is a question of interpretation that lingers in the background of Newton's dispute with Descartes. Descartes finds limitations and ambiguities in the ancient treatment of curves and presents his system as one that will clarify the relationship between geometry and algebra and thereby remedy the supposed faults of the ancient system. Newton, on the other hand, takes a much more positive view of ancient mathematics and sees in their work a pristine knowledge of mathematics that should be preserved. For both, then, ancient n1athematics serves as the starting point, but they clearly disagree on whether the foundations of ancient mathematics deserve emulation or modification. Unfortunately, it's difficult, if
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not impossible, to settle the issue of who has correctly interpreted ancient mathematics, in large part because there are ambiguities inherent in Pappus's Mathematical Collections (fourth century B.C.E.), the text on which both Descartes and Newton relied for their account of ancient mathen1atical practice. 25 As brought out by commentators such as Bos (2001), Cuon10 (2000), Jones (1986) and Molland (1976), Pappus's remarks concerning the ancient interpretation of the relationship between geometry and algebra and the classification of "geometrical" curves can be-and have been-read very differently. Even with the lingering problem of textual interpretation, the conflicting attitudes that Descartes and Newton take towards their historical ancestors in mathen1atics shed light on their conflicting attitudes towards the practice of philosophy and, in particular, the role they grant history in their philosophical investigations. I turn to this issue in the next section.
4. The Role of History in Philosophy The Geometrie, along with the Meteorology and Opticks, were included together at the end of Descartes's Discourse on Method, which has as its full title Discourse on the Method of rightly conducting one)s reason and seeking truth in the sciences (1637). Descartes refers to the shorter tracts as Essays in this Method, and, as he writes in a letter to Mersenne (February 1637), they contain truths that could not have been discovered without the method he describes in the six part Discourse (CSMIZ 53; AT I 349).26 In Part I of the Discourse, Descartes offers an autobiographical account of how he discovered his method for "rightly conducting" his reason and "seeking truth in the sciences." According to Descartes, his success in finding this method rested, in part, on his withdrawal froin academic subjects, among which included history. Though he initially claims that "the memorable deeds told in histories uplift [the mind] and help to shape one's judgment if they are read with discretion" (CSM 1 113; AT VI 5), as he embarked on his search for truth and certainty in the sciences, he distanced himself from the fields of study he embraced as a student and turned a skeptical eye towards the usefulness and accuracy of history. He writes, But I thought I had already given enough time to languages and likevvise to reading the works of the ancients, both their histories and their fables. For conversing with those of past centuries is much the same as traveling. It is good to know something of the customs of various peoples, so that we may judge our own more soundly and not think that everything contrary to our own ways is ridiculous and irrational, as
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those who have seen nothing of the world ordinarily do. But one who spends too n1uch time traveling eventually becomes a stranger in his own country; and one who is too curious about the practices of past ages usually ren1ains quite ignorant about those of the present. Moreover, fables make us imagine many events as possible when they are not. And even the most accurate histories, while not altering or exaggerating the importance of matters to make them more worthy of being read, at any rate aln10st ahvays omit the baser and less notable events; as a result, the other events appear in a false light, and those who regulate their conduct by examples drawn from these works are liable to fall into excesses of the knights-errant in our tales of chivalry, and conceive plans beyond their powers. (CSM I 113-14; AT VI 6-7) Based on this account, the study of history could not service Descartes in his search for a method by which to distinguish truth from falsehood. For as indicated above, historical reports not only distance us from our present state of affairs; they are inherently flawed and should not be trusted as accurate representations of the past,27 History, therefore, could not supply the firm foundations he was seeking. Having discredited the study of academic subjects as reliable sources for "true principles," Descartes chooses to undertake "studies within himself" and aims to construct his own foundations for the sciences. He withdraws from academic studies and worldly events in order to "uproot [his] mind from false opinions," and in doing so, he relies on the ability of the individual-the single man-to recognize clear and distinct perceptions that are presented to the mind. Via this turn towards the mind and away from the world, the firm foundations of knowledge were attained. The centrality of introspection is expressed in the first rule of Descartes's method for directing his reason towards true principles: The first [rule] was never to accept anything as true if I did not have evident knowledge of its truth: that is, carefully to avoid precipitate conclusions and preconceptions, and to include nothing more in my judgements than what presented itself to my mind so clearly and so dis'tinctly that I had no occasion to doubt it. (CSM 1 120; AT VI 18) The method described in this rule shares a close affinity with the method of doubt that Descartes famously invokes in his Meditations on First Philosophy (1641). Recall that in the Meditations it is only after he recognizes the most clear and certain principle of self-knowledge-the cogitothat Descartes can establish knowledge of God, mathematics, and the natural world. I<nowledge in general thus finds its ground on ideas presented clearly and distinctly before the mind, and as we see in the
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Discourse, it is only via withdrawal from worldly and scholarly pursuits that the single man (in this case, Descartes himself) is able to discover those principles that allow us to distinguish truth from falsehood in the sciences. The importance of principles for Descartes's philosophical pursuits deserves emphasis here, for ultimately, Descartes is aiming to establish criteria that enable us to clearly perceive truth and thereby place knowledge in the sciences on a firm foundation. The importance Descartes lays on pin-pointing such criteria emerges in his discussion of the ancient pagans' treatment ofvirtue. He claims that in their moral writings the ancients lack "the certainty and self-evidence" of mathematical reasoning, and he deems these writings inadequate, precisely because they "do not adequately explain how to recognize a virtue" (CSM 1 114; AT VI 8). Descartes thus compares "the moral writings of the ancient pagans to very proud and magnificent palaces built only on sand and mud" (CSM 1 114; AT VI 7-8). With this emphasis on criteria for distinguishing truth from falsehood, we can better understand the project Descartes adopts in the Geometrie. His classification of curves and his explication of the relationship between geometry and algebra are heralded as improvements over the ancient account of geometrical and mechanical curves, for as in the case of virtue, the ancients have not offered a clearly stated criterion for distinguishing curves that are truly geometrical from those that are not. Descartes's self-appointed task in the Geometrie is thus to offer the very criteria for truth-for the clear and distinct knowledge of geometrical curves-that the ancients fail to offer. 28 The same emphasis that Descartes places on the clear statement ofprinciples and the use of a method of introspection in philosophy is conspicu0usly absent in Newton's writings. Unlike Descartes, Newton does not find fault with the ancients for their lack of clearly explicated principles in mathematics. By surveying what they did in practice, and specifically, by taking into account what they did in conjunction with what they said, he gleans from their writings the principles that underwrite their mathematical practice, even at instances where they are not explicitly supplied. And rather than turn to "studies within himself" for the foundations of the mathematical sciences, Newton turns his glance backwards towards the ancients and adopts as his starting point the very historical sources that Descartes dismisses as a possible source of true principles. Historical knowledge, rather than self-knowledge, serves as the ground for n1athematical practice, and in turn, Newton replaces Descartes's method ofintrospection with a n1ethod of retrospection. Ultimately, it is Newton's optimistic and anti-Cartesian stance towards history and historical evidence that underwrites his alternative approach to philosophy. Whereas Descartes emphasizes the inaccuracy of historical accounts, Newton instead adopts a principle of charity in his readings of
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ancient texts and looks to his historical ancestors for the foundational principles of practice that Descartes argued could only be found before the mind's eye. This sort of historically-oriented philosophy was certainly characteristic of the Cambridge Platonism tradition in which Newton was schooled. Platonists such as More and Cudworth showed a deep respect for ancient wisdom and embraced the pristine knowledge of the ancients as the starting point of study.29 However, in the realm of mathematics, a greater influence should perhaps be attributed to Newton's predecessor as Lucasian Chair at Cambridge, Isaac Barrow, whose works illustrate the same historically-oriented strategy for engaging with philosophical disputes in mathematics that we find in Newton's writings. 30 As Ian Stewart (2000) has lucidly illustrated, Barrow turned to the actual practice of mathematics-the actual ancient practice of mathematics-in order to address the philosophical questions surrounding mathematical practice raised by Proclus in his commentary on Euclid's Elements. Whereas Proclus appealed to Plato's writings to argue for the certainty of mathematical knowledge, Barrow saw little value engaging with the philosophical arguments of Plato, Aristotle, or even Proclus himself. Their arguments, he claims, "are conversant about things very remote from the foundation of mathematics" (Barrow 1970, 237), and their questions do not scarce touch the outmost skin of geometry, much less reach its inward bowels. These rams do not shake its principles, nor overthrow, or at all disturb, its reasonings. While contentions and strifes, clamours and janlgings do arise, and make a noise, in its confines and suburbs, a deep peace and profound silence dwells within the walls, and in the citadel itself; nothing is there heard of controversy or opposition. (Barrow 1970, 237) For Barrow, it was only by recourse to the actual sciences-to the manner in which ancient mathematics was practiced-that questions concerning the truth and certainty of mathematical propositions could be settled. Thus, to assess the foundations on which mathematical knowledge rests, our gaze should turn to the secure mathematical foundations erected by Euclid, Archimedes and other practicing mathematicians, not towards the philosophical speculations of Plato, Aristotle, or even Proclus. 31 The ancient sciences also played a central role in Barrow's debate with John Wallis. The specific question at the heart oftheir dispute was whether the newly developed system of the algebra deserved a rightful place among the mathematical sciences. 32 Wallis argued affirmatively: he granted algebra a universalizing power over arithmetic and geometry, and esteemed it as the most superior of all mathematical disciplines. In sharp contrast, Barrow brings the certainty of ancient geometrical reasoning to his aid,
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and argues that any mathematical science must have a clear relationship to the natural world. With no evident connection between the abstract formalism of algebra and bodies in nature, Barrow invoked the ancient criterion against Wallis. And not only did Barrow thereby fend offWallis's claim that algebra is the loftiest mathematical science; he relegated algebra to the status of a logic, deeming it "an instrument subservient to the mathematics" (Barrow 1970, 28). To be sure, Barrow agreed with his analytically-minded contemporaries that modern mathematics included techniques that reached beyond what the ancients had achieved. However, he firmly disagreed that the use analytic methods entailed a reinterpretation of how the mathematical sciences are related or, more drastically, the dismissal of the ancient view of the mathematical sciences. 33 Those, such as Wallis, who were intent on distancing the new from the old, failed to recognize the compatibility of modern mathematics with ancient mathematics, precisely because they failed to acknowledge their historical indebtedness and admit that modern advances in mathematics were won by extending, not correcting, the ancient practice of mathematics. As he writes, To preserve ancient authors ... from destruction seems an important task for their modern followers, who would otherwise earn a reputation for ingratitude. True, their contents can in large part be derived more rapidly or constructed more concisely by modern techniques; yet reading them retains its value. First, it seems pleasant to examine the foundations from which the sciences have been raised to their present height. Second, it will be of some interest to sample the sources from which virtually all the discoveries of the moderns are derived; for it was by studying or imitating the clever and subtle methods of the ancients that the industry of the latter reached its eminence. Furthermore, I think that there is no better source for learning a pure taste for and skill in demonstration than those whose skill and elegance became especially illustrious in their deduction of theorems. (from Archimedes Opera, cited in Grafton 1990,299) On this interpretation of the link between the old and the new, Barrow can consistently maintain his classical interpretation of the mathematical sciences and his use of analytic methods. With the latter ultimately derived from ancient practice, it was unnecessary as well as disrespectful to censure the ancients even at moments when modern mathematicians were clearly treading new ground. The same generous and respectful attitude Barrow takes towards ancient mathematical practice is adopted by Newton in his commentaries, indicating a richer picture of Barrow's possible influence on Newton. 34 For
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in the spirit of Barrow's critique of Wallis's position on algebra, Newton levies ancient wisdom against Descartes and elicits ancient authority to lend credence to his critique of Cartesian analysis. Also, as urged by Barrow above, Newton makes the ties between his mathematical innovations-including his development of a fluxional calculus-and ancient mathematical practice explicit. 35 For instance, in the realm of geometry, Newton belonged to a tradition of modern mathematicians, such as Galileo, Toricelli, Wallis, Barrow, and Descartes, who focused their geometrical studies on the generation of curves in the plane and enhanced their systems of geometry with elements from kinematics, including motion, force, and time. 36 Though this kinematical geometry is an evident advance over the classical study of static geometrical figures, Newton saw fit to connect his use of this innovative style of mathematics in the Principia with the classical study of geometry found in Euclid's Elements. As he states in an unpublished preface to the Principia, The geometry of the ancients was indeed concerned with magnitudes; but propositions concerning magnitudes were sometimes demonstrated by means of local motion, as when the equality of two triangles in prop. 4 of book 1 of Euclid's Elements was demonstrated by transferring either triangle into the place of the other. But also the generation of magnitudes by continual motion was accepted in geometry, as when a straight line was multiplied by a straight line to generate an area, and an area was multiplied by a straight line to generate a solid. (MP, 8.455) Newton's reference to Proposition 4 in Book 1 of the Elements is elucidating, because in his proof, Euclid employs a method of superpositionof transferring a triangle on top of another-in order to prove the side-angle-side postulate. The proof runs as follows: given two triangles, ABC and DEF, that have two congruent sides and a congruent angle included between the two congruent sides, the problem is to show that the triangles are congruent. To prove this congruence, we place one triangle on top of the other in such a way so that the two sets of congruent sides coincide. As a result of superposing one triangle on the other in this way, we find that "the whole triangle ABC will coincide with the whole triangle DEF, and will be equal to it" (Euclid 1956,247-48). Historically, the proof of Proposition 4 has been a cause of worry, because it relied on a contentious axiom of congruence, included in the Elements as Common Notion 4. This common notion reads, "Things which coincide with one another are equal to one another" (Euclid 1956, 155). You'll notice that this rendering of congruence, and specifically the appeal to coinciding things, suggests the use of a method of superposition
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and the rigid displacement of figures through geometrical space. Later mathematicians, including Hilbert, found this method illegitimate because it was not clearly supported by the postulates of Euclid's geometry, and replaced Common Notion 4 with analytic definitions for congruence in their respective axiomatizations. 37 However, following Heath's reading, the fact that Euclid himself uses the method of superposition to establish his axiom of congruence indicates that motion was in fact an essential part of ancient Greek geometry (Heath in Euclid 1956,224-25).38 Certainly, this was a fact not questioned by Newton. For him, the contentious proposition served as solid evidence that the groundwork for modern kinematical developn1ents in geometry was laid by Euclid himself. Newton makes a similar appeal to the ancients in his priority dispute with Leibniz. For instance, arguing for the superiority of his geometricallybased system of fluxional calculus, Newton claims, Mathematical quantities I consider not as consisting of least possible parts [as Leibniz did], but as described by a continuous motion ... These geneses take place in the reality of physical nature and are daily witnessed in the motion of bodies. And in much this manner the ancients, by 'drawing' mobile straight lines into the length of stationary ones, taught the genesis of rectangles. (MP, 8.123) In a similar vein, and speaking of himself in the third person, Newton writes, We have no idea ofinfinitely little quantities & therefore Mr Newton introduced fluxions into his method that it might proceed by finite quantities as much as possible. It is more natural and geometrical ... Nature generates quantities by continual flux or increase, & the ancient Geometers admitted such a generation of areas & solids .... (MP, 8.597)39 Notice that there are two connections Newton draws attention to here: the connection between his fluxional calculus and ancient geometry, and the connection between ancient geometry and nature. As he indicates above, there is a broadly construed ontological fit between geometrical motions and natural motions, and therefore, with his fluxional calculus founded on ancient geometry, his mathematics has a clearer connection to nature than Leibniz's infinitesimal calculus. The arguments of Barrow clearly resonate in these arguments from Newton. As Barrow, Newton invokes the ancient method of mathematics as his model, emulating their techniques and sharing their concern for preserving a connection between mathematics and nature. And as we see in his arguments with both Descartes and Leibniz, the connection Newton shares to past practice serves a justificatory role: by appealing to ancient
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authority-by bringing his new mathematics together with the old-he highlights the philosophical merits of his projects and justifies his new methods as properly n1athematical. For Newton, then, history was a way of doing philosophy. Or if you want an 'ism' attached to Newton's approach, his philosophy is a historical foundationalism not a rationalistepistemological one. Whereas on Descartes's model, truth is gained through introspection and the inspection of ideas before our solitary minds, on Newton's historical model, truth is gained through retrospection-through a view of the past as offering as great a certainty as Cartesian clear and distinct ideas.
5. Conclusion Given the tradition of historically-oriented philosophy to which Newton belonged, what place is there for Newton in the canon of modern philosophy? Certainly, historical arguments are the rare exceptions rather than the rule among those we place in the modern canon. This perhaps comes as no surprise, since, as the story is commonly told, the moderns were not trying to preserve traditions; they made progress precisely because they broke with and corrected what came before: Descartes rejected the Scholasticism that characterized the n1iddle ages; Spinoza and Leibniz tried to remedy the ills of the Cartesian system, while Berkeley and Hume attempted to do the same for Locke's theory of ideas; and lZant was famously awoken into a transcendental state when he noticed the limitations of H ume's empiricist analysis of experience and causality. If we en1brace this view of modern progress as one marked by philosophical tradition-breaking' it remains difficult to find a place for Newton in his own philosophical context. Yes, Newton was highly critical of just about everything Descartes proposed; but his 'solutions' to the contemporary problems in modern natural philosophy-and especially in the philosophy of mathematics-were not presented by Newton as self-initiated solutions; they were solutions recovered and rediscovered from the Ancients. Though Newton's historical model of doing philosophy doesn't fit neatly into the framework we commonly adopt for the early modern period, it is, I think, no le~s worthy of our philosophical attention than Descartes's more "systematic" style of argumentation. On the one hand, ~~ewton's use of a historical argument as a philosophical argument broadens our perspective on Newton's anti-Cartesianism and, on the other hand, it broadens our perspective on the styles of philosophical practice that characterize the early modern period-and along with it, an alternative conception of how progress in philosophy can be won. For although progress in philosophy, or progress of any sort, is con1monly characterized
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by the distance between past and present, in Newton's case, whatever distance we find between him and his contemporaries is a distance achieved through an endeavor to reinstate a tradition that, to his mind, had been corrupted and misunderstood-a tradition that Barrow as well found worthy of emulation and extension. For them, respect for history is not antithetical to progress. Rather, as evidenced both by their remarks and their achievements in mathematics, progress could be gained by admitting historical indebtedness, ren1aining faithful to their philosophical past, and extending an underappreciated viewpoint to contemporary problems. Such a historically-grounded philosophy may not fit neatly with the model of philosophy more commonly attributed to the early moderns, but it is no less a form philosophy-and no doubt a form of philosophy that continues to shape and inform important attempts to bring history to bear on our own contemporary philosophical problems (see, among other works, Friedman 2000).
NOTES 1. My thanks to audience members at the University of New Mexico for their comments on an earlier version of this paper as well as to Tom Higgins for several stimulating conversations that helped me shape some of the ideas I present below. lowe a very special thanks to Niccol6 Guicciardini for reading an earlier version of this paper and offering comments and suggestions that helped me put the paper in more polished form. More recently, Marco Panza kindly provided comments that prevented me from making crucial omissions and errors. Certainly, though, I am responsible for any omissions and errors that remain. 2. Though it's generally agreed that De Gravitatione was composed prior to the 1687 publication of the Principia, its precise dating remains a matter of debate. When the document was originally published by Hall and Hall (Newton 1962), they placed the composition of the n1anuscript between 1664 and 1668, a period that they claim coincided with Newton's early, "juvenile" backlash against Descartes. The Halls' dating of the manuscript was later questioned by Betty Joe Teeter Dobbs (1991), who found reason to place the composition in the years immediately preceding the publication of the Principia. On her reading, the manuscript belongs to the early 1680s, not the late 1660s as the Halls contend. 3. Howard Stein (1967) persuasively argues that it is the Cartesian relationist, not the Leibnizian one, who is the target of Newton's criticisn1s in the Principia and elsewhere. 4. See Section 3.1 of Cohen's Introduction to Newton (1999). 5. See D.T. Whiteside (1982) as well as the commentaries that Whiteside includes in the seven volume The Mathematical Papers of Isaac Newton. The Mathematical Papers will be referred to as MP in the remainder of the paper.
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6. I'll discuss Descartes's reasons for banishing the infinite from geometry below. My thanks to Nicco16 Guicciardini for urging me to consider this point with greater care. 7. My approach to Newton complements the work of Frank Manuel, who has en1phasized the critical though anti-rationalist manner of Newton's work in religion and chronology. See the classic treatment in Manuel (1963). 8. Though not published until 1637, much of the work central to the mathematics of the Geometrie was completed by Descartes over a decade earJier, as detailed in his 1619 correspondence with Isaac Beeckman. For a brief overview of the early stages of Descartes's work in geometry, see Part II of Schuster (1980). 9. As urged to me by Marco Panza, a clarification of terminology is in order here. In general, 'algebra' refers to "those mathen1atical theories and practices that involved unknowns and/or indeterminates, employed the algebraic operations, involved equations, and dealt either with numbers or with geometrical il1agnitudes or with magnitudes in an abstract more general sense" (Bos 2001, 129). However, when I say that Descartes brought about a reinterpretation of how algebra and geolnetry are related, I am adopting only the broader account of algebra as dealing with magnitudes in general, not as it is already applied to numbers or geometrical magnitudes. As Bos clarifies, such an application of algebra (in this general sense) to geometrical magnitudes "presupposed (tacitly or explicitly) a redefinition of the algebraic operations so as to apply to such rnagnitudes" (ibid, 129). It is in this redefinition that I am claiming Descartes's innovation lies. 10. According to Descartes's reading of the ancients, it's unclear why they deem some curves "geon1etrical" and others "mechanical." I should note here that whereas Descartes equates "geometrical" with "geometrically intelligible," the ancients instead appear to equate "geometrical" with "curves constructible by straight line, circle, and solid." For a trenchant account of Descartes's supposed in1provement over the ancient classification of curves, see Molland (1976). I'll return to the questions surrounding the accuracy of Descartes's reading of the ancient classification of curves later in the paper. 11. Note, here, that, for Descartes, the curves of geometry were curves necessary to construct the solution to a problem, where solutions were generated by considering the intersection of curves. Only later, in the generation after Descartes, were curves themselves adopted as objects of study in mathematics. My thanks to Nicco16 Guicciardini for emphasizing this point to me. 12. See Guicciardini (2004) pp. 123-125 for further discussion of the inexactness that Descartes ascribes to "n1echanical" curves. 13. These are the only two "mechanical" curves cited in the Geometrie (Descartes 1925,44). To get a sense of the "two separate movements" necessary for construction of these curves, we can appeal to Archimedes's description of the construction of the spiral by the simultaneous rotation of a line and a point on the line: If a straight line is drawn on a plane and if after having rotated uniformly round one of its extremities which is held fixed, it returns whence it started, and during the rotation of this line a point is carried uniformly with respect to itself through the length of the line, beginning at the extremity that is held fixed, this point will describe a spiral in the plane. (Archimedes Spirals, cited in DeGandt1995,203)
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14. As opposed to Descartes, neither Newton nor Leibniz found any difficulty including indefinite equations or infinitesimal quantities in the proper domain of mathematics. For the latter, the treatment of infinite series in mathematics allowed them to provide a more rigid definition of the notion of a limit, which is, of course, essential for finding the area under a curve. For more on Leibniz and transcendent equations, see Mahoney (1990). For Newton's treatment of infinite series, see Guicciardini (2004). 15. See Bos (2001) for Descartes's failure to prove that 'geometric' and 'algebraic' curves are coextensive in the Geometrie. 16. See Westfall (1980) and Dobbs (1991) on the development of Newton's admiration for ancient wisdom in other disciplines, including alchemy and scripture. 17. The lectures on algebra were published anonymously as Arithmetica Universalis in 1707 by William Whiston. Newton edited his own Latin version of the lectures and published them with the same title in 1722. 18. Newton offers the same criticism in his "Researches into the Greeks' 'Solid Locus'" (ca. late 1670s) and "The 'Geometry of Curved Lines'" (ca. 1680), both found in Volume 4 of the Mathematical Papers. In the latter work, Newton writes, "Men of recent times, eager to add to the discoveries of the ancients, have united the arithmetic of variables with geometry. Benefiting from that, progress has been broad and far-reaching if your eye is on the profuseness of output but the advance is less of a blessing if you look at the complexity of its conclusions. For these computations' progressing by means of arithmetical operations alone, very often express in an intolerably roundabout way quantities which in geometry are designated by the drawing of a single line" (MP, 4.421). 19. With my focus on Newton's strategy for arguing against Descartes, I am unable to relate Newton's commentaries with his actual mathen1atical practice in a paper this short. For a trenchant account of the relationship between Newton's mathematical practice and his commentaries on ancient mathematics, see Guicciardini (1999), especially Chapters 2 and 4, and Guicciardini (2004). 20. Newton admits that geometrical solutions are not always simple and elegant. In the same treatise in which he censures the con1plicated solutions of algebra, he concedes that geometrical constructions can be tedious. Over-complicated solutions are defects, regardless of whether they are analytic or synthetic (i.e., whether they proceed by the treatment of equations or by the construction of curves in the plane). And since synthetic, geometrical solutions are "at least for the most part" simpler, he proceeds in a synthetic manner in his treatment of curved lines (MP, 4.423). 21. In a similar fashion, Newton claims, ". . . the arithmetically simpler is indeed that which is determined by simpler equations, while the geometrically simpler is that which is gathered by a simpler drawing of lines-and in geometry what is simpler on geometrical grounds ought to be first and foremost" (MP, 5.429). Newton points out that if we accept Descartes's "analytic" classification of geometrical curves, the simplicity of curves is determined by their corresponding equations rather than the description of their construction. Consequently, geometrical constructions by means ofparabolas would falsely be considered simpler than those by circles (cf. MP, 5.425). Descartes does in fact consider the circle to be simpler than the parabola even though they both belong to the "first and simplest class" of
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geometrical curves. He writes, "It should be observed, however, with regard to the curves of anyone class, that while many of them are equally complex so that they n1ay be employed to determine the same points and construct the same problems, yet there are certain simpler ones whose usefulness is more limited. Thus, among the curves of the first class, besides the ellipse, the hyperbola, and the parabola, which are equally cOlnplex, there is also found the circle, which is evidently a sin1pIer curve ..." (Descartes 1925, 56). The notion of simplicity within a class of curves thus appears to depend on the uses of a curve in construction. Descartes's intermingling of the geometric and algebraic criterion of curves is the prin1ary focus of Bos (1981). 22. For readings of Newton's ll1athematics that place greater emphasis on construction, see Dear (1995), Garrison (1987), Molland (1991), and Guicciardini (2004). My argument against Dear, Garrison, and Molland can be found in Domski (2003). 23. Admittedly, Newton at one point claims that the straight line, the circle, and the conic section are "the sole tools of geometry" and are the only lines "to be accepted into geometry." On my reading, the straight line, circle, and conic section are, for Newton, the tools by which we can determine the motions of construction with precision and exactness-a claim consistent with Descartes's construction criterion of "geometrical" curves. But as opposed to Descartes, Newton does not want to clain1 that only those objects constructed in such a manner can be accepted into geometry. Problems involving more intricate curves can be solved geometrically and, as such, preserve their status as geometrically intelligible. For the relationship of these remarks to the Preface to the Principia, see Domski (2003). 24. In addition to the examples provided earlier, Newton claims that a geometer assumes in practice that a beginner has learned the more basic construction procedures involving straight lines and circles; and moreover, a geometer uses "unnatural" linear curves and draws on the resources of algebra and mechanics in composing the solution to a problem (MP, 5.433). Newton's concern for the practice of geometry emerges again in 1706. He claims that if a fourth postulate is added to geometry to supplement Euclid's three construction postulates it should not be "to cut a cone by a given plane" as traditionally argued. Since in practice no geometer knows how to cut a cone nor should be required to know how, a conchoidal postulate should instead be added, for by means of this postulate, 'solid' problems could be reduced to the plane (MP, 8.181-183). In the Appendix to the Arithmetica Universalis, Newton in fact extends Euclid's postulates to include a conchloidal postulate, allowing him to use neusis constructions to find the roots of third degree equations (MP, 7.382-388). In doing so, Newton is stepping over the boundaries established by Descartes, who did not allow the use of higher degree curves to construct the roots of curves of lower degree. For Newton, the use of the conchoid is legitimate, because for him, the conchoid is a simpler curve; as he puts it, in the simplicity of its description, "it yields to no curve except the circle" (MP, 4. 429). As pointed out to me by Niccol6 Guicciardini, here we see the influence ofViete on Newton, for Viete also proposed to extend Euclid's postulates to mal(e room for neusis constructions.
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25. The Latin translation of Pappus's Mathematical Collections by COll1ll1andinus appeared in 1588 and was titled Pappi Alexandrini Mathematicae Collectiones. Jones (1986) has translated Book VII in its entirety, and translations of selected portions can be found scattered in recent works on ancient mathematics, including Cuomo (2000). 26. Following standard usage, 'CSMIZ' refers to Descartes (1991) and 'CSM' refers to Descartes (1985). 'AT' refers to the pagination of the Adam and Tannery complete works of Descartes which are included in the margins of CSM and CSMIZ. 27. See Collingwood (1993), 59-61, for a helpful account of Descartes's skepticisll1 towards history as expressed in the above passage. 28. To be clear, I don't mean to suggest that Descartes was not himself a part of a particular ancient tradition. As urged to me by Michael Friedman and emphasized in his contribution to this volume, we can understand Descartes as extending a sort of Platonist and Augustinian philosophy, one that itself emphasizes pinpointing principles and criteria for truth. Accepting this claim (which I do) is not, however, incompatible with my goal above to emphasize the way in which Descartes advertises his form of philosophy, namely, as one different from and an improvement upon certain kinds of ancient methods. Thus, whatever its ancient roots might be, the form of philosophy Descartes forwards does not make its own ancestry a matter of concern. 29. For the relationship between the Cambridge Platonism of More and Cudworth and Newton's respect for the ancients see the classic treatment in McGuire and Rattansi (1966). See Mandlebrote (1993), pp. 292-296, for important differences between Newton and the Platonists in regards to religion. 30. Though Benjamin Pulleyn was Newton's official tutor at Cambridge, Newton was undoubtedly aware of Barrow's work and was a possibly even a student at the public lectures Barrow delivered at Cambridge. These lectures were translated by Rev. John I
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together into certain Systems do compose particular Sciences. (Barrow 1970, 115) As a consequence, Barrow dismisses a hierarchy of the mathematical sciences that makes any discipline superior to the other. The only relevant question is ,vhether a discipline counts as a lnathematical science at all, and as indicated by the quote above, to answer this question we must consider whether our ideas share some connection with those things that may exist in nature. 34. For general accounts of Barrovv's influence on Newton, see Feingold (1993) and Dear (1995). 35. Newton also connects other innovations, including his proposal of a universal force of gravity, to ancient wisdom, claiming that the ancients held these ideas long before he did. See McGuire and Rattansi (1966). 36. See DeGandt (1995), 202-244, for a brief yet lucid overview of developlnents in "kinematical geometry" as well as Newton's use of this mathematical machinery in the Principia. 37. My account of the problems surrounding Proposition 4 and Comlnon Notion 4 rely primarily on Heath's commentary in Euclid (1956), Volun1e 1. Though I may not have addressed all his worries, I thank Marco Panza for encouraging me to clarify the relationship between congruence and the rigid displacen1ent of figures in space. 38. See Heath's commentary in Euclid (1956), Volume 1, 224-226, for an overview of the different opinions on this matter. 39. Since the passages I cite pertain to Newton's calculus dispute with Leibniz, I should note that there is a very interesting story to be told about the relationship between mathematics and metaphysics in the seventeenth century. Nico Bertoloni Meli (1993) has done important work on how issues of metaphysics helped fuel the debate about the superiority of one calculus over another.
REFERENCES Barrow, 1.([1734J 1970). The Usefulness ofMathematical Learning Explained and Demonstrated: Being Mathematical Lectures Read in the Publick Schools at the University of Cambridge. Translated by Rev. John lZirby. London: Frank Cass & Co. Ltd. Bechler, Z. (ed.) (1982). Contemporary Newtonian Research. Dordrecht: Reidel. Bertoloni Meli, D. (1993). Equivalence and Priority: Newton versus Leibniz. Including Leibniz's Unpublished Manuscripts on the Principia. Oxford: Clarendon Press. Bas, H. (1981). "On the Representation of Curves in Descartes's Geometrie." Archive for the History ofExact Science 24: 295-339. - - - . (2001). Redefining Geometrical Exactness: Descartes)s Transformation of the Early Modern Concept ofConstruction. New York and Heidelberg: SpringerVerlag.
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Collingwood, R.G. (1993). The Idea of History. Edited by J. Van der Dussen. Oxford: Oxford University Press. Cuomo, S. (2000). Pappus of Alexandria and the Mathematics of Late Antiquity. Cambridge: Cambridge University Press. DeGandt, F. (1995). Force and Geometry in NewtonJs Principia. Translated by Curtis Wilson. Princeton: Princeton University Press. Dear, P. (1995). Discipline and Experience: The Mathematical Way in the Scientific Revolution. Chicago: The University of Chicago Press. Descartes, R. ([1637] 1925). The Geometry ofRene Descartes. Translated from the French and Latin by D.E. Smith and M.L. Latham with a Facsimile of the First Edition, 1637. Chicago and London: Open Court. - - - . (1985). The Philosophical Writings of Descartes. 2 Volumes. Translated by J. Cottingham, R. Stoothoff, and D. Murdoch. Cambridge: Cambridge University Press. - - - . (1991). The Philosophical Writings of Descartes: The Correspondence. Translated by J. Cottingham, R. Stoothoff, D. Murdoch, and A. IZenny. Cambridge: Cambridge University Press. Dobbs, B.J,T. (1991). The Janus Faces of Genius: The Role ofAlchemy in NewtonJs Thought. Cambridge: Cambridge University Press. Domski, M. (2003). "The Constructible and the Intelligible in Newton's Philosophy of Geometry." Philosophy of Science 70 (5): 1114-24. Feingold, M. (ed.) (1990). Before Newton: The life and Times of Isaac Barrow. Cambridge: Can1bridge University Press. - - - . (1993). "Newton, Leibniz, and Barrow Too: An Attempt at Reinterpretation." Isis 84: 310-38. Friedman, M. (2000). A Parting of the Ways: Carnap) Cassire1) and Heidegger. Chicago and La Salle, IL: Open Court. Garrison, J.W. (1987). "Newton and the Relation of Mathematics to Natural Philosophy." Journal ofthe History ofIdeas 48: 609-27. Gaukroger, S. (ed.)(1980). Descartes: Philosophy) Mathematics and Physics. Sussex: The Harvester Press. Grafton, A.(1990). "Barrow as a Scholar." In M. Feingold (ed.), 291-302. Guicciardini, N. (1999). Reading the Principia: The Debate on NewtonJs Mathematical Methods for Natural Philosophy from 1687 to 1736. Cambridge: Cambridge University Press. - - - . (2004). "Geometry and Mechanics in the Preface to Newton's Principia: A Criticism of Descartes's Geometrie." Graduate Faculty Philosophy Journal 25: 119-59. Hill, IZ. (1996). "Neither Ancient nor Modern: Wallis and Barrow on the Composition of Continua. Part One: Mathematical Styles and the Composition of Continua." Notes and Records ofthe Royal Society ofLondon 50: 165-78. - - - . (1997). "Neither Ancient nor Modern: Wallis and Barrow on the Composition of Continua. Part Two: The Seventeenth Century Context: The Struggle Between Ancient and Modern." Notes and Records ofthe Royal Society ofLondon 51: 13-22.
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Jones, A. (1986). Pappus ofAlexandria: Book 7 ofthe Collection in Two Parts. New York, Berlin, Heidelberg, and Tokyo: Springer-Verlag. Lindberg, D. and R. Westman (eds.) (1990). Reappraisals of the Scientific Revolution. Cambridge: Cambridge University Press. McGuire, J.E. and Rattansi, P.M. (1966). "Newton and the Pipes of Pan." Notes and Records of the Royal Society ofLondon 21: 108-43. Mahoney, M.S. (1990). "Infinitesimals and Transcendent Relations: The Mathematics of Motion in the Late Seventeenth Century." In D. Lindberg and R. Westman (eds.), 461-91. Mandlebrote, S. (1993). '''A Duty of the Greatest Moment': Isaac Newton and the Writing of Biblical Criticism." British Journal for the History of Science 26: 281-302. Manuel, F. (1963). Isaac Newton Historian. Cambridge, MA: The Belknap Press of Harvard University Press. Molland, A.G. (1976). "Shifting the Foundations: Descartes's Transformation of Ancient Geometry." Historia Mathematica 3: 21-49. - - - . (1991) "Implicit Versus Explicit Geometrical Methodologies: The Case of Construction." In R. Rashed (ed.), 181-96. Newton, 1. (1962). Unpublished Scientific Papers of Isaac Newton. Edited and translated by A.R. Hall and M.B. Hall. Cambridge: Cambridge University Press. - - - . (1967-1981). The Mathematical Papers of Isaac Newton. 8 Volumes. Translated and edited by D.T. Whiteside. Cambridge: Cambridge University Press. - - . ([1687J 1999). Mathematical Principles of Natural Philosophy (third edition). Translated by LB. Cohen and A. Whitman with an Introduction by LB. Cohen. Berkeley: University of California Press. Pemberton, H. (1728). A View of Sir Isaac NewtonJs Philosophy. London: S. Palmer. Rashed, R. (ed.) (1991). Mathematiques et Philosophie de LJAntiquite a LJAge Classique: Hommage a Jules Vuillemin. Paris: Editions Du Centre National De La Recherche Scientifique. Stein, H. (1967). "Newtonian Space-Time." Texas Quarterly 10:174-200. Sasaki, C. (1985). "The Acceptance of the Theory of Proportion in the Sixteentl1 and Seventeenth Centuries." Historia Scientiarum 29: 83-116. Schuster, J.A. (1980). "Descartes's Mathesis Universalis, 1619-28." In S. Gaukroger (ed.), 41-96. Stewart, 1. (2000). "Mathematics as Philosophy: Barrow and Proclus." Dionysius 18: 151-82. Whiteside, D.T. (1982). "Newton the Mathematician." In Z. Bechler (ed.), 110--16.
[5J Newton's Forces in IZant's Critique l ANDREW JANIAIZ
Readers of Michael Friedman's [(ant and the Exact Sciences (1992) will rather naturally seek cases in which I
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This is clearly a rich and complex passage, one that raises numerous interpretive and philosophical issues. I(ant signals here that he takes attractive and repulsive force-which obviously have a Newtonian heritage-to be essential elements in our knowledge of ordinary objects. Following Friedman's lead in [(ant and the Exact Sciences, we might approach this complex topic by focusing specifically on the contrast that I
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of matter, but to a broader range of objects, or to material objects considered from a more abstract point of view (one that does not consider them qua Inaterial). That is, the Critique explicitly abstracts away from the very properties of objects-such as their mass-that would seem essential for conceiving of thein as falling under Newton's three laws of motion and law of universal gravitation. This is no accident: I
1. Is there a Tension in IZant's Critical Project? In the period before the publication of the Critique of Pure Reason, one aspect of the relation between science and metaphysics posed a deep philosophical problem for IZant. Since his earliest writings, I
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particular by explicating how it is possible for the sciences to provide us with such knowledge. 4 IZant's invocation of attractive force in the Amphiboly passage above wi11 lead me to focus in particular on one aspect of Newton's treatment of force in the Principia. One reason to do so, of course, is that from lZant's point of view the study of force is fundamental to the scientific understanding of nature. Consider this extren1ely rich, and strikingly Newtonian, passage from the Metaphysical Foundations of Natural Science: [A]11 natural philosophy consists in the reduction of apparently diverse given forces to a smaller nun1ber of forces and powers sufficient for the explanation [ErkHirung] of the actions of the former, but this reduction continues only to fundamental forces, beyond which our reason cannot go. And thus the investigation of metaphysics behind that which lies at the ground of the empirical concept of matter is useful only for the purpose of leading natural philosophy as far as possible in the investigation of the dynamical grounds of explanation [dynamischen ErkHirungsgrunde], because these alone admit the hope of detern1inate laws and thus of a true rational coherence of explanations. (4: 534; Friedman translation, IZant 2004)5 We might therefore reformulate one aspect of IZant's project as follows: a critical metaphysics ought to explicate how a science of forces is possible. We might narrow this topic further by noting that to achieve knowledge of some phenomenon we must be able to represent it using the canonical lZantian representational resources of concepts and intuition. In the most general terms, for instance, Kant argues in the Aesthetic that intuition is required for the representation of objects as bearing spatial relations, particularly the sorts of relations required for doing geometry. Hence many interpreters have taken lZant to argue that certain types of mathematical knowledge require singular, immediate representation (since these are characteristics of intuition). 6 This point suggests a potentially fruitful parallel: perhaps one way of explicating how a science of forces is possible is to characterize the resources required for representing objects as bearing the dynamical relations presented, say, by the relevant equations of the science. We might therefore ask: what is required for the representation of the dynamical relations described in the laws of motion, or the law of universal gravitation, in Newton's Principia?7 Unfortunately, the possible parallel between spatiotemporal and dynamical relations would appear to be immediately broken because of the distinct status of the Metaphysical Foundations of Natural Science-where we find the characterization of science in terms of discovering forces-and the Critique itself. As I mentioned briefly above, the Critique occupies a
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standpoint on our knowledge of objects that is abstract in certain important respects. For instance, whereas the Metaphysical Foundations "applies" the principles of the Critique to what I(ant understands to be the empirical concept of matter, the Critique itself presents a priori principles-such as the three analogies of experience-that are arrived at independently of a consideration of what I(ant takes to be the nlaterial properties of objects. From IZant's point of view, this abstract standpoint on our objects of knowledge does not prevent us from taking these objects to bear spatiotemporal relations-indeed, such relations are obviously the focus of the discussion in the Aesthetic. In that sense, the transcendental character of the critical project does not prevent one from taking the objects of our knowledge to bear the sorts of relation-such as spatial relations-attributed to them in mathematics. But as should be immediately clear, the consideration of Newton's dynamics within the context of the Critique is disanalogous to the case of spatial relations. As we know from Newton's second law of motion, for instance, impressed force is defined to be proportional to mass and acceleration. If one abstracts away from material properties, one presumably abstracts away from mass; that would seem, in turn, to exclude a consideration offorce in Newton)ssense. If one ofI(ant's tasks in the Critiquewere to explicate how it is possible, say, to know that the gravitational interactions among lllaterial bodies are directly proportional to their masses and inversely proportional to the square of the distance between them, then it seems I(ant would have to explain how it is possible to know that certain equations are true (for instance, the equation that expresses the law of universal gravitation). But ifI(ant's standpoint on our knowledge ignores the fact that objects have mass, then presumably his project cannot be to explicate the possibility of the knowledge of objects that Newton expresses in equations that relate mass to other variables, such as distance. In the very least, this presents a prima facie interpretive difficulty, and one, moreover, that apparently does not arise in the case of the spatial relations characterized by geollletry. Perhaps this should not surprise us-whereas IZant takes matter to be an empirical concept, he clearly takes space to be a priori. Since IZant takes matter-like motion-to be an empirical concept, another way to see the problem at hand is to investigate the treatment of empirical concepts and material objects in other sections of the first Critique. Consider the Second Analogy's famous discussion of our perception of a ship's passage down a river. Obviously, the concept ship is a perfectly ordinary empirical concept, and ships are obviously ordinary material objects; yet IZant just as clearly takes the causal law expounded in the Second Analogy to be an a priori principle. One difficulty, then, is to understand precisely which aspects of the ship example are salient for lZant's discussion, and which are irrelevant. It would seem that if lZant is
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to be consistent in his critical project, we are not to consider the ship as a material object-for instance, as an object with n1ass, or as made of wood, etc.-but rather more generally as a spatiotemporal object undergoing a certain type of (rule-governed) alteration. If you like, the causal law expounded in this section of the text does not apply to the ship qua n1aterial object, but rather qua object in a more abstract sense. If the causal law were to apply to the ship in virtue of the fact, say, that it bore mass, or that it was made of oak, then we would obviously be confronted with the question of how the presentation of such a law is compatible with the strictures of Kant's critical project. We might provisionally conclude that there is no general problem in the Critique of discussing material objects-such as ships and pieces of ice-as long as we emphasize two points: (1) the principles of the Critique, such as the analogies of experience, do not apply to them qua material objects; and, (2) these principles are not arrived at through a consideration of the material properties of objects. This provisional conclusion is sufficient to generate a contrast with the present interpretive difficulty. There is no general problem in invoking attractive force in the Critique any more than there is in discussing the motion of a ship. Indeed, we should expect I(ant to discuss attraction (and repulsion) in just the way that he handles other empirical phenomena, such as the cases of the ship moving downstream and of the ice melting. That is, we should expect him to abstract away from the fact that attractive force-as it is presented in Newton's Principia-is borne by objects qua material objects (indeed, qua massive objects). But now the interpretive problem is clear. Attractive force obvi0usly presents a difficulty that the ship and the ice do not: since Newton defines gravity to be an impressed force, he defines it to be generated by objects with mass; that is, if we abstract away from the fact that an object has mass, we presumably cannot consider it to bear gravity. If you like, it is as if I
2. Forces in the Metaphysical Landscape of the Critique Let's begin by searching for clues as to what Kant might intend by placing attraction at the center of his epistemic thesis in the Amphiboly passage with which we began. Since Leibniz's correspondence with Clarke was both one of I
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primary sources for Newtonian views in natural philosophy-in addition of course to the Principia and the Opticks-the correspondence is a reasonable place to search for clues. It seems to me that the Leibniz-Clarke correspondence will turn out to be a red herring, but it is worth exploring briefly because it helps to underscore the interpretive predicament I want to highlight. For his part, especially in his fifth and final rebuttal to Leibniz's charges, Clarke argues that the Principia)s mathematical treatment of gravity construes it as a "matter of fact" concerning certain tendencies to motion, rather than as a cause of motion. s Although Clarke does not explicitly use this language, he effectively denies that gravity is a real force, a move intended to eschew action at a distance. In response, Leibniz contends that Newton ultimately strays from this instrumentalist safe harbor into problematic territory by endowing attraction with a physical meaning, by invoking "an attraction properly so called" (Leibniz 1890, Volume 7: 397-98). By this, Leibniz clearly means that Newton invokes action at a distance? This discussion, it should be clear, does not illuminate Kant's invocation of attraction for at least two reasons. First, Kant rather famously does not reject, but in fact strongly endorses, action at a distance in the Metaphysical Foundations, arguing even that Newton himself wrongly rejected such action (see 4: 515 and cf. Friedman 1990). As far as the Leibniz-Clarke construal of the problem of attraction is concerned, then, Kant actually seems to side neither with Clarke, who denies that Newtonians must invoke distant action, nor with Leibniz, who rejects the very idea of such action. The second point is evident from the discussion above: the Critique abstracts away from the very elements-such as matter, and therefore mass-that seem necessary for even entertaining the question of whether gravity involves distant action between material bodies. It seems, therefore, that we must look elsewhere for clues. Perhaps the best way to proceed is to reconsider the textual context of the Amphiboly passage above. As we have seen, Kant invokes attraction in the Amphiboly in his discussion of "the inner and the outer," a section that characterizes the types of property borne by objects when they are considered from the perspectives of understanding and of sensibility. Very roughly put: when we consider objects from the point of view of sensibility-when we represent objects in intuition, i.e., when we consider our singular, immediate representation of objects-we find that they are characterized by various types of relational properties, including spatial and dynamical properties. In this context, it seems to me, attraction functions in a rather different way than in the canonical Newtonian contexts, such as Clarke's correspondence with Leibniz; here, it seems to represent a type ofproperty. But what type exactly?
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There are two general features of the metaphysical landscape in the Critique that are salient here. The first concerns I
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ment-should not obscure lunt's view that logical grounds involve analytic connections between concepts, where lunt understands the latter to involve the content of the concepts in a hypothetical judgment and not merely the form of the judgment itself. Hence for lunt, general logic allows us to characterize the twelve forms of judgment, but does not distinguish within a given logical form-such as the logical form of hypothetical judgment-between synthetic and analytic judgn1ents. General logic is purely formal, which is to say, it characterizes logical forms independently of the content of any concepts employed within those forms. I-Ience general logic treats, "If Socrates is Greek, he enjoys olive oil," and, "If Socrates is human, he is mortal," as hypothetical judgments, ignoring the fact that the former is synthetic and the latter analytic. And lunt is quite clear in the passage above that logical grounds involve analyticities: "Socrates is human" is the logical ground for "Socrates is mortal" precisely because human contains mortal. This n1ay be why lunt takes logical grounds to be self-evident; we need no explanation of how the one proposition grounds the other. And that provides us with a nice contrast: with real relations, we cannot analyze the relevant concept( s) and derive the relevant grounding relation from our analysis. For lunt, butter does not contain melts in sunlight.1 1 So we require something beyond conceptual analysis to explain why butter melts in the sun. The second feature of the metaphysical landscape of the Critique involves the types of property that lunt invokes in the Amphiboly. Just as Leibnizian monadic forces-in l(ant's understanding-involve logical grounds and Newtonian forces involve real ones, the former represent a particular type of property, and the latter a crucially distinct type. To see this, we can begin with ordinary relational and ordinary intrinsic properties; for the sake of simplicity, and because it is sufficient for our purposes here, I define the former to be incompatible with loneliness, and the latter to be compatible with it.1 2 But we will ultimately need to consider three types of intrinsic, and four types of relational, property. Consider here one aspect of l(ant's interpretation of Leibniz's monadology, which is crucial to the project of the An1phiboly. I(ant writes: Substances in general must have something il1ner, which is therefore free of all outer relations, and consequently also of composition. The simple is therefore the foundation of what is inner in things in themselves. The inner of their state, however, also cannot consist in place, shape, contact, or motion-which determinations are all outer relations-and thus we can attribute to the substances no other inner state than that through which we internally determine our sense itself, namely the state of representations. In this way, the n10nads are complete; they are to constitute the basic material [Grundstoff] of the
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entire universe, but their active force [tatige Kraft] consists merely in representations whereby they are essentially efficacious only within themselves. (A274/B330) Since monads are simple substances, they lack spatial constituents, which entails that their intrinsic properties do not depend on any such parts. IZant attempts to account for this aspect of Leibniz's view by distinguishing in the Amphiboly between two types of intrinsic property. I'll call monadic properties absolutely intrinsic to distinguish them from ordinary intrinsic properties; absolutely intrinsic properties do not supervene on the spatial parts of their bearers. Whereas shape is an ordinary intrinsic property, it would fail to be absolutely intrinsic because it presumably supervenes on the spatial parts of shaped things.l 3 A thought in your mind is supposed to be an example of an absolutely intrinsic property. We also have a third category, a subset of absolutely intrinsic properties, which I will call Leibniz-intrinsic. A monad bears such a property iffit is the sole bearer of an absolutely intrinsic property in the actual world. As we will see, this last notion might be important for Kant's construal of Leibnizian forces. This three-fold distinction enables us to articulate two points: first, since monads are not spatial and therefore lack spatial parts, all of a monad's properties must be absolutely intrinsic, although not all need be Leibniz-intrinsic.l 4 Second, as we will see, monadic forces-one type of monadic property-will turn out to be not simply absolutely intrinsic, but Leibniz-intrinsic. From IZant's point of view, as he clearly notes in the Amphiboly, there is an especially salient way of understanding the notion that monadic forces are absolutely intrinsic. He notes in particular that a monad's force "consists merely in representations." This brings us to the intersection of the two features of the metaphysical landscape I've been discussing, for it seen1S that on IZant's reading of then1, monadic forces involve only logical, and no real, grounds: As an object of pure understanding, however, every substance must have inner determinations and forces [IZrafte] that concern its inner reality [innere Realitat]. But what can I think of as inner accidents except those that n1Y inner sense offers me? Namely, that which is either itself thinking, or analogous to this. Thus because Leibniz represented them as noumena, removing in thought all that might signify outer relation, thus even composition, he made out of all substances, including the constituents of n1atter, simple subjects gifted with powers of representation, in a word, monads. (A265-66/B321-22)
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The states of a monad are analogous to thoughts, according to IZant, because the latter are the only sort of property that we can understand to be absolutely intrinsic (other intrinsic properties, such as shape, being merely ordinary). And of course one relevant feature of thoughts as they figure here is that they can bear logical relations to one another, which is to say that they bear the relations that propositions bear to one another. More precisely, monadic forces \vouJd seem to be logical grounds of n10nadic states: these forces "cause" a state only in the sense that they logically contain the state, and therefore entail it. Hence from a monadic force, we ought to be able in principle to deduce all of its states, since the force contains all of those states. IZant understands this containment, it seems, in the way that he understands the concept containments expressed in analytic judgments. Hence for a n10nad to bear its force and lack a state that the force contains (and therefore entails) would be logically impossible. Finally, although I(ant does not emphasize this, I take it that monadic forces are special sorts of monadic properties in virtue of the fact that they are Leibniz-intrinsic. That is, I take them to be akin to a monad's complete concept: all of a monad's states-not just some subset of its states-are contained within, and therefore follow logically from, its force. IS With this understanding of monadic force in place, we are in a position to consider attraction as a Newtonian counterpoint. If we think of the law of universal gravitation as characterizing the force of gravity, and we then abstract away from mass on the grounds that it is a property of matter and therefore not salient in the Critique, that leaves us with distance, a spatial property that is perfectly salient here. The (inverse) proportionality to (the square of the) distance suggests that attraction is not an ordinary intrinsic property. Newton himself is clear on this point, as we see in his third "rule of philosophizing" in Book Three of the Principia; after concluding that gravity is a universal force, he cautions: "Yet I am by no means affirming that gravity is essential to bodies. By inherent force I mean only the force of inertia. This is imn1utable. Gravity is din1inished as bodies recede from the earth" (Principia, 796). So let's presume that attraction as IZant considers it is a relational property. But that is simply the beginning. As with the intrinsic properties discussed above, we will need to distinguish several types of relational property in order to fill-in the metaphysical landscape. First, there are ordinary relational properties, which I have defined to be incompatible with the loneliness of the bearer. Second, there are what I will call "symmetric" relational properties, which can be defined as follows: a relational property P is symn1etric just in case, in some possible world where object 01 bears P, there is another object 02 that bears P (where P is a type, rather than a token). Hence being ten miles from Paris is an ordinary relational
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property-its existence presumably depends on Paris and is therefore incompatible with the loneliness of the bearer (since Paris cannot be any distance fron1 itself)-but being a sibling is symmetric-relational. Whereas Henri can be ten miles from Paris without Paris being ten miles from itself, Henri and Sophie obviously cannot be siblings unless each of them is. Third, there are spatial symmetric relational properties, that is, symmetricrelational properties that involve some kind of spatial relation; being ten miles apart is an example of such a property. The reason for introducing this third type of relational property may be obvious: gravity is what Howard Stein has helpfully called a force of interaction (Stein 2002, 288; cf. Newton 1731, 37), so we might think that attraction is a spatial symmetric relational property. That is, for body A to attract body B is ipso facto for B to attract A, where attraction must be understood in terms of the spatial relations between A and B.l6 But as I hope to indicate, this isn't quite what we need to understand attraction in IZant's hands. Newton's own language in the Principia indicates what we need to understand attractive force in the Amphiboly: it is not just the case that gravity is "mutual"-which might entail that it is a spatial symmetric relational property-it is also the case that it is "opposite in direction. "17 For two bodies, A and B, to interact gravitationally is for A to gravitate in the direction of B, and B to gravitate in the direction of A. Hence even if we abstract away from material properties such as mass ala I
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This allows us to articulate a difference between monadic (Leibnizian) and attractive (Newtonian) force as I(ant understands them: whereas monadic forces are Leibniz-intrinsic (qualitative identity entails numerical identity), attractive forces are Newton-relational (attraction is spatially symmetric, but directional). Since monadic forces are Leibniz-intrinsic, they are inherently non-spatial, and involve logical grounds alone; in contrast' Newtonian forces are inherently spatial, and involve real grounds. Or so I will contend. But before proceeding, there n1ay be an obvious objection to my discussion thus far. Is it really the case that attraction must be taken to be Newton-relational? In particular, is it the case that two attractive forces, as they are conceived of in the abstract environment of the Critique, can be distinguished solely via spatial direction? Is the attractive force of body A on body B distinguishable from that of B on A solely by their distinct spatial directions (which are derivative, of course, from the distinct spatial positions of A and B)? Why should that be? This question, it seems to me, requires us to reconsider the significance of the fact that, in the Critique, I
3. Lessening the Tension We are now in a position to recognize that the two aspects of the critical project discussed at the beginning are not in tension with one another after all; rather, the transcendental character of I
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dovetails nicely with his attempt to explicate the possibility of Newtonian dynamics. That is, we can now underscore the significance of the fact that Kant considers attraction in the Critique in abstraction from the specifically material properties of the objects that exert attractive force. For as we have seen, it is precisely in virtue of the fact that we are not considering objects qua material entities that allows us to understand how attraction is best understood as a Newton-relational property. Ifwe were to consider these objects as bearing specifically material properties, there would be no need to think of the property of attraction as Newton-relational, for forcestokens of the force-types-would be proportional to mass and therefore distinguishable independently of spatial direction. This leads to two salient points. First, attraction is inherently spatial, the spatial element in attractive force is central to lZant's explanation of the resources we require in order to represent objects as bearing attractive force, and the latter explanation will be an essential element in lZant's explication of the possibility of Newtonian dynamics. The story, perhaps, is a reasonably familiar one to readers of the Critique. If attractive force is Newton-relational, it seems, we cannot represent two objects as bearing that property without representing them as bearing particular spatial relations to one another. And as we learn from the argun1ents in the Transcendental Aesthetic, it is essential to IZant's view in the Critique that we cannot represent objects as bearing spatial relations with one another using concepts alone; rather, intuition must playa role. In the metaphysical landscape of the Critique, the dynamical property of attraction must be construed as an essentially spatial property, so it seems that we require intuition in order to represent objects as bearing this property. This would seem to be a precondition for understanding how it is possible to know that objects bear this property. This may strike readers of IZant as an in1portant aspect of his rejection of Leibniz's metaphysics in the Amphiboly-I(ant takes Leibniz's root error to involve his insistence on representing our objects of knowledge through conceptual resources alone. The second point may highlight the significance of another I
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2. Agreement and opposition. If reality is represented solely through pure understanding (realitas noumenon), no opposition between realities can be thought, i.e., a relation such that if they are bound together in a subject they negate one another's consequences, as with 3 - 3 = O. On the contrary, realities in appearance (realitas phaenomenon) can surely be in opposition with one another and, united in the same subject, one can destroy the consequence of the other partly or ,vholly, as with two moving forces along the same straight line in so far as they either pull or push a point [Punkt] in opposed directions, or as with an enjoyment that balances the scale against a pain. (A264-65/B320-21) And IZant adds the following discussion of real opposition later in the Amphiboly: [T]he principle, that realities (as mere affirmations) never logically oppose one another, is a perfectly true proposition concerning the relations of concepts [Verhaltnisse der Begriffe], but means nothing at all either in regard to nature, nor in regard to any thing in itself (of this we have no concept). For real opposition always obtains where A - B = 0, that is, where one reality, if combined in one subject with another, cancels out the effect of the latter-this is constantly placed before our eyes by all hindrances and countereffects in nature, which, since they rest on forces, must be named realitates phaenomena. General mechanics can even give the empirical condition of this opposition in an a priori rule by looking to the opposition of directions, a condition which the transcendental concept of reality absolutely does not know. (A272-73/B328-29) When considered within the Critique, the attractive forces of any two bodies will be in real opposition with one another, pulling the bearers of the forces in opposite spatial directions and balancing one another out. That is, the effects of any given attractive force will be "canceled" by those of the attractive force of the relevant member of what we might call the attraction pair. And as we might expect, IZant thinks that real opposition can be represented only through intuition-a purely conceptual representation of objects a la Leibniz will fail to represent a real opposition (see A274/B3390 and Warren 2001). Two final points might be in order. First, this episode may indicate an important difference between IZant's discussion of the possibility of mathematics and his discussion of the possibility of natural science, one that reflects the transcendental character of the Critique. If we take geometric knowledge as a paradigm case, the difference is particularly stark. Since geometry involves the representation of various figures in space, it is a
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particularly appropriate exemplar of mathematical knowledge in the context of the Critique, for the latter's transcendental character does not prevent us from conceiving of the objects of our knowledge as bearing spatial relationships (on the contrary, this seems to be central to the Transcendental Aesthetic.) The prima facie contrast with Newtonian dynamics is clear: the notion of gravity in the Principia would seem to require an employment of concepts that are unavailable to us in the Critique; as we have seen, this fact raises at least a prima facie question concerning the other aspect of the critical project that I have been discussing. So the contrast with the case of geometry may help to underscore the significance of the fact that IZant's procedure of abstraction requires hilll to conceive of the attractive forces of distinct objects as distinguishable only through distinct spatial directions. For that enables the construal of a dynamical notion in abstract spatial terms, and that is perfectly appropriate in the context of the Critique. The case of geometry does not require this extra step through the metaphysical landscape of the Critique. Second, we can also place my conclusion within the context of IZant's critique of Leibniz's monadology. I(ant obviously takes Leibniz to reject intuition as a genuine and independent source of clear representation and of knowledge. He also contends that if we represent objects solely from the perspective of "understanding," as Leibniz is said to do, we cannot represent them as bearing spatial relations with one another (this of course presupposes the argument of the Transcendental Aesthetic). It follows that we cannot represent them as having real oppositions, and that we cannot represent them as bearing attractive forces. In tandem, if we were to meet the strictures of understanding, we would represent objects only as bearing logical relations, which would include monadic forces, but exclude dynamical properties such as attractive force. From LeibnizJs point of view, of course, that would be entirely appropriate, given his rejection of attraction. But for Kant, Leibniz's metaphysics simply cannot explain how the science of forces presented in Newton's Principia is possible. It cannot do so because the very object of study of the science would simply disappear in a philosophically defensible conception of the natural world a la Leibniz.
NOTES 1. I first started thinking about this topic while talking with Michael in his office in Goodbody Hall, and I eventually wrote a dissertation on Newtonian themes in the first Critique under his direction. My views have changed over the years, but Michael's encouragement and guidance have always been essential components of my philosophical education. For that, I am grateful. For their helpful
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COlnments, I \vould also like to thank Michael Dickson, Mary Domski, Tad Schmaltz, Eric Watkins, and especially Paul Franks. Special thanks are due to Dickson and Domski for proposing and editing this volume to honor Michael. 2. All translations are my own unless otherwise noted. I cite the Critique by the usual AlB method; I have often consulted the new standard translation by Guyer and Wood, although my translations differ somewhat from theirs. Other references to IZant's works are to the relevant volume and page number of the Academy edition, !(antsgesammelte Schriften (Kant 1910-). 3. Euler's ([1748J 1911) paper was significant for IZant already in the pre-critical period: for instance, see the discussion in Negative Magnitudes (2: 168). Cf. the introduction to Friedman (1992). 4. Any interpretation of this topic would presumably need to account (at least) for the following passages in the Critique: A7 IB11, where IZant famously contends that "Bodies are heavy," in contrast to "Bodies are extended," is a synthetic judgment; B17-18, where he claims that natural science-"physica"-includes synthetic a priori judgments; and, A95-96/B127-28, where lZant contends that our a priori scientific knowledge conflicts with any attempted "empirical derivation" of the concepts treated in the transcendental deduction. 5. The coherence of this understanding of natural philosophy with Newton's own conception is immediately evident. As Newton writes in the preface to the first edition of the Principia: "For the basic problem of philosophy seems to be to discover the forces of nature fron1 the phenomena of motions and then to demonstrate the other phenomena from these forces" (Newton 1972, Volume 1: 16 and Principia, 382). Since Cohen and Whitman's translation of the Principia (Newton 1999) is now the standard, I follow it whenever possible, citing it throughout as "Principia." 6. Cf. Friedman (1992), Chapters 1 and 2. 7. A well-known comment from the Metaphysical Foundations might be salient here: "The general principle of the dynamics of material nature is: all that is real in the objects of outer sense which is not merely a determination of space (place, extension, and figure) must be seen as moving force [bewegende IZraftJ" (4: 523)." Cf. the following intriguing comment from the second edition of the Transcendental Aesthetic (B66-67), where lZant is discussing an aspect of his view of time: "everything in our knowledge [unserem ErkenntnisJ that belongs to intuition ... contains nothing but mere relations, of places in an intuition (extension), alteration of places (motion), and laws, according to which this alteration is determined (moving forces [bewegende IZrafte])." 8. See especially sections 118-23 of Clarke's fifth letter in Leibniz (1890), Volume 7: 439-40; cf. sections 110-16 of his fifth letter in ibid. (436-38). Of course, Clarke's treatment of this issue was not limited to his correspondence with Leibniz, although that was probably his most prominent discussion. In his famous "Newtonian" notes to Rohault (1697), he may endorse a distinct conception; the discussion appears to be too limited to decide matters. See the interpretation in IZoyre (1965), 171. 9. Beginning already in the late seventeenth and early eighteenth century, commentators disagreed strenuously on the question of whether Newton's physics postulates action at a distance between material bodies, and the related but distinct
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question of whether Newton himself could tolerate distant action. Newton expressed skepticism toward action at a distance in his famous correspondence with Bentley in 1692-93-see Newton (2004), 100, 102-103. In Janiak (2007), I argue that Newton ultimately rejected action at a distance on strong metaphysical grounds, and that this rejection was consistent with his understanding of the Principia. Some interpreters argue that, despite his professed skepticism, Newton accepted action at a distance in some circumstances-see especially Westfall (1971, 395) and McMullin (1978, 151 note 210)-whereas others take his views to be inconsistent-see, for instance, Heilbron (1982, 44-46). For a comprehensive historical account of action at a distance in physics, see Hesse (1961). 10. For an excellent discussion of Kant's distinction between real and logical relations, see Longuenesse (1998, 347-58), who cites the passage from Metaphysik Volckmann quoted in the main text. I've used Longuenesse's translation of this passage (1998, 355). Cf. also Cohen (1928), Volume 1: 302-307. 11. One should recall here that in the metaphysical deduction, Iunt "deduces" the twelve categories from the corresponding logical forms of judgment available to us in (Iunt's understanding of) general logic. For instance, Iunt deduces cause from the logical form of hypothetical judgment, and substance from the categorical form of judgment. It seems crucial that the analytic/synthetic distinction is irrelevant to this deduction, for as I note above, general logic distinguishes judgments into various logical forms, ignoring the conceptual containment relations that are necessary for understanding which judgments are analytic, and which synthetic. This should not confuse us into thinking that the notion of a logicalground is well-formed within general logic itself, since the former involves the containment relations that determine analyticities, and the latter is ignorant of such relations. It is relevant here that Iunt tal(es all causal judgments to be synthetic, and therefore any expression of a real ground to be unavailable to us through an analysis of concepts alone. 12. For discussion, see Langton and Lewis (1998). Various objections to the Langton-Lewis conception of intrinsic properties have been raised-including, of course, the fact that loneliness is compatible with itself, and yet is presumably relational-but they are not salient here. 13. The notion of an absolutely intrinsic property captures an important aspect of Leibniz's view of monadic properties, but it also leaves several questions unanswered; I don't have space to answer these questions here, although I can raise some of them. For instance, the contrast with ordinary properties may be partially unclear because it is unclear what type of supervenience is relevant for defining ordinary intrinsic properties. Similarly, is any property compatible with loneliness borne by a non-spatial entity absolutely intrinsic? That is, does Kant intend his definition of absolutely intrinsic properties specifically to cover monadic properties, or might other properties fall under the definition as well? 14. Consider Leibniz's analysis of Alexander's properties: the principle of the identity of indiscernibles prevents another substance in Alexander's world (i.e. the actual world) from having all of his properties, but not from having a property in common with him. For instance, Alexander need n.ot be the only king in the actual world. Since the principle of the identity of indiscernibles prevents two substances from bearing qualitative identity but not qualitative overlap, not all monadic properties need be Leibniz-intrinsic. However, Alexander's comple~e_~~c~0_'\l"~J~lst
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appear to be Leibniz-intrinsic. For Leibniz's discussion of Alexander, the reader can refer to Discourse on Metaphysics, section 13, (Leibniz 1989, 44-46) and Leibniz's later letter to Arnauld, May 1686 (Leibniz 1989, 69-77). 15. Nothing hinges here on this reading of monadic forces. As for the complete concept of a substance, Leibniz discusses the notion extensively in his letters to Arnauld-see particularly Leibniz (1989), 73-74. The place of indexicals in Leibniz's monadology is notoriously complex, so I am unsure whether a monadic force entails which world its bearer is a mernber of 16. This indicates that for Newton, attraction is not akin to true motion, for one body could be truly moving tOvvard a second body that is at absolute rest-that is, true motion is not inherently mutual. See the Scholium to the Principia, esp. 408-11. 17. Cf. the Scholium following Corollary Six to the Laws of Motion (Principia, 424-30), where Newton argues that the third law of motion holds also for "attractions;" cf. also his correspondence with Roger Cotes, the editor of the Principia)s second edition of 1713 (Newton 2004, 118-22). To understand what Newton means by attraction in the context of his so-called mathematical treatment of force in the Principia, see the Scholiun1 to section 11 of Book One (Principia, 588-89), and the discussion in Janiak (2007). 18. The abstraction away from mass is also evident in IZant's discussion of attractive force in the Dynamics chapter of the Metaphysical Foundations, for he discusses what he takes to be the two types of fundamental force independently of a consideration of mass; the latter is not introduced until the Mechanics chapter (4: 547 note). In a move that is, of course, foreign to Newton's procedure in the Principia, IZant introduces attractive force independently of his discussion of the laws of motion, which also does not appear until the Mechanics chapter. 19. See A264-65/B320-21, A273/B329, A282-83/B338-39, and Iunt, Negative Magnitudes, 2: 173-75. IZant discusses real oppositions in another text from 1763, the Only Possible Basis (2: 86). Gueroult (1967, 168) notes that IZant's understanding of real opposition reflects a decidedly Newtonian influence on his thinking, one complicating his earlier, more closely Leibnizian, views. See Warren (2001) for an outstanding discussion of real opposition in IZant.
REFERENCES Bricker, P. and R.I.G. Hughes (eds.) (1990). Philosophical Perspectives on Newtonian Science. Cambridge: Cambridge University Press. Cohen, I.B.and G. Smith (eds.) (2002). Cambridge Companion to Newton. Cambridge: Cambridge University Press. Cohen, H.(1928). Schriften zur Philosophie und Zeitgeschichte. Volume 1, Berlin: Al(ademie-verlag.. Euler, L. ([1748J 1911). "Reflexions sur L'Espace et Ie Temps." In Leonhardi Euleri Opera Omnia. Edited by F. Rudio, et al. Leipzig: Teubneri. Friedman, M. (1990). "IZant and Newton: Why Gravity is Essential to Matter." In P. Bricker and R.I.G. Hughes (eds.), 185-202.
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- - - . (1992). !(ant and the Exact Sciences. Cambridge, MA: Harvard University Press. Gueroult, M. (1967). Leibniz) dynamique et metaphysique. Paris: Aubier. Heilbron, J.L. (1982). Elements of Early Modern Physics. Berkeley: University of California Press. Hesse, M. (1961). Forces and Fields: The Concept of Action at a Distance in the History of Physics. London: Nelson. Holden, T. (2004). The Architecture ofMatter. Oxford: Oxford University Press. Janiak, A. (2007). "Newton and the Reality of Force." Journal of the History of Philosophy 45: 127-47. lZant, I. (1910-). !(ants gesammelte Schriften. Edited by koniglich preussischen [later: Deutsche] Akademie der Wissensclfaften. Berlin: Rein1er. - - . ([1781] 1956). !(ritik der reinen Vernunft. Edited by R. Schmidt. Hamburg: Felix Meiner Verlag. - - . ([1781] 1998). Critique ofPure Reason. Translated and edited by P. Guyer and A. Wood. Cambridge: Cambridge University Press. - - . ([1786] 1997). Metaphysische Anfangsgriinde der Naturwissenschaft. Edited by IZ. Pollok. Hamburg: Felix Meiner Verlag. - - - . (2004). Metaphysical Foundations of Natural Science. Translated by M. Friedman. Cambridge: Cambridge University Press. IZoyre, A. (1965). Newtonian Studies. Cambridge, MA: Harvard University Press. Langton, R. and D. Lewis (1998). "Defining 'Intrinsic'." Philosophy and Phenomenological Research 58: 333-45. Leibniz, G.W. (1849-). Mathematische Schriften. Edited by C. Gerhardt. Berlin: A. Asher. - - . (1890). Die Philosophischen Schriften. Edited by C. Gerhardt. Berlin: Weidmann. - - . (1989). Philosophical Essays. Edited and translated by R. Ariew and D. Garber. Indianapolis: Hackett. Longuenesse, B. (1998). !(ant and the Capacity to Judge. Translated by C. Wolfe. Princeton: Princeton University Press. Newton, I. (1731). A Treatise ofthe System of the World. London. Second edition. - - . ([1687] 1972). Philosophiae Naturalis Principia Mathematica. Third edition. Edited by A. Koyre and LB. Cohen. Cambridge, MA: Harvard University Press. - - . ([1687] 1999). The Principia: Mathematical Principles of Natural Philosophy. Translated by LB. Cohen and A.Whitman. Berkeley: University of California Press. - - - . (2004). Philosophical Writings. Edited by A. Janiak. Cambridge: Cambridge University Press. Rohault, J. (1697). Jacobi Rohaulti Physica. London: Jacobi IZnapton. Stein, H. (2002). "Newton's Metaphysics." In Cohen and Smith (eds.), 256-307. Warren, D. (2001). Reality and Impenetrability in !(ant)s Philosophy of Nature. New York: Routledge. Westfall, R.S. (1971). Force in Newton)s Physics: The Science of Dynamics in the Seventeenth Century. London: MacDonald, New York: American Elzevier.
PART II I(ant
[6J I(ant and Lambert on Geometrical Postulates in the Reform of Metaphysics ALISON LAYWINE
Johann Heinrich Lambert was probably IZant's most valued critic and collaborator. Though they never met, and though they exchanged letters only three times, their association was unusually close. It was based on deepseated mutual respect and shared purpose: to effect some kind of badly needed reform in metaphysics. The intermittent correspondence began in 1765 at Lambert's instigation; it can1e to a halt in 1770 after Lambert sent IZant detailed critical comments on his Inaugural Dissertation. IZant never wanted the correspondence to end. He explained in a letter of 1781 to the astronomer Johann Bernoulli that he kept trying to work out a reply to Lambert as fully as possible-so they would have a basis for further collaboration. The work was not in vain; it ultimately led to the publication of the Critique of Pure Reason in 1781. But to IZant's everlasting regret (and mortification), the delay was too long: Lambert died in 1777 without ever hearing from IZant again. "I regret this loss all the n10re," IZant wrote to Bernoulli, "because after thinking I had come into possession of what I had been seeking, Lambert was just the man ... [I wanted] to look over the claims subsequently presented in the Critique of Pure Reason in their whole context [sc. presumably in the context of the early exchange between the two men], to assess these claims and to reveal to me any errors they might contain ... " (10.278.9-16).1 IZant's regard for Lambert was so high and his sense of debt so acute that he even considered dedicating a work to him-probably the Critique ofPure Reason itself or some earlier incarnation of it from the mid-1770s. This is suggested by the following draft in IZant's hand, scribbled in the margin of his copy of Baumgarten's Metaphysics: (For the dedication). You honoured me with your letters. The effort to conceive a method for pure philosophy at your insistence led to a series 113
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of reflections aimed at developing the idea that still lay obscure within me; and, it continually postponed my replies [sc. to your letters] as the prospects of success improved with the progress I made. This work can serve in the place of a reply, so far as the speculative part is concerned. Since it [sc. the work] n1ust be credited to your tips and requests, it was my wish that it belong entirely to you through the effort to take account of you in my revisions. (R5024,18.64.2-10) The dedicatee is not named, but the talk of letters and a postponed reply recalls the circumstances reported to Bernoulli so vividly that it is hard to imagine 10nt was planning to honor anybody other than Lambert (Erdmann: 1884,1-2). When the Critique ofPure Reason finally came out, it bore a dedication not to Lambert, but to 10rl Abraham Freiherr von Zedlitz. We do not know precisely what prom.pted the change of dedication, but it clearly did not signal a change of mind about Lambert's philosophical significance in the years since his death. 2 This is plain from the 1781 letter to Bernoulli I mentioned earlier. 3 Hence, the question: what exactly did Lambert contribute to IZant's writing of the first Critique? A full answer to the question is beyond the scope of this paper. But I would like to make one concrete suggestion: Lambert gave Kant the interesting idea that geometrical postulates in Euclid's sense were somehow essential to the reform of metaphysics they both sought. I will begin by exploring what Lambert had to say about this. Then I will try to show how Kant may have adapted Lambert's insights.
1. Lambert and the Reform of Metaphysics
In his first letter to I
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finds in the writing of Wolff's most vehement detractors: as in Euler, for example. But Lambert's praise here is genuine. 4 His criticism for now is that Wolff expected far too much from definitions. His philosophical method proceeds by defining every term remotely relevant to the topic and then using these definitions to derive all the rest (Lambert 1771, 8). According to Lambert, this method is good for exposing errors of inference, but it can never settle the hard questions. Lambert says that "Wolff inadvertently thrust upon his definitions the doubts and difficulties previ0usly found in metaphysics" (Lambert 1771, 8-9). For Lambert, one of the most significant cases of this problem can be found in Wolff's treatment of parallel lines in the geometry of the Elementa matheseos universae. Before looking at the specifics, here are some historical facts to keep in mind.
B
---r-.-------:::-D c
By the mid-1760s, Lambert had developed a very keen interest in Euclid's fifth postulate, the so-called parallel postulate: if two lines CD and BD are intersected by a third line BC and both interior angles DCB and DBC taken together are less than two right angles, then the lines CD and BD will meet in the direction of D. Lambert had read what might well have been the first history of attempts to prove this postulate in a dissertation from 1763 by Georg Simon Klugel-a student ofAbraham Gotthelf Kaestner in Gottingen. Kaestner himself had become interested in the fifth postulate some years earlier and had begun to collect all the writings he could find on the subject. Under his direction and quite possibly with access to his library, Klugel examined no less than thirty attempteo proofs and showed that all of them misfire. s He concluded that we recognize the absurdity of counterexamples to the fifth postulate "not as the result of strict inferences or by means of distinct concepts of straight and curved lines, but rather through experience and through the judgement of our eyes" (Engel and Stackel 1895, 140).6 Lambert agreed that all of the attempted proofs surveyed by Klugel had failed, but he did not share Klugel's conclusion. "This principle," he wrote of the fifth postulate, "is indisputably not as clear and illuminating as the others-not by a long shot; and the impression that it naturally leaves is that one will not only
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demand a proof for it, but one even senses to a certain degree that it can be proved or that there must be a proof for it" (Lambert 1895,152-53). By September 1766, Lambert had written up his reflections on the fifth postulate-including his own attempt at a proof. He put aside the manuscript; Johann Bernoulli published it posthumously in 1786 under the title, The Theory of Parallel Lines. For our purposes, the manuscript is interesting, because it shows that Lambert was predisposed to examine critically the different ways people had dealt with parallels and more especially because it opens with SOUle very critical comments about Wolff on this subject. These con1ments turn on the difference Lambert sees between Wolff and Euclid in their use of definitions. Euclid defines parallels as "straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction" (Elements I. Def. 23). Wolff departs from Euclid. He defines parallels as straight lines that are always the same distance apart (Wolff 1742,128).7 He then uses this "definition to prove the equivalent of Proposition 29 in Book One of Euclid's Elements-the proposition that states: "A straight line falling on parallel lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side equal to two right angles."s Euclid's proof of this proposition depends in part on the fifth postulate. Indeed, it is in this very proof that the fifth postulate appears for the first time in Euclid's Elements. But the fifth postulate is left out of Wolff's proof of the equivalent proposition. In fact, it seems to have been left out of his geometry all together. At least there is no statement of it in the section of the geometry where Wolff lays out his first principles. Lambert is struck by the difference between Wolff and Euclid. He pretty clearly takes Wolff to be looking for a way to prevent the uncertainty of the fifth postulate fron1 infecting the proof of Euclid 1.29 and any other proof that ultimately depends on it. That is supposed to explain the key role played in the proof by Wolff's definition of parallel lines and indeed Wolff's motivations for revising the definition in the first place. Lambert writes in the manuscript of 1766: "Thus Wolff-with many oth.ers-believed that one could get rid of the difficulty posed by Euclid's eleventh principle [sc. the fifth postulate] by changing his definition of parallel lines" (Lambert 1895, 158-59). Lambert is highly critical of this move: [The difficulty associated with the fifth postulate] is neither gotten rid of [by changing the definition of parallel lines], nor is it avoided, nor dealt with in a clever way and as it were eliminated from behind. Rather, even if everything works out, the difficulty is shifted from the
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principles and relocated in the definitions; and indeed, as far as I see, without it being possible thereby to get rid of the difficulty more easily. (Lambert 1895,159) One of the underlying issues here is what definitions are good for. In a letter of 11 April 1765 to Baron Georg Jonathan von Holland, Lambert explained that Euclid treats his definitions as a "nomenclature": "[Euclid] does nothing more than what, for example, a watchn1aker or any other craftsman does when he sets out to teach his apprentice the names of his tools" (Engel and Stackel1895, 141). Thus Euclid's Definition 23 tells us only what kind of figures we may call "parallel lines. " It tells us that we may apply this name to straight lines that do not meet in either direction-provided that any such exist. But whether they do exist-indeed whether they are even possible-is unsettled by the definition all by itself. Hence Euclid does not even invoke Definition 23 until the end of the proof of Proposition 27, i.e., not until he has proved that, if a straight line falling on two straight lines make the alternate angles equal to one another, they will not meet in either direction. As long as two straight lines satisfy this condition, he says we may call them parallel. This gives him the last step in the proof of Proposition 27. Once Propositions 27 and 28 have established the conditions under which parallel lines are possible, Lambert says they discharge the hypothetical status of Definition 23, which Euclid may now regard as a real rather than a nominal definition and therefore use as a first premise in later proofs. "The definition of parallel lines," he wrote to Georg von Holland, "is nothing more than an hypothesis until the possibility of such lines is proven [sc. by Propositions 27 and 28] ..." (Engel and Stackel1895, 142).9 By contrast, Lambert takes Wolff to be leaning too heavily on his definition. Wolff's alternative is as much a nominal definition as Euclid's. Yet Wolff expects it not only to tell us how to use the name "parallel lines"; he expects it to do as much work proving the equivalent of Euclid 1.29 as Euclid himself assigned to the fifth postulate. But without the fifth postulate, he cannot get a proof for this proposition; or if he could, it would only be by somehow transforming his nominal definition into a real one: Lambert says that he would inevitably need the fifth postulate to do so (Lambert 1895, 159). Wolff did not get rid of the difficulty associated with this postulate; he simply substituted one nominal definition for another. Lambert now goes on to say that this has been bad not only for geometry, but also for philosophy. Many philosophers followed Wolff's example and tried to adapt Euclid's method to their discipline. They learned Euclid's method from Wolff, but failed to notice that his misguided dependence on nominal definitions left unsettled very substantial difficulties-as in his treatment of parallel lines:
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I said before that Wolffgranted too much to definitions.... It became a fashion for many to think that to have the concept of a thing is impossible until the thingJs name has been defined. Even the principles had to be preceded by definitions without which they could not be understood. Hence it is no wonder if the proposition that every definition) before it is proven) is an empty hypothesis-the very proposition that Euclid knew so well and took so seriously-was, if not completely lost, then pretty much forgotten. I call attention to this all the more urgently because it had such bad consequences for the presentation of philosophical science, and in particular because it is precisely that in which Wolf went wrong when he abstracted his method from Euclid, and finally because parallel lines provide the most striking example of the point that a definition stated in advance proves nothing until it itself has been proved. (Lambert 1895,157-58)10 Wolff should have used the fifth postulate in his geometry and followed Euclid's order of presentation. Had he presented his definition just as part of a "nomenclature" and then stated the fifth postulate with the other principles separately from the definitions, as Euclid does, Lambert says the hard problems would have been in plain view. The philosophers would have been confronted at the very beginning of the geometry with an explicit statement of the principle needed for the theory ofparallels and everything that follows from it. Like so many before them, they would-or shouldhave immediately recognized that this principle seems to require justification and that the theory of parallels itself is therefore in doubt (Lambert 1895,155).1 1 But following Wolff's example, they tried unsuccessfully to pass off nominal definitions as principles of philosophy and therefore refused to discuss the real principles and their attendant difficulties-just as Wolff himself had done in the theory of parallels. To the extent that Wolff recommended what he took to be Euclid's method to philosophers and met with their approval, he did as much harm as he did good. Indeed, he compromised philosophy as a scientific discipline. This is at least part of the reason that Lambert so urgently calls for a reform of metaphysics. Now the other side of Lambert's complaint about Wolff's adaptation of Euclid is that it neglects the postulates: in particular, it substitutes a nominal definition for the fifth postulate in the theory of parallels (Lambert 1771, 9). This observation leads Lambert in the Anlage zur Architectonic to reflect on postulates more generally and their function in proofs. We saw earlier that a Euclidean definition does nothing more than tell us the names of the different figures. Euclid defines a circle as a plane figure the circumference of which is everywhere equidistant from the center. This definition tells us only what kind offigure we may call "circle"; it does
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not tell us whether circles are possible or whether it is possible to form a lneaningful, non-empty concept of them. The function of Euclid's postulates is precisely to secure "universal possibility" both for the relevant figure and the corresponding concept. Lambert writes, "Euclid took care to address [sc. the universal possibility of the relevant concept]; and, for this purpose, he used his postulates which represent universal) unconditioned and in and of themselves conceivable possibilities, i.e., simple possibilities or things we can do [Thulichkeiten] ... " (Lambert 1771,10). The first thing to notice here is that Lambert characterizes postulates as "possibilities." He seems to mean that they are practical possibilities, since he also calls them" Thulichkeiten," i.e., things we are really able to do. Euclid's first three postulates straightforwardly meet this description. The first postulate tells us that 1we can draw a straight line from any point to any point. The second tells us that we can extend any finite straight line continuously in a straight line; the third that we can construct a circle of any center and radius. Hence the first three postulates assert the possibility of straight lines, extensions of straight lines and circles, respectively. If I can construct these figures, they must be possible-and so must the concept I have of them since, in geometry at least, I must know what I am doing in order really to be able to do it. 12 By contrast, Euclid's fourth postulate does not follow this pattern. It says that all right angles are equal to one another. Unlike the others, it is not constructive and hence it does not state a practical possibility.13 This leads me to think that Lambert would simply have denied that it is a genuine postulate, though I have yet to find a passage in which he explicitly does so. It is also interesting to note that the fifth postulate does follow the pattern of the first three. For it too is constructive. It tells us that we can construct two lines that will meet if extended far enough-on the condition that, if we let a third line fall on both of them, the angles so formed on one side are less than two right angles. Since we really can produce such a construction, it seems that the fifth postulate secures practical possibility for lines that meet in just the way that the other postulates do so for circles and the rest. But Lambert treats the fifth as a genuine postulate no more than he presumably does the fourth. Mter all, he sought a proof for it. That means he regarded it as a theorem-or rather, as a constructive problem. Euclid's postulates are supposed to require no proof. They are what a teacher is entitled to demand from his pupil-the Greek is aitemawithout any further justification than his authority as a teacher. This, I take it, is the force of Lambert's claim that Euclid's postulates are possibilities "conceivable in and of themselves." It is also the force of his claim that they are "simple." Following Locke, Lambert thinks that we have certain simple ideas that are the constituent elements of all our complex ideas. The truth and certainty of con1plex ideas depends on that of the simple ideas
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and whether they have been con1bined in the right way. Since the simple ideas have no internal complexity, they are evident in themselves. The Euclidean postulates are simple ideas-indeed that's what makes them elements of geometry. Hence, they are the starting point of this science; and they require no proof (Lambert 1771,20).14
But there is at least one n10re thing to emphasize in Lambert's remarks. The practical possibilities asserted by the first three Euclidean postulates are universal and unconditioned. The third postulate says that we can construct a circle of any distance and circumference. It doesn't place any restrictions on us. This is important, because it means that we can demand a circle of precisely the distance and circumference we need. Suppose we are trying to solve the problem of constructing an equilateral triangle on a given line segment AB. The solution to that problem depends on constructing two circles-one of center A, the other of center B-having the same distance, namely, equal to the length ofAB. Suppose that a sceptic is satisfied that we have solved the problem just for the given line segment AB, but doubts that the solution holds for other line segments of different lengths. We can put this doubt to rest by appealing to the fact that the third postulate is universal and unconditioned. No n1atter what the length of the given line segment, we can always construct the circle we need to solve this problem. As a result, our solution to the problem is just as universal and unconditioned as the postulate upon which it depends: no matter what the length of the given line segment, we can always construct an equilateral triangle upon it (Lambert 1771, 16-17).1 5 Of course, what holds for the third postulate holds for the first two as well. For the first tells us that we can construct a line between any two points, the second that we can extend any given line segn1ent continuously. There are no restrictions here. So these postulates may be used to put to rest the san1e kind of doubts as that which we considered in the case of the equilateral triangle. This just goes to show that the first three Euclidean postulates are not just universal and unconditioned in themselves; they also happen to be the source of unconditioned universality in any of the results in geometry that depend on them.
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Now the first three Euclidean postulates cannot immediately secure universal possibility for every kind of figure and the corresponding concept. Hence they are not enough on their own to discharge the hypothetical status of all twenty-three of Euclid's definitions and convert them into real definitions. But they can do so indirectly, by transmitting unconditioned universality to the relevant proofs and solutions. Thus we draw on the third postulate to give an absolutely universal and unconditioned solution to the problem of constructing an equilateral triangle (Elements 1.1). We can use this solution similarly to solve the problem of placing at a given point (as an extremity) a straight line equal to a given straight line (Elements 1.2). This solution in turn is used to prove the theorem that, in any triangle, two of the sides taken together are always greater than third (Elements 1.20). Lambert says in the letter to Baron Georg von Holland 1 mentioned earlier that this theorem states the condition under vvhich triangles of any shape and size are possible. "One would not have granted this to Euclid [sc. that triangles of any shape and size are possible ]," he says, "from the definition of the triangle alone" (Engel and Stackel 1895, 142). Lambert recon1mends in light of all this that philosophers follow Euclid rather than Wolff. He says in particular that philosophers must find a way to incorporate postulates, as he understands them, into their metaphysics: The conclusion is easily seen to follow from all this that metaphysics, which illuminates abstract concepts and propositions by laying down an individual case or well-chosen example, ought to have its universality and compass determined by postulates and axioms, and that especially the postulates at least ought to be setting out universal and unconditioned possibilities for forming concepts ....(Lambert 1771, 10) Just as Euclid's third postulate declares the universal and unconditioned possibility of circles and thereby our concept of them, so the postulates Lambert has in mind for metaphysics will declare the possibility of specifically metaphysical notions. The Anlage zur Architectonic states a whole bunch of postulates of this kind. 16
2. IZant on Postulates Though 1Zant early on shared Lambert's conviction that metaphysics had to be reformed, there is at least one reason to think he would have resisted the ideas 1 just ascribed to Lambert. They all rest on the assumption - - - - - - - - - - - - - - - - - - - - - - - - -
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Lambert shared with Wolff that metaphysics should adopt the method of mathen1atics, at least as it is exemplified in Euclidean geometry. I(ant rejected this assumption. In the Preisschrift of 1763, he approvingly quoted William Warburton as saying that "nothing is more harn1ful to philosophy than mathematics, that is, to think of imitating the method of nlathematics in philosophy where this method cannot possibly be used ..." (2.283.20-24). The reason he gives in support of Warburton is that mathenlatics is constructive, while metaphysics is not. Metaphysics proceeds by analyzing the relevant concepts into their constituent marks. Kant never seems to have seriously questioned this position. Indeed, he would defend it once again in the Critique of Pure Reason.!7 But as long as metaphysics is not constructive, there can be no reason to introduce Euclidean style postulates into it. Not surprisingly, there is no mention of such postulates in the specific methodological recommendations IZant makes for metaphysics in the Preisschrift or the parallel passages in the first Critique. Nevertheless, lZant seems to have believed that thinking about the function and significance of postulates can be useful-at a minimum for helping us get a fix on certain problems metaphysicians must confront. In what follows, I will try to tease out his line of thinking. No doubt it does not sit comfortably at first blush with his strictures against metaphysicians making use of mathematical methods. I do not know how ultimately to relieve the tension. Perhaps the thing to say for the purpose of this paper is just that lZant did not believe that his reflections on postulates had implications for the method of metaphysics. IS What sort of implications they did have in his eyes is the object of what follows. We may begin by noting first that lZant does occasionally discuss postulates-in contexts other than those where he discusses the right method for metaphysics. These discussions indicate that he clearly conceives them in precisely the way Lanlbert does. For example, he says in the Critique of Practical Reason that "pure geometry has postulates as practical propositions that however contain nothing more than the assumption that one can do something if indeed it should be required to be done .... They are thus practical rules under a problematic condition of the will" (5.31.2-4). In Lambert's way of putting it, they are "Thulichkeiten."19 As such, IZant is pointing out something we haven't yet considered, nanlely, that the postulates certainly tell us what we can do, but in themselves give us no reason for actually doing so. The third postulate tells me I can construct any circle I want, but unless I somehow need a circle, I just won't bother. Hence, postulates are practical propositions "under a problematic condition of the will." I will indeed construct a circle of a certain center and radius, if I am trying to solve a problem whose solution calls for nle to do so, e.g., the problem of constructing an equilateral triangle on a given line segment. For then I will need two circles-both having a radius equal
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to the length of the line I have been given and each having one extremity of the line as its center. In general, it will be some problem of construction-or some construction needed to prove a theorem-that will motivate me to act on the practical possibilities asserted by the postulates. Lambert himself certainly could have made this point. I(ant picks up on it to promote his agenda in the second Critique. The postulates of geometry are not simply practical principles; they are-again, as Lambert himself points out-a priori practical principles. To that extent at least, I(ant is struck by the fact that they resemble the so-called "principle of pure practical reason," which commands us to act in such a way that the maxim of our will can always at the same time hold as the principle of a universal legislation" (5.30.38-39). I(ant wants to distinguish this principle of the will from the postulates of geometry. He does so by arguing that this principle commands us unconditionally. It does not say, as the postulates do, "You can, if you want"; it says, "You must under any condition ...." There is another important passage for us to consider, namely the one in the Critique of Pure Reason where I(ant introduces the so-called Postulates of Empirical Thought. He is self-conscious in his choice of the term; so he carefully spells out what he takes to be a geometrical postulate in the technical sense. Thus he writes: Now a postulate in mathematics is a practical proposition that contains nothing other than the synthesis whereby we first give ourselves an object and produce its concept, for example, to describe a circle with a given line fron1 a given point on a surface. Such a proposition cannot be proved for the following reason: the procedure that it requires is precisely that whereby we produce the concept of such a figure in the first place. (B287) Once again I(ant makes it clear that postulates are "Thulichkeiten," i.e., practical propositions that tell us what we can do. But here he insists that they have a special status among such propositions. I cannot take for granted that every proposition giving me license to do something is legitimate. Hence I should ask for proof or justification. Euclid's third postulate admits of no such proof, because it gives me license to just the concepts and figures I would need to invoke in any proof I attempted on its behalf. As a result, any such proofwould be viciously circular-and similarly for the other constructive postulates. (I don't know what views, if any, I(ant had on the fifth postulate.) These remarks are interesting for a number of reasons. For our purposes, the thing to see first is that IZant is directing these remarks against "a few recent philosophical authors"-he does not name them-who have been using the word "postulate," for their own purposes,
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in a way somehow at odds with that of the mathematicians. I
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I(ant himself does take a look at Euclid's constructive postulates, whether in the first or the second Critique, the analysis he gives always coincides with Lan1bert's-for the most part. 20
3. How I(ant Puts Postulates to Work: Given that lZant has seen his way to Lambert's analysis, the next question is whether and how precisely he puts postulates to work in philosophy. This question can take us in a number of different directions. For example, we might consider why lZant thinks that the postulates of empirical thoughtthe modal principles of pure understanding-are just like the constructive postulates of geometry.21 But there is another question I propose to take up instead, namely, what role the analysis of postulates may have played in lZant's ideas about the transcendental deduction of the categories. Consider the difficulty that may well have first led lZant to feel the need for such a deduction-the difficulty raised in the famous letter to Marcus Herz of 21 February 1772. I(ant admits in this letter that he had characterized the intellect in his Inaugural Dissertation of 1770 in purely negative terms, i.e., as a faculty that represents objects a priori without being affected by them. But how are such representations possible, he asks. The answer apparently depends on being able to establish some kind of relation between these representations and their objects. Without any kind of relation, it would be very hard to understand how the intellect could represent anything at all and we would have to conclude that what gets passed off as representations of objects a priori is really just the result of the intellect allowing its cognitive wheels to spin. Now if the intellect were affected by its objects, it would be easy to understand how such a relation is possible, namely, as a result of the object's agency. But, of course, lZant says that the intellect is not affected by its objects and that, if it were, its representations would not be a priori, but empirical like those of sensibility. An alternative might be to consider whether the intellect creates its objects. But though the act of creation would indeed establish some kind of relation between objects and representations-namely, one of dependence-it is possible only for the intellect of a divine being. So the answer to lZant's question in the Herz letter requires some other way of showing how a relation between object and intellectual representation is possible for a finite being. Ifwe now return to Lambert's ideas, it is interesting to notice that the constructive postulates of geometry suggest a way of confronting this problem. They assert the practical possibility of circles, line segments and the extensions of line segments. They thus entitle us both to any of the relevant figures we may need and the corresponding concepts. The constructive
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procedure that I use to generate a circle of any center and radius at the same time generates the universal concept of a circle. To have this concept is not merely to be able to use the word "circle" in the way Euclid's nominal definition instructs us, namely, of any plane figure the circumference ofwhich is everywhere equidistant from the center. It is to understand how a circle may be produced; this understanding goes on display every time we exercise our right to the relevant procedure and construct a circle. Concept and object thus always go together. Indeed, they spring from the same source. That is what establishes a certain relation between them. The relation is such that there can be no talk of the object without invoking the concept, and no use of the concept without invoking the nature of the object. Notice too that the relation is invested with universality, for the constructive procedure upon which it depends allows no restriction on which circles can be constructed or thought. Hence, the concept applies universally-to all possible circles. By virtue of this relation we may indeed say that the concept of a circle really represents any and all circles to usin a way that the non-constructive, nominal definition cannot. That's at least part of the reason for Lan1bert's insistence-contra Wolff-that Euclid's postulates are the ultimate principles or foundations of geometry, not the definitions. Finally, if we follow Lambert we can say that the concept of a circle represents circles to us a priori, since the relevant constructive procedure is not grounded in experience. So it looks as if Lambert's analysis of postulates gives us a way of understanding how a finite intellect can have a priori representations of objects. It can have such representations if we may regard these representations as expressions of something like the constructive postulates of geon1etry. But however plausible this idea inspired by Lambert might be, one wonders whether it ever occurred to I(ant. I would now like to suggest that there is at least son1e circumstantial evidence that it did. In the text of the letter to Herz itself, for starters, I
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Lambert himself certainly stated something like it as a postulate in the arithmetic section of the Architectonic. In §77, he says that the object of arithmetic is "Unity together with the numbers that arise out of its repetition" (Lambert 1771,60). This remark suggests that Lambert is assuming IZant's idea as an ur-postulate of arithmetic: we can generate any given number by repeating unity as many times as we need to. But he goes on directly to state his own first arithmetic postulate in the following terms: "every number can be taken as many times as we like" (Lambert 1771, 60). It is easy enough to see why Lambert explicitly takes this principle to be a postulate. It tells me what I can do with numbers: I can keep adding them to themselves. Thus it expresses a practical possibility. Moreover, it does so without any restrictions: I can perform this operation on any number I please; and, I can repeat the operation as often as I please. 22 Hence it ensures, as a result of this operation, the universal possibility of a determinate quantity which we may think of as the multiple of some given number. Thus the act of generating the multiple of a given number and the act of conceiving the result are really one and the same thing: so too for the act of repeating one as many times as we need and the act of conceiving a number generated in this way. That's precisely what leads IZant to say in the passage I quoted from the letter to Herz that the concepts of quantity are "self-active" and hence capable of representing objects to us as numbers or determinate quantities a priori. Now there is just one problem with these ideas. They will not help IZant solve the problem he has raised in the letter to Herz unless the story he tells about the way mathematical concepts represent their objects a priori can be told n10re broadly. To be more specific, it is not obvious that it helps us understand how pure concepts of the intellect or understanding can represent a priori things given to us in experience. IZant is perfectly aware of this problem. Immediately after having discussed the arithmetic postulate in the passage I quoted earlier, he goes on to say: But in relation to qualities, how my understanding is itself supposed to form concepts of things for itself entirely a priori-concepts with which the things are supposed necessarily to agree-and how my understanding is supposed to work out real principles of their possibil-, ity-principles with which experience must faithfully agree while nevertheless being independent ofit [sc. experience ]-this question always leaves darkness in respect of our power of understanding, namely whence comes this agreement between it and the things themselves. (10.131.15-22) But this difficult question seems in context quite naturally to lead to another. Isn't it possible that our power of understanding represents
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empirically given things a priori in a way somehow analogous to the mathematical examples we have been discussing? In other words, isn't it possible that the understanding has certain resources that are to experience what Euclid's constructive procedures are to circles and line segments? If so, might it not be the case that we can represent nature to ourselves a priori, because son1ehow we construct a model of it using these resourceswhatever precisely they might be? Now there seem to be indications in the Critique of Pure Reason that I(ant would have been unwilling to go the route suggested by these questions. He is ultimately going to say that some of the resources the understanding uses in representing nature a priori-here we are talking about the so-called Analogies of Experience and the Postulates of Empirical Thought-have to do with the existence of appearances; and, he explicitly denies that such existence can be constructed (B221-22). This remarkand others like it-suggests that I
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structed only according to a rule and serves as a rule to all. (R4684,17.670.13-671.2) The details of the passage are obscure. 23 But the main idea is not. The claim is that it is possible for us by the power of our understanding to represent a priori objective ten1poral relations in the order of succession. How? By appealing to some kind of universal rule that allows us to construct what follows what in time, just as we appeal to a certain rule to construct a triangle. This rule will be a postulate of objective timedetern1ination and thus a fundamental principle we use in constructing a priori our model of nature. I would like to suggest that IZant kept trying to develop this idea in the years leading up to the Critique ofPure Reason and that they playa role in the Transcendental Analytic, and even in the Transcendental Deduction. But that suggestion leaves us with a host of questions. First, what are the constructive procedures the power of understanding uses to construct our model of nature? Are the so-called categories IZant claims to have discovered the procedures we are looking for? All of them? Or perhaps only the modal ones governed by the so-called Postulates of Empirical Thought, which are-after all-explicitly characterized as constructive postulates? Either way, isn't it remarkable to think that any of the categories might be, or license, constructive procedures? For they are supposed to be mere "logical functions of thought" which have-as I(ant says-"no meaning and [are] completely empty in content" (A239/B298). If that's the case, wouldn't we need an argument to show that these functions are constructive of anything at all? Nothing could be more surprising than to think, for example, that the logical function of thought by which we distinguish subject and predicate-the category that gives us the structure of every categorical judgment-could be capable of determining any object a priori. The ancient geometers did not have to be convinced that the procedure for producing a line segment is constructive of line segments; they had only to be convinced that this procedure is simple or primitive enough that we may treat it as a starting point in geometry. By contrast with the ancient geometers, doesn't I(ant have to show that the logical functions of the understanding, however primitive they may be, are indeed constructive procedures? But if that's right, maybe we should take the project ofa transcendental deduction to be to furnish just such an argument. By way of a conclusion, I will raise one last question. If the project of a transcendental deduction is as I have suggested, then it seems that lZantlike Lambert-believed that geometric postulates were to play some important part in the future of metaphysics. For understanding what a postulate is and how it contributes to progress in geometry seems to offer a way of answering the question how, by the power of our understanding, we can
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have knowledge of objects a priori. In the preface to the first edition to the Critique ofPure Reason, Kant explicitly ties the answer to this question to "the decision concerning the possibility or the impossibility of a metaphysics as such and the specifying of the sources as well as the extent and boundaries of this science-but all this from principles" (AXIl). As I have been suggesting, Lambert may well have played an important part in leading lZant to the framing of this decision. It is not that I(ant adopts Lambert's specific recommendations for the reform of metaphysics, which-as we saw-rested on methodological assumptions lZant just didn't share. Rather, lZant apparently accepted Lambert's analysis ofpostulates and applied it in a quite original way to solve the problems that arose out of his own, quite idiosyncratic attempt to reform metaphysics. 24
NOTES 1. Cf. Kant's letter to Gotthilf Christian Reccard of 7 June 1781 (10.270.35-271.24). All citations to lZant's works refer to [(ants gesammelte Schriften (the "Academy Edition"). Where applicable, I give the volume, page and line number. Where applicable, I also give the "Reflexion" number, prefixed by "R". Following custom, citations from the Critique ofPure Reason refer to the pagination of the first edition of 1781 (A) and that of the second edition of 1787 (B). All translations from the German of Kant and of Lambert are my own. 2. Perhaps the reason is just that Lambert was dead by 1781. It was, I think, already quite unusual for lZant to have thought of dedicating a work to an individual who was not some kind of patron. (The actual dedication of the first Critique is thus a lot more conventional. Zedlitz was a potential patron; indeed, he would actually have been Kant's patron had I(ant been willing to accept his offer of a recommendation to a well-endowed chair at the University of Halle. See Iunt's correspondence with Zedlitz from the late 1770s in volume ten of the Academy Edition.) I don't know of examples of eighteenth-century dedications to the memory of people already dead. 3. And also from the letter to Gotthilf Christian Reccard of 7 June 1781 (10.270.35-271.24). 4. Precisely because the praise is genuine, one infers that Lambert was deeply read in certain philosophical authors, but not widely read in philosophy as a whole. Hobbes and Spinoza had both insisted on taking geometry as son1e kind of model for philosophical reasoning. Yet Lambert passes over them in silence when he singles out Wolff as a pathbreaker. 5. The title of the dissertation was Conatum praecipuorum theoriam parallelarum demonstrandi recensio) quam publico examini submittent Abrah. Gotthelf [(aestner et auctor respondens Geor;gius Simon [(lugel) Gijttingen 1763 (Engel and Stackel1895, 140). That Lambert had read this dissertation is clear from remarks he makes in §3 of his Theorie der Parallellinien (Lambert 1~2~L
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6. Engel and Stackel report that IZaestner too seems to have doubted that the fifth postulate could be proved (Engel and Stackel 1895, 140). 7. Definition 44 of Wolff's Elementageometriae (Wolff 1742). 8. See Wolff's Elementa geometriae, Theorem 37 (Wolff 1742). The proof depends on some basic theorems of arithmetic and theorems about the equality of triangles as well. I should add that the proof of TheorelTI 37 depends on Wolff's definition of parallel lines to the extent that this definition is used to prove the Second Corollary to Theorem 34. The proof of Theorem 37 explicitly appeals to this corollary. 9. Earlier in the letter, Lambert says that "Euclid does not at all use the expression per definitionem in proofs in any other way than the expression per hypothesin. For until the possibility of the concept is proven, the definition is still just an hypothesis" (Engel and Stackel 1895, 142). 10. It is interesting to note that these pages from Die Theorie der Parallellinien have a parallel in Lambert's second letter to lZant of February 1766. Lambert wrote, "Wolff accepted nominal definitions gratis, so to speak, and thrust or concealed unwittingly every difficulty in them" (10.64.31-33). 11. Given that Lambert never published his theory of parallel lines, one has to wonder whether he changed his mind about the possibility of proving the fifth postulate, or anyway what his considered view of the matter was. Unfortunately, there seems to be no way to learn the answer, because any papers of his on the matter may well have been destroyed when the Bernoulli house in IZopenick near Berlin burned down in 1807 (Engel and Stackel1895, 150). 12. As the editors of this volume remind nle, it is important to specify the sense in which these postulates are "practical possibilities," because, in actual practice, we cannot produce the constructions tlley license-at least not with any actual instruments like a straightedge or compass. No circle I construct with an actual compass will be a perfect circle; I cannot construct a line segment of infinite extension with any straightedge actually available to me. But Lambert says explicitly that the operations of geometry are idealizations carried out in thought. Speaking more specifically of the application of lines and areas, he says that "all this is taken in an ideal sense" (Lambert 1771, 62). What holds for the application of areas will also hold quite naturally on his account for the constructions licensed by the first three postulates. 13. W.W. Tait points out to me that the fourth postulate seems more like a conlmon notion than a postulate: but while the common notions are about magnitudes (of some given kind) in general, the fourth postulate is about plane angles. He also points out that the fourth postulate adnlits the following proof. Let PQR be a right angle. Suppose it is equal to PQS such that Q bisects the line RS. Let p' Q'R' be another right angle equal to p' Q' S' such that Q' bisects R'S'. Apply R'S' to RS with Q' at Q and p' on the same side ofRS as P. IfP, Q and p' lie in a straight line, all four right angles are equal. Otherwise, we can show there is a contradiction. Suppose p' Q lies in the angle PQS. Then p' QS < PQS, and therefore p' QS < PQR, since PQS = PQR by hypothesis. But PQR < PQR + p' QP, and therefore PQR < p' QR, since PQR + p' QP = p' QR. Hence, p' Q'R' = p' QR. Therefore p' Q'R' is not a right angle, as we initially supposed. You can wonder whether Euclid would have been comfortable with this proof: it depends on rigid
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motions, i.e., translations and rotations; and, it assumes that such motions preserve angles. (But, as Tait observes, Euclid himself appeals to such things in the proof of 1.4.) Still, whatever Euclid himself may have felt, the important thing is that Lambert often appeals to rigid motions in the arguments in his manuscript on parallel lines. He would certainly have accepted this proof. Since genuine postulates cannot be proved, he would not have accepted the fourth postulate as a postulate for this reason, and the reasons n1entioned above. 14. I should say explicitly that I don't know what view Locke had, if any, about geometrical postulates. 15. In the same vein, Lambert writes: "The universality of the principles renders their application sure and dependable; and, the postulates must indicate universal possibilities so that the worry concerning restrictions does not make their application uncertain" (Lambert 1771, 59). See also Lambert's letter of 11 April 1765 to Baron Georg Jonathan von Holland (Engel and Stackell895, 142). 16. I cannot discuss Lan1bert's metaphysical postulates. But I will say that they' do not always seem to me to do a better job of settling hard problems in philosophy than Wolff's nominal definitions. They often just state what we are able to conceive. Thus one of the postulates governing the Good states that, for any good, you can conceive one still greater (Lambert 1771, 82). But this "postulate" simply stipulates that there is no highest Good; it would never satisfy the philosopher who, unlike Lambert, thought he or she can conceive of a highest good. 17. Thus compare the Preisschrift on this with the Transcendental Doctrine of the Method (B756-59). 18. Elsewhere I have argued that runt reflected on the function and significance of both geometrical postulates and problems of construction as a way of clarifying the difference between reason and the understanding (Laywine 1998). 19. But Kant, lilce Lambert, is surely taking the postulates to be "practical possibilities" in an ideal sense. See footnote 12. 20. runt never says that the postulates are simple ideas; indeed, he seems studiously uninterested in entering the debate concerning which of our ideas are simple and which are composite. 21. That was one of the questions of an earlier paper. See Laywine (1998). 22. The iterability at pleasure of either the unit or a given number is something that these arithmetic postulates quite plainly have in common with the geometrical postulates we were considering earlier: not only are we able to construct a circle of any radius and circumference, we can do so as many times as we please. Such iterability is one aspect of the practical possibility stated by a postulate. 23. Of course, the story ultimately lives in the details. But I cannot pursue them here. For some of the details, see Laywine (1998) and Laywine (2003). 24. r would like to thank an audience at the Philosophy Department at the University of Western Ontario and the participants of the workshop held at the University of South Carolina in Columbia in preparation for this volume. Their stimulating discussion helped me pull this paper together. Likewise, I must thank the editors of this volume. Finally, I would also very much like to thank W.W. Tait whose critical con1ments of the penultimate draft helped me clarify things. The remaining murkiness is my fault-and perhaps also Kant's.
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REFERENCES Euclid (1956). The Thirteen Books of the Elements. Edited and translated by Sir Thomas Heath. New York: Dover Publications. Engel, F. and P. Stackel (1895). Die Theorie der Parallellinien von Euklid his auf Gaufl. Leipzig: Teubner. Erdmann, B. (1884). Reflexionen ](ants zur ](ritik der reinen Vernunft. Leipzig: Fue's Verlag [R. ReislandJ. Reprinted in Erdmann (1992), Reflexionen ](ants zur kritischen Philosophie. Edited by N. Hinske. Stuttgart: FrommannHolboog. lZant, 1. (1902- ). ](ants gesammelte Schriften in 29 volumes. Berlin: Walter de Gruyter and predecessors. Lalnbert, J.H. (1771). Anlage zur Architectonic. Riga: Johann Friedrich Hartknoch. As reprinted in volume three of Lalnbert (1965), Philosophische Schriften, edited by Hans-Werner Arndt, Hildesheim: Georg Glms. - - - . (1895). Theorie der Parallellinien. In Engel and Stackel, Die Theorie der Parallellinien von Euklid his auf Gaufl, 152-207. Leipzig: Teubner. Laywine, A. (1998). "Problems and Postulates: I(ant on Reason and Understanding." Journal ofthe History ofPhilosophy 36 (2): 279-311. - - - . (2003). "IZant on Sensibility and the Understanding in the 1770s." Canadian Journal ofPhilosophy 33(4): 443-82. Wolff, C. (1742). Elementa matheseos universae) tomus 1. Halle: Regnersche Buchhandlung. Reprinted in Wolff (1968), Gesammelte Werke, 2. Edited by J.E. Hofmann. Abteilung, Band 29, Hildesheim: Georg Glms.
[7J Two Studies in the Reception of lZant's Philosophy of Aritll1netic CHARLES PARSONS 1
The present paper takes its point of departure from a thought I have had at various times in thinking about interpretations of IZant's philosophy of mathematics in the literature, in particular that offered by Jaakko Hintikka. That was that if the interpretation is correct, shouldn't one expect that to show in the way that Iunt's views were understood by others in the early period after the publication of the first Critique? That reflection suggests a research program that might be of some interest: to investigate how Iunt's philosophy of mathematics was read in, say, the first generation from 1781. I have not undertaken such a project. However, I will make some comments about a two examples of this kind. In doing so I haven't always kept my eye on IZant, because the figures involved are of interest in their own right. The first is Johann Schultz (1739-1805), the disciple of Iunt who was professor of mathematics in IZonigsberg. The second is Bernard Bolzano (1781-1848), who in an early essay of 1810 offered a highly critical discussion of IZant's theory of construction of concepts in intuition. In one way, I think the result of this little experiment is negative, in that it does little toward settling disputed questions about the interpretation of IZant. On the other hand, I think it brings out some problems of Iunt's views that could be seen either at the time he wrote or not long after. We might recall some of the disagreements in the literature on Iunt's philosophy of mathematics. One might see these as arising from challenges to a traditional and natural view, that what is synthetic in mathematical truths is entirely reflected in axioms from which they are derived. In opposition to this tradition, E.W. Beth and Jaakko Hintild
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of intuition itself. It is characterized by IZant as a singular representation in immediate relation to its object (e.g. A320/B376-77). The meaning and significance of the immediacy criterion "vas debated, with the main issue being whether its significance is epistemic and whether it implies some analogy with perception. 2 Michael Friedman's work is in the tradition of Beth and Hintikka, in that he regards intuition (at least in mathematics) as playing mainly a logical role and its role as making possible both the mathematical inferences that the logical resources available to IZant could not analyze and constructions that not only witness what we would formulate as existence statements but even give Ineaning to mathematical staten1ents. 3
I Let me turn now to Schultz. Schultz is explicit about some mathematical matters about which IZant is not. This has made him of value to interpreters of IZant, but it has led to disagreement about the extent to which what he says reflects IZant's views or work or is original with him. The view I defended many years ago is that there is no convincing reason to believe that the mathematical material that Schultz brings to bear in defending IZant, where it is not found in Kant's writings, is not original with hin1. 4 On the whole I still uphold this view; see the Appendix below. But in any case my present strategy is to treat Schultz as a figure in his own right and ask how he understood IZant. Although his Prufung der kantischen ](ritik der reinen Vt:rnunft5 is not specifically a work on the philosophy of mathematics, that subject occupies a prominent place in it, no doubt in part because the author was a mathematician, and in part because it deals almost entirely with the Introduction and the Aesthetic. 6 A natural question to put to Schultz is how he understood the term 'Anschauung'. What was his conception of intuition? So far as I could determine, there isn't an explicit discussion of the meaning of this term in the Prufung. That leaves not as clear as one would wish where he stands on the singularity and immediacy of intuition. IZant's discussion of mathematical proof brings out the importance of the singularity of intuitions, and the third argument of the Metaphysical Exposition of the Concept of Space is generally read as arguing that the original representation of space is singular, although in the characterization of intuition at the beginning of the Aesthetic only the immediacy criterion is mentioned. The fourth argument seems to be to the effect that the representation is immediate, but as we have noted, the force of this in IZant's philosophy of mathematics has been controversial. What can be found in Schultz bearing on these questions is disappointing. The most informative passage is probably the following:
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If, however, the representation of space is ... not a product of any concept' but an immediate representation, that, as e.g. the representation of color, precedes the concept and must first offer to the understanding the material for the formation of the concept, then it [the representation of space] is undeniably a sensible representation, or, as I
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metical judgments are synthetic. Interesting as this is, it was unsatisfying to me in my earlier work because it left Schultz with little to say about the evident difference from Kant's point of view between arithmetic and geometry. Schultz could not have simply missed Kant's claim that arithmetic has no axioms, since it is repeated in IZant's well-known letter to him of 1788. 8 So with regard to the axioms, we have a clear disagreement. There is at least a difference also about postulates, since IZant in speaking of postulates of arithmetic does not seem to have in mind general principles like those stated by Schultz. In "IZant's Philosophy of Arithmetic," I wrote: IZant does not seem to have had an alternative view [to that of Schultz] of the status of such propositions as the commutative and associative laws of addition. He can hardly have denied their truth, and it seen1S that if they are inden10nstrable, they must be axioms; if they are demonstrable, they must have a proof of which IZant gives no indication. (1969, 123) Some recent writers, beginning with Michael Friedman, have suggested what view IZant might have held about the status of such principles as associativity and commutativity.9 If something along the lines they propose is correct, then there is a disagreement between Kant and Schultz, and for reasons I will explain shortly, Schultz seems to me to have on the whole the better case. It is possible that Schultz did not understand IZant's view well enough to see this disagreement clearly. But very likely Schultz was not inclined to advertise disagreements with IZant; when he expressed some criticism of the Transcendental Deduction in an anonyn10US review in 1785, the episode seems to have caused severe strain between them.1° The interpretation proposed by Friedman seems to amount to the claim that for IZant these laws are not propositions at all, so that the question of their truth should not arise. They are "procedural" or "operational" rules. The magnitudes that arithmetic and algebra are applied to come from elsewhere, in the first instance from geometry but not only from geometry. Arithmetic and algebra are quite "independent of the specific nature of the objects whose magnitudes are to be calculated" (113). They merely "provide operations ... and concepts ... for manipulating any magnitudes there may be" (ibid.). This general character n1ust already be possessed by the singular propositions (such as '7 + 5 = 12') on which IZant focuses attention, so that it is not itself sufficient to make what for us would be truth-value-bearing propositions not such for IZant. Evidently the idea is that the associative, commutative, and related laws function as rules of inference. Given that genuine propositions must
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occur as premises and conclusions of these inferences, the question of their soundness can hardly be evaded, at least once attention is called to them as Schultz did. In her discussion of algebra, Lisa Shabel (1998) seems to attribute to IZant the view that in the case of the application of algebraic methods to a geon1etric problem, it will in the end always be possible to cash in the result of the algebraic manipulations by a geometric construction. That would allow algebraic rules to have a nonpropositional character, but then their soundness would be a problem for particular domains of application. It would be solved for the case of applications to Euclidean geometry by the well-known constructions of arithmetic operations. Beyond this geon1etric setting, how generally was this problem solved in the eighteenth century? Schultz distinguishes "general" from "special" mathematics; instances of the latter are concerned with a specific kind of quantum, as is geometry. In contrast, general mathematics abstracts completely from the different qualities of quanta, so it deals only with quanta as such and their quantity, and it only examines all the possible ways of combining the homogeneous, by which the magnitude of a quantum in general is generated and can be determined.!l Schultz then describes addition and subtraction as the two main ways of "generating quantity by combining the homogeneous." Multiplication as iterated addition he seems to regard as derivative, although essential to giving a number as answering the question how many times (1,214-15). Schultz's conception of general mathematics is developed at length in the first part of his Anfangsgriinde der reinen Mathesis (1790),12 published between the two parts of the Priifung. The subject begins with the general concept of quantity and the most general combinations of quantities. He writes that things are called different 13 insofar as there is something in the one that is not in the other, and the same 14 insofar as they are not different (§1). Things are called homogeneous, insofar as one looks to that in them that is the same, inhomogeneous [or] heterogeneous insofar as one looks to that in them which is different. (§3)15 The determination, how many times something homogeneous with it must be combined with itself in order to generate [a thing] is called a quantity [Quantitat]. (§4)16 A thing in which quantity [Quantitat] takes place is called a quantity [Quantum]. (§5)17
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Mathematics is (conventionally) defined as the science of quantity. Here Schultz uses the term GrOfle, indicating clearly that both Quantitat and Quantum are n1eant to be included. The general theory of quantities (mathesis universalis) investigates the generation of quantities in general (§8). The most general types of such generation (Erzeugung) are addition and subtraction. However, about this Schultz writes: But through this the quantum is not yet determined as a quantulTI, that is with respect to its quantity (Quantitat), but the latter requires the determination, how many times just the same homogeneous is combined with itselfin order to generate the quantum (§4). The determination of the how many times is possible only through a number. Therefore all further generations of quantities except general addition and subtraction rest on numbers. Since, however, every number is again a quantum that is generated from numbers, the general theory of quantities, except for general addition and subtraction, consists merely in the science of numbers or arithmetic. IS
I
Schultz assumes something that lunt does not state and conflicts with the view that arithmetic has no axioms. That is that a science that deals generally with quantity, applying, as Friedman says, to whatever quanta there may happen to be, will have general principles statable as propositions. But one of the principles (his first postulate) is that quanta can be added: To transform several given homogeneous quanta through taking them together successively into a quantum, that is into a whole.1 9 Since it gives a closure property, this seems to put a constraint on what quanta there are. 20 The same would be said of the second postulate: To increase and to decrease every given quantum in thought without end. 21 To increase, and to decrease, any given quantum as much as one wants, i.e. to infinity.22 For adding small numbers, such as 7 and 5, I have to imagine the units out of which the number 5 is composed, according to the series individually; then I have to add one after the other onto the number 7 and so generate the nun1ber 12 by means of successive combining. (Priifung, I, 223)23
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The postulate seems to allow the mathematician to treat 7 + 5 as defined, but the procedure described (essentially that of IZant, B15-16) reduces the defined character of '+5' to that of '+1'. So it appears that only that special case of the postulate is used. But so long as one talks generally of quantities as Schultz does, and does not single out and deal separately with the natural numbers, the rational numbers, and the real numbers, one can't derive the generally detlned character of addition in this way. Schultz remarks later that a "laborious synthetic procedure" is needed to see that 7 + 5 = 12 (ibid., I, 232). An interesting feature of Schultz's procedure is that in the Anfangsgriinde he undertakes to treat multiplication as derived. So it is not an accident that he does not state in either work any axioms concerning multiplication. His definition seems to presuppose that the second argun1ent is a whole number: Multiplying a quantum a by any number n means finding a quantum
p that is generated from the quantum a in just the manner in which the number n is generated from the number 1. (p. 61)24 Schultz has in a rudimentary way the idea of multiplication as iterated addition. He offers a proof of the distributive law for multiplication by a nun1ber (p. 63) by a step-by-step procedure that, to become a proper proof by our lights, would have to proceed by induction, and a similar proof of the commutativity of multiplication of numbers (p. 64).25 Friedman states that IZant's view is that in arithmetic and algebra "there are no general constructions" analogous to the basic Euclidean constructions (1992, 109 n. 24). This would serve to account for a difference between his remarks in his letter to Schultz and the Priifung: when he talks of postulates, he clearly has in mind numerical formulae. Son1ething like Schultz's postulates seem to be needed in actual mathematical practice, as it was before the modern axiomatic treatments of the nun1ber systems. And even in those, there is a functional equivalent in treating certain functions at the outset as defined or in making explicit existence assumptions. And perhaps the postulates do amount to general constructions. IZant, as interpreted by Friedman, still has a point: addition' for example, does not always function in mathematics as a construction that serves as a building block for other constructions, although I think it possible that Schultz thought of it that way in relation to multiplication, without getting far in thinking through the problems involved. But his own remarks about '7 + 5 = 12' illustrate the fact that sometimes the result of an addition is the result of a potentially complex procedure, which can be mirrored by a proof.
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On this latter point there may be a clear disagreement with Kant, since in the letter he says that '3 + 4 = 7' is a postulate because it requires "neither an instruction for resolution nor a proof' (Ale 10: 556). Schultz does not say explicitly that a proof is necessary, but he does seem to say that one is possible, and he uses the example as a reason for assuming associativity and commutativity as axion1s. 26 So I don't think that Schultz rejects the idea of proving such propositions, and he clearly did not go along with IZant's regarding them as postulates. Schultz clearly saw something that IZant did not acknowledge: that proofs in arithmetic, and therefore in higher mathematics built on it, require general principles. Even if the "procedural rule" interpretation gives lZant a stronger position than it seems to me it does, one quickly comes to the proof of general theorems, as lZant hardly denies. Though mathematical induction had been identified as a distinctive method of proof a long time before, the whole problem posed by rules of inference in mathematics really only came to consciousness some time later. On the whole, the Kantian way of thinking was not favorable to this consciousness-raising. lZant may have seen clearly that the existing logic was not adequate to mathematical inference. There is in modern forn1ulations a trade-off between axioms and rules of inference, so that with at least some principles (most familiarly induction) there is a choice as to whether to formulate them as axioms or as rules. Arithmetic is a clear case where one cannot just rely on constructions (which we could formulate as existence axioms) and parametric reasoning that could be rendered by propositional logic with operations on variables and function symbols. Schultz identified associativity and commutativity as principles that had to be used. Beyond saying (apparently under Kant's prodding) that they are synthetic, he does not offer a philosophical account of them, and we have mentioned difficulties for some proposals of a IZantian view of them. Although Schultz was not as explicit about induction as some other mathematicians of his time and earlier, he implicitly appeals to it in his treatment of multiplication. This is a case where granting more to concepts than Kant's philosophy of mathematics provides for is something interpreters might agree about.
II My second example is an early writing by Bernard Bolzano, Beitriige zu einer begriindeteren Darstellung der Mathematik, published in 1810, only six years after lZant's death. 27 This essay contains an appendix on rZant's conception of construction of concepts in intuition, to which attention was drawn not long ago by a French writer, the late Jacques Laz, whose 1993 comments on it extensively. Bolzano has often been mentioned as a pio-
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neer in a way of thinking about logic and mathematics that in the long run undermined many aspects of a IZantian view. What is of interest to us, however, is his understanding of IZant at a tilue that was still historically close to that of IZant. Early in the main text of the Beitriige (I §6, p. 9), Bolzano expresses the view that there is an internal contradiction in the concept of pure or a priori intuition. The argument must be contained in the early sections of the appendix. In §1 Bolzano writes that IZant posed the question: What is the ground that determines our understanding to attach to a subject a predicate that is not contained in the concept of the subject? And he believed he had found that this ground could be nothing other than an intuition, which we connect with the concept of the subject, and which at the same time contains the predicate. What he says about IZant's concept of intuition is brief; he describes it as representation of an individual. In §4, speaking for himself, he describes intuition as the representation occupying the place of X in judgments of the form "I perceive X," where clearly there is no room for a priori intuition. Evidently the object of a perception is a representation; it does not have to be sensible. 28 But it does seem to be a representation as a particular event, so that "I perceive X" is unavoidably empirical. An implication of this formulation is that an intuition can be a constituent of a judgment, contrary to IZant's stated view. 29 It's not very clear how Bolzano thinks intuition is meant to be related to perception on !(ant)s conception. About a priori intuition he writes in §2: Ifwe finally ask what an a priori intuition should be, I think that here no other answer is possible than: an intuition that is connected with the consciousness that it must be so and not otherwise. Only thus can intuition give rise to the necessity of the judgment based on it. On balance I am inclined to think Bolzano understands IZantian intuition generally on a perceptual model. Why else should he think that for intuition to be the basis of a judgment of necessity, the intuition itself should contain "consciousness" of necessity? Even with the help of Laz's commentary, I am not able to see clearly what Bolzano's argument against a priori intuition is. He complains that IZanthas not given a clear definition even of the a priori-empirical distinction, and rightly observes that necessity is properly a property of judgments. Since an intuition is not a judgment, it cannot be necessary. But Bolzano's own account surely doesn't imply that an intuition does not have content that would have to be spelled out in propositional form.
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Bolzano turns more directly to the role of pure intuition in mathematics beginning in §7. He attributes to Kant the following reasoning: If I connect the general concept, e.g. of a point, or of a direction or distance, with an intuition, i.e. represent to myself a single point, a single direction or distance, then I find of these individual objects, that this or that predicate applies to them, and feel at the same time, that this is equally the case for all objects that fall under these concepts. How, asks Bolzano, can we come to this "feeling"? Is it through what is single and individual, or through what is general? Obviously through the latter, that is through the concept and not through the intuition. A I
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that one in the case of an arithmetical sum attends only to the collection of the terms, not to the order (a concept certainly wider than sequence in time). This proposition excludes the concept of time rather than presupposing it. Bolzano is not concerned with the question whether the associative law or '7 + 5 = 12' is synthetic, but rather with whether it depends on intuition. Kant, in the Introduction to the second edition of the Critique, is naturally read as deriving the former from the latter. Bolzano is an opponent of a priori intuition but not of the synthetic a priori, so that for him it is at least a possibility that arithmetical judgments should be synthetic a priori judgments of a purely conceptual character. 33 It is not easy to say where the disagreement with Kant (or for that matter Schultz) lies here, although undoubtedly there is one. Bolzano could be saying no more than that in the content of a statement like '7 + 5 = 12' there is no reference to time, something with which Kant apparently agrees. But he evidently thinks it possible to reason mathematically with more general concepts such as that of order, without representing them by succession in time. That something like that is his quarrel with Kant and Schultz is indicated by the general remarks about mathematics in the main text, where he characterizes mathematics as the science dealing with "the general laws (forms) that things must conform to in their being (Dasein) " (I §8, p. 11). But he glosses the latter by saying that mathematics does not give proofs of existence but concerns only conditions of the possibility of things. This is where he draws a contrast between mathematics and metaphysics. He is (and remained) critical of Kant's claim that the methods of mathematics and philosophy are essentially different. 34 The difficulty I had in understanding Bolzano's quarrel with Kant over this issue arose from the fact that Bolzano's remark doesn't clearly say more than that the concept of the associative law doesn't directly involve time, and this seems to be something Kant agrees with. One might infer, though, that Bolzano thought that if we have to represent the succession of numbers by succession in time, that is just a subjective condition of our consciousness of the relations of numbers, and would not detract from the purely conceptual character of arithmetic even if it were shown that time is an intuition, which would anyway be hardly compatible with Bolzano's own conception of intuition as expressed in §4. Some of Kant's own statements encourage the idea that time is only a subjective condition, for example that of the Schematism that number is the unity of the synthesis of the manifold of a homogeneous intuition in general "in that I generate time itself in the apprehension of the intuition" (A143/BI82). Why, Bolzano might well ask, is a condition of the apprehension of the intuition part of the characterization of the relation of number to the category of
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quantity? One might ask this question even if one accepts the transcendental point of view that pervades I
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why I(ant insisted that mathematics required intuition does not, it seems to me, get as much indirect support froill Schultz's writings he might hope for. One controversy about I(ant's philosophy of mathematics was whether intuition plays a necessary role in mathematical inference and not merely at the stage of axioms and postulates. I regard that controversy as largely settled in Friedn1an's favor, but Schultz says so little about inference in mathematics in his writings that they hardly strengthen the case. The text of Bolzano that we have discussed does not directly address this issue, but his rejection of a priori intuition was of a piece with the procedure noted above in his mathematical work, to go ahead with the kind of reasoning whose analysis in the end required polyadic logic, possibly trusting that in the end logic would catch up.
Appendix The investigation made here of Johann Schultz's work and views offers an occasion to reconsider a question originally raised by Martin, what the revision might have been that Schultz made in part I of the Priifung after receiving Iunt's letter of 25 November 1788 commenting on his draft and then discussing it with him. It is clear that the draft maintained that such arithmetic statements as 7 + 5 = 12 are analytic and thus that I(ant succeeded in convincing Schultz on this point. Martin makes the further claim that the mathematical material relevant to this issue, the axioms and postulates stated in the published Priifung, were not in the draft and were either contributed by I(ant or worked out in discussion between Iunt and Schultz. 36 As noted above, I questioned this claim in my 1969, pp. 121-23. My view was and is that Schultz could well have argued that the axioms are analytic, as Leibniz did in the case of commutativity.37 It also seems a pri0ri unlikely that I(ant would have proposed axioms that would contradict his own thesis (reaffirmed in the letter) that arithmetic has no axioms. Concerning the postulates, matters are somewhat more complicated. The idea that arithmetic might have postulates of the sort that Schultz states was not original with either I(ant or Schultz, since similar principles are regarded as such in Lan1bert's Anlage zur Architectonic (1771, §76). 38 It could also have been more difficult for Schultz to admit postulates, in formulation somewhat modeled on Euclid's, and still argue that they are analytic. Therefore the conjecture that they were already in the manuscript on which Iunt was commenting is less likely. Kant certainly knew Lambert's book, and one possibility is that he pointed out its relevance to Schultz. But it is also possible that Schultz was directly influenced by Lambert, who introduces his postulates without invoking an analytic-synthetic distinction. 39
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Evidently we do not have firm evidence concerning either the axioms or the postulates. 4o The view that both were added at the last minute does not square well with Schultz's remark at the beginning of the preface to the Anfangsgriinde that the book is "the work of a laborious reflection of many years." Why should one revisit this question, when it apparently cannot be resolved definitively? One reason vvould be Martin's broader thesis, that books by disciples of IZant presented arithmetic axiomatically, and that this had an influence on subsequent developments leading in the end to the late nineteenth-century axiomatization of arithmetic. This thesis and the work beyond Schultz's that he cites would be worth further examination. Schultz's disagreement with IZant about whether arithmetic has axioms is a reason independent of the above discussion for giving Schultz a ll10re autonomous role in this development than Martin credits him with. Martin himself cites another indication of this: In 1791 Kant's pupil J.S. Beck defended as one of the theses for his habilitation, "It can be doubted whether arithmetic has axioms. "41 Even if Beck's intention was to defend IZant's position, the formulation leads Martin to conclude that this was a matter of dispute in the IZantian school. Martin makes another interesting observation about Schultz, which is apart from the main concerns of this paper but which connects him with Bolzano. He says that Schultz was quite clear on the point "that arithmetic, in particular of irrational numbers, and infinitesimal calculus should be cut loose from all geometric accessories" (1972, Ill). He is relying on the fact that Schultz puts these subjects in general mathematics and explicitly says that its proofs should be conducted independently of geon1etry (Anfangsgriinde §21, 10-11). This aspiration may give Schultz sonle historical in1portance. It may be a reason why Bolzano in citing this work says of Schultz that he "deserves much credit for the foundation of pure mathematics. "42
NOTES 1. A rough version of this paper was presented to the conference on I
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2. For my own presentation of different views on this issue, see the postscript to my 1969, pp. 142-47. However, my most considered view is presented in 1992, section I. 3. More recent work on IZant's philosophy of mathematics has in many ways moved beyond these issues. However, it is about them that I will interrogate Schultz and the early Bolzano. 4. 1969, 121-23. I was criticizing Gottfried Martin's dissertation, subsequently published in expanded form as Martin 1972. I can't forbear to comment that chapter 6 of that book seems to me to show distinct influence from my paper, although Martin does not cite it. (I had sent him a copy before publication.) It was added by tl1e translator to the bibliography of Martin 1985. 5. Schultz 1789, 1792. Much of part II is devoted to replies to articles in Eberhard's Magazin; the systematic discussion of the Aesthetic that the reader might expect is not presented. Translations from this work are my own, altl10ugh passages from part I devoted to arithmetic are translated in Martin 1985. 6. I was struck by the fact that the phrase 'Philosophie der Mathematik' occurs in the preface to part II (p. v). But it already occurs in the Critique, A730/B758. 7. It seems possible that Schultz at one time attended IZant's lectures on logic. But I do not know of definite evidence on the matter. 8. However, in the treatment of this claim in Schultz's exposition of the Axioms of Intuition in Erliiuterungen, this claim is not mentioned. In the Priifung, it is possible that Schultz has the passage of the Axioms of Intuition in mind when he writes, "It seems initially as if arithmetic knew of no axioms" (I, 218). He then proceeds to discuss principles that IZant considers analytic. 9. Friedn1an 1992, 112-14; see also Longuenesse 1998, 282. Shabel 1998 seems to express such a view with reference to algebra but is silent about aritl1metie. 10. See Beiser 1987,206-207 and 360 n. 57, and IZuehn 2003, 321. The review is the one to which IZant responds in the well-known footnote in the preface to the Metaphysical Foundations of Natural Science (Ale 4: 474 n.). 11. I, 212., Wubnig's translation. 12. Because this work is almost unknown, I have included a fair amount of quotation from it. 13. verschieden (diversa) 14. einerley (eadem) 15. Dinge heiBen gleichartig, homogen, so fern man auf das sieht, was in ihnen einerley ist; ungleichartig, heterogen, so fern man auf das sieht, was in ihnen verschieden ist. 16. Die Bestimmung, wie vielmal zur Erzeugung eines Dinges ein ihm gleichartiges mit sich selbst verknupft werden muB, heiBt eine GroBe oder Quantitat. 17. Ein Ding, in welchem Quantiat statt findet, heiBt eine Gro~e, ein Quantum. (All these quotations are from p. 2.) 18. Allein hierdurch wird das Quantum noch nicht als Quantum, d.i. in Ansehung seiner QJtantitiit bestimmt, sondern diese erfordert die Bestimmung, wie vielmal eben dasselbe Gleichartige mit sich selbst verkniipft werden mufl, um das Quantum zu erzeugen (§4). Die Bestimmung des Wievielmal aber ist nur durch eine Zahl moglich. Also beruhen, ausser der allgemeinen Addition und Subtraction, alle ubrigen GroBenerzeugung auf Zahlen. Da aber jede Zahl wiede
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ein Quantum ist, da~ aus Zahlen erzeugt wird, so besteht die allgemeine Gr6~enlehre, ausser der allgemeinen Addition und Subtraction, blo~ in der Zahlwissenschaft oder Arithmetik. (§9, 3) 19. Mehrere gegebene gleichartige Quanta durch ihr successives Zusammennehmen in ein Quantum, d.i. in ein .ganzes zu verwandeln. (Anfangsgriinde, §7, 32) A different formulation occurs in Priifung, I, 221. 20. Kant, in his draft of comments on Iastner's essays in Eberhard's Philosophisches Magazin, makes a comment that relates to Schultz's first postulate. He says of the statement that a line can always be extended, That does not mean what is said of number in arithmetic, that one could increase it, always and without end, by the appending of other units or numbers (for the appended numbers and quantities that are thereby expressed are possible by themselves, without its being the case that they may belong to a whole with the previous ones). (Ale 20: 420) Schultz takes this comment into his review almost without change, although it may appear to conflict with his first postulate. I
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both useless and deceptive, for its validity was derived froin the very operation "vhose validity it was supposed to ground. (1998, 282) I don't have an argument to the effect that IZant did not think of the matter in the "vay Longuenesse claims. But why should one not try to state the rules she refers to precisely and derive some from others? Then one can see if the circularity suggested by the second quoted remark actually obtains. Longuenesse n1ight reply that this procedure is incompatible with denying that that the rules in question express "properties of an object" rather than "pertaining to the very act of generating quantity." But IZant did apparently think that such acts could be represented symbolically and enter into reasoning in algebra. 27. Translations of quotations from this "vork are my own. 28. See the n1ain text, II §15, p. 76. Bolzano in this text holds a theory of perception according to which the existence of an outer object has to be inferred from my representations, just the theory that 1Zant opposes in the Refutation of Idealism. 29. Laz appears to attribute this view to Bolzano's interpretation of IZant; see 1993) 74. 30. Suggested by some con1ments of Laz, 1993,75. 31. The Anfangsgriinde der reinen Mathesis is cited in the main text (I §5, 9). But the detail of the discussion of '7 + 5 = 12' is not in that work. Bolzano certainly knew the Priifung later; it is discussed in Wissenschaftslehre §79 and §305. 32. Bolzano's simplification means that he does not reach the point at which Schultz had to appeal to commutativity, and therefore we do not see whether Bolzano knew how to avoid that assumption. 33. Later, in Wissenschaftslehre §305, Bolzano does argue that '7 + 5 = 12' is analytic. He relies on an explanation of a sum as "a totality ... in the case of which no order of the parts is considered and parts of parts are regarded as parts of the whole." He says explicitly that associativity is analytic; evidently he would have said the same about commutativity. His argument could be criticized on grounds like those on which, according to Alison Laywine's essay in this volume, Lambert criticized Wolff: associativity and commutativity are in effect packed into the definition of addition. 34. Although this matter is hardly at issue in the present paper, see Laywine's essay for J.H. Lambert's disagreement with IZant's view, and Frederick Beiser's essay in this volun1e for an interesting history of lZant's thesis in post-lZantian idealism. 35. This point is well made, with earlier examples than those I mention, in Rusnock 2004, pp. 433-35. Regarding the main argument of Rusnock's paper directed against Friedman, it should be said that it concerns Friedman's assessment of IZant's philosophy of mathematics given his interpretation, not the interpretation itself. 36. Martin 1972, 65. Martin makes the further claim that in the latter case the axioms should be credited to IZant "since he would doubtless have had the leadership in these discussions." That this would be so about a mathematical matter is surely far from evident, particularly since in proposing his axioms Schultz contradicts lZant's claim that arithmetic has no axion1S. 37. See Parsons 1969, 123 n. 13.
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38. See Alison Laywine's essay in this volume, to which I am indebted on this point. 39. Martin points out that postulates of arithmetic also occur in the earlier Neues Organon (1764); see the quotations in Martin 1972, 52, from Alethiologie §26 and §74. 40. Beatrice Longuenesse seems confused on this matter, surprising in so careful a scholar. She writes (1998, 280) that Martin had seen the manuscript on which IZant comments in his letter, a claim for which I can find no warrant in Martin's text. Although she cites my criticism of Martin, she adopts without comment a claim that I questioned, that the mathematical material in the Priifung was not present in the earlier draft. 41. Martin 1972,65. 42. I §5, Russ's translation. Martin attempts to trace this attitude of Schultz back to Kant as well. He does not mention the letter to A.W. Rehberg of September 1790, which is at least a problematic text for this view.
REFERENCES Beiser, F.C. (1987). The Fate of Reason: German Philosophy from !(ant to Fichte. Cambridge, Mass.: Harvard University Press. Balzano, B. (1810). Beytrtige zu einer begriindeteren Darstellung der Mathematik. Prague: C. Widtmann. Reprinted with an introduction by H. Wussing, Darmstadt: Wissenschaftliche Buchgesellschaft, 1974. Translated by S. Russ in W. Ewald (ed.) (1996), vol. I, 176-224. - - - . (1817). Rein analytischer Beweis des Lehrsatzes) dass zwischen je zwei Werten) die ein entgegengesetztes Resultat gewtihren) wenigstens eine reelle Wurzel der Gleichung liege. Prague: Gottlieb Haase. Translated by S. Russ in W. Ewald (ed.) (1996), vol. 1,227-48. - - . (1837). Wissenschaftslehre. 4 vols. Sulzbach. Ewald, W. (ed.) (1996). From [(ant to Hilbert: A Source Book in the Foundations of Mathematics. 2 vols. Oxford: Clarendon Press. Friedman, M. (1992). [(ant and the Exact Sciences. Cambridge, Mass.: Harvard University Press. Guyer, P. (ed.) (1992). The Cambridge Companion to !(ant. Cambridge: Cambridge University Press. IZuehn, M. (2003). !(ant: A Biography. Cambridge: Cambridge University Press. Lambert, J.H. (1764). Neues Organon) oder Gedanken iiber die Erforschung und Bezseichnung des Wahren und dessen Unterscheidung vom !rrtum und Schein. Leipzig: J. Wendler. Edited with an Appendix by G. Schenk. Berlin: Al
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Laz, J. (1993). Bolzano critique de [(ant. Paris: Vrin. Longuenesse, B. (1998). [(ant and the Capacity to Judge. Translated by C.T. Wolfe. Princeton: Princeton University Press. Martin, G. (1972). Arithmetik und [(ombatorik bei [(ant. Berlin: de Gruyter. Expanded version of a 1934 dissertation. Translated in Martin (1985). - - - . (1985). Arithmetic and Combinatorics: [(ant and his Contemporaries. Translated by J. Wubnig. Carbondale: Southern Illinois University Press. Morgenbesser, S., P. Suppes, and M. White (eds.) (1969). Philosophy) Science) and Method: Essays in honor ofErnest Nagel. New York: St. Martin's Press. Parsons, C. (1969). "I
[8] Philosophy, Geometry, and Logic in Leibniz, Wolff, and the Early I(ant DANIEL SUTHERLANDl
In his Inquiry Concerning the Distinctness of the Principles of Natural Theology and Morality of 1763, IZant argues that there is a deep difference between the methods of mathematics and philosophy that has in1plications for the certainty of each. Most of the Inquiry argues against attempts to imitate mathematics in philosophy, above all by starting with definitions, because doing so in philosophy will lead to uncertainty. But IZant also argues against an influence in the other direction, that is, against using philosophical definitions in mathematics. In particular, he argues against Christian Wolff's attempt, inspired by Leibniz, to reform Euclidean geometry based on a philosophical definition of similarity.2 It is this latter argument that is the focus of my paper: what are IZant's views of philosophical definitions, and why are they not admissible in mathematics? What is driving IZant's criticism? On a quite natural reading of the Inquiry, IZant's argument seelllS to be that the uncertainty of philosophical definitions will undermine mathematics. I will argue, however, that IZant's Inquiry offers all the resources needed to sidestep this problem. A close examination of Leibniz's and Wolff's use of the philosophical definition of similarity in geometry, and an understanding of its role in their wider projects, shows that IZant's best argument is quite different. I will argue that IZant's reaction to Wolff has more to do with his understanding of Wolff's and Leibniz's projects and his understanding of the mathematics and logic of his day than with his concerns about certainty. 3 This paper has four closely related aims. The first is to provide part of an historically sensitive account of the relation between philosophy and geometry in the 17th and 18th centuries. Second, the paper will show that IZant's argument that there is a deep difference between the methods of mathematics and philosophy supports his view that there is something
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special about mathen1atical cognition and suggests that intuition plays an important role in it. This view eventually deepens in the critical period into his doctrine of construction in pure intuition; it also broadens to make intuition a requirement of any cognition. Thus, I(ant's distinction between methods plays a crucial role in the development of Kant's critical philosophy and bears close scrutiny. Third, and more particularly, the paper aims to explain I(ant's Inquiry views on the nature of philosophical definitions, their certainty, and their banishment from mathematics. Finally, the paper will help solve a puzzle about IZant's critical views of mathematical cognition. On the one hand, IZant's critical philosophy provides a theory of human cognition, which includes mathematical cognition, and in that framework, I(ant appears to accept the very philosophical definition of similarity proposed by Leibniz and Wolff. On the other hand, I
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1. Leibniz's Analysis Situs and the Logico-Philosophical Method Based on Similarity Leibniz's analysis situs was an attempt to give geometry a new method and new foundations, and was inspired in large part by the success of algebraic symbolization. Part of the power of algebra derived from its symbolization of magnitudes in general, but Leibniz also saw this generalization as a weakness, for it precluded the direct symbolic representation of geometrical properties, in particular relative position, or "situation." Leibniz's primary approach to the analysis situs was based on the relation of congruence, but he also introduced another approach based on the relation of similarity. Both were described as belonging to the analysis situs, but in fact, the two approaches are not only unrelated but incompatible. Leibniz's work on the former was lost until the 19th century; his work on the latter influenced Wolff and was known to lZant through him. This paper will therefore focus on Leibniz's use of similarity.4
1.1 Proofs based on a philosophical definition of the similarity relation Leibniz uses the similarity relation to show that several theorems difficult to prove in the Elements can be easily derived. Leibniz's demonstrations appeal to his definition of similarity, a lemlna of similarity, and an axiom of determination: 5 Definition of Similarity: "Two presented figures [are] similar if nothing can be observed in one viewed in itself, which cannot be equally observed in the other" (L393). Lemma of Similarity: "Two similar figures have the same ratio of proportion among the parts in each figure" (L393-94). Axiom of Determination: "Things which cannot be distinguished through their determinants (or through data adequate to define them) cannot be distinguished at all, since all other properties arise from these data" (L394). Leibniz uses these propositions to demonstrate, for example, that circles are to each other as the square of their diameters, a result Euclid does not prove until Book XII, Proposition 2, and then only by reductio ad absurdum. Leibniz proceeds as follows. Suppose a circle with a diameter AB and a square CD with sides equal to AB circumscribing the circle. Suppose
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another circle with diameter LM and a square NO with sides equal to LM circumscribing the circle. (See Figure 1.) N
rn
C
L
A
B
M
D
o
FIGURE 1. Leibniz's example of a similarity proof
Leibniz's proof is succinct: The detern1inants are similar in both figures, circle to circle, square to square, and relation of circle to square, so by the above axion1 [of determination], figures ABeD and LMNO are not distinguishable at all, and hence by definition are similar. By the lemma of similarity, the circle AB is to the square CD as the circle LM is to the square NO. Hence the circle AB is to the circle LM as the square CD is to the square NO (L395-96). Similarity plays a key role in these demonstrations; this is not, however, the traditional geometrical notion of similarity between rectilinear figures, which Euclid defined in Book VI of the Elements as two rectilinear figures "having equal [corresponding] angles and the [corresponding] sides about the equal angles proportional."6 Leibniz instead provides a philosophical definition of similarity in terms ofwhat can be observed in a thing "viewed in itself." This notion of similarity rests on Leibniz's conception of quality and quantity, and to understand it, we must look more closely at Leibniz's philosophy.
1.2 LeibnizJs larger proJ·ect: mathematizing logic
and logicizing mathematics Leibniz saw the analysis situs as just one part of a larger project to bring clarity to human thought through symbolic notation, and he aimed to fit mathematics into it by means of his definition of sin1ilarity. Leibniz articulated this project very early in his career, and though his conception of it changed over the years, it deeply influenced his thinking throughout his
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life. 7 His dissertation of 1666 describes an ars combinatoria, or art of combination, which he hoped would represent the logical relations among ideas. Leibniz assumed that ideas were related by containment and combination relations and that the job of logic was to uncover these relations. In the canonical example, the concept "human" contains the concepts "animal" and "rational," and logic should represent the combination of these concepts in the concept "human." Leibniz wished to represent ideas with symbols and combined ideas as combinations of symbols in a way that would Inake their relations perspicuous; doing so would allow for logical inferences based on these relations. The ars combinatoria was inspired by mathematics, and in particular, early modern developments in algebra, which revealed the power of symbolization and demonstrated a fundamental connection between symbolic and mathematical reasoning; the ars combinatoria is nevertheless a part of logic and is intended to encon1pass all human thought, not just mathematics. The fact that the ars combinatoria represents the logical relations between concepts is crucial to understanding IZant's rejection of Leibniz's project, and should be kept in mind. Because it drew its inspiration from mathematical symbolization, one can characterize Leibniz's symbolic representation of concepts in order to reveal their logical containn1ent relations as an attempt to mathematize 10gic. 8 Leibniz's larger project, however, includes much more than this. In order to distinguish two pillars of Leibniz's project, I will limit the expression "ars combinatoria" to what I have described so far: the syn1bolic representation of concepts in order to reveal their combination relations. While the first pillar of Leibniz's larger project concerned the use of symbols to represent the general logical relations between thoughts, the second pillar concerned the particular content of our thoughts. Leibniz's view ofideas as composed of simpler ideas leads naturally to the notion that one ought, if possible, identify and symbolize the simplest ideas. On the crucial assumption that there is one set of primitive ideas common to all human cognition, the notion of a universal language of thought naturally arises. Symbolizing such primitive ideas would yield a "universal characteristic," that is, an alphabet of all human thought. 9 On the further crucial assumption that the prin1itive ideas reflect reality, the universal characteristic promises to make all human knowledge perspicuous and secure. A universal characteristic combined with an ars combinatoria would reveal the logical relations among primitive ideas, allow·perspicuous reasoning about them, and exhibit the organization of ideas in a way that mirrors reality. Leibniz sometimes refers to the combination of a universal characteristic with an ars combinatoria as a "lingua philosophica," where that is understood not as a language particular to philosophical topics, but a language of all human thought that permits lucid philosophical reasoning from the
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primitiVe ideas corresponding to reality. The lingua philosophica would have allowed an encyclopedic compendium of human knowledge based on the underlying logical relations between primitive ideas. While the ars combinatoria is the form of Leibniz's project, the universal characteristic supplies the content. The two are closely linked: the ars combinatoria quite naturally leads to the thought of uncovering the primitive ideas, and the universal characteristic aims to discover the primitive ideas and then symbolize them in a manner suitable for the ars combinatoria. Leibniz does not usually distinguish between them and he often refers to the entire project as the "universal characteristic"; most commentators follow hin1 in doing so as well. Since "universal characteristic" connotes an alphabet of all human thought, I will reserve this expression just for the part of Leibniz's project which aims to discover and symbolize primitive ideas; I will use "lingua philosophica" for the combination of a universal characteristic with an ars combinatoria. IO Leibniz searched for a universal characteristic for his entire career, but his views on it changed considerably. Early on, he hoped to find symbols that directly reflect the content of the primitive ideas-symbols that would make it perspicuous to a person of any language which thought corresponded to a symbol. Leibniz referred to Chinese ideograms as a step toward the universal characteristic, because they compress and combine symbols for various ideas into an ideogram of a more complex idea. Leibniz later suggested that the primitive ideas could instead be represented by the numerals of prime numbers and the combination of these ideas by their products. Since prime factorization recovers the unique set of primes that constitute a product, one can recover the simple ideas out of which a complex idea is composed. On the one hand, the exploitation of prime factorization lifts Leibniz's mathematization of logic to a much higher level; not simply symbolization but number theory are exploited in the representation of complex ideas. On the other hand, the symbol for a complex idea no longer wears its simple ideas on its sleeve, since recovering the simple ideas requires factorization and decoding of the (arbitrary) assignment of primes. This approach abandons the immediate perspicacity Leibniz originally hoped the universal characteristic would have. II However the primitive ideas are symbolized, the universal characteristic presupposes discovery of the primitive ideas that are to be symbolized. In this respect as well, Leibniz's views of the universal characteristic fundamentally changed, for he came to think that identification of even a relatively small number of truly primitive ideas might be beyond what was attainable in his lifetime.l 2 He did not think that this limitation required abandonment of the universal characteristic, however; he believed that a limited set of complex but relatively primitive ideas could be taken as primitive until further analysis revealed absolutely primitive ideas. 13
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Leibniz's search for primitive ideas that correspond to reality was influenced by Aristotle's Categories and its long history in western philosophy, from Porphyry to Melanchthon. 14 While Leibniz's ideas for an ars combinatoria are quite fan1iliar, his search for primitive ideas and his attempt to reform the categories has been neglected. IS By late scholasticism, a treatment of categories was not limited to the particular Aristotelian categories such as substance, quality and quantity, but encompassed the ordering of fundamental concepts under a highest genus. 16 Leibniz's own efforts to identify primitive ideas, or at least relatively primitive ideas, included quality and quantity. In notes taken around 1680, for example, Leibniz explored the relations among a variety of fundamental concepts, which included coincidence, similarity, quality, congruence, and quantity,!7 As is well known, the concept of identity is central to Leibniz's logic and metaphysics: Leibniz took the logical statements that reveal the containment relations among ideas to be identity statements. It is therefore unsurprising that the concept of identity plays a role in Leibniz's attempts to reform the Aristotelian categories. In particular, he defined similarity as identity of quality and equality as identity of quantity, thereby making similarity and equality dependent upon his account of quality and quantity.18 Leibniz did not regard quality and quantity as themselves primitive, however, if being primitive implies that they cannot be defined. Leibniz defined them as two kinds of properties of things distinguished by how they are known: Quantity can be grasped only when the things are actually present together or when some intervening thing can be applied to both. Quality represents something to the n1ind which can be known in a thing separately and can then be applied to the comparison of two things without actually bringing the two together either immediately or through the mediation of a third object as measure. 19 Defining quantity and quality in this way entails that two con1pletely similar objects cannot be distinguished from each other by their size unless they are compared with each other or are each compared to some other object taken as a common measure. Thus, if two similar things are viewed in isolation from each other, they cannot be distinguished. The crucial feature of this account is that Leibniz's metaphysical definitions of similarity and equality include geometrical sin1ilarity and equalityas a special case: two geometrical figures are similar if and only if all the properties internal to each figure are identical, that is, if the ratios among the sides of one rectilinear figure are identical to the ratios an10ng the corresponding sides of the other and the ratios between the angles and two rigrlt angles in one figure are the same as those ratios in the other.
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The fact that metaphysical similarity includes geometrical similarity as a special case furthered Leibniz's attempt to integrate geometry into metaphysics. He believed that his metaphysical definitions would improve geometry; he also believed that they would· allow him to assimilate the analysis situs into his universal characteristic and ars combinatoria, and hence into a comprehensive system of human knowledge based on the logical containment relations between concepts. 20 Geometrical similarity will only be a special case of metaphysical similarity if Leibniz's notions of quantity and quality are understood properly. Whether a property is quantitative or qualitative is relative to what one is taking as an object; quantitative properties of the parts of an object might be qualitative properties of an object as a whole. Suppose that a triangle has a side three times the length of another. This ratio is a quantitative property of each of the two sides, since it can only be known with reference to something outside of each, i.e., with reference to each other or some measure common to both. At the same tin1e, it is a qualitative property of the triangle as a whole since it can be known without reference to anything outside the triangle. Geometrical similarity requires that the ratios among the sides of a triangle be quantitative properties of the sides. Nevertheless, these ratios can be known without reference to something outside the triangle, and hence count as qualities of the triangle as a whole. Hence, Leibniz can state that two triangles are similar if they have the san1e qualities, which conforms to his metaphysical definition of similarity.21 Leibniz hoped that proofs based on similarity would place Euclid's geometry on an entirely new footing and lead to a radical reordering of geometrical demonstrations. While his examples are suggestive, Leibniz never carried out the reformation of Euclidean geometry he proposed. Leibniz's pursuit of a universal characteristic included a search for at least relatively primitive ideas out of which other ideas were composed and an attempt to reconstruct the relations between ideas, a task that belongs as much to logic as metaphysics. 22 Insofar as defining mathematical ideas in terms of the logical containment relations between more primitive ideas is a part of logic, one could characterize this part of Leibniz's broader project as an attempt to logicize mathematics. There are thus two distinguishable strands in Leibniz's attempt to find a lingua philosophica. On the one hand, Leibniz attempted to mathematize logic by means of symbolizing containment relations between ideas, and, in one version, to exploit the properties of prime numbers in the representation of those relations. On the other hand, Leibniz attempted to logicize mathematics, or at least the geometrical part, by defining fundamental geometrical concepts in terms of the logical containment relations between even more basic metaphysical concepts. Unfortunately, these two strands of Leibniz's thought were never successfully interwoven. As has been widely emphasized, Leibniz's ars combi-
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natoria anticipated the development of symbolic logic in the second half of the 19th century. At the same time, it was inherently limited, since for Leibniz logic only reflected silnple combination relations among monadic concepts; his conception was still far removed from Frege's treatment of relations and rethinking of the nature of logical objects. The inherent limitations of Leibniz's conception of logic left an unbridgeable gap between mathematics and logic, and it is the vision rather than the details for which Leibniz is rightfully lauded. Leibniz's universal characteristic cannot be judged as favorably. His search for primitive concepts that are comn10n to all human cognition and accurately reflect reality did not bear fruit. In particular, his atten1pt to logicize geometry by absorbing fundamental geometrical concepts into a hierarchy of metaphysical concepts proved both philosophically and mathematically unproductive. Leibniz's attempt to provide Euclidean geometry with a new foundation based on the similarity relation were part of his project to find a universal characteristic for a lingua philosophica. Both Leibniz's use of similari ty and his hopes for a universal characteristic influenced Christian Wolff's geometry, which dominated Germany in the 18th century and provoked IZant's reaction in the Inquiry.
2. Wolff's Geometry Wolff published his four-volume Foundations ofAll Mathematical Sciences [Anfang{qriinde] in 1709 and 1710. It was written in German, making it vvidely accessible in German speaking lands; it was also written with the student reader in mind, and focused on providing the foundations for practical applied mathematics. For all these reasons, it quicldy became a standard student text that went through seven editions in Wolff's lifetime. 23 In fact, Wolff's work replaced the Elements as the standard introduction to geometry in Germany. The general order of proofs loosely follows Euclid, but his practical focus led him to abandon Euclid's restriction to straightedge and collapsible compass and allow the use of a ruler and protractor. Furthermore, his geometry is organized around establishing about 30 practical "problems," that is, practical constructions, such as calculating the height of a tower in the distance or the volume of a keg. 24 Despite his pragmatic emphasis, Wolff insisted on clarity and rigor in exposition, which he strove for by means of copious definitions and references to previous results. While the development of Wolff's views can be traced in his German geometry, his Latin Geon1etry best articulates his views, and we will focus on it. Wolff published Elements of Universal Mathematics [Elementa Matheseos Universae] in two volumes in 1713 and 1715, three years after
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first publishing his German Anfangsgriinde; he expanded it into five volumes bernreen 1730 and 1741. In contrast to his German geometry, Wolff intended this work for a more scholarly audience; practical applications were still included, but they did not dominate the work, and he set higher standards of rigor. 25 The Elementa is generally much more thorough than the Anfangsgriinde. 26 But Wolff's views of the structure ofhun1an knowledge is crucial to his understanding of what will bring rigor. Wolff believed that if one organized properly defined concepts according to their logical relations, all human knowledge would be in principle derivable (a point to which I will return below). As a consequence, he believed that the geon1etrical axioms of the Anfangsgriinde should be replaced with definitions and derivations from them, which is what he atten1pts in the Elementa. This work contains a few axioms, but Wolff thought that, in principle, they too could someday be replaced by definitions and derivations from them. Wolff explicitly acknowledges that Leibniz had described to him the use of similarity in the analysis situs. 27 Wolff credits Leibniz in the introduction to the Elementa Matheseos Universae: I will define things which are not normally defined, and ... demonstrate things which are presupposed by others without proof . . . . Euclid and all those who have hitherto followed his example have demonstrated everything by means of the principle of congruency alone. Since, however, the most learned Leibniz imparted the concept of similarity to me and showed me that it has an extensive application in geometry, and since I have meditated upon it and come to recognize its importance, I have not hesitated at all to introduce the principle of similarity into geometry. You will . . . understand many things which are demonstrated with the greatest ease by means of that principle [of similarity] which are not normally demonstrated, except indirectly . . . by means of the principle of congruency.28 By remarking that he had to meditate on Leibniz's concept of similarity to appreciate its importance, Wolff acknowledges his debt to Leibniz while making it clear that he thought through the use of similarity on its own. 29 At the same time, he echoes Leibniz's own claims and raises expectations by stating that similarity demonstrations would greatly ease the demonstration of propositions otherwise difficult to establish. Despite employing a parallel definition of similarity in the Anfangsgriinde and the Elementa, Wolff's use of similarity in the Elementa is radically different. The Anfangsgriinde did not use the definition of similarity until more than halfway through the geon1etry, and only after many of Euclid's results had been proved by appeal to congruence and eq~~j!y_
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relations. In contrast, the very first theorems of the Elementa geometry all appeal to similarity. Wolff's definition of similarity in the Elementa also departs from Leibniz and his own definition in the Anjangsgriinde in several ways crucial to evaluating his demonstrations. We saw that Leibniz defines similarity with respect to those properties of each figure "viewed in itself. ,,30 The Elementa gives a somewhat different account, articulated in a different definition and two corollaries. Definition A12: 31 §A24. Those things are similar which are the same with respect to that by which they should be distinguished from one another. Those things are dissimilar which are different with respect to that by which they should be distinguished from one another. And, as such, similarity is the identity and dissimilarity the difference of that by which the things should be distinguished. Corollary 1: §A25. Therefore, nothing that is able to be understood without assuming the other is discovered in one similar which is not equally discovered in the other. Corollary 2: §A26. Since quantity, without another being assumed, is not able to be understood in itself, but only given (§A13, A14), similar things, without loss of similarity, are able to differ in quantity (§A25), and as such quantity is internally distinct from similarity. Wolff's corollaries make it clear that he thinks the Leibnizian conception of silnilarity simply follows from the definition. Wolff takes care to argue for the connection between understanding son1ething in itself and quantity, which he explicates in §13 and §14. Nevertheless, he makes a crucial slide fron1 the definition to the first corollary-from properties that "distinguish two objects" to properties "understood without assuming the other," i.e. the properties of an object "viewed in itself." Why should only those properties of an object viewed in itself be the properties that distinguish it from another? Wolff frequently extracts substantive claims from a definition in his effort to reduce all knowledge to definitions and their consequences. Unfortunately, doing so undermines his enterprise, since the definitions are by no means as fruitful as he represents them; the conclusions he draws simply do not follow. This difficulty can be sidestepped if we understand Corollary 1 as further defining what kind of properties ought to be used to distinguish things, and hence as a part of the definition of similarity rather than as a consequence of it. Setting this difficulty aside, then, Wolff's philosophical definition of similarity taken together with its corollaries corresponds to Leibniz's conception of similarity. Wolff defends his departure from
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Leibniz's definition in §27, but that need not concern us here. 32 In both Leibniz and Wolff, qualities are those properties ofan object that can be distinctly known or understood when viewed in isolation from other objects, and the similarity of two objects consists in identity of such properties. The Elementa also includes a proposition corresponding to the Axiom of Determination. In an effort to replace axioms with definitions, Wolff defines "san1e manner of determination," and then derives the desired result as a corollary: Definition G79: 33 §G119. They are said to be determined in the same manner if the data by which one is determined is similar to the data by which the other is determined; in other words, if in both cases, from similar data, by the same rule or principle further things are determined. Corollary: §G120. Things determined in the same manner coincide in those things by which they should be distinguished, and as such they are similar. (§A24 [Wolff's definition of similarity]). §G120, which I will call the "Corollary of Determination," connects determination with similarity; it forges the link between philosophical and geon1etrical similarity and plays an important role in Wolff's demonstrations. Unfortunately, Wolff's definition of "determined in the same manner" simply shifts the burden of the definition onto "determined by the same rule [or principle]." Without granting further content to the notion of "san1e manner or rule," it does not follow from two things being determined in the same way from similar data that they will coincide with respect to what might distinguish them. Once again, the definition cannot support the clain1s Wolff extracts from it, a failure which puts paid to his aim of grounding all human knowledge in definitions. Despite these problems, Wolff's inability to ground all human knowledge in definitions can be distinguished from his attempts to provide geometry with a new foundation based on similarity. It is therefore important to look at how Wolff uses these definitions and corollaries in geometrical proofs. I noted earlier that similarity plays no role at all in the first half of Wolff's Gern1an geometry. In contrast, the Latin geometry begins with a chapter on fundamental propositions that contains 17 theorems and 7 problems; every theorem save one depends either directly or indirectly on the philosophical conception of similarity. The following chapter on straight lines and triangles establishes a further 34 theorems and 21 problems which rely on the previous more fundamental theorems and problen1s. I noted in Section 1 that Leibniz n1erely suggested a geometry based on similarity and illustrated it with a few examples. Wolff attempted to realize Leibniz's suggestion; the Elementa radically reorders geometry and truly departs from Euclid's Elements.
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The very first theorem aims to prove that a diameter divides the circumference and area of a circle into equal parts. Theorem G1: §G135. Diameter AE of circle divides the circumference equally and divides the circle itself into two equal parts. A
B E
FIGURE 2. Demonstration of Theorem Gl
Demonstration: Determine the parts of the circumference ADE & ABE and also the circle ADECA & ABECA by moving the straight line AC around the center until it lies straight with itself(§G131). Since the parts of the circumference, arcs ABE & ADE, and the two parts of the circle, ABECA & ADECA, are determined in the same manner, they are similar (§G120). And for this reason, the arc ABE has the san1e ratio to the circumference as arc ADE, and part ABECA has the same ratio to the circle as part ADECA (§A170), consequently the arcs are equal to each other and the parts of the circle are equal to each other. (§A177) Wolff's proof rests on the claim that one determination of a circle contains two determinations of the same manner, one for each arc ABE and ADE. 34 By the Corollary of Determination (§G120), the arcs and areas of the two figures are similar. Wolff then appeals to the claim that if two parts are silnilar, then they have the same ratios to their wholes, citing §A170 Theorem A16 of the Arithmetic. Since the wholes to which the parts are related are one and the same circle, the arcs and the areas of the semicircles are not only similar but equal, by §A177 Theorem A19. Wolff appears to be off to a good start. There is a serious difficulty with this demonstration, however. Theorem §A170 A16 states that if two parts are similar, then they have the san1e ratio to their corresponding wholes, where the wholes are not necessarily the same; as Wolff puts it: "Similar parts P & P have the same ratio to the wholes T & t." But this claim is simply false. Consider two similar figures, such as the squares ABCD and HIJIZ in the following diagram.
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:ttJF
~EJ I
D
K
C
G
M 1
~J---------_-J
N
FIGURE 3. Similar Parts of Different Wholes
The two parts ABCD and HIJIZ are similar to each other ,;yhile they have different ratios to the whole areas to which they belong, DEFG and I
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suggestions into practice, and over time he tempered his views about what might actually be attained. 37 In contrast, Wolff remained dedicated to fulfilling his vision even in the face of difficulties he ought to have recognized. Wolff's con1mitment to his ideal leads him to overstep the means available to him and draw conclusions to which he was not entitled. 38 Wolff's vision of how human knowledge is organized and his attempt to incorporate geometry into it provide the background for IZant's reaction in the Inquiry.
3. IZant's Inquiry Distinction between Philosophy and Mathematics I(ant wrote the Inquiry for a prize con1petition sponsored by the Prussian Royal Academy in 1761 with a deadline at the end of 1762. The question posed by the Academy asked what kind and what degree of certainty was attainable in metaphysics (especially in natural theology and in morality), and how it con1pared to the certainty of the proofs of geometry. 39 IZant's essay focuses on the difference between philosophical and mathematical methods and how they attain certainty. IZant takes a strong stand on the difference between the methods appropriate to each and the dangers of mixing them; he concludes that certainty is much more difficult to achieve in metaphysics, but if one does not imitate mathematics and proceeds carefully according to his guidelines, certainty is in some cases achievable. In answer to the particular question posed by the Academy, IZant states that the certainty required for conviction is achievable for some fundamental principles of natural theology. Although the main theme ofIZant's essay is to discourage the imitation of mathematical method in philosophy, our focus will be on the much less emphasized theme discouraging the intrusion of philosophy into mathematics. This theme also springs from his views on the differences in method; a closer look at them and lZant's guidelines for doing philosophy will reveal why lZant reacts to Wolff in the way he does. IZant's Inquiry is divided into four "reflections"; the first describes four differences between mathematical and philosophical methods and their relation to certainty. I will discuss each in turn.
3.1 Synthetic versus analytic definitions According to IZant, mathematics arrives at a definition by the "elective" synthesis of characteristics to create a new concept, which he calls a "synthetic definition," while philosophy arrives at a definition through analysis
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of a given concept, which he calls an "analytic definition. "40 In elective synthesis one is unconstrained in which characteristics to choose to create the defined concept, while in philosophy the definition must answer to the given concept. In his example of a mathematical definition, I(ant first describes thinking of four straight lines bounding a plane surface so that the opposite sides are not parallel, and he then names the object defined by this concept a trapezium to emphasize that the concept first comes into existence through the definition. IZant's example of a philosophical definition is the definition of the concept of time, a concept which everyone has prior to defining it and is in that sense given. 41 This first difference between mathematics and philosophy plays by far the n10st important role, and lZant returns to it repeatedly throughout the Inquiry. Because mathematical definitions are elective, we can be completely certain of what is contained in the defined concept; personifying mathematics, lZant states that "it can say with certainty that what it did not intend to represent in the object by means of the definition is not contained in that object ... the concept has no other significance at all apart from what is given to it by the definition. "42 In philosophy, in contrast, one cannot be sure that one has not overlooked some characteristic mark that belongs to the given concept. This can easily lead one to think that a definition has been completed when it has not and to judge that some characteristic mark does not belong to the concept when it does. . . . if one should fail to notice some characteristic mark or other, which nonetheless belongs to the adequate distinguishing of the concept in question, and if one judges that no such characteristic mark belongs to the complete concept, then the definition will be wrong and misleading. (Third reflection, §1, 2:291) As a consequence, philosophical definitions are not only less certain, they have led to "numberless errors.,,43 This is the most important shortcoming for philosophical definitions; it is also the most damaging consequence of imitating mathematical method in philosophy by attempting to begin with definitions.
3.2 The use ofsigns in concreto versus in abstracto The second important difference between mathematics and philosophy is in the way that they employ signs. Arithmetic and algebra use signs to stand for magnitudes, relations between them, and operations on them. This allows for rule-governed manipulations of the individual signs without regard for what they stand for until the manipulations are complete.
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And as geometry demonstrates, the use of signs allows us to establish universal rules governing all figures from a consideration of just one figure; doing so is to consider the universal under signs in concreto. 44 In contrast, philosophy cannot employ signs in this way; in philosophy, the signs are words which simply stand for concepts and thoughts without revealing anything about their composition or the relations among them. Hence, in philosophy, one must simply think of the thing itself, e.g. body, and the universal concept of that thing in abstracto rather than in concreto. In IZant's view, the advantages of symbolization demonstrated so dramatically by the development of algebra in the early modern period simply aren't available to philosophy. IZant insists that the manipulation of signs is of no use in philosophy because they cannot represent the content of concepts or thoughts in philosophy, which can only be represented in abstracto by words, and ... words can neither show in their composition the constituent concepts of which the whole idea, indicated by the word, consists; nor are they capable of indicating in their combinations the relations of the philosophical thoughts to each other. (2:279) Later in the Inquiry, IZant states that the use of signs in concreto increases certainty, for by means of them, we can know things with the same confirmation as "seeing something with one's own eyes":45 ... since signs in mathematics are sensible means to cognition, it follows that one can know that no concept has been overlooked, and that each particular comparison has been drawn in accordance with easily observed rules, etc. And these things can be known with the degree of assurance characteristic of seeing something with one's own eyes.... (2:291-92) IZant identifies two distinct advantages of representing under signs in concreto: it allows us to know easily observed rules have been followed, and it also allows us to know that no concept has been overlooked. In philosophy, in contrast, we can easily overlook a characteristic belonging to a given concept. Thus, one important role for symbolic representation is to aid us in not overlooking what is contained in a concept-the same potential problem that IZant emphasized in the distinction between mathematical and philosophical definitions. In the Third Reflection, lZant identifies one additional source of error arising fron1 the use of words, which should be mentioned. The meaning ofwords for given concepts is determined by linguistic usage, but language is full of ambiguities; the same word is often used for different concepts in
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different contexts. It is part of the philosopher's task to be sensitive to multiple, closely-related meanings of a word and not to be led astray. Kant's claim that symbolic representation and manipulation cannot be exploited in philosophy is a clear rejection of an ars combinatoria, and has implications for Leibniz's use of similarity. Mter giving a philosophical demonstration that all bodies consist of simple substances, he states: In this case, neither figures nor visible signs are capable of expressing either the thoughts or the relations which hold between them. Nor can abstract reflection be replaced by the transposition of signs in accordance with rules, the representation of the things themselves being replaced in this procedure by the clearer and the easier representation of signs. (2:279) Kant opposes Leibniz's goal of mathematizing logic by means of an ars combinatoria, because he believes that signs cannot fruitfully symbolize the content of thoughts and concepts or the relations between them. I
3.3 Few versus innumerable fundamental concepts
and propositions The third important difference between n1athematics and philosophy concerns the number of unanalyzable concepts and indemonstrable propositions in each. Since the unanalyzability of concepts is at issue, Kant begins this section by returning to the topic of definitions. Significantly, I
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only be one straight line between two points" might be provable "elsewhere," but claims that they are immediately certain in mathematics. The point to which IZant is building is that the number of unanalyzable concepts and indemonstrable principles in mathematics is much smaller than for philosophy. lZant does not list all of the unanalyzable concepts of mathematics, but he mentions the concepts of unity, plurality, and space, which he calls "concepts of magnitudes in general." IZant clearly has in mind Euclid's Elements and the ability to derive countless propositions from a small set of axioms. In contrast, philosophy has "uncommonly many" fundamental concepts because "it is impossible that universal cognition of such great complexity should be constructed from only a few fundamental concepts.,,46 IZant gives examples of unanalyzable concepts or only partially analyzable concepts: the concepts of representation, being next to each other, time, the sublime, pleasure, and desire are sufficiently unrelated to one another that their diversity speaks against reduction to a few simple concepts. Most important, however, is IZant's clain1 that philosophy contains innumerably many indemonstrable propositions. In his view, they are the result of every analysis of a concept. lZant states: . . . no matter what the object may be, those characteristic marks, which the understanding initially and immediately perceives in the object, constitute the data for exactly the same number of indemonstrable propositions .... (2:281) lZant emphasizes that these indemonstrable propositions rest directly on what the understanding "initially and immediately perceives," and states that they could not be derived from analytic definitions; in fact, IZant claims that analytic definitions, or attempts to achieve them, are based on these indemonstrable propositions. lZant's position has important implications for the certainty of mathematics and philosophy. The fact that mathematics has fewer unanalyzable and indemonstrable propositions undoubtedly makes mathematics as a whole a more manageable discipline, and there will be fewer opportunities for error if one continually returns to the same previously established foundations. On the other hand, if a particular philosophical proposition is founded with care on what the understanding "initially and immediately perceives," then in lZant's view it will also be immediately certain. If philosophy only appeals to such indemonstrable propositions, it is not condemned to uncertainty.47 The real difference between the certainty of philosophy and mathematics en1erges in IZant's discussion of the unanalyzability of the concepts of philosophy. When IZant states that philosophy has innumerable
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unanalyzable concepts, he does not simply mean that it may not be useful to further analyze them, as in the case of mathematically unanalyzable concepts. After listing some concepts that are only partially analyzable, he states that they need an analysis that we often cannot supply: Without exact kno\vledge and analysis of these concepts, the springs of our nature will not be sufficiently understood; and yet, in the case of these concepts, a careful observer will notice that the analyses are far from satisfactory. I admit that the definitions of pleasure and displeasure, of desire and aversion, and of numberless other such concepts, have never been furnished by means of adequate analyses. Nor am I surprised by this unanalyzability. For concepts which are as diverse in character as this must presumably be based upon different elementary concepts. (2:280) Note that the difficulty here is not that the judgments of the understanding concerning initial and immediate characteristic marks lack immediate certainty. The difficulty is that we cannot satisfactorily carry out the analysis in philosophy that would be useful for it, an analysis which I
3.4 Ease versus difficulty of the obJ'ect ofstudy The fourth distinction I
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ter is difficult and con1plex. In IZant's view, the object of mathematics is magnitude, and all that n1athematics concerns is "how many times something is posited." Because of this: ... mathematics must rest on a few, very clear fundamental principles of the general theory of magnitudes (which, strictly speaking, is general arithmetic). There, too, one sees the increase and decrease of magnitudes, their reduction to equal factors in the theory of roots-all of them originating from a few simple fundamental concepts. And a few fundamental concepts of space effect the application of this general cognition of n1agnitudes to geometry. (2:282) IZant is here alluding to the Eudoxian theory of proportions and its the0ry of magnitudes, whose foundations are presented in Books V and VII of Euclid's Elements. 48 According to IZant, the basis provided by the theory of magnitude is particularly easy and simple. He cites as evidence the relation of a trillion to one; the fact that the "how many times" is extremely large does not prevent us from having a distinct understanding of it. In contrast, philosophy concerns countless concepts that have proved difficult to explain, such as the concept of freedom. The difficulty lZant sees here is providing a definition of freedon1, which echoes what he has said in the previous three sections. IZant also makes a claim about analysis n10re generally: ... there are infinitely many qualities which constitute the real object of philosophy, and distinguishing them from each other is an extremely strenuous business. Likewise, it is far more difficult to disentangle complex and involved cognitions by means of analysis than it is to combine simple given cognitions by means of synthesis and thus to establish conclusions. (2:283) This passage affirms that analysis itself may be difficult, and hence that the inden10nstrable propositions of philosophy n1ay be difficult to reach. It does not state, however, that the indemonstrable propositions we do reach in philosophy are less than certain.
3.5 I(ant)s lessons for metaphysics IZant's account of the deficiencies of philosophical method might lead one to think that certainty is not achievable in philosophy. Iunt, however, exploits the divergence between the difficulty of arriving at complete definitions in philosophy and the immediate certainty of the indemonstrable
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propositions on which they are based. One need only reject the ten1ptation to imitate mathematics by beginning with definitions and instead use indemonstrable propositions as one's starting point: In philosophy and in particular in metaphysics, one can often come to know a great deal about an object with distinctness and certainty, and even establish reliable conclusions on that basis prior to having a definition of that object, and even, indeed, when one had no intention of furnishing one. In the case of any particular thing, I can be immediately certain about a number of different predicates, even though I am not acquainted with a sufficiently large number of them to be able to furnish a completely determinate concept of the thing, in other words, a definition. (2 :284) Using appetite as an example, lZant demonstrates the kinds of things philosophy can learn with certainty: Even if I had never defined what an appetite was, I should still be able to say with certainty that every appetite presupposed the representation of the object of the appetite; that this representation was an anticipation of what was to come in the future; that the feeling of pleasure was connected with it; and so forth. Everyone is constantly aware of all this in the immediate consciousness of appetite. (2:284) In both these passages, I(ant emphasizes that we can have immediate certainty of indemonstrable propositions. I<.ant's strategy allows him to give guidelines for attaining the highest possible level of certainty in philosophy, a level sufficient for conviction. I(ant sets out two rules, and then reiterates and expands on them; they can be distilled into IZant's main lesson of the Inquiry and four rules. The main lesson is that one should not attempt to imitate mathematical method in philosophy, above all by attempting to start one's investigations with definitions. The four rules of philosophical method are: 1. Do not attempt to provide a full definition of your concept.
2. Begin instead with immediately certain judgments, and use them as axioms. 3. Make sure that the concept has not undergone change whenever its application is changed due to the ambiguities of linguistic usage. 4. Experiment with uncertain cognitions for purposes of discovery if you wish, but do not confuse them with certain cognitions. 49
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By following his precepts, I(ant claims that certainty can be attained in metaphysics. I(ant draws an analogy between his method and that of Newton, who "sought out the rules by which certain phenomena in nature occur," without attempting to discover "the fundamental principle of these occurrences" (2:286).50 I(ant adds: Likewise in metaphysics: by means of certain inner experience, that is to say, by means of an immediate and self-evident inner consciousness, seek out those characteristic marks which are certainly to be found in the concept of any general property. And even if you are not acquainted with the complete essence of a thing, you can still safely employ those characteristic marks to infer a great deal from them about the thing in question. (2:286) I(ant holds out hope for philosophy, as long as it is pursued according to his precepts.
4. IZant's Views
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Method and the Analysis Situs
4.1 The doctrine of the Inquiry and the philosophical
definition ofsimilarity We are now in a position to consider what the doctrines of the Inquiry imply for Wolff's and Leibniz's views. First, it is clear that lZant is taking a hard line against Wolff in the debate that lies behind the Academy's question: in rejecting the imitation of mathematics in philosophy, and in particular, the attempt to derive metaphysics from definitions, lZant is rejecting the entire Wolffian approach to philosophy. Second, philosophical definitions have no place in mathematics. Thus, both Leibniz's and Wolff's use of the philosophical definition of similarity will have no place in geometry. But why, by lZant's lights, should their definition of similarity be excluded? The natural conclusion is that inserting the philosophical definition of similarity into geometry will undermine its certainty. But is this so? As we have seen, the most significant problem with philosophical definitions is that one cannot be sure that one has not overlooked a characteristic that belongs to the definition, and it is tempting to slide on that basis to the judgment that the concept excludes the characteristic. But this difficulty with philosophical definitions hardly seems fatal to the use of philosophical definitions in mathematics, for lZant has diagnosed this potential error, and we can take heed not to commit it. Moreover, lZant has outlined how to attain certainty in metaphysics, and we can borrow his guidelines
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here. Instead of proposing that similarity is fully and completely defined as the identity of quality, we can simply recognize it as an initial and immediate characteristic, and hence as an indemonstrable proposition, that simi1arity entails identity of quality and vice versa. We can also recognize the further indemonstrable proposition that equality entails identity of quantity and vice versa. Indemonstrable propositions concerning quality and quantity will fill out the account. If in addition we are careful that the meanings of these terms do not shift in shifting contexts, then we can use these immediately certain propositions without undermining the certainty of mathematics. One might argue on IZant's behalf that we do not in fact have an initial and immediate consciousness of the characteristic marks of the concepts of similarity and equality. This would be a difficult line of argument for IZant. Mter all, he holds that there are innumerable indemonstrable propositions resulting from the analysis of concepts. Leibniz and Wolff could reasonably claim an immediate and self-evident consciousness that similarity entails identity of quality, and vice versa. In a passage cited above, Kant claims that we have an immediate and self-evident inner consciousness that the concept of appetite includes a representation of an anticipated object, and Leibniz's and Wolff's claims concerning similarity seem at least as secure. Furthermore, there is some evidence that at least in the critical period, IZant accepted the Leibnizian philosophical definition of similarity for philosophical purposes. 51 On the other hand, this line of argument might be more plausible against Leibniz's and Wolff's definitions of quality and quantity, which seem much less arguably self-evident. A distinction between qualities and quantities based on what we can perceive in an object regarded on its own is not a principle that in1n1ediately impresses itself on someone who reflects on it. In fact, Kant developed quite a different conception of quality and quantity in the critical period. 52 Whether or not IZant would have viewed these possible objections favorably, it is significant that they do not turn on the clain1 that the philosophical notion of similarity undern1ines the certainty of mathematics. With this understanding of the doctrine of the Inquiry in mind, let us look more closely at IZant's actual objections to Wolff's use of similarity.
4.2 !(ant)s criticisms of Wolff and indirectly ofLeibniz
on similarity In his initial description of the difference between analytic and synthetic definitions, IZant states that just as philosophers should not use synthetic definitions, mathematicians should not use analytic definitions. He adds:
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Mathematicians ... have sometimes offered analytic definitions. But it must also be said that for them to do so is always a mistake. It was in this way that Wolff considered similarity in geometry: he looked at it with a philosophical eye, with a view to treating the geometrical concept of similarity under the general concept. But he could have spared himself the trouble. If I think of figures, in which the angles enclosed by the lines of the perimeter are equal to each other, and in which the sides enclosing those angles stand in identical relations to each otherthis could always be regarded as the definition of similarity between figures, and likewise with other similarities between spaces. The general definition of similarity is of no concern whatever to the geometer. It is fortunate for mathematics that, even though the geometer from time to time gets involved in the business of furnishing analytic definitions as a result of a false conception of his task, in the end nothing is actually inferred from such definitions, or, at any rate, the immediate inferences which he draws ultimately constitute the mathematical definition itself. Otherwise this science would be liable to exactly the same wretched discord as philosophy itself (2:277) We can distinguish three possible criticisms I(ant makes against Wolff in this passage. First, Iunt states that even when such definitions are introduced, they are not employed. This is a complaint that had been traditionally leveled against some of Euclid's definitions. For example, there is no proof in the Elements that turns on the definition of a point as that which has no part or the definition of a line as a breadthless length. Even if these definitions were to provide some insight into the nature of the fundamental geometrical objects of geometry, they are mathematically otiose. Against this line of argument, both Leibniz and Wolff could rightfully counter that the philosophical definition of similarity does in fact playa role in the similarity demonstrations we examined. Thus, the definition of similarity is not idle, and this argument n1isses its mark. The second criticism claims that even if the philosophical definitions are employed, the immediate inferences drawn from them constitute the mathematical definition itself. We found, however, that the demonstrations did in fact rely on the philosophical definition of similarity, not just the geometrical definition which it subsumes; both Leibniz and Wolff appeal to the fact that two figures determined in the same way will be indistinguishable taken each on their own. Thus, this line of argument also fails against Leibniz and Wolff. In fact, Iunt may not have had similarity in n1ind when he makes this point in his general criticism of the use of philosophical definitions in mathematics. In An Attempt to Introduce the Concept of Negative Magnitudes into Philosophy, written immediately after the Inquiry, lunt states:
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The concept of negative magnitudes has long been en1ployed in mathematics ... Nonetheless, the representation which most people have formed of the concept ... and the elucidation which they have given of it, have been strange and inconsistent. This has not, however, resulted in any errors of application, for the particular rules governing its employment took the place of the definition and guaranteed its correct use. And whatever error there may have been in the judgment about the nature of this abstract concept it remained fruitless and without affect.(2:170) IZant likely had the philosopher Crusius in mind, against whose account of negative magnitudes I
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As our examination of their broader projects makes clear, however, Leibniz and Wolff also had a third, programmatic advantage in mind. We noted earlier that for Leibniz, introducing the definition of similarity served his project of incorporating mathematics into a universal language of all human thought, a lingua philosophica; in fact, this program is the driving force behind his similarity demonstrations. For Leibniz, that program included both the ars combinatoria and a universal characteristic. For Wolff, that program did not include the ars combinatoria, but it did encompass an attempt to discover the single true hierarchy of concepts beginning with the most general. In either case, if their program were to succeed, the use of the philosophical definition of similarity in geometry would be vindicated by its ability to unify all human knowledge. As we have seen, lZant's criticism of beginning with definitions in philosophy is an implicit criticism of Wolff's views of human knowledge. More particularly, I(ant rejects the idea of subsuming mathematical knowledge under metaphysical knowledge by means of definitions that appeal to increasingly general concepts, and hence Wolff's attempt to assimilate geolnetry into metaphysics by means of the philosophical definition of similarity. The passage quoted above begins" Wolff . .. looked at [similarityJ with a philosophical eye, with a view to subsuming the geometrical concept of similarity under the general concept. But he could have spared himself the trouble" (2:277).53 I(ant does not accept that a universal characteristic will bring the broader advantages hoped for by Wolff. As we saw above, I(ant argues that philosophy contains innumerably many unanalyzable concepts; he therefore does not believe that a universal characteristic can define concepts in terms of fewer and increasingly more general concepts' as in Wolff's view: The error, committed by son1e, of treating all such cognitions as if they could be completely analyzed into a few sin1ple concepts is like the error into which the early physicists fell. They were guilty, namely, of the mistake of supposing that all the matter of which nature is constituted consists of the so-called four elements-a view which has been discredited by more careful observation. (2:280) lZant does not shut the door con1pletely on the hopes for a universal characteristic, however. First, despite his objections to beginning with a search for analytic definitions, lZant allows that we may be able to arrive at them. We saw above that when lZant describes the proper procedure of metaphysics, which should begin with immediately certain propositions rather than definitions, he uses inden10nstrable propositions concerning appetite as an example. 54 He adds:
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One might perhaps eventually be able to arrive at a definition of appetite on the basis of such remarks as these [i.e. immediately certain propositions], once they had been compared with each other. But as long as it is possible to establish what one is seeking by inference from a few immediately certain characteristic marks of the thing in question, and to do so without definition, there is no need to venture on an undertaking which is so precarious. (2:284) The definition of complex concepts in terms of unanalyzable fundamental concepts is precarious, but not completely ruled out. Kant addresses this possibility directly at the close of the Third Reflection: Metaphysics has a long way to go before it can proceed synthetically. It will only be when analysis has helped us towards concepts which are understood distinctly and in detail that it will be possible for synthesis to subsume compound cognitions under the simplest cognition, as happens in mathematics. (2:290) I(ant expresses a sin1ilar view in A New Elucidation ofthe First Principles of Metaphysical Cognition, written in 1755. I(ant states there: For my own part, I do not deny that, once one has arrived at absolutely first principles, a certain use of the art of signs may be legitimate, for one has the opportunity there of employing the concepts and consequently the simplest terms, as well, as signs. However, when compound cognition is to be expressed by means of signs, all the n1ind's perspicacity finds itself suddenly stranded, so to speal<:, on a reef, and in1peded by difficulties from which it is unable to extricate itself. (1:390) While I(ant will not entirely rule out the possibility of a universal characteristic, he differs from Leibniz and Wolff in his assessment of its prospects. Faced with the difficulty of attaining absolutely prin1itive ideas, Leibniz n1aintained that the discovery of a few relatively primitive ideas could still be useful. Kant rejects this more modest proposal, for the symbolization of compound rather than simple ideas forsakes perspicacity. Moreover, at the close of Negative Magnitudes, Kant expresses doubts that the relation between a real ground and its consequent can be made distinct by reducing it to simple, unanalyzable concepts of real grounds; there is something that will escape the account (2:203-204). Furthermore, in the Inquiry, I(ant has ruled out a universal characteristic based on a few simple concepts, and he believes that the unanalyzable concepts of philosophy are innumerable; hence, the possibility he leaves open is a universal characteristic based on innumerable fundamental concepts. In that case, a complete
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universal characteristic, and a lingua philosophica based on it, will not be humanly achievable, even in principle. Finally, even if a limited universal characteristic were achievable, it would not be because we begin our philosophizing by defining our concepts; rather, the definitions will emerge as we successfully pursue philosophy according to his recommended guidelines.
5. Conclusions We have seen that Wolff's philosophical definition of similarity differs in details from Leibniz's definition, and that Wolff offers different similarity demonstrations in his Latin Geometry than Leibniz offers in his letters. Nevertheless, Wolff's definition is effectively equivalent to that given by Leibniz, and his demonstrations are in the spirit suggested by him. Kant's criticisms of Wolff therefore amount to indirect criticisms of Leibniz as well. We have seen that Kant's primary objection to Wolff is not what one might at first think, that is, that a philosophical definition of similarity will undermine the certainty of mathematics. The problem of uncertainty can be circumvented by adhering to Kant's guidelines and appealing to indemonstrable propositions. Kant's primary objection to Wolff is that use of the philosophical notion of similarity does not bring any advantage to geometry. As our analysis revealed, Wolff's own fundamentally flawed attempt to reform geometry spoke against any such advantage, and even if Kant had known of Leibniz's sound similarity proofs, they were too few and unsystematic to have bolstered the case. Kant's reasons for rejecting the use of a philosophical definition of similarity to reform geometry therefore rests on his assessment of its prospects, which was based on his acquaintance with the details of Wolff's geometry and its contrast with Euclid's geometry. But Kant did not simply oppose introducing the philosophical definition of similarity; he opposed all use of philosophical definitions in geometry. His position, I think, is based on his repudiation of Wolff's broader project to incorporate mathematical knowledge into a framework of human knowledge, a framework based on the logical relations among concepts. Kant thereby disavows Leibniz's and Wolff's attempts to logicize mathematics by means of a universal characteristic. In fact, advances in mathematics did not come by defining mathematical concepts in terms of philosophical concepts such as quality and quantity, which vindicates Kant's judgment. More importantly, as Michael Friedman has rightly emphasized, Kant was sensitive to the limitations of early modern Iogic. 55 At about the time of the Inquiry, Kant came to believe that much of
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human knowledge, and in particular mathematical knowledge, could not be fit into a Wolffian logical framework. 56 In fact, it seems likely that in Kant's view, Wolff's failed reformation of geometry-his fallacious inferences from philosophical definitions and his unsound demonstrationsvividly demonstrated the unbridgeable gap between a Leibnizian-Wolffian logical framework for human knovvledge on the one hand and mathematical knowledge on the other. 57 We have also seen that, according to IZant, signs cannot symbolize the content of thoughts or concepts or the relations among them, and hence that an ars combinatoria of philosophical knowledge is not possible. I
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conception of a universal characteristic in particular, which is based purely on the logical relations among primitive concepts. This negative point is a central insight of IZant's Inquiry, but by the time of writing the Critique, lZant had developed a positive account of how the categories, supplemented by intuition, make knowledge possible. IZant's expanded conception of human knowledge therefore has resources that the Leibnizian and Wolffian frameworks did not. It is interesting, nevertheless, that with these extra resources, IZant was willing to entertain the possibility of an ars combinatoria and a universal characteristic based on the categories. It is also noteworthy that he accepted the very notions of similarity and equality whose use in mathematics he had earlier rejected. His acceptance of them also does not show that he changed his position, however. For it is one thing to give a philosophical analysis of similarity and equality in an account of human cognition; it is quite another to employ them in geometry to prove theorems. It is only the former which IZant allows in the critical period. One would still justly wonder about the relation between the philosophical concepts of similarity and equality and their purely geometrical counterparts. But as we saw in §1.2, on a suitable understanding of quality and quantity, the philosophical concepts of similarity and equality entail the traditional geometrical concepts of similarity and equality as a special case. IZant crucially altered the conception of quality and quantity, but he still thought of them in such a way that the philosophical concepts of similarity and equality included the traditional geometrical concepts. 60 Thus, the connection between the philosophical and geometrical notions is quite natural. We might nevertheless find a conflict between his earlier and later views, if lZant had objected to the use of philosophical definitions because of their inherent uncertainty. For if the philosophical concepts of similarity and equality figure into the correct account of human cognition and hence are in fact the concepts we employ in mathematical cognition, it seems that their uncertainty would taint our mathematical demonstrations after all. As I have argued, however, IZant's real objection to the use of these philosophical concepts in mathematics is that they do not bring any advantage. And this is a position he would have held in the critical period as well.
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1. I would like to thank Michael Dickson and Mary Domski for their invitation to present an early version of this paper at a conference at the University of South Carolina in October 2004, and Gholamreza Avani, Shahin Avani and Zia Movahed for the opportunity to present this paper at a conference at the Iranian Institute of Philosophy in Tehran in December 2004. I \vould also like to thank the participants of both conferences for their insightful comments, especially Michael Friedman and Domenico Bertoloni Meli, who pressed me to look more closely at the way Wolff's geometry mediated Kant's understanding of Leibniz, and to Friedman for more recent discussions. I'm particularly indebted to Dickson and Domski for their helpful comments on a penultimate version. In addition, I would like to thank Brian Dean for his help in translating the passages from Wolff, though final responsibility for the translations rests with me. 2. I
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11. Rutherford (1995), 235. 12. There were also other fundamental changes in his view of the universal characteristic that I will not discuss here. For example, for a tin1e Leibniz thought that there might be only two primitive ideas, that of God and of nothing. This universal characteristic lent itself to an ars combinatoria based on the binary number system, which demonstrates the close connection between a universal characteristic and a corresponding ars combinatoria. It also reveals another Leibnizian conception of how logic could be mathematized. For more on this topic, see Rutherford (1995), p. 236ff. 13. Rutherford (1995), pp 232-33,237-38. 14. Schepers (1969), 38. 15. See Rutherford (1995), p.233. As he notes, Chapter V of Couturat's La Logique de Leibniz discusses Leibniz's larger aim of an encyclopedia of human knowledge, but it is the work of Heinrich Schepers on Leibniz's attempts to reform the categories that has most helped to bring this side of Leibniz's work to the fore; see Schepers (1966) and (1969). 16. See Schepers (1969), 38. I am setting aside Leibniz's belief for a time that there were only two absolutely primitive ideas, that of God and nothing. See fn. 12 above. 17. See Schepers (1969), 45 for this and other examples of Leibniz's attempts to reforn1 Aristotle's categories. 18. See, for example, "On Analysis Situs," Leibniz (1971), V 179, translated in Leibniz (1956), 392. The tradition of defining equality and similarity in this way appears in Aristotle; see Aristotle (1984), Met. V, 1021 al0. 19. Leibniz (1971), V 180. Translated in Leibniz 1956, I 392. 20. Couturat (La Logique de Leibniz, 412, fn. 1) states that Leibniz's philosophical approach to reforn1ing geometry occurred to him as early as 1677. 21. For more on this issue, see Sutherland (2005). 22. Cf. Schepers's discussion of the relation between logic and metaphysics in Leibniz's project; Schepers (1969), pp. 45ff. 23. See I.E. Hofmann's introduction to Christian Wolff Gesammelte Werke, 2. Abt., Band 29, p. v ff. and Charles A. Corr's introduction to Christian Wolff Gesammelte Werke, 1. Abt. Band 3, pp. 4-5. 24. The term "problem" indicates those propositions that describe some construction that can be completed, such as "To construct an equilateral triangle on a line segment," in contrast to theorems, which state a geometrical fact, such as "The greater side of any triangle subtends the greater angle." Euclid did not mark this distinction between two types of proposition, but later geometers did, including Wolff whose term for problem is "Aufgabe." In Euclid, problems are chosen for their usefulness in carrying out theoretical proofs, not for their practical applications. 25. As noted in J.E. Hofmann's introduction to Christian Wolff Gesammelte Werke, 2. Abt., Band 29, p. v ff. 26. For example, the latter simply used ratios and proportions without explaining them; in contrast, the Arithmetic of the Elementa devotes an entire chapter to explaining ratios and proportions and another chapter to the rules governing them.
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27. Strikingly, there is no mention of the congruence approach to the analysis situs here or elsewhere in Wolff's work. Either Leibniz did not impart it to him, or Wolff failed to appreciate its potential. 28. Elementa matheseos universae (1742), Christian Wolff; Gesammelte Werke, Part II, Vol. 29, pp. 119-20. As translated in I(ant)s Theoretical Philosophy 1755-1770, p. 443, fn. 11, with emendations. 29. He also expresses indebtedness to Leibniz in a scholium following his definition of the relation, in which he says that "Leibniz first elicited the distinct notion of similarity." Gesammelte TvVerke, II. Abt., Band 29, p. 25. 30. See §1.2 above. 31. Wolff's Elementa is divided into the Arithmetic and Geometry. His references to sections within each of these do not specify the work from which they come, which can lead to confusion when they are removed from context. I have therefore prefaced each section, theorem, and problem number with A for the Arithmetic and G for the Geometry. 32. Wolff's explication there is deficient for various reasons. For more on this issue, see Sutherland (2005). 33. See footnote 31 above. 34. Alternatively, perhaps, Wolff may be assuming that determining each semicircle, once in each direction, counts as the same manner of determination that results in a full circle. The determination of a circle is described in problem §GI31. 35. Wolff's demonstration states: "Suppose that P & p are similar but that P has a different ratio to T than p to t. Parts p & P can be distinguished from one another by means of differing ratios to the wholes. (§A132). They are, as such, dissin1ilar (§A24). This is absurd and contrary to the hypothesis; hence P is to T as p is to t. This was the first thing to be demonstrated" (Elementa, p. 63). The definition of similarity simply does not apply to things with differing ratios to the wholes of which they are a part because these relations are external to them. In fact, even if they are parts of the same whole, as is the case in the demonstration of Theorem G1, the proof fails for the same reason. The fact that Wolff simply defines similarity in terms of the sameness of properties by which they ought to be distinguished makes his error possible. Whether we consider the corollaries as consequences of the definition of similarity (as Wolff seems to ), or whether we consider the corollaries to the definition as further refinement of the definition (as I suggested above), the oversight is fatal to Wolff's derivation of §A170 Theorem A16. 36. For a helpful and stimulating discussion ofWolff's hierarchical conception of knowledge and the difficulties of incorporating mathematics into it, see Anderson (2005). I owe much of my background understanding of Wolff's logic to this work. 37. Leibniz seems to have emphasized the ars combinatoria over the universal characteristic in later years. See Rutherford (1995), 227, 237ff. 38. Wolff's problen1 in overstating the fecundity of his starting point is not limited to definitions. In his Metaphysics, for example, he derives the principle of sufficient reason from d1e principle of contradiction. See §§ 30 and 31 of Wolff's German Metaphysics, Gesammelte Schriften, Part I, Vol. 2, 16-18.
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39. The Prize question: "One wishes to know whether the metaphysical truths in general, and the first principles of natural theology and morality in particular, admit of distinct proofs to the same degree as geometrical truths; and if they are not capable of such proofs, one wishes to know what the genuine nature of their certainty is, to what degree said certainty can be brought, and whether this degree is sufficient for complete conviction." Translated in I(ant)s Theoretical Philosophy: 1755-1770, p. lxii. 40. The term I translate as "elective" is "vvillkurlich," vvhich is normally translated as "arbitrary" or "voluntary," neither of which quite fits IZant's sense; I(ant does not mean that one chooses any old characteristics, or that one's act of choosing is free rather than compelled, but that there are no constraints on which characteristics one may choose to synthesize. IZant uses the term "characteristic" or "characteristic mark" [MerkmalJ to mean a conceptually represented property; it is unrelated to Leibniz's use of "characteristic" to mean alphabet. See footnote 9 above. 41. Inquiry, First reflection, §1 (2:276-77). 42. Inquiry, Third reflection, §1 (2:291). 43. Ibid. 44. IZant switches from arithmetic and algebra in making his first point about Il1anipulating signs to geometry in malting his second point about representing the universal under signs in concreto. Presumably, the signs employed in geometry, e.g., lines and circles, are manipulated according to Euclid's rules of construction; one wishes IZant had directly addressed what is particular and what is universal in 5 + 7 = 12. 45. Inquiry, Third Reflection, §1 (2:291). In the First Reflection, IZant refers to "visible signs" when introducing his distinction in methods, but it is in his discussion of certainty in the Third Reflection that IZant first refers to the role of intuition. In fact, intuition figures in the definition of certainty: the more intuition involved in knowing that it is impossible for a cognition to be false, the more subjective certainty one has in that cognition. IZant states that in matheIl1atics the greater clarity of impression and the resulting greater exactitude [Richtigkeit] of intuition results from representing the object under sensible signs in concreto, which ties these epistemic advantages to the difference in method described in the First Reflection. 46. Inquiry, 2:280. 47. One might argue that the indemonstrable propositions of philosophy are not really capable of the same degree of certainty as those of mathematics. Since IZant relativizes both the unanalyzability of concepts and indemonstrability of propositions to a discipline, what counts as immediately certain in philosophy may not in mathematics, and vice versa. The proposition "the whole is equal to all its parts taken together" is indemonstrable in mathematics because it does not serve mathematics to ask for a proof of it, even if it is provable elsewhere. On this basis, one might argue that the needs and ends of philosophy require less certainty than mathematics, and hence that even if they are immediately certain relative to phi10sophy' they are not as certain as the indemonstrable propositions of mathematics. This line of argument runs into difficulties, however. It is surely an aim of philosophy, as I(ant understands it, to attain as high a level of certainty as possible;
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indeed, one of the reasons for the Academy's question was precisely that philosophy had not achieved the certainty one would hope it to have. Thus, the standards of certainty set by philosophy should be at least as high as for mathematics. 48. This theory begins with a general account of continuous and discrete magnitudes and their relations, which was applicable in geometry but which I(ant also thought provided a foundation for arithmetic and for algebra ("general arithmetic"). For n10re on this issue, see Sutherland (2004) and (2006). I
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REFERENCES Adams, R. (1994). Leibniz: Determinist) Theist) Idealist. Oxford: Oxford U niversity Press. Anderson, L. (2005). "The Wolffian Paradigm and Its Discontents." Archiv fur Geschichte der Philosophie 87 (1): 22-74. Beth, E.W. "Uber Lockes 'Allgelneines Dreieck'." !(ant-Studien 49 (1956-57): 361-80. Carson, E. (1999). "IZant on the Method of Mathematics." Journal ofthe History ofPhilosophy 37 (4): 629-52. Couturat, L. (1901). La Logique de Leibniz. Paris: Ancienne Librairie Germer Bailliere et cie. Euclid (1956). Euclid: The Thirteen Books ofthe Elements. 3 vols. Edited and translated by Sir Thomas Heath. New York: Dover Publications. Friedman, M. (1992). !(ant and the Exact Sciences. Cambridge, MA: Harvard University Press. - - - . (2000). "Geometry, Construction and Intuition in IZant and His Successors." In Sher and Tieszen (eds.), 186-218. Hartschorne, R. (2000). Geometry: Euclid and Beyond. New York: Springer Verlag. Hintikka, J. (1969). "On IZant's Notion of Intuition (Anschauung)." In Penelhum and MacIntosh (eds.), 38-53. - - . (1974). !(nowledge and the !(nown. Dordrecht: D. Reidel. Ishiguro, H. (1990). Leibniz}s Philosophy of Logic and Language. CalTlbridge: Cambridge University Press. Jolley, N. (ed.) (1995). The Cambridge Companion to Leibniz. Cambridge: Cambridge University Press. IZant, I. (1902). !(ant}sgesammelte Schriften. Hrsg. von der koniglich preussischen Akademie der wissenschaften. Berlin: W. de Gruyter, 1902-. - - . (1992). Theoretical Philosophy 1755-1770. The Cambridge Edition of the Works of Immanuel !(ant. Edited and translated by D. Walford. Cambridge: Cambridge University Press. - - - . (1997). Prolegomena to Any Future Metaphysics. Translated by G. Hatfield. Can1bridge: Cambridge University Press. - - - . (1997b). Lectures on Metaphysics. Edited and translated by IZ. Ameriks and S. Naragon. Cambridge: Cambridge University Press. - - - . (1998). Critique of Pure Reason. Translated by Paul Guyer and Allen W. Wood. Cambridge: Cambridge University Press. Leibniz, G.W. (1956). Philosophical Papers and Letters. Vols. I and II. Translated and edited by L. Loemker. Boston: IZluwer Publishing. - - - . (1961). Opuscules et fragments inedits de Leibniz: extraits des manuscrits de la Bibliotheque royale de Hanovre. Edited by L. Couturat. Hildesheim: G. Olms. - - . (1971). Mathematische Schriften. Vols. I-VII. Edited by C.I. Gerhardt. Hildesheim: Georg Olms. - - - . (1995a). La caracteristique geometrique. Translated and edited by J. Echeverria and M. Parmentier. Paris: J Vrin.
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(1995b). Philosophical Writings. Translated and edited by G.H.R. Parkinson. Rutland, Vermont: Totowa, NJ : Rowman and Littlefield. Mates, B. (1986). The Philosophy of Leibniz: Metaphysics and Language. Oxford: Oxford University Press. Penelhum, T. and J. MacIntosh (eds.) (1969). The First Critique. Belmont, Calif.: Wadsworth. Rutherford, D. (1995). "Philosophy and Language in Leibniz." In N. Jolley (ed.), 2224-69. Schepers, H. (1966). "Leibniz's Arbeiten zu einer Reformation der I
[9J I(ant on Attractive and Repulsive Force: The Balancing Argument DANIEL WARREN 1
Introduction Michael Friedman has, more than anyone else working in the field of IZant studies, brought us to see the importance of the Metaphysical Foundations of Natural Science (MP) for a proper understanding of IZant's theoretical philosophy. By pursuing the idea that it is in the MF that IZant gives content to the transcendental philosophy, Friedman has given us both a new way of approaching the Critique ofPure Reason and a new way of seeing how IZant's engagement with the development of physical theory anchored his critical philosophy in some of the most compelling intellectual concerns of the time. In particular, Friedman's work on the MF has given us a detailed and systen1atic picture of how this work is attempting to come to terms with the central ideas of Newtonian physics. 2 Part of that picture concerns IZant's rejection of a "mechanistic" conception of matter. A number of interpreters have seen IZant's arguments in this regard as aimed against Newton's theory of matter. But, as Friedman has argued, this is extremely unlikely and the "dynamical" conception of matter, which IZant defends, is much more of a piece with what is novel in Newtonian physics, as IZant understands it. I follow Friedman on this issue. However, I am perhaps more inclined to focus on the more backward-looking elements in IZant's views on physics. I have often tended to concentrate on the role of the metaphysical views developed by IZant's predecessors in shaping how Iunt understands Newton's significance and how he thinks it can be integrated into a systematic philosophical picture. 3 And, I think, IZant's philosophy of science is burdened, for better or worse, with metaphysical commitments and with conceptions of philosophical understanding that often drive it in directions that diverge from the more "progressive" trends in the
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development of physics. I hope that this paper can serve to illustrate that idea. IZant's Metaphysical Foundations of Natural Science-written in 1786, in the middle of the "critical" period-is intended as a defense of a broadly Newtonian physics, though it is also meant to diverge from Newton's picture in some respects. Officially, IZant presents this work as founded, in some way, on the lines of thought developed in the Critique of Pure Reason. But in fact it is an odd mixture of arguments. SOIne are clearly of an epistemic nature and are explicitly intended to connect up with the transcendental themes of the first Critique. Other arguments, though they are harder to tie to concerns with the conditions ofknowledge, can be thought of as having epistemic concerns lying beneath their surface presentation. But there are son1e arguments in the MF that appear to rest at least partly on considerations of a definitely physical or metaphysical character, which are particularly difficult or, perhaps, impossible to explain by appeal to the conditions of experience and knowledge. At the center of chapter 2 (the "Dynamics" chapter) oflZant's MF, there is an argument, sometimes called the "balancing argument," which seems to be of this latter sort. That is not to say that connections with the conditions of experience are wholly absent from the balancing argument, just that it also relies on considerations that seem to be independent of these transcendental themes. Such considerations will constitute the focus of this paper. IZant's argument is intended to show that matter must possess both repulsive and attractive force, if it possesses either, and it consists of two component arguments. The first half moves from the supposition of repulsion to the need for attraction; the second moves in the converse direction. IZant clearly considers the balancing argument crucial to his grounding of Newtonian physics in the MF and, in particular, to the defense of a gravitational attraction which is universal and acts at a distance. Moreover, even very early on in the pre-critical period-in the "Physical Monadology" of 1756-IZant presents a close variant of the first argument (1:483-84), and informally he summarizes both of them (1:476). So Kant's commitment to these lines of thought was one that was long standing. However, these arguments are in a variety of ways problematic. And what I would like to do in this paper is to articulate a group of related objections that these arguments face, and to consider the extent to which IZant might be able to address these problems. In the earlier parts (sections 1, 2, and 3) of the paper, I will focus particularly on Kant's reliance upon considerations of a seemingly physical character, and I'll be raising a series of related doubts-which in their essentials could have easily been articulated by Kant's contemporaries (in fact, from the point of view of a broadly \Newtonian mechanics )-concerning the adequacy of Kant's descriptions of \the behavior of matter, descriptions which playa crucial role in the argu~_
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luent he is putting forward. In the later parts (sections 4a, b, and c), I consider some responses that could be made on I
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which can counteract the repulsive force. It must have a tendency toward compression which can check the tendency toward expansion. The second argument closely parallels the first, but in reverse. We are asked to consider what would follow if matter were endowed only with a force of attraction, without any repulsive force to counteract it. And what I(ant tells us is that "all parts of matter would approach one another without hindrance and diminish the space that matter occupies" (4: 511). Eventually, the parts of matter would "coalesce [zusammenflie£en] in a mathen1atical point." And thus the unchecked action of an attractive force would have as its result that, in I
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to the balancing argument that I will present are meant to motivate these questions in more concrete terms. Now some people have found IZant's arguments to possess a high degree of plausibility. If a body (e.g., a table) were endowed with repulsive force alone, it would expand indefinitely; but since this doesn't occur, it lnust possess a countervailing force. But this sense of plausibility often rests on assumptions which are of a definitely empirical character, that is, we can observe that matter does not expand indefinitely. For example, L. Pearce Williams, in The Origins ofField Theory, puts IZant's argument as follows: But the repulsive force is not all that is involved; the table, after all, does not swell to fill all space as it would do if it were merely a repulsive force in space. There is clearly some force that holds the table together as is clearly evidenced by any attempt to pull the table apart. The table resists such dissolution. Therefore, an attractive force must also be associated with the table if the table is to exist as an entity recognizable as a piece of matter with a distinct shape and size. (Williams 1966, 40) One problem with this formulation of the argument is that the force it shows the need of is cohesion, which IZant would consider a derived contact force, rather than the fundamental Newtonian force of gravitation acting at a distance. The more important problem with his formulation is that Willian1s is here appealing to a clearly empirical claim ("the table, after all, does not swell ...," a fact Willian1s later refers to as the body's "observed individuality") . However, lZant would not regard it as permissible to rely on these empirical claims in this context. I(ant meant the balancing argument to fill a gap he believed to exist in Newton's picture of a world of mutually attracting bodies. He felt that Newton needed much stronger grounds for thinking of gravitation as a universal property of matter than merely inductive or empirical grounds would allow. lZant's intention was to prove that matter (insofar as it is impenetrable and thus endowed with repulsive force) is necessarily endowed with attractive force as well, that this attractive force is part of the essence of matter, and that it is therefore absolutely universal to it. 14 I(ant was not just making claims based on observation of the way matter actually behaves. lZant's idea here is not: Given that matter is endowed with repulsive force, then it must also be endowed with attractive force if it is to behave as it is observed to behave. Rather, his point is: Given that matter is endowed with repulsive force, then it must also be endowed with attractive force, and for this reason it must (in the relevant respects) behave as it is observed to behave. To be sure, IZant is starting the MF with an empirical concept (the concept of matter 15 ), but beyond that, he excludes any empirical knowledge from the resources he may draw
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on in the arguments of the MF.16 Thus, the claims Kant is trying to establish are, on account of their necessity, absolute universality, and apriority, much stronger than many readers (like Williams) have assumed them to be. However, once we recognize how strong Kant intends his claim to be, serious worries arise about the adequacy of his arguments in support of them, and I would like to articulate and pursue a nun1ber of interrelated objections which can be distilled from these worries. I'll do this by discussing cases in which we assume just one of the two forces, but where doubts may be raised about the consequences I
1. Objection to the First Half of the Balancing Argllment The objection to the repulsion-entails-attraction half of the balancing argument is straightforward and very easy to state. I said that, according to
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Kant, if Inatter were endowed with repulsive force alone, it would "disperse itself to infinity," and as a result, "all spaces would be empty; and hence, strictly speaking, there would be no matter at all" (4:508). A natural objection, however, is that this consequence does not in fact follow from the assumption that n1atter has only repulsive force. Such matter vvould not disperse itself to infinity with the result that all spaces are empty: it would no more do so than one object moving away from another with constant velocity would become infinitely distant fioln it. Rather we should say that in the latter case there is no limit to how distant the objects can be. And in the forn1er case, we should say that there is no limit to how dispersed the matter can be. But of course that is compatible with the degree of dispersion being always finite. At no time does the space become empty; the most we could say is that the situation approaches that of empty space as the length of time approaches infinity. However, this is not a state that would arise after any finite time, no matter how long. And thus we need not posit a countervailing (attractive) force in order to ensure against it. This objection can be put more strikingly by asking why ruling out a state that can never actually arise should be of such significance that it determines what essential properties a thing has. It takes time for matter to disperse. That it takes time, one might think, is due to the fact that matter is endowed with inertial properties opposing the action of the repulsive force of expansion. This point will come up again as the paper continues. The above objection can be amplified as follows. By using the argument as it is presented by IZant, it would seem that we could equally well claim, by an argument of the same form, that a corresponding consequence would follow even in the case where there is no repulsive interaction. Such an argument would begin: (i) if a body17 A moves with constant velocity away from another body B, the distance between them would increase to infinity, and as a result, bodies .A and B would be infinitely far from one another. And yet, we must surely grant that (ii) each body can only be a finite distance from every other body. However, what are we supposed to conclude from this argument? Are we meant to infer, even in this case, the existence of an attractive force to avoid this supposed problem? This would be an absurd argument; there is no problem that needs to be ruled out. Bodies A and B never are an infinite distance fron1 one another. Now, on lZant's behalf it might be replied that his argument is not of the same form as the one just given, for in IZant's argument we draw implicitly on the claim that the cause of the dispersion (the repulsive force) is essential to the matter itself. That is, the matter as such is the cause of its own dispersion. However, it isn't at all clear how IZant's argument is in any better shape vis-a-vis the original objection once one grants this feature of his original argun1ent. The point of the objection was that the matter never does become infinitely rarified. Whether the cause lies in the matter or is
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instead in some way contingent or external, that point will still hold. IS To anticipate a bit, I should say that I do not believe that anything I will say can, in the end, respond adequately to this first objection, although it will be seen in a different light once the responses to the other objections are spelled out. A further problem, which can be directed at the first half of the balancing argument, will be added in section 3.
2. Objections to the Second Half of the Balancing Argument There are two closely related objections (which I'll call "objection 2a" and "objection 2b") that I wish to raise against the second half of the balancing argument (the argument fron1 the supposition of attraction to the need for repulsion). Both pose doubts about the claim that matter, endowed with attractive force alone, will coalesce in a point. The first objection will require a somewhat lengthy discussion, and it is very general in character. The second objection concerns a special case of the first. It can be presented very briefly, but it presents a particularly clear-cut case in which collapse can be avoided without appealing to a countervailing force. These two objections (2a and 2 b ) will differ from that just raised against the first half of the balancing argument (which I'll now call "objection 1") in that the process of collapsing to a point, unlike the process of infinite dispersion, can be completed in a finite time. I begin presenting the first objection (2a) by asking how, more exactly, I(ant is imagining the physical process of coalescing or collapsing to a point. Kant may somehow have in mind the idea that all the parts of matter, because they are subject to this mutual force of attraction, would, so to spealc, become "trapped" in a single point where they will all eventually have landed. However, it seems to me that there is no reason to assume that the process goes this way. 19 For it needs to be remembered that as they approach each other under the influence of this attractive force, these parts of matter will gain more and more momentum. One might plausibly think that, as each part of matter reached the center, it would not just becon1e trapped in the center, but would instead pass right through the center. For, at the center, as along its whole course, it encounters no resistance from in1pact with any other matter, since, ex hypothesi, there is no repulsive force, and hence no impenetrability. And even if the force of attraction approaches infinity as these parts get closer (which I(ant might take to support the idea of entrapment), so too could the momentum of each bit increase without limit. We cannot determine how the system will behave through a simple and intuitive analysis of the sort lZant provides. The behavior of matter at a given point depends not just on the forces to which
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it is subject, but also on the momentum it has, that is, on the fact of its inertia. Let us consider a case of a homogeneous ball of fluid matter, in which the parts all start off at rest, but then pick up more and more speed as a result of the attraction they exert on one another. Mter arriving at the center' each piece of matter, with all its accumulated momentum, might simply continue in a state of motion. But now, Inoving away from the center, it will experience a drag, and thus a deceleration, due to its continuing attraction toward the center, eventually slowing down until it momentarily comes to rest, and then the process would be repeated. Thus, the motion, one n1ight well think, could be cyclical, 20 rather than a case of entrapment (in which the matter would end up sitting there at the center21 ). Now for IZant, it is incompatible with the essence of matter (qua something that fills a space) that a finite amount of it occupy a point. And therefore it might seem of little consequence whether collapse results in entrapment or in a cyclical pattern of behavior. However, such indifference might be warranted only in certain special cases, e.g., when we assume that all the matter arrives at the center at the same time. When the parts of matter do not all arrive at the center at the same time, significant differences relevant to lZant's conclusion can arise. These will be discussed shortly. The alternative conception of collapse I have been sketching-cyclical (with a merely momentary collapse)-has arisen out of an attempt to describe the behavior of matter based on mathematical laws of motion and force laws, in abstraction from the more philosophical claim that it is not a real possibility that matter occupy a mere point. The value of making such an abstraction rests not so much on its leading to a correct solution to the mathematical problem, as on its making us more sensitive to the idea that the behavior of matter depends crucially on its momentum, and thereby on its inertial tendencies, not just on the forces it is subject to. By focusing on this, a certain way of generalizing the objection comes into the foreground. This more general and fundamental objection can be seen as follows. We have considered, for example, a spherical region of space homogeneously filled with matter endowed with attractive force alone, and we have gone on to ask how it will behave. However, before we can answer this, we need to specify something that may not have been in the foreground in the preceding discussion: we need to specify the initial velocities of the different parts of this sphere of matter. 22 We have tended to make the assumption that all the parts of this matter are initially at rest. But of course many alternative initial conditions are possible. The problem is that if all the parts of the original sphere of matter are to get to a given point at the same time, this would depend on the exact coordination between the initial speeds and directions of the parts. Matter will collapse to a point
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only if certain very special initial conditions prevail; in other circumstances, the matter would never be found collected in the central point all at once. Let us assume that each part of the matter in the sphere gets to the center at some time or other. But let us say that no finite amount of it arrives there all at once. It just streams into and out of the center continuouslybit by bit. This is a case where the difference between the entrapment view and the cyclical view of collapse is important. If it is assumed that the bits of matter in the original sphere just sit there, trapped in the center once they arrive there, then the bits of matter that get to the center at any time could all be found there together at some later time. And then I
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inverse square attractive force, the closer son1e bit of n1atter is to the central axis, the greater the rate of revolution will have to be (just as in the case of the solar system), so the matter will not all rotate like a solid mass. But still, the tendency to collapse to a point will have been checked, and that is the important thing. The counterbalancing centrifugal tendency in this case is not a repulsive force; in fact, it is not any kind of a real force (a power to produce a change of state, in this case, a change in the state of motion), but is rather a manifestation of matter's inertia. And this possibility seems to undermine IZant's claim that the only way the collapse to a point can be prevented is by ascribing to matter a force of repulsion (assun1ing that it is already endowed with attraction).24 What needs to be noted about the alternative way of preventing collapse, however, is that it involves ascribing to the parts of matter a special set of initial n10tions, and this will be important in the discussion which follows. This most recent objection (objection 2b, charging IZant with overlooking an alternative way of avoiding collapse by appeal to a spinning motion) again brings out the fact that IZant seems to be excluding from consideration a factor that is essential in thinking about the effects of forces: bodies have inertial mass. And it is in virtue of this inertia that, when we determine how a body is going to behave at a given time, we must consider not only the forces it is subject to, but also its momentun1 at that instant. 25
3. Two Clarifications of these Objections and a New Worry First, I wish to note that the objections have in common a concern with the fact that matter has inertia. It tends to continue in its state of motion or rest unless it is acted on by a force. So it might seem tempting right away to defend IZant by pointing out that in Chapter 2 of the MF (the "Dynamics"), where IZant presents the balancing argun1ent, it is in fact IZant's intention to abstract from the fact that bodies have inertia. For he suggests that it is only in the next chapter of the MF, Chapter 3 (the "Mechanics"), that he will consider this further property of matter, viz., its inertia. But it is very difficult to spell out what we are proposing to do when we set aside the fact that matter has inertial mass. To be sure, IZant may be able to make this abstraction for certain limited purposes. And he can certainly defer explicit and systematic discussion of a topic that comes in only implicitly and unsystematically at an earlier stage. But if he is considering the behavior of matter over tin1e (in particular, if he is considering where it is headed and whether it will ever actually get there) under the influence of various forces, which is what he is doing in the "Dynamics"
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chapter, can he simply abstract from the fact that matter has inertial properties? Could he coherently treat n1atter as though it simply lacked these properties-as though it had an inertial mass of zero? If we were to defend him by saying that, in chapter 2 of the MF, lZant assumes that matter has no inertial mass, we would, it seems, be saddled with spatiotemporal discontinuities in the behavior of matter as it jumps in an instant from one location to another. These inertial features seem to be essential to characterizing precisely those elements in the description of matter which are relevant to its behavior in the cases he is considering. So I want to put this line of defense aside for the moment, although in the last section of the paper I will come back to it and endorse it in a special form. Secondly, one might suggest, in lZant's defense, that by positing a repulsive force in the second half of the balancing argument, he wants to ensure that no set of initial conditions will result in the matter's collapsing to a point, and thus (so he claims) in a state incompatible with matter's essence. That is, it's not that IZant wants to show that without an attractive force there must inevitably be a collapse to a point; rather (says IZant's defender), he wants to rule out the mere possibility of such a collapse. So the existence of initial conditions for which collapse does not occur, which is all that the above objections show, is beside the point. IZant's intention in positing a countervailing repulsive force is to rule out the possibility that any initial conditions will lead to collapse, or to put the point another way, to guarantee that no possible initial conditions will lead to collapse. However, this response cannot provide us with a successful defense of IZant's argument. For according to this response, repulsive force is posited because it is a sufficient condition for avoiding collapse. However, Kant's argument can only work ifit shows that repulsive force is a necessary condition, that is a condition, without which, collapse must occur. Only then would one be able to argue that, if matter is endowed with attractive force, it must have a repulsive force as well. But I have argued that a collapse that results from attractive force can be avoided even if one doesn't posit repulsive force. Perhaps I
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Maybe the positing of repulsive force in the second half of the balancing argument is a condition of a more complicated sort. Perhaps the positing of repulsive force is a (necessary) condition without which the collapse to a point cannot be ruled out. However this is extremely hard to evaluate and to state with sufficient precision. For in a sense, couldn't we say that there are other ways of ruling out collapse: perhaps, spinning motion plus an inverse square law of attraction? One is inclined to respond that we can't build a specification of the force law into the necessary condition itself. But it is unclear what motivates this restriction on what can and what cannot go into the content of the condition, and it is unclear what will motivate other such restrictions that one might need to appeal to, were we to consider a full range of alternatives. Now, this is tied to a further problem that I want to raise in this connection, and this is a point that can be made about both halves of the balancing argument. I have said that what we needed to show in the balancing argument is that positing a countervailing force is a necessary condition for avoiding the unlimited collapse/dispersion. However, this must also function as a sufficient condition of some sort as well. For it is not really enough to show that there must be a countervailing force because, without it, we've got a problem. In addition to satisfying this necessary condition, we need to know that adding the countervailing force makes a difference, i.e., that it ameliorates the problem. We need to know that, once we add repulsive force to check the effect of the attraction, or in the first half, once we add attractive force to check that of repulsion, the problem is resolved, or, at least, partly resolved. However, this makes the line of thought a lot less clear-cut. It makes it much harder to determine what we expected of the argument. There are a number of things that could count as successfully resolving the problem or partly resolving the problem. Consider the first half of the argument, where we argue to the need for attractive forces to check the effects of the assumed repulsion. One might think that the problem is counted as "resolved" once one adds attractive force, because in that case we are guaranteed that unlimited dispersion is not inevitable. There are some initial velocities for which dispersion will not occur. Yet, even after adding attractive force, some initial velocities still do lead to unlimited expansion. That is, if the parts of matter start out with sufficiently high speed directed away from the center, they n1ay have, so to speak, "escape velocity" and so these parts will spread apart, overcoming the attractive pull and dispersing without limit. And one might argue that unless we show that total dispersion is precluded, we cannot count the problem as "resolved." And if this stringent condition is our criterion for success, then the positing of a countervailing attractive force does not count as a "successful resolution." I am not saying that there is no way to
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determine which criterion is more appropriate in this context. All that I need to point out now is that a whole new layer of difficulties has arisen, and that we will need to ask ourselves whether we aren't making the argument more complicated than it should be, if properly interpreted. I propose that in order to understand Kant's argument, we think of him as assuming, from the outset, that only a restricted (perhaps unique) set of possible initial velocities are the relevant alternatives, at least, within the context in which the argument occurs. That is, my strategy will be, in effect, to propose restrictions on which sets of initial velocities can con1e up for consideration when we say things like "for all possible assignments of initial velocities," thereby allowing a more plausible case to be made for the "inevitability" of collapse absent a countervailing force. But this must not be an ad hoc limitation; the set of "possible" assignn1ents of initial velocities will have to be limited in a principled way, and I will be exploring various ways in which the n1otivation for the restrictions might be seen to flow from the context in which the balancing argument is presented. 26
4a. First Response on I(ant)s Behalf and
Further Criticism Early on in this paper, I gave an objection (objection 1) to lZant's first argument, the argument from the supposition of repulsive force to the need for attractive force as well. After that, I raised two objections (objections 2a and 2b) to Iunt's second argument, the argument that moved in the converse direction. The first (2a) concerned Kant's claim that matter endowed with attractive force alone would collapse to a point. I argued that we should not expect such a collapse except perhaps under very special circumstances. The second worry (2 b) was really a further development of the idea that such collapse seems avoidable even without an appeal to a repulsive force. In the third section, I added a further worry concerning what is to count as a "successful" resolution of the problems arising from unlimited expansion or contraction. I now wish to propose a first response to son1e of the worries I raised concerning Iunt's second argument, the attraction-presupposes-repulsion argument. I begin by bringing out a consideration which may be helpful in addressing the second objection (2b) I raised against it, and then I will extend it to some of the questions raised by the first (2a). In the kind of case associated with this second objection (2b), there seemed to be no obvious conceptual or physical impossibility in the idea that matter, endowed with attractive force alone, would be prevented from collapsing if its parts were all revolving around a central point. Why then 111ight Iunt, nevertheless, take this to fail as a counterexample to his view
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that matter, if it is endowed with attractive force, must be endowed with an opposing, repulsive force as well? I want to suggest that part of the answer may lie in the explanatory role that the concept of matter, according to Kant, is meant to play here, and in the special significance he ascribes to that role. Let us grant Kant the claim that if matter were to collapse to a point, it would (per impossibile) be in a state incompatible with one of its essential properties (filling a space). Let us grant him, further, that this shows that something which opposes the action of attractive force is needed for the "possibility" of matter. By insisting that it is a repulsive force that must oppose the attraction, what Kant is rejecting, in effect, is the idea that a motion of a certain sort, e.g., a spinning motion, be sufficient to ensure the "possibility" of matter (qua space-filling), even in the absence of a repulsive force. In asking why Kant rejects this alternative, it is worth stepping back to recall the most central themes of the MF"Dynamics" chapter. Kant's main purpose there is to defend and elaborate what he calls a "dynamical," as opposed to a "mechanical," conception of matter. Crucial to the characterization of the former is the appeal to moving force, i.e., a power to produce motion. Kant writes: The universal principle of the dynamics of material nature is this: all that is real in the objects of our external senses and is not merely a determination of space (place, extension, and figure) must be regarded as moving force [bewegende Kraft]. (4:523) I believe that we can shed light on our question by considering the reasons Kant gives for favoring this "dynamical" conception of matter. The contrast between dynamical and mechanical views, in the first instance, concerns two conceptions of impenetrability, i.e., of what it is to fill a space. The dynamical view (which Kant favors) explains space-filling in terms of a force of repulsion, in virtue of which matter opposes compression by exerting an outwardly directed force of some finite degree. By contrast, the mechanical view (which Kant rejects) explains space-filling as an all-or-none property, an "absolute" solidity, on account of which matter admits of no compression whatsoever, no matter how large a force is applied to it. 27 Kant appeals to a number of connected considerations in making out his case for a dynamical, rather than a mechanical, approach to "material nature." Many of them are presented at various stages in the "Dynamics" chapter, and most explicitly in the long "General Observation on Dynamics," which makes up the last third of that chapter. One such consideration that is relevant to our present question, though touched on several times in that section, is not systematically developed in the
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"Dynamics" chapter at all. However, it figures centrally in Kant's discussion of the mechanical-dynamical contrast in his Lectures on Metaphysics. It involves an idea that I(ant presents, in essentially the same form, at various stages in his development. In one version of the Lectures (dating from 1792-93); Kant puts it this way: The method of physics can be divided into the mechanical and the dynamic; the first derives all alterations of the world from the figure ofits basic particles; it assumes these as determined, indivisible-atoms [atomi], corpuscles [corpuscula], themselves still matter, but unsplittable. Descartes gave them certain figures (e.g., hooks in nitric acid). All motions are derived from others, which one already assumed. (28:664) Although much of this quotation concerns the mechanist's appeal to atoms of various shapes, I will want to focus on the point I
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The general properties of bodies are: impenetrability, connection, and shape. These general properties of bodies are the ground of all physical explanation of body. If something is explained by means of a con1municated motion, then that is the mechanical mode of explanation. (28:210)29 lZant immediately goes on to contrast this to the "dynam,ical mode of explanation." Here the reference to explanation "by means of a communicated motion" serves as the distinguishing feature of the mechanical approach. As in the lectures cited earlier, IZant here singles out Newton as the representative of the "dynamical mode," characterizing his originality thus: "Newton was the first one who suspended the mechanical mode of explanation and attempted to explain by physical powers" (28:210). And, here too, lZant makes clear that the positing of attraction is what he has chiefly in mind. With regard to one phenomenon, I(ant says: "Those who explain it physically assume basic powers; those who explain it mechanically assume an initial motion and a basic shape of a basic matter, and that is what Epicurus did" (28:210). (Sometimes IZant uses the word "physical" and its cognates as another way of referring to the "dynamical.") This contrast between the "mechanical" force of a body, which produces motion only by "communicating" it, and its "dynamical" forces, by means of which it can produce motion even when it is at rest, is of central importance for IZant, and it will come up again in the last section of this paper. IZant opposes the "mechanical mode of explanation" not because it has recourse to "mechanical" forces (which IZant himself appeals to in the positive view he presents in the "Mechanics" chapter of the ME), but because the mechanist repudiates any appeal to "dynamical" forces. Now, what is lZant's dissatisfaction with the mechanical approach? There are a nun1ber of them, as I said before. But one that comes out particularly clearly in the Lectures derives from the need, if one is to provide a mechanistic explanation, always to appeal to prior motions. This is how lZant puts the problem in a passage from the Lectures on Metaphysics dating from 1782-83: Here [in mechanical explanation] I explain not the origin, but rather the alteration of motion from motions already present. E.g., one explains how vinegar dissolves chalk mechanically this way: vinegar has in its atoms [atomis] sheer pointed particles, which look like spears. Now these drill through the chalk and thus it is dissolved. This power which drives the vinegar spears into the chalk comes from the warm outer air. But where does that come from? There we are stuck. (29:935 )30
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I(ant then says that the dynamical mode of explanation is "more satisfactory and complete" than the mechanical (29:935). Here one might think that IZant's problem with mechanical explanation is that one can keep asking, as he did above, "But where does that come from?" and thus, that mechanical explanation never comes to an end-and that this is why he prefers a dynamical physics. But in fact, IZant says this is true of both the mechanical and the dynamical approaches: Thus to explain something mechanically means to explain something according to the laws of motion, dynamically, from the powers of bodies. With either explanation one never comes to an end. (29:935-36) It may not be clear precisely what IZant has in mind when he says that dynamical explanation "never comes to an end." What is clear is that this is not a liability specific to mechanists. Thus, when Kant says in this passage that "[t]his [the dynamical mode of explanation] was first discovered by Newton, and is more satisfactory and complete than the former [the mechanical mode of explanation]" (29:935, my emphasis), he cannot simply have in mind that dynamical explanations are fully satisfactory or complete, while mechanical ones are not. Rather, the point is, at least in part, that the dynamical approach allows us to explain the origin of n1otion in matter. A more or less mechanical approach may contribute to understanding the communication of motion between parts of matter. But the dynamist's forces, acting between different parts of matter, allow us to understand how each part "without any motion of its own, even at rest, puts all others into motion." The dynamist's approach allows us to explain, as IZant attempted, e.g., in his writings on cosmogony (specifically, in the Universal Natural Histor:Jl), how a physical system whose parts are at rest can develop into one whose parts are in motion. The point here is that on the dynamical approach, the presence of properties of a certain sort, nan1ely n10tion 32-more specifically, the presence of a motion in the world or in (what we treat as) some causally isolated part of it-can be explained by appealing to properties of a wholly different and causally more fundamental sort, namely, the dynamist's moving forces (powers to produce motion). Thus on IZant's view, it is possible for the dynamical approach to provide explanations which are in this way "more satisfying and complete" than ones which are purely mechanical. 33 In light of this we may see more clearly why I(ant might not have been n10ved by an objection like 2b. He may take it that we undermine this central explanatory ambition and advantage of the dynamical approach if we make motion a "condition for the possibility of" matter-for example, when the tendency of matter to collapse is counterbalanced by a spinning n1otion rather than by an opposed repulsive force. 34
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The rejection of this alternative way of counterbalancing the attractive force of matter fits well with the fact that we would usually regard being in motion as an accidental property of matter. That is, we usually take it that the possibility of matter does not require that it be in motion-matter may be endowed with motion, but equally well it n1ay be at rest. This is an assumption shared by those who tal(e the mechanistic approach, at least as I(ant understood it. The specification of certain causal powers or forces, however, was intended as a central component in characterizing a thing's essential properties. (Certainly this was IZant's view of the matter.) Thus, the explanatory project that is open for the dynamist but not for the mechanist-the project of explaining the origin of motion-is one of accounting for the existence of properties that both approaches assume to be accidental in terms of properties that are, at least in lZant's view, essential. However, it seems to me that, in explaining the origin of motion, it is misleading to say that we are simply explaining an accidental property (motion) in tern1S of essential properties (the dynamist's forces). Rather we are explaining an accidental property (motion) in terms of (i) essential properties (the dynamist's forces) and (ii) an accidental property (rest). I think that what this shows is that I(ant is not treating an appeal to a state of motion (in the center of mass frame) and an appeal to a state of rest (in that frame) as on a par within explanation. This does seem intuitively quite natural and it is in no way peculiar to lZant. (One sees it in various forms in Locke, for exan1ple; in fact it was extremely widespread. However, in many of the most striking developn1ents of theoretical mechanics in the eighteenth century, this special way of treating rest vis-a-vis motion played an ever smaller role.) Yet it is unlikely that lZant could have been satisfied simply by the fact that it is intuitively appealing. I take it that I(ant would have felt that the asyn1metry between motion and rest would need to be explained in more fundamental terms by placing great weight on the contrast between positive properties ("realities") on the one hand and merely privative properties ("negations") on the other. Yet it is hard to see how he could have done so without drawing on commitments to traditional n1etaphysical doctrines that would strike us as wholly unjustifiable. I have focused on the idea that, on lZant's view, the dynamist, but not the n1echanist, has the resources to explain the origin of motion, to explain how matter at rest can develop into matter. in motion, and I have said that this explanatory ambition can help us see why I(ant might have been loath to appeal to a spinning motion to prevent matter from collapsing to a point assuming it is endowed with attractive force alone. That was the second (2b) of the two objections I raised for the argument from attraction to repulsion. I believe that the same set of considerations help us to see how lZant might have thought about the problem described at the end of section 3, and (at least in part) how he might have thought about the first objection
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(2a) to the attraction-entails-repulsion argument. One point that turned out to be central to these worries was the fact that the behavior of a piece of matter will depend on the initial velocities assigned to its different parts. For reasons of the sort presented above, however, I(ant might believe that we need not consider all possible initial conditions, but can instead restrict ourselves to the cases in which the parts are all initially at rest, at least when we are determining what does and what does not belong to the essence of matter. As far as section 3 is concerned, this would have the effect of eliminating differences between the different "criteria of successful resolution" we were supposed to choose between. In objection 2a, we saw that not every assignment of initial velocities should be expected to lead to collapse to a point. However, perhaps I(ant assumes that the parts of matter will collapse to a central point if they are all initially at rest. (After all, many of the most intuitively compelling counterexamples to the claim that matter will collapse to a point involve an assignment of initial motions.) Now, given what has been said about the character of the dynan1ical conception of 111atter and the advantages I(ant attributes to it vis-a-vis the mechanical conception, we can see why I(ant would be concerned to account for the behavior of matter when we restrict consideration to cases with precisely those initial conditions, viz., perfect rest. For, if the point of a dynamical conception of matter is to be able to account for the origin of motion in a physical system, the state in which all parts of the n1atter are at rest will be precisely the initial condition that we will want to be focusing on. As I indicated in section 2, I suspect that lZant has not actually succeeded in coming to terms with the difficulties in characterizing the behavior of matter endowed with attractive force alone. But in light ofI
4b. Second Response on I(ant)s Behalfand
Further Criticism There is another point to be made which might help us to see why I
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lation concerning the larger picture of physical explanation that lZant may have had in mind. Preliminary to presenting this point, I need to distinguish two considerations that could motivate the argument that if matter is endovved with one of the two forces, it must also be endowed with the other. There is the consideration that I have been focusing on: the idea that without the countervailing force, the matter will undergo unlimited dispersion or unlimited collapse. That this n1ust not occur I will call the "limitation requirement." This is the consideration that appears explicitly in the balancing argument as actually presented in the MF. However, the other consideration, which I(ant seems to have had in mind many passages, is as follows: that without a countervailing force matter will not have a stable or fixed degree of density or space-filling-this being possible only when a tendency to expand is exactly balanced by a tendency to collapse in such a way that expansion is countered by a net inward force and contraction is countered by a net outward force. I will call this the "stability requiren1ent." But given the way I(ant switches between these two considerations, it is not always clear that he distinguished them, and he may have taken them to amount to the same thing. And in fact they are very closely related to one another if we help ourselves to some intuitively plausible background assumptions. lZant may have thought that the only way of satisfying the limitation requirement (i.e., by positing a countervailing force) would in fact satisfy the stability requirement as well. 36 However, there are advantages to focusing on the satisfaction of the stability requirement. For the stability requirement, if defensible, might provide us with a much more direct defense of the claim that attractive force must be balanced by repulsive force, and repulsive by attractive. That is, it might provide us a clearer route that avoids many of the objections we have been discussing, including that raised early on (objection 1) concerning the first half of the balancing argument, i.e., the argument from the supposition that matter is endowed with a repulsive force to the need for an attractive force as well. 37 Yet the question which unavoidably arises is this: why should we think that the stability requiren1ent must be satisfied? As regards the limitation requirement (that is, the requirement that matter not collapse to a point or be dispersed to infinity), lZant connected this to issues concerning the compatibility of a state with the conditions on the existence of n1atter, which will then be closely tied to questions concerning something's essence. But what are we to say about the stability requirement? Why would lZant assume that some version of the stability requirement must be satisfied? I do not believe that we can go very far in answering this question, but as I said earlier, we can perhaps discern the outlines of a picture of physical explanation that I(ant may be operating with, a picture on which the stability requirement n1ight seem a natural one.
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To begin with, I want to discuss two ways of defending the stability requirement that might initially look promising, but seem to n1e to be difficult to justify. They concern what I
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Certain sorts (species) of matter cannot be transformed into certain other sorts; the atomists had a way of understanding how this could be so. And in lectures from 1792-93, lZant again makes the same point: "Corpuscular philosophy arose in order to make the perdurability of the species [specierumJ conceivable" (28:664). The important point is that we can think of matter as made up of stable parts of different sorts. Only then do we go on to think of these parts as interacting by communication of n10tion. In that sense, the stability of the parts is what affords them a kind of explanatory priority. Changes are explained in terms of the rearrangements of smaller parts that do not themselves change. Stability is required not by the bare requirement that a property have a determinate value, but by the requirement-presupposed by this explanatory approach-that there be a determinate value the property has throughout a period of time (a period of time during which a change is occurring). One obvious problem in seeing the relevance of these reflections on stability for I(ant's physics is that he did not believe that matter was made of ultimate parts having various shapes. In fact, in the MF he argues that matter is "divisible to infinity," and he makes a point of saying that this is not just mathematical divisibility, but the physical separability of its parts, i.e., physical divisibility (4:503-504). So, for lZant, the "perdurability of species" could not be due to the stable shape of physically indivisible constituents. However, an essential physical property of matter that can be coherently ascribed to matter, even as we consider ever smaller parts without limit, is its degree of density or space-filling. And in the long "General Observation on Dynamics," with which I(ant ends the second chapter of the MF, he suggests that the variety of species or sorts of matter, which the mechanists explain in terms of the various shapes of their constituent particles, can be explained (in a limited sense) by the dynamist in terms of the various degrees of space-filling resulting from different balances of attractive and repulsive forces intrinsic to different bits of matter. In this way, the stability of the degree of density or space-filling could play the same role in the dynamical conception of matter as the stability of shape played for the mechanists. So this central idea about the stability of species in mechanistic explanation is one that I(ant preserves even in his dynamical conception of physics, which is in many other respects so different from mechanism. Moreover, though applied to problems about species in the passages cited above, this basic idea is clearly of a much more general character. I(ant's idea of a fundamental science of matter is one in which we can first characterize stable parts (in a way that is independent of their size and shape), and can then go on to characterize a range of situations in which they are set in relative motion and in this way interact with one another. I will make a brief comn1ent on the conception of interaction at work here, and then
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a brief comment on the concept of stability in order to make clearer the framework that lunt has retained from the mechanical philosophy, broadly construed, and how he has transformed it. Both Kant and the mechanists think of these interactions in terms of the communication of motion among parts (e.g., in collisions, though IZant himself is not limiting himself to cases in which collision is involved, for he wants to include interactions by means of attractive force as well). And this communication of motion is conceived in terms of special forces the parts have just insofar as they are in motion. (Earlier in the paper I characterized them as "mechanical" moving forces.) What is different about Iunt's physics as compared to the mechanists' is that he has embedded this mechanistic picture of interaction within a dynamical framework of moving forces (like attraction and repulsion) which always mediate these interactions and which matter exercises independent of its state of motion and thus even when at rest. Moreover, both Iunt and the mechanists emphasize the idea that motions (or perhaps relative motions) and their consequences, e.g., rearrangements of parts, are fundamental to all of the changes to which objects in space are subject. And so the stability of a part is conceived in terms of stable relations among its own parts, and that entails a stable absence of relative motion among them. For mechanists, it is atomism that guarantees stability in this sense. (The "parts" of an atom are not in relative motion.) For lunt, as I said earlier, the stability is conceived in the first instance as stability in the degree of density resulting from a stable balance of attractive and repulsive force. One might ask what this stable degree of "space-filling" or balance of force has to do with stable relations among parts. But what must be kept in mind is that the whole burden of the earlier propositions of the "Dynamics" chapter of the MFis to show how the degree of the force associated with space-filling is to be understood precisely in terms of the relations among parts of matter-the closer the parts come together, the more intense the force (a bit like a spring or an elastic fluid). Thus, here too, stability amounts to the stable absence of relative motion an10ng parts. But again we have to remember that Kant, unlike the mechanist, views matter as infinitely divisible. There will be no particular level of analysis into parts where perfect stability will be assumed to exist. For IZant, it would probably be sufficient to assume only that we approach perfect stability in the limit, as our analysis becomes more and more fine-grained. I have been spelling out my suggestion that the stability requirement comes out of a general picture of physical explanation that lZant shares with the mechanists. But of course there are many other more specific features of the mechanistic approach that Kant rejects. And because the mechanist, on account of those features, is Kant's main opponent in the "Dynamics"
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chapter of the MF, the comInon ground will tend to recede to the background. But it is important that it be brought forward and made explicit. For it allows us to see that a commitment to this two-tiered explanatory schema of building-block and interaction (even when we relativize it to accommodate infinite divisibility) is substantive, controversial, and (I believe) ultimately dispensable, once we think of the schema more determinately as requiring that the building-blocks be, in the relevant sense, stable. That doesn)t mean that as far as the building-blocks are concerned, we should instead say that anything goes. It just means that whatever regularity, or conservation, or ll1inimal stability conditions actually are required, there is no reason to think that they will support a stability requirement strong enough to underwrite the balancing argument. Moreover, IZant's primary concern in the MF is not with the possibility of species of matter, but rather with the possibility of matter per se. It is significant that IZant only discusses the possibility of species of matter in a "General Observation," not in presenting the major line of argument in the "Dynamics" chapter. And IZant's major point in those discussions about species is that the dynamical conception of matter cannot by itself account for the possibility of species. The dynamical conception cannot assure the "real possibility" of species. At most, this conception permits us to assert the "logical possibility"38 of the schema on which different species of matter correspond to different balances of attractive and repulsive force. In this respect, IZant concedes, the dynamical approach does significantly less well than the mechanistic approach (though, for IZant, the latter has disadvantages which are far ill.ore damning).
4c. Third Response on !(ant)s Behalf and
Further Criticism There is a further response we might make on IZant's behalf. It is, like the others, limited in its success. But ultimately, I believe that this third response gives the most plausible account of the presuppositions that are at work in the relevant texts, and it incorporates some of the central ideas that came out in the previous two responses. It draws on a distinction lZant makes between the "Dynamics" chapter and the "Mechanics" chapter of the MF. 39 The contrast that I have in mind is suggested in a passage early in the "Mechanics" chapter of the MF which has tended to receive little attention: one might far more appropriately call those forces with which matter (even if one entirely abstracts from its proper motion or even its effort to move itself) acts on other matters dead forces, and hence call the
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original forces of dynamics dead forces. And one might, on the other hand, call all mechanical moving forces, i.e., by means of motion proper, living forces. (4:539, my emphasis) Here I(ant returns to a topic that had interested him very early in his career,40 namely, the contrast between living force and dead force. In this passage, lZant is associating (perhaps with qualifications41 ) the conception of force presented in the "Dynamics" chapter with the idea of dead force, and I believe that this gives us further clues for understanding the point of view lZant intends to adopt in the "Dynamics" chapter, where the balancing argument is presented. The contrast between living and dead forces has its locus classicus in an essay Leibniz published in 1690, the "Specimen Dynamicum." This contrast is in certain ways obscure, and as it evolved we find differences in detail and emphasis from one writer to another but, roughly, the contrast amounts to the following. "Living forces" are those forces a body has which are in a special way associated with its being in motion. Matter, insofar as it has a certain bulk and a certain speed, can be regarded as the cause of a range of characteristic effects: it can bring about a state of motion in a body previously at rest, e.g., by colliding with it; it can rise against the resistance of gravity; it can depress a piece of putty when it hits it. All of these phenomena are regarded as manifestations of living force. 42 Dead forces, by contrast, are forces of push and pull,43 mere "endeavors" or "strivings" toward motion, and can be exerted by a body independent of whether or not it is in motion, and can thus be exerted even when at rest. Leibniz's examples of dead force include weight, elasticity, and centrifugal force. 44 A further feature of this account is that dead force and living force are causally connected. The continuous application of a dead force over a finite period of time results in real finite motion and therefore in living force. Now, this way of formulating the distinction and the connection between living and dead force is clearly discernible in lZant's chief formulation, located at the beginning of the "Mechanics" chapter, of the contrast between the forces treated in that chapter and those treated previously in the "Dynamics" chapter. The contrast between these "dynamical" and "mechanical" forces (which was discussed in section 4a) is presented in the first part of the quotation [1]; the connection between then1, in the second [2]: [ 1] The merely dynamical concept [of matter] could also regard matter as at rest. The moving force that was then [namely, in the "Dynamics" chapter] taken into consideration concerned n1erely the filling of a certain space, without one's being permitted to regard the matter that filled the space as being itself n10ved.... On the other
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hand, in mechanics, the force of a matter set in motion is regarded as a force for communicating its motion to others. [2J But it is clear that the n10veable would have no moving force through its motion if it did not possess original moving forces whereby it is active in every place where it exists before all proper motion. And it is clear that no equivalent motion would be impressed on another matter by a matter whose Inotion lay in the path of a straight line in front of this other matter, unless both possessed original laws of repulsion . . . . Hence all mechanical laws presuppose dynamical ones; and a matter as moved can have no moving force except by means of its repulsion or attraction, upon which and with which it acts directly in its motion and thereby communicates its own motion to another matter. (4:536-37)45 Now, Leibniz, in his characterization of dead force, adds that dead force correctly describes the functioning of machines, and that this had already been done in antiquity. "So far as one can establish," says Leibniz in the "Specimen Dynamicum," the ancients had knowledge only of dead force and this is what is commonly called mechanics, which deals with the lever, the inclined plane (where the account of the wedge and the screw belong), the equilibrium of bodies and the like. There we treat only the first conatus of bodies before the bodies have received impetus through acting. (Leibniz 1989,122) ("First conatus" should be understood as a striving or endeavor toward motion, while "impetus" is an actual, finite motion.) Two sides of a lever in equilibrium, for example, are characterized in terms of two applied dead forces (weights) which stand in the appropriate ratio. This can be regarded as typical of the kind of case we can characterize within the science of statics. 46 When we want to go beyond the mere endeavor toward motion, by contrast, we need the modern conception of how to treat bodies in motion, and for Leibniz, that will mean that we must appeal to living force. In light of this, we might be tempted to associate dead force, and therefore the forces of the "Dynamics" chapter, exclusively with those situations in which there is no actual motion. And in fact IZant seems to say things which would encourage such an association. In the passage just quoted from beginning of the "Mechanics" chapter, IZant writes, "The merely dynamical concept could also regard matter as at rest. The moving force that was then [namely, in the "Dynamics" chapterJ taken into consideration concerned merely the filling of a certain space, without one)s being permitted to regard the matter that filled the space as being itself
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moved" (4:536, my emphasis). Now, to be sure, the "Mechanics" chapter deals with forces matter has insofar as it is in motion, but that falls short of not being "permitted," in the "Dynamics" chapter, "to regard the matter that filled the space as being itself moved." Moreover, this prohibition is hard to square with what that chapter actually says and with the exposition of the central concepts there, because discussion of motion, in fact, pervades the "Dynamics" chapter. For Kant begins his proof of Proposition I (of the "Dynamics" chapter) with the words, "Penetration into a space (the initial moment ofwhich is called the endeavor to penetrate) is a motion. The resistance to motion is the reason why motion diminishes or even changes into rest" (4:497). Now, it might be thought that such motion is restricted to the intruding bodies, rather than the resisting body, which is chiefly under consideration, and that the ban on regarding matter "as being itself moved" was only meant to apply to the latter (namely, the resisting body). But this way of squaring the ban with the appeals to motion seen in lZant's text will not work for the line of thought he is then about to pursue. For, in Proposition 2, Kant applies what he had established and shifts the discussion to the reciprocal relations of repulsion among the parts of a body as a way of explaining the cause of the impenetrability (here understood as incompressibility) essential to matter. One cannot simply say at this stage that the resisting part is not itself regarded as moved. Compression is the mutual approach of the parts. Moreover, on the same grounds (mutual repulsion) that we understand the essential impenetrability of n1atter, we must also, according to IZant, think of matter as having a tendency toward expansion. And this constitutes the starting point for the balancing argument. 47 lZant is thinking of the parts as withdrawing from one another. It is no good saying at this point that lZant is talking only of tendencies toward motion, not actual motions. In order to evaluate IZant's argument, we need to know how, within the context of the "Dynamics" chapter, to think and to reason about these motions (mutual approach and withdrawal of parts of matter, con1pression and expansion), whether these are actual or are merely possible effects which must be considered for the purposes of the argument. If we are then told to think of this motion just as one thinks of n10tion in the "Mechanics" chapter (but without assuming that they are actual), then we are faced with precisely those objections concerning inertial motion that I was concerned to emphasize in the earlier parts of this paper. It was by regarding the motion of matter described in the "Dynamics" chapter in this way that we generated those problems to begin with. In this paper, I have mostly focused on the examples of expansion and compression presented in the balancing argument. However, some light is shed on the conception of motion at work in the "Dynamics" chapter ifwe exan1ine with a fresh eye the n10re concrete descriptio~_~f2Aell_011!~Xl'!Jj.k~__
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compression given by IZant in other parts of the chapter, temporarily putting the worries about the balancing argument to the side: There can be found beyond every extensive force a greater moving force which can work against the former and would thus diminish the space that the extensive force is striving to expand ... for every matter there must be able to be found a compressive force capable of driving this matter fron1 every space it fills into a smaller space. (4:500) When the piston of an air pump's barrel that is filled with air is driven continually closer to the bottom, the air matter is compressed. (4:500) [F]or any given extensive force of matter there can be found a greater compressive force that drives this n1atter into a smaller space. (4:501) It seems to me clear, clearer than in the presentation of the balancing argument itself, that these forces are not being conceived as causes of acceleration or increases of momentum proportional to the magnitude of the force, but rather as causes of change of volume, change of configuration, or change in the distance between parts. There is no role played, for example, by the thought that, because of the impulse (momentum) imparted to the air-mass in being compressed, that mass will continue to decrease in volume (overshooting, in other words) even after the compressive and expansive forces reach equilibrium. Also, we should note how I(ant characterizes attractive and repulsive force a bit earlier in the "Dynamics" chapter: Attractive force is that moving force whereby a matter can be the cause of the approach of other matter to itself (or, equivalently, whereby it resists the withdrawal of other matter from itself). Repulsive force is that whereby a matter can be the cause of making other matters withdraw from itself (or, equivalently, whereby it resists the approach of other matter to itself). The latter we shall also sometimes call driving force, and the former, drawing. (4:498) Again, these forces are being characterized as those whereby matter causes or resists a motion, i.e., a change of spatial configuration (approach or withdrawal), rather than as a cause of an increase or decrease in momentum, the way it would be according to a Newtonian conception of force. Consider the compression of the piston in more detail. What we have is a case in which one force (that of compression) overcomes another force (the expansive tendency of the air in the cylinder), not a case of a force overcoming the inertia of the body48 it acts upon (leading to an increase or decrease of its momentum). The compressive force overcomes the opposing expansive force, and the piston is driven in, with a consequent
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decrease in the volume of the gas. 49 Such situations are not treated as cases where a net inward force determines the inward acceleration of the air in the piston, in accordance with a Newtonian F=ma type of law. Rather, it is conceived the way machines are generally conceived, when, for example, you pull the rope on one side of a pulley-system to raise a weight at the other end. This is not, in the first instance, conceived as a case in which the force exerted causes a continuing acceleration; it is a case in which we merely think of the force as the cause of a continuing displacement (Le., a change of position). Of course all of this can in fact be accommodated within a Newtonian conception, once we bring in the idea that there is frictional resistance to motion (e.g., in the pulleys themselves). What I want to note is that this is not the way I(ant seems to be treating it in the passages of the "Dynamics" chapter we've been examining. And this is not surprising, since IZant does not think that friction should be treated at the very fundamental level of theorizing about matter, i.e., the level on which the Metaphysical Foundations is supposed to be operating. IZant (like us) would consider friction to be a kind of derived force, a higher level phenomenon, rather than one which is fundamental. And yet he appears to consider phenomena in which a force is regarded as a cause of an approach or withdrawal, rather than as a cause of an acceleration, as having some important place in an analysis of the foundations of science. In IZant's examples in which a fluid (like air in a piston) is reduced in volume, the fluid is not conceived as acquiring some momentum that would keep it (the fluid) moving inward even after the motion of the piston itself (under the influence of the compressive force) came to an end. This inward motion is conceived as lasting only as long as the compressing force is applied. However, friction is not being brought in as a fundamental component of the picture. We are instead thinking of force and the effect it produces in a special way. For we are thinking of matter as if it had no inertial mass, and the notion of force we are bringing in allows us to consider the behavior of matter under this assumption. Of course the assumption is false, as I(ant knows. 5o But that doesn't mean this doesn't constitute a legitimate idealization for the purposes of analysis, with the inertial mass to be introduced only at a later point, namely, in the "Mechanics" chapter. 51 It might be objected that if a massless object is subject to a force, then, in the absence of any countervailing frictional forces, it will immediately attain an infinitely large speed. Because I(ant does not consider this kind of question, we can only speculate about the kind of answer he would give to it. I only want to point out that if we limit ourselves to cases in which the two forces are just barely out of balance, e.g., where the con1pressive force exceeds the expansive force by a vanishingly small degree, this problem is
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not forced on us. For we can in1agine a case in which a massless object is subject to a finitely large net applied force (resulting from the difference between two opposed applied forces), exactly balanced by frictional resistance at a finite speed v. And then we can imagine decreasing the net applied force as well as the frictional force, but in such a way as to keep the speed v the same. Continuing this sequence to the limit of vanishingly small net applied and vanishingly small frictional force, we can see that we need not be forced to build infinite speeds into the picture. The idealization is intelligible even if it can't be realized with the actual massy materials we have to work with. Apart from this last more speculative point, the view that I have been advocating about how force and motion are treated in the "Dynamics" chapter receives strong confirmation from IZant's earliest pre-critical work, The True Estimation ofLiving Forces,S2 in which the contrast between dead force and living force is the focus of discussion. There he sets out what he calls a "twofold classification of motion"S3 (§15; 1:28, my translation s4 ): Therefore I classify all movement into two main kinds. The one has the property that it maintains itself in the body to which it is communicated, and continues to infinity if no obstacle stands in the way. The other is a constant effect of a continuously driving force, an effect which needs no obstacle in order to destroy it, but which instead depends on the external force and disappears as soon as this stops maintaining it. Bullets that have been fired and all bodies that have been thrown are examples of the first kind; examples of the second kind are the motion of a ball that is gently pushed forward by the hand, or otherwise any bodies that are conveyed or pulled with a moderate speed. The former motion, says IZant, is associated with living force, the latter with dead force. (Cf. §§16-17.) In terms of the framework that I have set up, we can say that the former is the motion of a body with inertial mass, a motion which will, for that reason, continue even after the force has been removed; the latter is the motion of a body conceived as lacking inertial properties, and as being in motion only as long as the force is applied. The idea of motion at work in the "Dynamics" chapter is motion of the second kind. More specifically, IZant is concerned with cases of approach and of withdrawal, and he is conceiving of them as the effects of a force, which can be intelligibly discussed even without bringing in the concept of inertia. Statics is one such situation, but the point here is that what we know from statics can be applied even in a situation in which there is motion, as long as we think of it as being n10tion of the second kind. It is by not yet bringing in considerations of inertia that the concept of dead force is applied, not just to static situations, but to ones in which there is motion
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as well, such as those situations which are considered in the "Dynamics" chapter. And when IZant says at the beginning of the "Mechanics" chapter that we had previously not been "permitted" to consider bodies as moved, he is, I believe, restricting the ban to what, in the pre-critical work we've just looked at, he calls "motion of the first kind. "55 Force, conceived as the cause of what Kant called "motion of the second kind," presupposes abstraction from the inertial properties of matter, and it is not regarded as something which overcomes inertial resistance and thereby brings about changes of momentum. This is not Newton's conception of force; rather, it is much more pre-Newtonian in character. 56 But neither is the notion of moving force officially introduced in the "Mechanics" chapter to be identified with the Newtonian conception. In that chapter of the MF, which he begins with the words "Matter is the movable insofar as it is something having a n10ving force" (4:536), he has in mind a force (which I earlier called "mechanical force") that a body can exert on others insofar as it is in motion (e.g., by smashing into them), and it is more closely analogous to a Leibnizian conception of living force (though with the appropriate caveats, mentioned earlier), than it is to force on the Newtonian conception. But this certainly does not mean that Kant rejects a Newtonian conception offorce or excludes it from his system. For although it is not the conception offorce at work in the "Dynamics" chapter itself, and it is not the notion of a "mechanistic" force that is the central focus of the "Mechanics" chapter, it does have a place in the MF, and this place also is in the "Mechanics" chapter. It corresponds to the role played by the dead forces when inertia is added to the picture. The same push and pull that was responsible for withdrawal and approach in the "Dynamics" chapter is also regarded, as I mentioned earlier, as causally connected with living force. For within a context which assumes that matter has inertial properties, we can think of the continuous application of a dead force over a finite period of time as resulting in motion (of the first sort, rather than the second), and therefore, as resulting in living force. The point I have been malung is that it is important not to read this step back into the conception of force at work in the "Dynamics" chapter, and that the argun1ents and lines of thought in that chapter cannot be properly understood or assessed if we do. 57 I want to make two further points about motion of the second sort, motion which is conceived as the effect of dead force in an inertia-less world. The first point is that this lund of analysis of a physical system does not tell us how quicldy the resulting change of location or configuration will be. It may tell us the direction of motion and the path taken (assuming the force at any given location is not varying with time). Of course, when, for example, we analyze a complicated system of pulleys, we may find out about the relation between the speed of a body on one end and
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the speed of that at the other. However, what is left totally indeterminate is the actual magnitude of the speeds themselves. In a more general form, this kind of explanatory approach is very familiar, and in fact it is seen all the time in science. Many of the changes in certain physical systems 58 can be adequately explained simply by reference to a decrease in potential energy that these changes bring about. And chemical changes, for example, are very often explained as transitions from high energy59 states to lower energy states. And these explanations are able to function, without requiring that we consider the "kinetics" of the reaction; we need make no appeal whatsoever to the rate at which the transition occurs. Such explanations also tell us nothing about the time it takes for the process to reach completion. And this brings me to the second point I wanted to make. In the case of motions of the second sort, the processes of approach or of withdrawal come to an end when the force is no longer applied. 6o In machines, e.g., pulley systems, this may depend on the structure of the machine. Or it may be left undetermined by the character of the machine itself; it may be up to whoever is doing the pulling, or maybe it just depends on when we run out of rope. But in addition to these machine examples, we also have available to us some very different ways of describing physical situations in terms of the dead forces that are in play. Most prominent in the "Dynamics" chapter are descriptions of the degree of dead force as a function of the position of the piston or as a function of distance from a given body. In these cases, too, where what we know is not the form of some machine, but rather a force curve, that is, where we know how the force varies as a function of spatial location or configuration, we may at least be able to say what the intermediate and final locations or configurations will be, even if we cannot say how long it takes to get there. We may, so to speak, know the shape of the path and where it ends. It can terminate at a place where the force at that location acts to prevent the body's further progression. We find the path and we locate the potential terminating points by analyzing the function correlating dead force with spatial location, and it might well seem that time considerations (other than those just dependent on the order in which the points on the path are traversed) can drop out of the analysis altogether. In the specific kind of case we are concerned with in the balancing argument, cases in which we are focusing on changes of spatial configuration (collapse or dispersion) under the influence of various forces, we are meant to abstract from the fact that a system under the influence of a certain force may gain momentum with time. We think of the presence of attractive force simply as a tendency that makes the parts of matter assume a new spatial configuration, i.e., a more collapsed state, not as a tendency to make the parts assume a new velocity, a tendency to increase the momentum these parts have in a certain direction. My suggestion is that
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this way of thinking about forces and their effects is the one Iunt thought appropriate in the "Dynamics" chapter of the MF, where we are meant in some way to abstract from matter's having inertial motion-from motion of the first kind, that is. If this is so, how well would it handle the objections we focused on earlier in the paper? This kind of analysis seems to do very well in the case in which a piece of matter endowed with attractive force alone is said to collapse to a point, in that it allows us to see why Kant might have ignored the objections I have raised against his treatment of it. The attractive force is regarded as the cause of the mutual approach of the parts, which comes to an end as soon as the parts get to the center. Because motion is treated as motion of the second kind, we are not to tal<e into account the inertial n10mentum a part acquires as it moves toward the center, and we are not to take into account an initial momentum that the part might have started with. I also want to note that the extra level of difficulties (discussed in section 3) which are involved in assessing what is supposed to be accomplished once we add a countervailing force just drops out of the picture. This holds for both halves of the balancing argument. For example, we were worrying about the question of whether the addition of a countervailing attraction in the first argument was meant to make total dispersion avoidable (i.e., for son1e initial velocities it won't happen) or whether it was meant to preclude even the possibility of such dispersion (Le., for all initial velocities it won't happen). But, on the kind of analysis that is at work in the "Dynamics" chapter, the initial velocities do not make the kind of difference they do on a Newtonian analysis. Because differences in the initial velocities make no difference to the later behavior of the system, there really won't be a difference between the weal<er and the stronger criteria for a successful resolution of the problem. From the point of view of the "Dynamics" chapter, the attractive force simply blocks the repulsive, and the idea that, with sufficient outward velocity, matter could still disperse without limit cannot come up until inertia is brought into the picture. I said earlier that this way of thinking about the "Dynamics" chapter allows us to abstract from the question of how quickly a process goes forward and how long it talces the process to finish, and so this approach might seem initially promising as a way of dealing with the first objection to the case in which matter endowed with repulsive force alone is said to expand without limit. For that objection focused on the time it would take matter to disperse. Nevertheless I think that it does not in fact deal successfully with this objection. For the explanatory approach I have just been describing is not without limits. And even if we are meant to abstract from the rate of change and from the time the transition takes, that does not mean we can abstract from whether it can be completed at all within a finite time. In abstracting from the rate at which matter disperses and fron1
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hovv long it takes to do so, it still seems to me misguided to take that to mean that we are to abstract from whether a certain extreme state of dispersion ever occurs. And even from the point of view of the "Dynan1ics" chapter, where force is considered as a cause of change of relative location rather than of a change of momentum, the state of total dispersion can never actually be reached. And, once we have recognized the point of view IZant is taking in the "Dynamics" chapter, ,ve can see that the problem in the first objection need not have been based on considerations of inertia in the first place,61 and probably a completely different tack should be taken with this objection. The present response on lZant's behalf can be seen as, to a certain extent, superseding those presented in sections 4a and 4 b. From the first of these responses it preserves the idea that we are to think of the matter as at rest prior to the application of force. From the second it puts aside the particular focus on the stability of species, but preserves the more general idea of a two-stage approach to the science of matter. I hope that what I have said will help to us to see why the kinds of objections I raised, which concern inertial motion, would not, from IZant's perspective, have posed themselves. They arose because we assumed that the picture of force and motion at work in the "Dynamics" chapter was basically Newtonian in character. The picture that is actually at work there is pre-Newtonian and the inertial considerations, central to the conception of force as causing a change in motion, have not yet been brought into play. However, even granting that this gives us some insight into how I(ant might have meant the "Dynamics" chapter to be understood, there is a more general problem associated with defending I(ant as I have just been doing. I have been appealing to a special way of interpreting what it is we are doing when, in the "Dynamics" chapter, we are meant to consider matter apart from its inertial properties. Aside from the last of the objections I've just discussed, my worries have focused on the following questions: Are there ways of avoiding total collapse or total dispersion other than by positing a countervailing force? And has such collapse or dispersion really been successfully avoided even when we have a countervailing force? In either case, the problems are seen to arise only when we draw on considerations concerning inertial motion. The response on IZant's behalf was that, for example, he can still say that unless we posit the countervailing force, the state of total collapse or total dispersion cannot be avoided-if we are limited to the resources of dynamical analysis alone. But the more general problem is that we are not in fact liluited to dynamical resources alone. We can consider appealing to inertial momentum. It is only a certain way of stratifying physical explanation in such a way that dynamics is in some sufficiently strong sense prior to mechanics that leads us to think that we cannot. There is not necessarily
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a basis for insisting that problems of total collapse and total dispersion, assun1ing that they are real problen1s, must be solved solely on the limited basis of what I have been calling "dynamical resources," and that this be assured prior to availing oneself of the full resources of physical explanation. To insist on this seems a much n10re substantive application of lZant's architectonic ordering of dynamical (corresponding to quality) and mechanical (corresponding to relation) levels than seems warranted. lZant's architectonic structuring, which stratifies the levels of quality and relation, itself reflects a deeper commitment to the priority of the inner determinations (quantity and quality) of a thing, vis-Ii-vis its outer (relational) determinations,62 and to the idea that it is through its inner determinations that we can think of a thing as independent of others, i.e., as a substance. Kant seen1S to think that we must take care of certain problems (e.g., ensuring against unlin1ited collapse or dispersion) within the resources of the dynamicallevel before we can advance to the mechanical level. However, it is difficult to see how this way of working with the architectonic structure, and the stratification of substance and composite (of interrelated substances) which underlies it, might really turn out to be required of us Of, more specifically, to be required by the "conditions of possible experience." We have been focusing here on the necessary conditions of the possibility of avoiding unlimited collapse or dispersion of matter. Yet son1ething might be a necessary condition of the possibility of something when one's resources are limited in some way, and then turn out not to be a necessary condition when one can avail oneself of a fuller range of resources. We need to know which senses of "necessary condition of the possibility of" are relevant in considering what is and what is not essential to matter. !(ant thought that the sense this expression takes on when we limit ourselves to dynamical resources had a special significance for the concept of matter. We tend to find some of the architectonic structures lZant offers and the deeper grounds for them less compelling, and we not infrequently find that some of the so-called "necessary conditions," which seemed robust within those structures, do not survive when they are fished out of them.
NOTES 1. I am grateful to Randall Amano, Stephen Engstrom, Dina Emundts, Michael Friedman, Daniel Goroff, Peter McLaughlin, Seana Shiffrin, Eric Watkins, and audiences at the University of Potsdam, the University of Pittsburgh, and UCLA for extremely helpful discussions and comments. I thank the editors of this volume for their patience and their helpful comn1ents. Some of the work on this paper was done while I was a guest at the Max-Planck-Institut fUr Wissenschafts-
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geschichte, and I greatly appreciate their support. I also thank Martin Schonfeld for making available a draft of his translation of Kant's Gedanken von der lvahren Schiitzung del' lebendigen Krafte. Above all, lowe an especially great debt of gratitude to Hannah Ginsborg for discussion, advice, and support. 2. See especially Friedman (1992), Ch. 3 and Ch. 5, pp. 213-43. 3. See especially Warren (2001). 4. Unless otherwise noted, I have used James Ellington's translation (1985) of the MF, Norman Kemp Smith's translation (1986) of the Critique o/Pure Reason, Stanley L. Jaki's translation (1981) of the Universal Natural History, and Karl Ameriks and Steve Naragon's translation (1996) of the Lectures on Metaphysics. 5. Expansion of a piece of matter is the mutual withdrawal of its parts (or their centers). 6. Here I might add that, for Kant, if some matter is spatially separable from all the matter outside it, it constitutes a distinct material substance (Explication 4). 7. The first part of Proposition 2 states, "Matter fills its space by the repulsive force of all its parts, i.e., by its own force of extension ... " (4:499). And the idea here is that, because the parts of matter repel one another, it has a tendency to expand. 8. I take it that the first half of the balancing argument is meant to show (assuming merely that matter is endowed with repulsive force, on account of its essential impenetrability) that a force of attraction is part of the real essence of matter, and is thus absolutely universal and necessary to matter. It is appealed to again in proving tlle next proposition, Proposition 7, which states that this attraction is an immediate action at a distance, not reducible to some complex mechanism involving only action by contact. For in the proof of that proposition, Kant draws on the claim that "[t ]he original attractive force ... contains the ground of the very possibility of the contact of matter" (4:512). This is because the contact between two pieces of matter requires that each of them "fills a space in a determinate degree," and in accordance with the first half of the balancing argument, tllis in turn is only made possible by the possession of an attractive force. The role of the second half of the balancing argument (Proposition 6) in the succeeding arguments is harder to discern. It is, perhaps, important to Kant that in accordance with the second half of the balancing argument, the possession of attractive force is itself sufficient to guarantee that matter is made up of parts that repel one another, and therefore lie spatially outside one another. This will permit Kant to tllink that the "quantity of matter" is what he calls an "extensive" magnitude. The idea that possession of an attractive force entails possession of a repulsive force might also be relevant in drawing on the fOurtll ("Phenomenology") chapter of the MF, where Kant concerns himself wi th the special status of circular motion (4:556-57). For from a circular orbit, what is in the first place inferred is an attractive force. And in principle Kant might want to go on to infer that what is attracting is extended and impenetrable as well. The problem with these suggestions is that it is not clear that Kant actually does rely on such a line of thought in the MF. However that may be, it should also be remembered that the results oftllis second half are repeatedly used in one of the most common and fundamental kinds of mechanical explanation that Kant wants to underwrite and employ: that matter
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(e.g., air, ether/caloric) just insofar as it has weight, or just insofar as its parts attract one another, nlust also exert an outward pressure. Moreover, my intention in this paper is not so much to defend the two parts of the balancing argument, and thereby to save something important that follows. My concern is rather to examine the arguments themselves in order to reveal IZant's underlying assumptions about natural science and about mechanics (broadly conceived). 9. I note in passing that, if we can use "physical process" and "natural process" interchangeably in this context, the idea of something that can neither come into or out of existence "by nleans of any physical process" ties the characterization of the balancing argument to a traditional conception of substance. 10. Other ways of presenting this incompatibility, focusing instead on Kant's idea that we only know outer relations, are also possible. 11. I have in mind the "Axioms of Intuition" section of the first Critique. 12. This is meant only as a rough sketch and is primarily intended to locate within the balancing argument the central worries I will be concerned with in this paper. 13. It should be noted that the notion of substance has an extremely important role in the "Dynamics" chapter, as do the other categories of relation, causality and community. The notion of causality is evident throughout, insofar as that chapter is in general concerned with causal powers, and it is especially important in Explication 1 and in Proposition 1. Later in the "Dynamics" chapter, the idea of community among substances is perhaps most important in the discussions of action by contact and action at a distance in Explication 6, Proposition 7, Explication 7, and Proposition 8. One might have thought that consideration of the relational categories is restricted to the "Mechanics" chapter, judging from the kind of architectonic considerations Kant raises in the "Preface" to the MF. But the correspondence between the chapters of the MF and the table of categories is not as direct as that. Rather all three relational categories mal(e an appearance in the "Dynamics" chapter and then again in the "Mechanics" chapter. It is the way they are treated that distinguishes the two chapters. 14. At 4:514 (MF ch. 2, Prop. 7, Obs. 2), I
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an empirical concept is given in such a way that besides what lies in this concept, no other empirical principle is needed for cognizing things" (4:470). 17. I vvill use the expressions "body" and "piece of matter" interchangeably. 18. A further objection may be raised as well. It is not clear that at the limit of infinite time we would want to say, as Iunt does, that "there would be no matter at alL" For infinite dispersion, even if it nleans that we approach a density of zero, is compatible with conservation of the total quantity of matter. Matter could disperse without end over all of space, becoming infinitely thin in the limit, but the sum of all that matter, taken over all of space, could be the same as it was at the beginning. So even if we grant that, at the limit, the density is zero, it is not clear that we will also want to grant that "there would be no matter at all," for at the limit, the volume increases to infinity. However, in the rest of this paper I will not be drawing on this further objection. 19. Consider this from the point of view of classical mechanics. If the matter, after collapsing, just sits there, its velocity (in the center of mass frame) has become zero. One reason for thinking that this is not what classical mechanics would predict is that it is incompatible with the conservation of energy. For, using a later terminology, this process would involve what could be described as involving an anomalous drop in "kinetic energy." It would be anomalous because it would occur at a time when the conservation of total energy would have required a rise in kinetic energy. It should also be noted (employing, again, a somewhat later terminology and conceptual apparatus) that such a process would be hard to reconcile with the time-reversibility of the classical equations of motion. 20. I am certainly not taking this actually to constitute an argument that the motion must be cyclical. It is meant to show that there are prima facie plausible alternatives to the supposition that after collapsing to a point, the matter would just sit there at the center. It is instructive to compare the system I've just discussed to a much simpler system, which is similar in that attractive force increases to infinity as the nlatter approaches the center and the momentum will increase without limit as well: two equal point masses attracted to one another by an inverse square force. How will this system behave as the two masses approach one another? It would seem that whatever reasons Iunt might have for thinking that a homogeneous ball of matter must simply collapse would apply equally well to the case of these two particles. We might be able to see ways one is led astray in thinking about the homogeneous ball of matter by following out certain lines of thought as applied to the simpler system. Assume the two particles start out at a distance 2r from one another. Call this line initially connecting the two particles "the central axis." Now, what initial velocities are we going to assign to them? We will be especially interested in the case in which they both start with zero velocity (relative to the center). But let us start by considering cases in which the initial velocities of the particles are (i) each perpendicular to the central axis, (ii) equal in speed, and (iii) parallel but opposite in direction. For some initial speed, this will mean that they are both revolving clockwise (for example), on opposite sides of a single circle. For smaller initial speeds, each particle will trace out its own ellipse. (The ellipses be equal in size and shape; they will partly overlap; their major (i.e., longer) axes will both lie on the
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central axis of the system; and exactly one of the two focal points of the one ellipse will coincide with one of the focal points of the other, and this point will be located at the system's center of mass.) If each of the initial velocities is very small, then the ellipses will become very thin in relation to their length, with major axes each just larger than r-ellipses almost approximating line segments (each of length r) set end to end on a single line (the central axis)-and the motion would be oscillatory, with each particle getting closer to the other and then receding from it. But what about the case when the initial speed of each particle is actually zero? In that case, the equations of motion do not predict that the particles will simply meet and stop (as I(ant would seem to have thought). Rather, they would seem to predict that each particle will move back and forth on the central axis, toward one another until they both meet in the center, and then away from one another until they are separated by a distance of 2r, in a cyclical manner, as one might expect by extrapolating from what has just been said about very small velocities. (However, this account of the motion of the particles may have some strange elen1ents to it, for it needs to be allowed that each time the particles return to the center, their transverse speeds (i.e., their speeds perpendicular to the central axis) will tend to infinity, but their direction of motion along the transverse will be indeterminate, and so their state of motion at that point will be in this special way undefined. This may at first seem a bit bizarre, but again it may turn out to be the mathematically correct description of their trajectories.) In considering the behavior of the more cOll1plex system-a homogenous ball of matter, all the parts ofwhich begin at rest-it needs to be recognized that its state 'of motion might similarly turn out to be oscillatory (and in the same sense, undefined, when it converges at its center). 21. The latter is apparently IZant's view in his pre-critical work on the formation of solar systems and galaxies, the Universal Natural History, where he rehearses a similar line of thought, in which he considers how matter would behave if it possessed attractive force alone: The scattered elen1ents of heavier kind gather, through attraction, from a sphere around [them] all matter of lesser specific weight; they themselves, together with the material with which they have been united, gather into points where particles of even denser kind are found .... [0]ne easily perceives that all results of this process would in the end consist in putting together of various lumps [of matter] which, after the completion of their formation, .would be at rest and forever unmoved. But nature has still other forces in store which especially evidence themselves when matter is diluted into fine particles, whereby they repulse one another .... (I :264-65, emphasis mine) 22. This is actually an oversimplification. There are other ways to fix this supplementary information. Nevertheless, for simplicity of presentation I will continue to talk in terms of "initial" values or velocities or conditions. 23. I also want to emphasize that lZant is not yet malting any assumptions about the character of the law this attractive force is supposed to obey. In particular, he is not making any assumptions about how the force varies with the distance from the attracting object. But in the absence of this kind of specific information about the form Dfthe law, there is no way ofknowingE-~~!E~_~~!~f!l_~jllJ2(~Jl
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as its parts are drawn toward each other. However, for the purpose of arguing against IZant's balancing argument, it is certainly legitimate to consider an inverse square force law for the purpose of constructing counterexamples. 24. Note that in fact there are many other ways of preventing collapse by ascribing appropriate motions. For example, all the parts of a sphere of matter could be given an outwardly directed motion that is greater than or equal to, so to speak, escape velocity. An advantage of the spinning case is that it seems to lead to a relatively stable system, which does not then become liable to the "unlimited dispersion" worry that lZant brings up in the first (from-repulsion-to-attraction) arguluent. 25. I wish to note that this is in no way to suggest that a circular motion could be essential to matter. Rather, the point is that the balancing argument does not lead to a determinate answer to the question of the properties essential to matter. 26. I note at this point that there are two quite different senses of "inevitability" of collapse/dispersion which are both operative here in conjunction. One component is the idea that all possible assignments of initial velocities lead to collapse/dispersion. The other is that, any given assignment leads, as a matter of causal necessity, to collapse/dispersion. One feature of the last of the responses I will examine (in section 4c) is that the "inevitability" can be restricted to the second sense alone. 27. Note that on the mechanistic conception of space-filling, the balancing argument cannot be made, because without repulsive forces between the parts, there will be no tendency toward expansion that needs to be checked. And so a mechanist is deprived of an argument for the universality of attractive force. 28. Thus IZant is not saying that, in dynamics, a power is assigned to a motion; rather the power ("to motion"), e.g., the power of attraction, is assigned to a body or to matter. 29. Here IZant's list of the general properties of body includes "connection [zusammenhang]." By "connection," I take it that lZant has in mind the fact that the parts of a body are connected to one another. On an atomistic view, this might be regarded as resting ultimately on an absolute cohesiveness of the parts of the aton1 itself (where these atoms could perhaps interlock in some way to form cohesive composites). On lZant's own view, the parts of a body are connected to one another by a force of attraction. T'he fact that this is a "general property" of bodies is something IZant argues for in his 1756 "Physical Monadology," in an early version of the balancing argument. What is made more explicit in later work (from the "critical" period) is that, although connection (construed as above) is a general property of matter, it is not part of the content of the concept of body, which includes shape and impenetrability, but not connection. The relation between the concept of body and the concept of connection is thus regarded as synthetic. The fact that lZant mentions shape and not extension should not surprise us. This is because he is considering body rather than matter. Both body and matter are necessarily extended, but shape seems to be treated as specific to body. But since extension is implied by shape, it need not be mentioned explicitly when characterizing body, once one has attributed shape to it. 30. I take it that when lZant asks "But where does that come from?" he is not asking where the air came from. As I understand it, the point is that the power of
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the spear-like particles comes from a power in the air, and what I(ant is asking is where that power came from. 31. See especially 1:264, where Kant writes, "the universal standstill lasts only for an instant. The elements have essential forces to set each other in motion ...." 32. I take this to mean motion relative to a center of mass frame. 33. Of course, the possibility of such explanations (i.e., the possibility, given the nature of matter) doesn't guarantee that the presence of motion in the world in fa-ct evolved from a universe at rest, under the influence of the dynamists' moving forces. But-assuming that (given what we know about the world) we also cannot guarantee that this is not the case-I(ant will take the claim to be sufticient justification for seeking such dynamical explanations, and for framing one's conception of matter in such a way that they are, at least, not ruled out. 34. The point is not that the proposal to balance attractive force with a spinning motion amounts to talcing a mechanical approach. For, insofar as it appeals to a fundamental power (attraction), it falls squarely within the dynamical camp. Radler the point is that it is a version of the dynamical approach that would rule out a kind of explanatory project, the possibility of which constituted a chief reason for favoring the dynamical approach over the nlechanical to begin with. 35. In addition, I want to note that this way of defending Kant's theory of matter requires an important supplementary story. For I have motivated our putting aside considerations of motion by focusing on the significance of certain explanatory goals or ideals-the possibility of explaining the origin of motion from the forces inherent in matter. But it needs to be remembered that these are explanatory goals and explanatory ideals. Nothing guarantees us that nature is such that they can be realized. In particular, nothing about the conditions of experience guarantees this. At most what we can say is that we are always rationally entitled to pursue them. They serve as what I(ant calls "regulative" principles. More would need to be said about how we get from this to the assumption that nature can be explained in this way, that is, how we nlove from the claim (i) that nature would be particularly intelligible if it can be explained in this way to the claim (ii) that nature is such that it can be explained in this way. For it seems that it is the latter claim that is needed if we are to defend the balancing argument along the lines I've just been sketching. 36. However, it seems to me that the requirement associated with the second consideration (the stability in density requirement) is stronger than that associated with the first (the limitation of expansion or collapse requirement). Since density will change with expansion or contraction, having a stable density will guarantee that the limitation requirement is satisfied. However, the converse is less straightforward. For it seems to me that the limitation requirement, taken by itself, is compatible with all lcinds of variation in the density. Minimally, the limitation requirement entails that at some distance from the center there must be a net force inward, and that, at some distance closer to the center, there must be a net force outward. But by itself this does not rule out, for example, that there is an extended finite region of space (say, a three-dimensional spherical shell comprising the space further than a certain distance from the center but closer than some other larger distance) where the force is zero over that whole region. And that seems to me to be incompatible with the stability requirement as IZant probably understood it, i.e.,
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that the stabilization is toward a determinate density. However, if we add some further assumptions IZant might have taken for granted about the possible functions that could have served as force laws (assumptions we would express in tenus of the function's being "analytic" or "well-behaved"), we can perhaps formulate a "strengthened" limitation requirement that IZant could have rightly taken to be equivalent to the stability requirement. (I leave several questions entirely out of consideration here, e.g., whether there could be multiple (but isolated) stable points.) 37. I do not believe that the stability requirement would help with the second objection to the from-attraction-to-repulsion argument (the objection involving matter endowed with attractive force alone which is prevented from collapsing by its rotation around a central axis), for this already has built into it the means for satisfying this requirement. What I have to say in response to this objection I have already said in the previous section or I will say in the next. 38. In other words, it is not self-contradictory. 39. We need to distinguish between two senses of "mechanics" in the MF. There is the "mechanics" (the "mechanical conception of matter") that I(ant rejects in the "Dynamics" chapter, especially in the "General Observation." But there is also a broader sense of "mechanics" and this is what I(ant has in mind in the "Mechanics" chapter of the MF. What is dropped in the n1echanics IZant endorses in the "Mechanics" chapter of the MF are the assumptions of atomism and of absolute solidity, in terms of which mechanists had typically understood the differences between filled and empty space. I(ant intends to show that, once it is freed of these assumptions, n1echanics (as an account of the interactions in which motion is communicated from one body to another) can be embedded within a fully dynamical physics, in which space-filling is understood in terms of original forces (viz., atu action and repulsion). In the first two responses I discussed, we were concerned with the dynamics-mechanics contrast, in the sense in which dynamics is favored as an alternative to mechanics. In this third response I focus on the contrast between the two not as mutually incompatible alternatives, but as distinct components of a physics IZant means to support. 40. The 1747 Gedanken von der wahren Schiitzung der lebendigen ](riifte [True Estimation ofLiving Forces] (1: 1-182). 41. The reader may have noted that I(ant says, "one might far more appropriately call those forces ... dead forces." This is a significant endorsement for using the designation "dead forces" in this way, but perhaps it suggests the qualification that, though this description is far more appropriate than a proposed alternative, something else might be even better. This suggestion is supported by the last sentence in the passage which reads: "All this holds providing these designations of dead and living forces deserve to be retained at all" (4:539). IZant's point here is that he is not proposing to revive the traditional Leibnizian contrast between living and dead force. And so from IZant's perspective, clarity of exposition would not be served by officially reviving the corresponding Leibnizian terminology. The problematic aspect of the traditional contrast, which I(ant is probably worried about in the passage from which the above quotation was drawn, concerns a Leibnizian way of contrasting living and dead force, through the contrast between (respectively) the force something possesses in virtue of having a 4
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finite degree of motion and the force something possesses in virtue of having an infinitesimal degree of motion. IZant believes that, insofar as, in either case, one is dealing with the force something has in virtue of being in motion, this difference between the two kinds force does not have the significance that the Leibnizians claimed for it. Specifically, with respect to the appropriate measure of their magnitude (mv versus mv2 ), they should not be treated differently. From the perspective of the "Mechanics" chapter, where lZant is arguing for measuring moving force through the measure mv, both cases (the finite and the infinitesimal) concern the force a body possesses in virtue of its motion: "there is no difference between living and dead forces if moving forces are regarded mechanically, i.e., as those which bodies have insofar as they are themselves moved, and the velocity of their motion may be finitely or infinitely small (mere endeavor toward motion)" (4:539). This echoes IZant's more detailed and explicit argument to this effect in his pre-critical True Estimation ofLiving Forces, chapter 2, sections 25-26, where the assignment of one n1easure (mv) of force to infinitesimal motion and of another (mv 2 ) to finite motion is said to be incompatible with continuity conditions. This is a point lZant makes repeatedly in the second chapter of this work. In this light, perhaps one can resolve the relation between lZant's claim that "one might ... call the original forces of dynamics dead forces. And one might, on the other hand, call all mechanical moving forces, i.e., by means of motion proper, living forces" and his claim that "there is no difference between living and dead forces if moving forces are regarded mechanically." For, although the distinction between infinitesimal and finite motion does not (on lZant's view) mark a significant contrast between two kinds afforce, there are further elements involved in characterizing the difference between living and dead force which do mark a philosophically important contrast, and they correlate "far more appropriately" to the contrast between the "Mechanics" and the "Dynamics" chapters. I also note that, although Leibniz assigns the quantity mv2 as the measure of living force, Kant does not appear to be taking this measure for granted in the MF passage described above. However, in some of the arguments in his True Estimation 'ofLiving Forces, Kant seems to treat living force simply as that force that has mv2 as its measure. (Cf. §28, where he argues that because mathematics cannot prove the existence of a force with measure mv2, it cannot prove the existence of living force.) 42. Note that there is a fundamental difference between the way Leibniz understood these phenomena and the way IZant did. This is perhaps another reason lZant seems to qualify his endorsement of using the language of dead and living force to contrast the "Dynan1ics" and "Mechanics" chapters of the MF. For Kant will think of these phenomena, which are associated with matter's being in motion, as manifestations of its inertia. But Leibniz thinks of inertia simply in terms of matter's tendency to remain at rest (or perhaps, its resistance to an increase of motion/speed). (Cf. Leibniz (1989), pp. 120, 161, 172-3.) And so he will consider these manifestations of force associated with matter's maintaining itself in motion as essentially different in kind. 43. The talk of "push" and "pull" need not be interpreted as excluding action at a distance. This will be important in Kant's understanding of the fundamental forces.
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44. To include centrifugal force in this list we should modifY the preceding phrase to make it explicit that this force is exerted independent of whether or not it is in motion in the direction ofthe dead force in question. In any case, Kant would not have granted cenu'ifugal force to be an example of dead force. 45. This is a slightly revised version of Ellington's translation. 46. Michael Friedman has emphasized to me in conversation the centrality of hydrostatics in understanding the theory of matter presented in "Dynamics" chapter. The proposal I am putting forward in this section is, I believe, a close kin of that view. However, in the view I will now spell out, the central concern will be how, lVithin the «Dynamics» chapter, we are to treat bodies in motion. 47. To be more precise, Kant first argues for the infinite divisibility of matter and then goes on to the balancing argument. 48. Very early on, Kant thought of inertia as a special kind of force (vis inertiae). However, this is an idea he rejected after his "New Theory of Motion and Rest" (1758). Cf. also MF 4:550 for a passage describing what he thought was wrong with the idea of vis inertiae. 49. In his proof of the first proposition of the "Dynamics" chapter, Kant says: "Penetration into a space (the initial moment of which is called the endeavor to penetrate) is a motion. The resistance to motion is the reason why motion diminishes or even changes into rest." (4:497) The second sentence of this quotation might be thought to suggest that Kant does think of force as the cause of change of velocity (the "reason why motion diminishes or even changes into rest"), rather than change of configuration or relative position. I think that we should not interpret the passage in this way. We can understand this example on the analogy of the case of the piston (rather than, say, the analogy of billiard balls). We apply a force to the piston and compress the air it contains. As we do so the air offers resistance. This resistance is the "reason why motion diminishes." However, this does not mean that resistance of the air in the piston is being understood in terms of an F=mass times rate of change of velocity (F=ma) law. To be sure, Kant is thinking of a later motion being less than an earlier motion. Two motions are being compared, not two relative locations. However, the resistance associated with matter's impenetrability is a very specific sort of repulsive force. It is the reason an object pressing on the matter with a given force cannot press beyond a certain limit. It is conceived in terms of a limit on compression, a limit on change of relative position or configuration. It is because the resistance associated with impenetrability has this specific character that we can say that the motion diminishes (i.e., that the change of position over a later time interval is less that the change of position over a comparable earlier time interval). The specific character of this resistance means that it is not simply a cause of change of position, but a cause of a change of position of a specific sort, viz., one in which motion must diminish as well. This has nothing to do with an appeal to an F=ma law or a conception of force as a cause of acceleration or deceleration. 50. In a footnote to the "Second Analogy" (Al07/B252 fn.), Kant says, "It should be carefully noted that I speak not of the alteration of certain relations in general, but of alteration of state. Thus, when a body moves uniformly it does not in any way alter its state (of motion); that occurs only when its motion increases or diminishes." This might seem to go against the idea that, in the "Dynamics"
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chapter, Kant is considering the alteration of spatial relations as themselves having a cause. However, the apparent problem is dissolved if we attend to the context in which IZant adds this footnote. For there I
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limits on our explanatory ambitions) prior to any appeal to inertial considerations (see Warren 2001, Ch. 3). In Proposition 7 and its proof proper, inertial considerations are not brought in. Only in the appended "Observation" does Kant appeal such considerations, and there, only to set up an alternative conception of am'action which he goes on to reject. 52. See especially §§15-18; 1:28-30. 53. I am not claiming that Kant had this work explicitly in mind when he wrote chapter 2 of the MF. However I think that we can fruitfully draw on the earlier work in interpreting the later, ifwe systematically take into account the changes in Kant's views over time when we appeal to docn-ines, arguments, and (what is significant here) distinctions he made earlier on. In this context specifically, what is relevant is that the changes in Kant's views on inertia presented in his New Theory of Motion and Rest (1757) do not undermine the distinction Kant malces (about ten years earlier) between the two kinds of motion in The True Estimation ofLiving Forces. If anything, it provides the means for interpreting the distinction in a way that is in more accord with the standpoint of the MF. The value of drawing on The True Estimation ofLiving Forces in interpreting the MF is further confirmed by the very specific help it renders for us in interpreting the otherwise obscure point Kant makes in tlle discussion of living and dead force, when he says, "there is no difference between living and dead forces if moving forces are regarded mechanically, i.e., as those which bodies have insofar as they are themselves moved, and the velocity of their motion may be finitely or infinitely small (mere endeavor toward motion)." (4:539) This point was discussed in footnote 41 early in this section of the paper. 54. Here I have drawn on tlle draft of a n-anslation being prepared by Martin SchOnfeld. 55. Kant distinguishes a "phoronomic" conception of motion, which is introduced in chapter 1 of MF, and a "mechanical" conception of motion, inn-oduced in chapter 3. I want to suggest that we distinguish a further conception of motion, which is characteristic of chapter 2. Early in chapter 2 (the "Dynamics" chapter), Kant says that "in this straight line [between two points each representing a piece of matter] only two kinds of motion are possible: one by which the above points recede from one another and a second by which they approach one another" (4:498). So perhaps we should posit a "dynamical" conception of motion, and this would be the motion of withdrawal or approach, or more specifically, the motion of expansion or conn-action of a piece of matter, considered as motion of the second sort (while the motion of the third ("Mechanics") chapter should be conceived as motion of the first sort). 56. One may well ask why force, as the cause of motion of the second kind, rather than force as cause of motion of the first kind (e.g., force conceived on a more Newtonian model), should be the more appropriate focus for the "Dynamics" chapter. I think it helps to recall that tlle chapter is concerned to account for the "space-filling" property of matter (its impenen-ability). This is in the first instance understood in terms of exclusion from a certain region of space. A given body mls a space, in other words, in virtue of a force which prevents certain (inn-usive) changes of location in other bodies. To account for this we do not yet need to bring in inertia and the idea of changes in momentum. Moreover,
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according to IZant, it is the sense of feeling (touch) that first presents the space-filling property of matter to us. This, again, does not require that we bring in inertial properties. One can simply think of the given body as being pressed in from all sides (analogous to the piston examples). In this way, much can be established with respect to the concept of an object in space prior to bringing in considerations of inertia and motion of the first kind. "By means of the sense of feeling," says IZant, "this property [of filling space] provides us with the size and shape of an extended thing, and hence with the concept of a determinate object in space" (4:510). This makes possible "the first application of our concepts of quantity to objects whereby there first becomes possible the transformation of our external perceptions into the experiential concept of matter as object in general" (4:510). 57. It might be objected that one of the things IZant wants to do in the "Dynamics" chapter is to establish the proportionality of attractive force to inertial mass, and so he couldn't be abstracting from inertial properties there. However, what IZant wants to establish in that chapter is the proportionality of attractive force to quantity of matter. And it is only in the next chapter that the quantity of matter is correlated with its inertial properties. I take it that, in the "Dynamics" chapter itself, the notion of quantity of matter is to be understood in terms of the notion of a quantity of material substance and the fact that material substances must lie spatially outside one another. This gives the notion content sufficient for the purposes of the "Dynamics" chapter. The linear dependence of attractive force on quantity of matter is then said to follow from the fact that attraction is a "penetrative" force, and so the part of the total effect contributed by any part of matter is unchanged by the presence of any other parts of matter in the vicinity: "Therefore [the attractive force] is a penetrative force and for this reason alone is always proportional to the quantity of matter" (4:516). 58. E.g., systems in which kinetic energy is quickly dissipated. 59. Here it may be what is called "free energy" that is at stal<:e, rather than potential energy, strictly speaking. 60. We are therefore putting aside inertial phenomena like overshooting, which enter only when we consider motion of the first sort. 61. However, we can do away with the "amplified" form of objection 1, in which we consider two bodies moving away from one anotl1.er in the absence of repulsive force. For, from the standpoint of the "Dynamics" chapter, we will regard the bodies as being in (relative) n10tion only insofar as and as long as a force is being applied. 62. IZant rejects the idea that "absolutely" inner determinations can figure in our knowledge. However, he allows for knowledge of inner determinations if they are "comparatively" inner, i.e., if they are reducible to outer (relational) determinations at a more fundamental level. (See the "Amphiboly" section of the first Critique, especially A265/B321, A277/B333, and A285/B341.) I(ant does not so much reject the priority of the inner determinations of a substance with respect to its relational determinations; rather, he relativizes it. (See Warren 2001, Ch. 2.)
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REFERE~CES
Friedman, M. (1992). I(ant and the Exact Sciences. Cambridge: Can1bridge University Press. Iunt, I. (1902-). I(ants gesammelte Schriften. Edited by IZoniglich Preussischen [later: Deutsche] Akademie del' Wissenschaften. Berlin: de Gruyter and predecessors. - - . ([1755J 1981). Universal Natural History and Theory of the Heavens. Translated by S.L. JaId. Edinburgh: Scottish Academic Press. - - . ([1786J 1985). Metaphysical Foundations of Natural Science. Translated by J.W. Ellington. Indianapolis: Hackett. - - . ([1781/1787J 1986). Critique ofPure Reason. Translated by N.IZ. Smith. London: Macmillan. - - - . (1996). Lectures on Metaphysics. Translated by IZ. Ameriks and S. Naragon. Cambridge: Cambridge University Press Leibniz, G.W. (1989). Philosophical Essays. Edited and translated by R. Ariew and D. Garber. Indianapolis: Hackett. Warren, D. (2001). Reality and Impenetrability in I(ant)s Philosophy of Nature New York: Routledge. Williams, L.P. (1966). The Origins ofField Theory. New York: Random House.
[10] Mathematical Method in IZant, Schelling, and Hegel FREDERICIZ C. BEISER
1. Triclqr Business in Post-IZantian Philosophy In a remarkable passage from the 'Methodenlehre' of the f(ritik der reinen vernunft-perhaps the only one in the entire book to use an exclamation mark-IZant complains how difficult it is to think about mathematics (B 753).1 It is, he exclaims, "a tricky business" (ein schweres Geschiifte!). What brought about such a rare and surprising exclamation? We do not have to think too long or too hard about what exasperated IZant. He was referring to an issue that had deeply troubled him and his entire generation: What is the proper relationship between n1etaphysics and mathematics? Should the n1ethod of metaphysics be the same as mathematics? Or is such imitation, given that the methods of these disciplines are different toto caelo) impossible? Since the end of the seventeenth century, these questions had gripped Gern1an philosophy. They had been an important bone of contention between Leibniz and Descartes. They also had been the chief source of friction between the Leibnizians and Newtonians in the Prussian Academy. They were the subject of the famous 1763 essay competition, for which IZant won the accessit and Mendelssohn the prize. As all IZant scholars appreciate, the issue was crucial for IZant from the very beginning to the very end of his philosophical career. It was the central problem behind his first publication, the 1746 Wahre Schiitzung der lebendigen f(riifte; and it was a crucial issue even in the final fascicles of the Opus postumum. Rightly, scholars ofIZant and pre-IZantian philosophy have focused on this issue and stressed its importance. 2 There is still much to be done to explain these subtle issues; but no one would question their fundamental importance. It is all the more remarkable, therefore, that this central issue virtually disappears from scholarship on post-IZantian philosophy. If one were to 243
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judge from the formidable body of secondary literature on this period, it seems as if post-IZantian philosophers had a completely new agenda. It is as if the question of the relationship between mathematical and philosophical method were no longer relevant, as if it were a defunct legacy of the long-lost days of rationalism which IZant had finally put to rest. After all, the n1ethodological paradigms of post-IZantian philosophy seen1 to move away from ll1athematics and toward art and the dawning biological sciences. Some recent scholarship-most notably the work of Dieter Henrich and Manfred Frank-has helped to rectify this picture, at least somewhat. 3 Their work has focused upon the fascinating debates surrounding Reinhold's and Fichte's foundationalism in the 1790s. Although this work promises to shed some new light on how the classical issues were pursued in post-IZantian philosophy, it so far still has not done so. Research on these debates is still in its infancy; and so far it has not focused on how much the debates about foundationalism involve the classical issue of the proper relationship between metaphysics and mathematics. Since foundationalism can take many forms-it can be empiricist and phenomenological as well as mathematical-we cannot simply equate the disputes about foundationalism with issues about the applicability of the mathematical method in philosophy. What I want to do is to show one way in which the classical questions about mathematical method were still of decisive importance in the development of post-l(antian philosophy. I want to connect post-IZantian philosophy with pre-Kantian philosophy by showing how thinkers from these apparently distinct periods were still, despite appearances to the contrary, deeply engaged in the same conversation about a single fundamental issue. This is not to say that I(ant changed nothing, as if the terms of the dispute were entirely the same. There is at least one important shift: it is away from the Leibnizian and toward the Kantian conception of mathematics. The question now becomes whether the method of construction, as IZant understood it, is applicable to philosophy. It would be wrong to suggest,ho\vever, that Kant's conception of mathematical method became canonical for the post-I(antian generation. Maimon and some die-hard rationalistsEberhard, Ma~ and Schwab-hotly contested it and challenged it in interesting ways. Still, the Leibnizian conception of mathematical method had ceded its prominence and dominance to the IZantian. I shall focus on two very important and interesting-and very much neglected-chapters in the post-IZantian dispute about mathematical method. The first chapter concerns Schelling's and Hegel's use of the mathen1atical method in their Naturphilosophie and their system of absolute idealism, which they developed together from around 1801 to 1804. Prima facie it might seem surprising that Schelling and Hegel were
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ever champions of the mathematical method in philosophy. Schelling is famous as the arch-romantic who gave pride of place to art in his System des transcendentalen Idealism us; and Hegel is notorious for his diatribe against the mathematical method in his' Vorrede' to the Phanomenologie des Geistes. What, we might ask, did they know about Kant's tricky business? But this is another one of those cases where history proves much more interesting and complex than all the legends of scholarship. During their brief collaboration in Jena from 1801 to 1804, Schelling and Hegel were stalwart defenders of the mathematical method in philosophy, which they employed in many of their writings during these years. One of the reasons this phase ofSchelling's and Hegel's intellectual development is so little known is that it takes place in some rather obscure writings, so obscure that they are rarely studied even by Schelling and Hegel specialists. I have in mind here first and foremost their co-written Fernere Darstellung des Systems der Philosophie and Schelling's important little essay 'Uber die Konstruktion in der Philosophie.' Most scholars know about the method of construction only through Hegel's tempestuous polemic against it in the 'Vorrede' of the Phanomenologie des Geistes. They overlook the fact that Schelling and Hegel jointly conceived, ardently defended, and extensively used the method from 1801 to 1804. Not surprisingly, then, the most basic questions about this method remain unanswered. What does construction mean? To what extent was this method inspired by Kant? To what extent does it deviate from him? Why did Schelling and Hegel believe themselves justified in applying such a method to philosophy, despite Kant's ban against its use in philosophy? The second chapter of my story is much better known. It concerns Hegel's later attack upon the method of construction in his Phanomenologie. Although Hegel's break with Schelling has been much studied, it is still fair to say that few scholars have adequately understood it, not least because they have scarcely examined the methodology that Hegel was reacting against. Usually, Hegel's critique of intellectual intuition is portrayed as a rejection of a kind of religious mysticism; but this is a misconception of intellectual intuition, which is very much modeled around the method of mathematics. When Hegel attacked intellectual intuition in 1806, then, he was also questioning the role of mathematical method in philosophy. Hegel's rejection of that role-so I shall arguewas the crucial negative step toward the development of his own phenomenological methodology. What we find here is a surprising parallel between Kant and Hegel: it was only in his later Jena years that Hegel learned to appreciate the force and importance of Kant's early distinction between the methods of philosophy and mathematics in the 1763 Prize Essay. Strangely, the ghost of Kant proved to be an unspoken ally in Hegel's battle against Schelling.
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2. The Method of Construction in Early Schelling and Hegel In the early 1800s Schelling's and Hegel's methodology revolved around their strange and difficult concept of intellectual intuition. Schelling and Hegel made great claims for this faculty and placed great hopes upon it. Intellectual intuition was meant to be the new organ of speculative knowledge. "Without intellectual intuition no philosophy!" Schelling proclaimed in his Vorlesungen iiber die Methode des akademischen Studiums. 4 "Without transcendental intuition one cannot philosophize," Hegel declared in his Differenzschrift. 5 Explaining the precise meaning of this concept would require a treatise. But to provide a basic idea it will suffice to indicate its ultimate source of inspiration, a source familiar to almost all philosophers: namely, the sixth and seventh books of Plato's Republic. Following Plato, Schelling and Hegel think that reason consists in a faculty of intuition or contemplation, and that it is more than, even though it also includes, a discursive power of conception, judgment and inference. Again with Plato, Schelling and Hegel contrast intellectual intuition with sensible intuition, reason with sensibility: whereas the senses intuit the ectypes or appearances of things, reason intuits the forms or prototypes of things, objects as they really are and in themselves. Finally, yet again with Plato, Schelling and Hegel maintain that the power of intellectual intuition is best exemplified and exercised in mathematics. 6 Summa summarum) Schelling and Hegel understood intellectual intuition as rational insight into the archetypes, forms or prototypes of things. In appealing to such a faculty, they were selfconsciously and deliberately reviving a classical Platonic doctrine. I stress the Platonic provenance of Schelling's and Hegel's concept of intellectual intuition chiefly because it reveals its deep rationalist dimension. It is necessary to stress the point because the most popular interpretation of this concept regards it as a form of irrationalism and mysticism. 7 Hegel scholars have been especially active in supporting this interpretation because they see Hegel's later rejection of intellectual intuition as solid proof of his own rationalism. But this interpretation is, I believe, deeply mistaken. 8 The very opposite is the case. If intellectual intuition suffers from any obvious failing at all, it is not its irrationalism but its hyperrationalism, the very strong claims it makes on behalf of the powers of reason. One might acknowledge this Platonic heritage but still insist that, because it sees reason as a kind of insight or vision, intellectual intuition is a form of mysticism. I will not quarrel here with this use of the term 'mysticism.' But ifwe use the term in such a broad sense then we have to admit that there are all kinds of mysticism. There is indeed a very big difference between a rational mysticism, which identifies its insights Wi~~!~!~9_I!L~~sl
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which regards its objects as archetypes; and an irrational or non-rational mysticism, which denies the existence of all entia rationis and equates its experience with something that transcends all conception, judgment and demonstration. 9 In this respect it is important to see that Schelling and Hegel, unlike most mystics, stress that an intellectual intuition must be expressible in systematic form. lO Although an intellectual intuition is not reducible to concepts, judgments and syllogisms, its insight or perception still must be able to embody or realize itself in them; it is indeed what makes it possible for us to create concepts, make judgments and draw inferences in the first place. The relation of an intellectual intuition to its discursive elaboration is like that of the knowledge of a whole to the piecemeal elaboration of its parts. What we see at once and as a whole in an intellectual intuition has to be explained successively and discretely. In this regard it is noteworthy that Schelling and Hegel distanced themselves from some of their romantic contemporaries-most notably, Friedrich Schlegel and Schleiermacher-because they shunned expressing their inspiration in discursive form. l l Against Schlegel and Schleiermacher, they demanded that the knowledge of metaphysics be methodological and systematic, and they stressed that an insight is only as good as its conceptual elaboration and demonstration. When Hegel later criticizes Schelling and his disciples on these grounds he was really making a rather cruel tid hominem point, turning against his erstwhile colleague their old polemic against the romantics. Nothing shows more clearly the rationalist dimension of Schelling's and Hegel's concept of intellectual intuition than the fact that they made mathematics into their standard of knowledge. In their Fernere Darstellung they contend that mathematics is the only science that provides an example of intellectual intuition in philosophy.l2 True to this belief in the paradigmatic status of mathematics, Schelling in his 1801 Darstellung meines Systems proceeded more geometrico, beginning with axioms and definitions and drawing consequences from them. It is striking that Schelling's and Hegel's conception of the mathematical method is fundamentally I
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Schelling maintains that whether a proposition is synthetic or analytic depends on its precise role within a system, so that a proposition that is a synthetic in one system might be analytic in another.l 5 Why did Schelling and Hegel regard mathematics as proof of the possibility ofintellectual intuition? Their best account appears in the first chapter of the Fernere Darstellung, in Lecture V of Schelling's Vorlesungen uber die Methode des akademischen Studiums, and in Schelling's very interesting but neglected essay, "Uber die IZonstruktion in der Philosophie." In making their case they appeal to at least three features of a ITlathematical demonstration, all of them stressed by IZant. All these features show, they argue, that we have an intellectual intuition, a kind of knowledge of archetypes, which consists in the perfect unity of universal and particular, possibility and reality, essence and existence. First, in a mathematical demonstration we construct our objects in intuition; we create them through an act of mind, so that we are not dependent on some given object. Hence in this case there is no dualism between possibility and reality, form and matter; rather, we make the object simply according to our definition. Second, in a mathematical demonstration we construct a single object that serves as a proof of a universal theorem. We draw in intuition, for example, two parallel lines on a chalkboard; although these lines are completely particular, having all the determinate features of any particular thing, they still embody the universal truth that parallel lines never intersect. To prove the proposition, we do not have to draw more lines or consult further instances; these two lines alone suffice, thus showing the perfect unity of content and form characteristic of an archetype. Third, in a mathematical demonstration we construct an object in intuition and do not simply calculate according to the laws of identity. There is a profound irony, of course, in Schelling's and Hegel's argument here. Their concept of mathematical method is very I
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depart from Kant on this important score. On the whole, they content themselves with pointing out that the intuition involved in construction cannot be a sensible intuition, and therefore can only be an intellectual intuition. The closest they come to taking issue with Kant is in Schelling's important essay 'Uber die Konstruktion in der Philosophie.' Although this essay is filled with much irrelevant and ineffective polemic, there are still a few interesting and suggestive passages. One of these is where Schelling counters Kant's point that philosophy cannot be like mathematics because it does not have axioms and definitions. Schelling's reply is that it is a moot point whether axioms and definitions are really essential to the mathematical method after all. Axioms and definitions are not true principles, he argues, but simply the limiting points (Grenzpunkte) that have been attained by a science so far. 16 Philosophy should not use definitions and axioms in the same sense as mathematics because it should never set limits to the creativity of construction. In another interesting passage Schelling directly takes issue with Kant's distinction between mathematical and philosophical methods.l 7 In the first Kritik Kant had very quicldy and schematically distinguished between the methods of philosophy and mathematics by stating that philosophical cognition considers the particular in the universal whereas mathematical cognition treats the universal in the particular (B 742). By this distinction Kant seems to mean that the medium of philosophy is essentially discursive and abstract, whereas the medium of mathematics is intuitive and concrete. Since the discursive and abstract deals with universals or concepts, philosophy considers the particular in the universal; and since the intuitive and concrete concerns something individual, it treats the universal in the particular. Whatever the precise meaning of Kant's distinction, Schelling objects that it is arbitrary and ineffective as a distinction between philosophy and mathematics because it falls within mathematics itself. Kant's distinction perfectly describes, he contends, the distinction between arithmetic and geometry: arithmetic treats the particular in the universal because its concepts (numbers) are purely abstract, while geometry treats the universal in the particular because it constructs objects in intuition. Furthermore, Schelling contends, Kant's own analysis shows that mathematics does not simply apply the universal to the particular, because when we construct a concept we see the universal in the particular. IS Schelling's argument here rests upon an understandable but flawed interpretation of Kant. He reads the difference between mathematics and philosophy along the lines of a distinction between determining judgment and reflective judgment in the third Kritik: the philosopher considers the particular in the universal because the particular is given and the universal is sought (reflective judgment); the mathematician considers the universal in the particular because the universal is given and the particular is sought
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(determining judgment). This is a plausible interpretation if one approaches I
3. A IZantian Interlude and Complication The tricky business concerning the role of construction in philosophy becomes even trickier when we consider one striking fact: that Iunt himself used the method of construction in his Metaphysische Anfangsgriinde der Naturwissenschaften. Incredible sed verum: it was I
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an apparent inconsistency here, given that he had already proscribed the method of construction in philosophy in the first Kritik. Kant's attempt to resolve this apparent inconsistency appears in the ' Vorrede' to the Anfangsgriinde. Here he continues to insist that philosophy cannot use the mathematical method; hence he states explicitly that a pure philosophy of nature, because it consists in a priori principles, cannot provide constructions for these principles (IV, 469, 29). Nevertheless, Kant makes an important distinction between a pure and specific doctrine of nature. While a pure or general doctrine of nature develops synthetic a priori principles and concepts, a specific doctrine of nature applies these principles and concepts to specific objects in experience. Of a specific doctrine of nature Kant insists that it can be a science only to the degree that mathematics is possible for it, so that its principles and concepts must be constructible (470, 13-15). Kant therefore states that while a pure philosophy of nature is possible without mathematics,21 a specific doctrine of nature is not (470, 27-32). Since the Anfangsgriinde is meant to be not only a pure but also an applied philosophy of nature, Kant thinks that he is justified in applying to it his methods of construction. On this account, then, it would seem that the impetuous young Schelling, in his haste toward metaphysical speculation, had failed to heed his master's subtle distinction, and so he had confused Kantian levels of discourse. But this is much too simplistic, not least because Kant's distinction remains problematic. Although he is intent on a sharp distinction between a pure and specific philosophy of nature, Kant muddies the waters himself when he states that, though its principles cannot be constructive, the pure philosophy of nature stlll determines the principles for the construction of concepts in the specific philosophy of nature (472, 1-7). Pure philosophy establishes "die Prinzipien der Konstruction)) and therefore «die MiJglichkeit einer mathematischen Naturlehre)) (473, 9). But it seems strange that pure philosophy of nature can provide the principles for the construction of a concept when it cannot construct concepts itself. The pure philosopher of nature is something like a tango teacher who can tell us the principles of dancing the tango but who does not dare to do the tango himself. Finally, one might question the whole distinction between a pure and empirical philosophy of nature, given that the pure philosophy of nature has to formulate laws of motion, where motion is an essentially empirical concept. 22 However salvageable or problematic Kant's distinction might be on its own grounds, it is also important to note that Schelling himself had his own reasons for rejecting it. Since he denies that there is a hard and fast distinction between the a priori and the a posteriori, he would not have accepted Kant's distinction between a pure and applied philosophy of nature in the first place. Hence he removes one obstacle to using the
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method of construction in philosophy. If there are no absolute distinctions in levels of philosophy, and if we can apply it on one level, it seems that there should be no reason in principle that we cannot apply it on all levels. We are now in a better position to understand the logic that motivated Schelling to apply the n1ethod of construction to philosophy. It is only necessary to add one further point to make this development seem inevitable. Namely, in the beginning of the 1800s, Schelling had already made Naturphilosophie the center of his system of absolute idealism. 23 It had become the crowning part of the system of identity, of which even transcendental philosophy had become a subordinate part. So if we can apply the method of construction in Naturphilosophie) then ipso facto we are applying it within the realm of philosophy in general.
4. Hegel's Critique of Construction in the Phanomenologie Scholars have often been misled by Hegel's attitude toward the mathematical method in his early Jena years. They think that Hegel had already renounced that method because of his criticism of first principles in his 1801 Differenzschrift. There Hegel had criticized Reinhold's and Fichte's method of beginning philosophy with a single self-evident first principle; and he stated bluntly that philosophy cannot make any worse a beginning than when it begins with axioms and definitions, as Spinoza had. 24 It is for this reason, as well as the polemic in the 'VOrrede' to the Phiinomenologie, that Hegel has been so often seen as a stalwart opponent of the mathematical method. But this interpretation of the Differenzschrift is mistaken. It commits two confusions that Schelling and Hegel warned against. First, it conflates the mathematical method with axioms and definitions; Schelling and Hegel insist, however, that what is essential to the method is construction and not the mere use of axioms and definitions. Second, it equates the mathematical method with first principles and deduction from them; but Schelling and Hegel made a clear distinction between deduction and construction. Where the deduction of a concept consists in deriving it from principles, in subsuming it under son1e higher universal, the construction of a concept consists in presenting it in intuition. While deduction is a strictly formal method that proceeds by reasoning according to the law of identity, construction is a method of acquiring and explicating an intuition. Hence Hegel's objection against the deductive method in the Differenzschrift is not directed against the mathematical method per se. To read Hegel in this manner is simply to confuse a Leibnizian with a I
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In general, there should be no doubt that, in his early Jena years, Hegel was still very far from rejecting intellectual intuition and the mathematical method of construction. In his Differenzschrift he proclaims the necessity of intellectual intuition; and in several drafts of his system, most notably the 1802 System der Sittlichkeit, he follows the method of construction. To be sure, the early Jena "vritings also reveal important hints about Hegel's en1erging dialectic, a method that will eventually replace construction; but these hints were not developed into a full-blown methodology until 1804, at the earliest, after Schelling's departure from Jena and the collapse of their alliance. Why, then, did Hegel eventually reject the method of construction? To answer this question we must first take another brief look at the method and the concept of intellectual intuition behind it. In their Fernere Darstellung Schelling and Hegel define an intellectual intuition as "the capacity to see the universal in the particular, the infinite in the finite, and indeed to unite both in a living unity. "25 An intellectual intuition consists in seeing a particular as part of a whole, in grasping how its identity depends upon the whole of which it is only one part. Schelling and Hegel equate the standpoint of intellectual intuition with reason, whose task is to comprehend something by showing how it depends on a whole. This is how they explain the task of reason in the Fernere Darstellung: Only for reason is there one universe; and to grasp something according to reason means: to conceive it as an organic member of an absolute whole, in a necessary connection with that whole and therefore as a reflection of absolute unity.26 When I have an intellectual intuition of an object, Schelling and Hegel write, I do not explain it, and I do not deduce it, but I contemplate it. To explain an object is to show how other objects act upon it and how they cause it to act as it does; it is not to grasp the object in itself but only how it stands in relation to other objects. To deduce an object is to derive it from higher principles; it is to show how the particular is an instance of some universal, but where the universal applies to many other particulars and where the particular instantiates many other universals. When I contemplate an object, however, I consider the object for its own sake, as it is in itself, apart from its relations to other objects. Here I see a unity of universal and particular, where the universal is entirely and fully embodied in the particular, and where the particular instantiates no other universal. Through contemplation of an object I recognize it as a unique, indivisible unity, an organic unity whose whole precedes and makes possible all its parts.
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But if this is an intellectual intuition, how do we acquire it? What precisely must we do to contemplate the object? Though Schelling and Hegel provide us with no recipe book, it is not difficult to see what they have in mind. We must first consider the object for its own sake, apart from any ends it serves. We must also focus on the object in itself, apart from its relations to other objects; we must try to grasp it as an indivisible whole or unity. We therefore must abstract from all properties or determinations of the object, because they relate it to other things; they consider it from one angle, under one aspect, never treating the thing as a whole or unity. Ifwe abstract from all these properties, then, Schelling and Hegel assure us, we will grasp its essential identity with the whole universe, how it is one with everything else. We will see the entire universe within that thing. This is because all objects in themselves, apart from their properties, are one and the same, or form a single whole or unity; they differ from one another only in virtue of their properties, and it is precisely these from which we abstract in intellectual intuition. Hence the net result of intellectual intuition is that we must negate the particular as a particular to grasp its pure identity with the absolute. 27 Assuming that we have acquired an intellectual intuition, we then proceed to analyze it, to divide it into its parts. Since each particular is identical with the universe as a whole, what is true of the whole is also true of each of its parts. Since the whole is the unity of the ideal and the real, the universal and particular, each of its parts will reflect this; each of them will have to be analyzed into moments of ideality and reality, universality and particularity. Since, however, reality in itself is indivisible, and since each part reflects the entire universe, it follows that each of these moments will be divisible in turn into ideality and reality, universality and particularity. And so on. The analysis or division will continue until we see the entire universe reconstructed before our eyes. Such, very crudely, is the method of construction developed by Schelling and Hegel in their early Jena years. When Hegel later rejects the method in the 'Vorrede' to his Phiinomenologie, his main charge against it is that it is guilty of formalism. Formalism consists in imposing a form upon an object, forcing it into a preconceived mould, and failing to consider the object for its own sake. Rather than deriving the universal from the object by examining its own inner nature, one applies the universal to the object a priori according to what it ought to be a priori. It is important to note that Hegel virtually equates construction with formalism;28 it is not that formalism is a misapplication or an abuse of the method; rather, it is an inevitable result of following the method. After our crude sketch of the method of construction, it is not too difficult to see why this so. The problen1 is that the method of construction demands that one abstract from the determinate nature of the particular; one does not consider it for
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its own sake and examine its inner structure or logic. Having abstracted from what makes particulars the very particular things that they are, one can attribute to them nothing more than what is involved in the general structure of the whole. This consists in nothing more than such abstract properties as 'ideality,' 'reality,' 'indifference,' and so on. The method generates determinate results only by covertly reintroducing the very materials from which it claims to abstract. We cannot reintroduce determinations through analysis when we have already abstracted from them in the first place. From a purely formal and undifferentiated unity nothing concrete and differentiated can come. This critique of the method of construction was ultimately the decisive factor, I believe, in Hegel's break with Schelling. The crucial disagreement in fracturing their alliance was not, therefore, ontological but methodological. The traditional account stresses their alleged ontological differences. Supposedly, Hegel held that the absolute is the unity of unity and difference, whereas Schelling maintained that it consists in pure undifferentiated unity alone. But this reading fails to take into account the development ofSchelling's metaphysics during his collaboration with Hegel. To be sure, in the 1801 Darstellung meines Systems Schelling did see the absolute as pure identity; but it is noteworthy that Schelling, perhaps under Hegel's influence, soon corrected this view. In Bruno and the Ferenere Darstellung he maintains that the absolute is not simply identity but the identity of identity and non-identity.29 Here Schelling argues explicitly-as Hegel will later do-that an absolute that does not include the finite within itself will be a relative absolute, a finite infinite. So Schelling and Hegel were in principle at one on the crucial issue of the nature of the absolute. For Hegel, the problem was that Schelling did not have a methodology to secure this metaphysics. The method of construction did not see the particular in the universal, the finite within the infinite' except by abstracting from those very features of the particular and finite that were to be included within it. If my reading of Hegel's break with Schelling is accurate, then it was chiefly a result of his critique of the method of construction, and so ultimately his belief that the method of mathematics is really different from that of philosophy. There is a remarkable analogy here to Hegel's method010gical views in the Phiinomenologie and those of Kant in the Prize Essay of 1763. It was Kant's central contention in the Prize Essay that the method of philosophy had to be more like the natural sciences. Rather than beginning from definitions and deriving theorems from them, it would have to begin with an analysis of the meaning of terms. It would have to proceed from the particular to the universal. That too was Hegel's essential conclusion in the Phiinomenologie. But I do not think that the analogy was entirely an unwitting one. Hegel probably never read the
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Prize Essay; but he knew his Kant very well, and he endorsed in principle IZant's polemic against the Romantics' appeal to an intellectual intuition. When Hegel describes "the labor of the concept" (die Arbeit des Begriffs) in the' VOrrede' of the Phanomenologie he was essentially rephrasing Kant's argument against the laziness of intellectual intuition, which attempted to avoid the hard labor (Arbeit) involved in discursive thinking. 30 So, in the end, we have come full circle. Hegel in 1807 fully appreciated the depth of a position Kant first held in 1763. l'hrough his own dialectic he had come to learn what I(ant con1plained about in the first ](ritik: that thinking about mathematics is a tricky business. Whatever one makes of this very tricky business, I will be happy here if I have at least established n1Y main point: that it never stopped after Kant and continued to trouble philosophers well into the post-IZantian era.
NOTES 1. All references to the [(ritik der reinen Vernunft are to the first (A) and second (B) editions. All other references to IZant are to the Al,ademie Ausgabe, [(ants gesammelte Schriften, ed. PreuBischen Al,ademie der Wissenschaften (Berlin: de Gruyter, 1902ff). Roman numerals refer to volume numbers, Arabic numerals to page numbers, and italicized Arabic numerals to line numbers. 2. For recent treatments of this problem in IZant, see especially Michael Friedman (1992), 1-54; Alison Laywine (1993); and Martin Schonfeld (2000). 3. See Dieter Henrich (1991); and Manfred Frank (1997). 4. Schelling, Samtliche Werke) ed. IZ.F.A. Schelling (Stuttgart: Cotta, 1859), V, 255. 5. Hegel, Werke in zwanzig Banden, Werkausgabe, ed. Eva Moldenhauer and Karl Michel (Frankfurt: Suhrkamp, 1970), II, 42. 6. See Schelling, Fernere Darstellung) Werke IV, 345. 7. See, for example, Rudolf Haym (1859), 655-56, and Lukacs (1973), II, 657. The thesis of Lukacs (1955), 103-24, is that Schelling's concept of intellectual intuition is nothing less than the birthplace of modern irrationalisn1. The best antidote to these interpretations is offered by Ernst Cassirer (1923), III, 253-74. 8. I have taken issue with this interpretation in more detail elsewhere. See "Friihromantik and the Platonic Tradition" in Beiser (2004), 43-55. 9. I have made this distinction in more detail in Beiser (2003). 10. See Fernere Darstellung aus dem System) Werke IV, 399-400. 11. See, for example, "Uber das Wesen der philosophischen I
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12. Schelling, Werke IV, 435. It is noteworthy that Schelling no longer sees art as the perfect embodiment of intellectual intuition, as he once did in his 1800 System des transcendentalen Idealismus, Werke III, 625. In his 1802 Philosophie der Kunst he demotes art to a lesser role. While philosophy presents as an archetype, art presents in the form of an image. See Werke V, 369. See too §§14-15, V, 380-81. 13. "Uber die Konstruktion in der Philosophie," Werke V, 125. 14. Schelling and Hegel do not think that mathematics actually gives us direct knowledge of archetypes but only of the reflections or appearances of the archetypes in the sensible world. See Fernere Darstellung, Werke IV, 345-46. Still, the knowledge of mathematics is the most perfect kind of ectypical knowledge; it shows us the possibility of an intellectual intuition of the archetypes themselves. 15. See Schelling's "Uber die Konstruktion," Werke V, 138-39 and his "Einleitung zu dem Entwurf eines Systems der Naturphilosophie," Werke III, 278. 16. Werke V, 136-37. 17. Ibid., V, 130-31. 18. Ibid., V, 132. 19. See Untersuchung iiber die Deutlichkeit der Grundsiitze der natiirlichen Theologie und der Moral, in Schriften II, 279, 291. 20. Schelling's and Hegel's attempt to cross the ditch, which greatly troubles them, is not convincing. See the verbal and question-begging explanation in Fernere Darstellung, Werke IV, 405-406. The "nasty and broad ditch" was Lessing's famous metaphor for the gulf between the empirical and rational. See his "Beweis des Geistes und der Kraft," in Gotthold Emphraim Lessing, Werke und Briefe, ed. Arno Schilson (Frankfurt: Deutsche Klassiker Verlag, 1989), VIII, 443. 21. Given that its principles cannot be constructed, Kant should in fact say that the pure philosophy of nature excludes mathematics. But he weakens the inference to make it possible for the pure philosophy of nature to establish the principles for the construction of concepts. 22. KrV, B 48, 58. It is also strange that Kant explains that the pure philosophy of nature is possible without mathematics (IV, 470, 30); for he should really say that it excludes mathematics because its pure a priori concepts cannot be constructed. 23. For the details of this development, see Beiser (2002), 488-90. 24. Hegel, Werke II, 37. 25. Schelling, Werke IV, 362. 26. Ibid., IV, 390. 27. This point is much emphasized in the Fernere Darstellung, Werke IV, 393, 407,408. 28. Hegel, Phiinomenologie des Geistes, ed. J. Hoffmeister (Hamburg: Meiner, 1952),42: "diese auikrliche und leere Anwendung der Formel die Konstruktion genannt [wird]." 29. Werke IV, 235-36, 407. 30. See Kant, "Von einem neuerdings erhobenen vornehmen Ton 111 der Philosophie," Schriften VIII, 389-90.
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REFERENCES Beiser, F.C. (2002). German Idealism: The Struggle against Subjectivism) 1781-1801. Cambridge, MA: Harvard University Press. - - - . (2003). "Two Concepts of Reason in German Idealism." Internationales Jahrbuch des Deutschen Idealismus I: 15-27. - - - . (2004). The Romantic Imperative: The Concept of Early German Romanticism. Cambridge, MA: Harvard University Press. Cassirer, E. (1923). Das Erkenntnisproblem in der Philosophie und Wissenschaft der neueren Zeit. Second edition. Berlin: Cassirer. Frank, M~ (1997). "Unendlichen Annaherung": Die Anfange der philosophischen Fruhromantik. Frankfurt: Suhrkamp. Friedman, M. (1992). I(ant and the Exact Sciences. Cambridge, MA: Harvard University Press. Haym, R. (1859). Die romantische Schule. Berlin: Gaertner. Hegel, G.W.F. (1952). Phanomenologie des Geistes. Edited by J. Hoffmeister. Hamburg: Meiner. - - - . (1970). Werke in zwanzig Banden, Werkausgabe. Edited by E. Moldenhauer and IZ. Michel. Frankfurt: Suhrkamp. Henrich, D. (1991). I(onstellationen: Probleme und Debatte am Ursprung der idealistischen Philosophie (1789-1795). Stuttgart: I
PART III Logical Positivism and Neo-Kantianism
[IIJ Validity in the Cultural Sciences? JOHN MICHAEL I
1. Introduction In his book A Parting ofthe Ways Michael Friedman showed how the main strands of twentieth-century philosophy, represented by Carnap, Cassirer, and Heidegger, differently transformed IZantianism. More orthodox "Neo-IZantians" such as Cohen or Rickert sought simply to distill what they considered to be lZantianisn1's essence, purifying it, so to speak, by stripping it of the contingent aspects of its presentation that it received at IZant's hands. But all of these thinkers, no matter how unorthodox, addressed what has usually been considered the most central issue in the effort to redefine or refine IZantianism: the problem of Geltung (Liebert 1920). The term "Geltung" is usually rendered in English as "validity." Whereas validity in English-language philosophy calls the topic of logical inference to mind (IZneale and IZneale 1962, If.), Geltung for the NeolZantians concerned first and foremost the generality of concepts and values. Hence, Geltung in German philosophy is more akin to the legal use of the word "valid" in English, the notion that some claim "holds" on the basis of a generally accepted authority. In the following I want to focus upon the topic of validity in the cultural sciences and, in particular, upon Michael Friedn1an's criticisms of Cassirer on this point. The term validity is meant here in the sense of the German "Geltung/' and unless otherwise noted, "validity" should be taken here in that broader sense. Formal logic provides us with a paradigm of truly universal intersubjective communicability, the ideal that informed the work of Carnap and the Vienna Circle generally. Cassirer, as Friedman writes, also wants to maintain a complementary but still universal intersubjective validity in the essentially non-mathematical cultural sciences. He never satisfactorily explains, however, how these two characteristically different types of
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validity are related-even if both, in the end, are conceived as purely regulative ideals (Friedman 2000, 156). This is no small matter, for, as Friedman explains, this problem ultimately leaves us with the unhappy choice between either holding fast to formal logic alone as the sole ideal of universal validity (Carnap) or renouncing the ideal of truly universal validity itself (Heidegger). I agree with Friedman that it should be the task of philosophy to overcome this impasse. However, I think that Cassirer made more progress in this direction than Friedman contends. Although he is generally sympathetic with Cassirer's intentions, Friedn1an makes two criticisms: (1) Cassirer does not satisfactorily explain how validity in the mathematical natural sciences and in the non-n1athen1atical cultural sciences are related to one another, and (2) he does not show how the latter can survive comparison with the former, which offers universal intersubjective validity. In the following I want to show how Cassirer could answer these objections, by showing how Cassirer established validity in the cultural sciences and by indicating how this validity compares to validity in the natural sciences.
2. Interpretations of Validity in the Cultllral Sciences Heinrich Rickert popularized the use of the term "values" (Werte) to characterize what was at issue in the discussion of Geltung. For Rickert, "Geltung" referred to the normative basis of theoretical, ethical, and aesthetic judgments. What makes true sentences true?; why do ethical and aesthetic judgments "hold" for others?-these were the questions of Geltung. The concept of" Geltung" was the residue of IZant's philosophy, with all the specifics abstracted. What are the conditions of the possibility of objectivity, regarded abstractly? Rickert claimed, "Values are not realities, neither physical nor psychological. Their essence consists in their validity, not in their factuality" (Rickert 1910, 89).2 The chief purpose of Rickert's doctrine of the Geltung of "values" was to establish the "Wissenschaftlichkeit" of the Geisteswissenschaften, or as he also preferred to call them, the" ]Culturwissenschaften": the historical disciplines studying literature, art, religion, and even the natural sciences, considered historically. In the late 19th century, as the natural sciences gained ever more prestige, the Geisteswissenschaften appeared to have no claim to being sciences at all. Dilthey's attempt to solve the problem by emphasizing the differences between the N aturwissenschaften and Geisteswissenschaften seemed to aggravate the situation even more. Dilthey's efforts to secure autonomy for the Geisteswissenschaften came at
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what many thought was too high a price. In his early work, at least, Dilthey appeared to give in to purely psychological or subjectivistic standards: the act of Verstehen or "understanding," which he declared was the basis of historically-oriented Geisteswissenschaften, seemed to have no general validity. The famous formula in Dilthey ([1894] 1924, 144) was "we explain nature, but we understand the life of the soul.,,3 But one person's act of understanding seems not to possess necessary validity for another. The reason we are able to understand documents from ancient cultures was ultimately, for Dilthey, the assumption that the documents' creators were all members of the same human species. Hence, the basis for our understanding was that there is a single general "human nature" (Dilthey [1900] 1924,329). The jump from this assumption to the concept of "validity" seemed too great for Dilthey's Kantian critics. Rickert and other Neo-Kantians sought to overcome this naturalistic approach with a logical conception of validity. Instead of distinguishing radically between the Geisteswissenschaften and the Naturwissenschaften as Dilthey did, the Neo-Kantians sought to characterize both in terms of their "validity." While the objects of Naturwissenschaft and Kulturwissenschaft could not be equated, Rickert reasoned that it should be possible at least to conceive of them as sharing the same standards of objective validity. Rickert (1910, 27) defined "culture" as "the totality of real objects, to which generally recognized values adhere and which are cared for in accordance with these values.,,4 According to Rickert, the natural sciences study the totality of real objects while the cultural sciences examine the "values" that "adhere" to them. These values are ideal forms, not psychological "evaluations." Rickert explicated his understanding of truth as a matter of Geltung: Truth is a system of statements with an unconditional right to be recognized as valid (giiltig). For Rickert, truth refers to those judgments (Urteile), that we have a kind of imperative duty to accept (Rickert 1904, 110-16). Rickert says nothing about truth itself except that it is normative. Geltung itself remains empty and inexplicable. All we know is that values can be said to "hold" or have Geltung, and the reasons that they possess Geltung can never be further explicated because of their very ideality. In response to this situation, Heidegger rejected the notion of Geltung, calling it a "word idol" (Wortgiitzen) mixing the notions of ideality, objectivity, and intersubjectivity (Heidegger 1927, §33, 155f.). Moreover, Heidegger rejected the possibility that there could be any such thing as eternal truths at all; as he put it, "That there are eternal truths will not be adequately proved until someone is successful in demonstrating that Dasein has been and will be for all eternity" (Heidegger 1927, §44, 227). This claim itself of course presumes what it denies, since it was presented in the form of an eternal truth.
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3. Cassirer on Validity in the Cultural Sciences Cassirer rejected both Rickert's and Heidegger's position. Rickert cannot arrive at his system of values without some kind of Platonic metaphysics, while Heidegger's thought possesses no means even to grasp the problem. s Heidegger's equation of the general with the superficial (das "n1an") in contrast to the "authenticity" of finitude meant that there could be no solution to the problem of Geltung forthcoming from Being and Time. Cassirer steered a course between Rickert's Idealism and Heidegger's Existentialism, focusing upon symbolism as something both concrete and "logical" at the same time. At first glance, Cassirer's approach to the concept of "Geltung" seems much like that found in German philosophy today, in which "validity claims" (Geltungsanspriiche) are related explicitly to the use of language as a concrete social practice (Habern1as 1981, 410-27). Cassirer initially focused upon language (Cassirer [1923b J 2001), but he did not take it as the prototype of symbolism (IZrois 1999, 531-47). His conception of symbolism includes language, but much else as well, including gesture and other types of expressive symbolism. Cassirer praised Heinrich Rickert for attempting to establish the validity of the non-mathematical cultural sciences (I(ulturwissenschaften), but he repeatedly took issue with his approach. Cassirer preferred the term "I(ulturwissenschaften" to "Geisteswissenschaften" but this did not indicate a preference for Rickert's axiological approach to the study of the human sciences over Dilthey's historical method as Maldaeel ([1969J 1997) has claimed. Actually, Cassirer found Dilthey's understanding of history preferable to Rickert's static systen1 of values, which Cassirer said suffered from a "double illness" of both positivism and formalism (Cassirer 2002, 92-93).6 When Cassirer used the term "I(ulturwissenschaft" he did not have the Neo-IZantian conception of Rickert or his predecessor Windelband in mind at all, but rather the I(ulturwissenschaft of his friend and associate Aby Warburg (to whom I will return later), in whose Hamburg library (the "Kulturwissenschaftliche Bibliothek Warburg") Cassirer actually worked out his philosophy of symbolic forms. 7 Cassirer first visited this library on November 27, 1920. 8 In 1920, Cassirer had been involved in a discussion with Moritz Schlick about the nature of the apriori. Schlick asked Cassirer in a letter to say exactly what he meant by "apriori." Cassirer's response to Schlick probably left him as uncertain as before and shows that Cassirer had still not formulated the conception of the apriori that would be central in his philosophy of symbolic forms ("symbolic pregnance"), which will be dis-' cussed below. Cassirer replied to Schlick on October 23, 1920: The point on which we differ lies, as far as I can tell, already in the concept of the Apriori, which I understand somewhat differently from you:
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namely not as some constant content that is fixed once and for all and material 'intuitions' (Anschauungen) or concepts, but as a function that has been defined according to some law and therefore remains identical with itself in its direction and form, but which can experience different transformations in the progress of cognition. I would only recognize as apriori, in the strict sense the notion, the 'Unity of Nature', i.e., the lawfulness of experience in general, or, perhaps more succinctly, 'univocal correlation' (Eindeutigkeit der Zuordnung). But as to the way this thought becomes specified in particular principles and presuppositions: that only occurs in the course of scientific experience, although even here I nowhere recognize rigid schemata, but rather only uniform basic motifs (Grundmotive) of cognition, i.e., of research and inquiry.9 This is quite vague, and it develops the notion of the apriori only in a regulative sense. Initially, Cassirer approached the topic of validity via the study of concept formation in modern science as it developed after Galileo, or what Cassirer called "mathematical science." He explained his change in outlook in 1928: Now the logical problem of the concept seems to me to be connected much more closely with the general problem of meaning than was the case in my earlier publication [Cassirer 1910]. The theory of concepts, it now seems to me, can only be sufficiently worked out and well founded in the framework of a systematic 'theory of meaning' (Bedeutungslehre). What I now see more sharply and clearly than was apparent to me in my earlier publication is this: that for such a 'theory of meaning' mathematics and mathematical science are of course always going to remain an important and indispensable paradigm, but that they in no way exhaust its content. 10 The 'Bedeutungslehre' mentioned here refers to Cassirer's theory of the three types of symbolic function that he first set out in Cassirer ([1927] 1978) and explicated in Cassirer ([ 1929] 2002): the expressive function (Ausdrucksfunktion), representative function (Darstellungsfunktion) , and pure signification (reine Bedeutungsfunktion). "Pure signification" refers to non-intuitional types of symbolization, exemplified by uninterpreted calculi in modern formal logic (ignoring the matter of reference) and mathematics. Cassirer's conceptions of pure signification and his notion of representational symbolism-typified by ordinary language assertions, i.e., declarative statements-were not original or specific to Cassirer. Cassirer even borrowed the designation "Darstellungsfunktion" from Buhler as he
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explains in Cassirer ([1929] 2002), 122 [trans. 110]. But Cassirer's conception of the first type of symbolism, the expressive function, did not follow any established conception. The expressive function of symbolism, as Cassirer understood it, was not so much a cultural phenomenon as a kind of natural symbolism that is ordinarily beyond our conscious control. The expressive function of symbolism, he claimed, is found in our perception of emotional and qualitative phenomena, Le., qualia, which traditionally were taken to be immediate "sense data," something not synlbolic in any way. Cassirer took this claim to its logical conclusion, and held that expressive symbolism (nl0re precisely, symbolism insofar as it is exercising its expressive function) is typified by our sensations of our own body, which we come to know in the same way that we experience sensations of touch such as roughness. He argued that if we consider the most basic qualities of touch-'hard' and 'soft', 'rough' and 'smooth'-we must grant that these arise through motion, i.e., the process of touching. If the sensation of touch were limited to a single instantaneous moment, then within that instant these qualities could no longer be found as data (Cassirer [1929] 2002, 206 [trans. 178]). The sensation of roughness emerges from a process. Whatever we feel or sense-within us or without-is symbolically pregnant with 'expressive meaning', i.e. it exhibits something, say roughness, which depends upon contrasts in time. This link in time, is how Cassirer conceived of the body-soul (Leib-Seele) relation, which he called "the prototype and model" of symbolic relationships (Cassirer [1929J 2002, 117 [trans. 100J). This is an original, and very untraditional conception of symbolism. This conception ofexpressive meaning was intended to supplenlent and complete the traditional conceptions of symbolization as referential or purely significative. Cassirer sought to demonstrate these claims by thought experiments and empirical findings, especially from neurological pathology. But his fundamental philosophical argument was formulated as a phenomenological doctrine, according to which nothing could be perceived as a phenomenon at all unless it exhibited some symbolic function. He ternled this function "symbolic pregnance" (symbolische Pragnanz). Unlike Iunt's notion of schematization, which he called an "art hidden in the depths of the human soul" (Critique ofPure Reason, A 141/B 181) that mediated between the pure forms of understanding and the pure forms of sensibility, symbolic pregnance was a phenomenological doctrine intended to characterize actual perception. It referred to the fact that perception always exhibits an organization, or to be more precise, phenomena exhibit different kinds of symbolic value: expressive, representative, or purely significative. Cassirer defined cultural "symbolic forms" as the result of intelligent activities: "Under a 'symbolic form' should be understood every energy of thought by means of which an intellectual meaning content is connected to a con-
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crete sensory sign and inwardly attributed to this sign. "11 But his definition of "symbolic pregnance" does not appeal to subjective or intersubjective activities at all. Symbolic pregnance is simply "the way in which a perception as a sensory experience contains at the same time a certain non-intuitive meaning which it immediately and concretely represents. "12 Cassirer cautions that it can be reduced neither to merely reproductive processes or to mediated intellectual processes-it must ultimately be recognized as an independent and autonomous determination, without which neither an object nor a subject, neither a unity of the thing nor the unity of the self ,vould be given to us. (Cassirer [1929J 2002,273 [trans. 235J) He concludes that this "relatedness of the single perceptive phenomenon, given here and now, to a total meaning"13 is a "genuine apriori, an essentially first factor."14 In other words, symbolic pregnance is the transcendental element in Cassirer's philosophy. The notion of symbolic pregnance is a "genuine apriori" because the inherent relational nature of phenomena it describes is, to use the IZantian distinction, constitutive and not regulative, for it concerns "the manner in which son1ething is apprehended in appearance" (Critique of Pure Reason, AI7S/B221). Yet symbolic pregnance, considered logically, is not a category. It is a triadic relation: the (1) n1anner (die Art) in which a perceptual experience as (2) something sensory (sinnlich), concretely represents (3) a particular n1eaning (Sinn). This "symbolic pregnance" can have three different values: expressive, representative, and significative. This "Bedeutungslehre" takes the place of IZant's transcendental logic. Symbolic pregnance, I hope to show, offers the key to Cassirer's understanding of validity in the cultural sciences. Beyond this, the entire edifice of Kantian terminology is absent fron1 Cassirer's work. Terms such as "reason" (Vernunft) or "understanding" (Verstand) are rarely used and then only because Cassirer refers to thinkers who rely upon them. "Reason," Cassirer once remarked, is a vague term which only makes sense when we provide its "differentia specifica" (Cassirer [1932J 2003,6 [trans. 6J). This is the thought behind the philosophy of syll1bolic forn1s: "reason" in language, in art, in science, really means different things. Cassirer drops the distinctions between reason and understanding and between the "zwei Stiimme" of knowledge, the sensory and understanding, because the sensory itself is not a simple given but en1bodies expressive syn1bolic pregnance. By developing this point we have sought to show how it is possible to meet Friedman's first criticism of Cassirer: that he does not satisfactorily explain how validity in the mathematical natural sciences and in the nonmathematical cultural sciences are related to one another. The link is to be found in his concept of "symbolic pregnance." His second criticism still ren1ains: that Cassirer does not show how validity in the non-n1athemati-
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cal cultural sciences can survive comparison with the former, which offer universal intersubjective validity. I shall now address that criticism.
4. Cassirer on Objectivity in the Cultural Sciences Unlike Carnap, Cassirer did not think that cultural phenomena were simply a matter of subjective feelings. The validity of cultural science cannot be based upon Emotivism, and Cassirer criticized this view repeatedly, with reference to Carnap (Cassirer 1995, 119-22 [trans. 122-26]), Julius I
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vate feelings without general validity, for they are recognizable in depictions as well. Cassirer explained his conception of the objective validity of expressive meaning this way: 'Objectivity' is found here, as in nature, among other things with temporal endurance, in a permanence of form and in the continued effectiveness of form (self-development of form). Every objectified expression displays this particular kind of permanence-Here we meet with the Warburgian problem, the problem that Warburg recognized as the central problem of all cultural sciences-the independence of the 'language of forms'; once a particular 'Pathos formula' (expressive formula)17 is created, it then retains a kind of independent 'endurance' (Bestand), it does not disappear with the passage of time, it unfolds, undergoes change; but it remains nonetheless 'the same'-it becomes a kind of 'entity' (Wesenheit) in the history of culture, which, like every entity, makes itself known only in its continued effects, the values it evokes, and its transformations. 18 For Warburg, the repetitive imitation of gestures marked the beginning of human culture-ritualized action (cult, dance )-not language. Cassirer followed Warburg and focused upon what he called myth as a Lebensform or "form of life" (Cassirer [1925] 2002, part 3). Warburg's conception of the transmission of "pathos formulas" through the history of image-making (i.e., poses and other visual formulas giving expression to particular emotions in pictorial art) was generalized by Cassirer to include the study of the "formation and transformation of cultural figurations" of any kind. The mathematical natural sciences and the non-mathematical cultural sciences therefore both depend upon the use of objective symbolisms, one based upon notational systems, the other upon gesture, depictions, or other symbolism using configurations (such as music). The fact that the mathematical natural sciences attain high exactitude does not mean that the non-mathematical cultural sciences have no objective validity at all. Their meanings range, however, in their validity, as these contents are reiterated over time and are translated from one culture to another. As Cassirer explained it, We find "permanence," objectivity in this metamorphosis-Once such a "pathos formula" has been coined, then the experience of expression becomes oriented to this formula over and over again for centuriesusing it as an already existing container, into which new contents are put over and over again-and these new contents lead, slowly and steadily, to a transformation of the form, which none the less remains constant in its basic features, in its 'structure. '19
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This is not the place to enter into a discussion of this conception, for the point is that Cassirer sought to give a more definite answer to the question of the comparability of the objects of the natural and cultural sciences than his previously published writings reveal. These formulations provide a way to respond to Friedman's other criticism, namely, that Cassirer does not show how the cultural sciences can be compared with the natural sciences, which offer universal intersubjective validity.
5. Historical Features of Exact Sciences I agree with Michael Friedman's forn1ulation that "the whole point of the philosophy of symbolic forms is that objectivity as such, intersubjective validity and communicability, is by no means confined to the significative function" (Friedman 2000, 126). Friedman, however, finds it difficult to accept Cassirer's contention that his philosophy of symbolic forms actually makes good on this clain1. As Friedman puts it, it is unclear, in particular, how the philosophy of symbolic forms can accomplish what it proposes: "Indeed, since 'objective validity' in general here finds its 'highest' expression in the purely formal significative meaning characteristic of modern abstract mathematics, it is radically unclear how practical, as opposed to theoretical principles can share in such 'objective validity' at all" (Friedman 2000,144). The problem here is not that a particular type of cultural symbolism can or cannot be measured by the standard of mathematical validity, which Cassirer calls the "highest" (meaning the most genuinely universal form of knowledge), but only whether it is possible to share validity at all. There are two ways to link significative and expressive phenomena-to show that there is a practical, historical dimension even in purely significative symbolisms, and to demonstrate the objectivity of the expressive function of symbolism. On the first point: mathematicians today can understand algebraic equations taken from ancient Babylonian n1athematics, but this does not insure understanding them in the way the Babylonians did. Babylonian mathematics did not use an exclusive decimal system, but a sexagesimal one, which has been transmitted to us today in our 60-minute hour and the division of the circle into 360 degrees. When we raise the question why the Babylonians used such a base, we need, on Cassirer's view, to look beyond the sphere of pure significance. According to Cassirer, ancient mathematics was related to astrological beliefs. Cassirer (1940) explored such connections. Astrology personifies heavenly bodies, taking them to possess benign or benevolent powers. Babylonian astrology and mathematics were entwined. 2o Only over time did mathematics become separated from such
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astrological and magical beliefs. The point here is that the history of mathematics is closely linked historically with practical matters and even with expressive phenomena. Mathematical thought attains a level of pure significance only at the end of a long history and not at the beginning. This is even the case in logic. Aristotle is usually said to have created formal logic, but this creation did not really occur until the 19th Century. Logical validity is formal only if the sphere of inference is treated in a su-ictly formal way. But within the history of logic it took considerable effort to distinguish genuine formal validity from material matters. Aristotle introduced the use of variables into logic, yet his inauguration of abstraction into logic was not completely "formal." Aristotle, acknowledged no valid syllogistic inferences in the fourth figure even though they exist, such as in the AEE mood: All Pare M No Mare S > No
S are P.
Aristotle did not fail to see the formal validity of such an argument, but such a syllogism made no sense in terms of his conception of reality. Given that the category of substance guided Aristotle's thought, this concept could not be ignored even in logic. 21 To recognize a fourth figure syllogism like the above demands that, in the premises, we regard a substance (the major term, S) to be the predicate of some characteristic (the minor term, P). That would be an absurdity in Aristotle's metaphysics, yet the fourth figure exhibits such an order. Hence, from the standpoint of Aristotle's metaphysics, there was good reason not to include the fourth figure in logic. While Kant could still claim that logic was complete-a sphere in which there were neither steps ahead nor backwards since the time of Aristotle-the sphere of logic in fact underwent a transformation when Boole, Peano, Peirce, Russell, and Whitehead reinterpreted logic in genuinely formal terms as a system of possible relations. Valid inference was no longer interpreted with any reference to existing things. This transformation of logic stands behind Cassirer's first revisions of Kantianism (Cassirer [1907] 2001 and [1911] 2001). The problem of the validity of non-mathematical cultural forms is not how to unifY or even to integrate them with the kind of validity found in mathematics, but rather to understand how such different kinds of meaning can be related at all-how the expressive, representational, and purely significative functions hang together. This, I take it, is what stands behind Friedman's claim that Cassirer never satisfactorily explains his outlook.
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6. Concluding Observations Friedman's criticism is more fundan1ental than he intends it to be. For he points to the fact that Cassirer left his most fundamental conceptions inadequately expressed. The most important topics in Cassirer's philosophy of symbolic forms are symbolism and form, but one must look long and hard for his explications of these notions (Krois 2002a). His views can be reconstructed, but they are not obvious. In Cassirer's recently published lecture on "Probleme der IZulturphilosophie," he indicates explicitly that the cultural sciences are based upon the concept of "form," not "value" as Rickert claimed (Cassirer 2004, 101). He goes further and claims that philosophers have failed so far to come to grips with cultural concepts because they lacked a proper conception of their "logical place" (logischer Ort). This, he says, is the concept of style: "All cultural concepts are concepts of style" (Cassirer 2004, 103). Instead of seeing particular phenomena in reference to a causal law, they are exemplary of a style. Hence, a particular object-an unknown drawing-is attributed to a time (baroque), a place (Netherlands), an artist (Rembrandt), and a period (late Rembrandt) by a series of style concepts. The point of the cultural sciences is not to subsume particular phenomena under universal value headings such as the good or the beautiful, but to make sense of these particulars, either in regard to their place in history as representatives of particular "forces" or changing styles. 22 The meaning of an individual action (Cato's suicide) or of an object (Raffael's school of Athens) is determined by the extent to which it is possible to relate it to different narratives, or to concepts of style (Cassirer 2002, 14-17). These permit us to "characterize" it as somehow "exemplary" of a style, a political conception, or a worldview (Cassirer 2004, 63f.). This characterization does not aim to derive the phenomenon in question from any law (Cassirer 2004,73), but to elucidate it in terms of its organization, which can be differentiated from comparable styles or worldviews. Hence, historical concepts such as "medieval" or "renaissance" make use of idealized characterizations that allow us to describe a type of historical personality (the typical renaissance man), but the individual figures that illustrate such a characterization (Leonardo, Machiavelli, Cesare Borgia) cannot be subsumed under any kind of law (Cassirer [1942] 2000,70-73). Cassirer says that such notions can be "universally communicable" without being necessary (Cassirer [1942] 2000, 118). Objectively valid knowledge in the cultural sciences depends upon the objective validity of the phenon1ena studied by these sciences: the styles or formulae used. The objectivity of mathematical relations is clear, but the objectivity of specific cultural concepts is always open to debate. These difficulties appear to increase insofar as cultural phenomena involve the trans-
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ll1ission of feelings, which seem to resist even this kind of formal understanding. Yet Cassirer claimed that even expressive phenomena possess objective validity. Cassirer's success or failure in treating the cultural sciences ultimately depends upon his treatment of the objectivity of the expressive function of meaning. This brings us back to symbolic pregnance, which is supposed to mediate between every type of meaning: expressive, representational, and purely significative. Phenomena possess different kinds of symbolic value: qualitative (say, a perceived blue), representational (the color type "blue" as named by the English word), or significative-a scientific object not directly visible to the naked eye (a particular wavelength in the spectrum of electromagnetic waves). The problem is how the first kind of, seemingly private, phenomenon can be objectively valid. Cassirer cautioned that the usual examples of perception in philosophy, such as a "red patch" or "blue spot," are not representative of actual perception. In reality, we normally perceive colors expressively, say as a "cold blue" or a "hot red." Cassirer pointed to empirical research that showed that expressive phenomena are more primitive than the philosophical concept of a simple perception (such as a blue spot) (Cassirer [1929] 2002,76, [trans. 65]). Phenomena such as perceiving a "cold blue" color are actually so commonplace that they go unheeded by philosophers. According to Cassirer, this type of multisensory perception is actually the rule, not the exception-a claim that has been confirmed in the most recent research. 23 Cassirer claimed that the spheres of visual and auditory sensations, of smell and taste, are more closely linked together than we usually recognize because of our tendency to isolate specific 'qualities' in thought (Cassirer [1929] 2002, 41 [trans. 34f. J). Cassirer asserted that the separation of the senses is a derived, not a primitive phenomenon. We only arrive at the data of distinct discrete sensation-such as light or dark, warm or cold, rough or sn100th-by setting aside a fundamental and primary stratum of multisensory perception and eliminating it, for theoretical purposes (Cassirer [1929] 2002, 85 [trans. 73J). This stratum-the perception of expression (Ausdruckswa,hrnehmung)-is composed of phenomena such as what we describe in phrases like a "cold blue," a "sharp odor," or a "deep tone." This kind ofperception, Cassirer stresses, is "at first a mere passivity, a being-acted-upon rather than an acting" (Cassirer [1929J 2002, 88 [trans. 75J). The phenomena themselves appear agitating or soothing, gloon1Y or joyful, pacifying or terrifying. This perception of expression can be diffuse, such as the perception of a mood or emotional atmosphere, or it can be focused, such as in the recognition of a physiognomic character-a sn1iling or frowning face, but it is above all an objective phenomenon. Naturally, an individual with a "lively imagination" is able to see what others do not, recognizing faces in clouds or hearing voices in the moaning of the wind. But others
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could perceive these phenomena as well. Cassirer's point was that the perception of expression as such was a type of symbolic phenomenon with a potential generally validity. Cassirer did not complete his philosophy of the cultural sciences nor have today's cultural theorists managed to develop a science of culture recognizing the objectivity of expressive syn1bolism. The recent reintroduction of the study of feeling in neuroscience and the second generation of cognitive science with its emphasis on embodied intelligence, are perhaps indications that the time has come for an interdisciplinary research program like Cassirer's which sought to link natural and cultural sciences. The general recognition in cultural studies of the fundamental importance of symbolic processes is a step in this direction. Cassirer believed that objective validity in the cultural sciences hinged upon understanding the symbolic character of expression. Many of his writings on the subject have only recently appeared and some the most important, especially his texts on "symbolic pregnance," are still forthcoming (Cassirer in preparation). It is still too early for a final judgment on his success or failure in showing the comparability of the natural and cultural sciences. But I hope to have shown the direction in which to look for a solution to the question: in the concept of symbolic pregnance.
NOTES 1. I want to thank the editors of this volume, Mary Domski and Michael Dickson, for providing me with many helpful criticisms of the first version of this paper. 2. "Werte sind keine Wirklichkeiten, weder physische noch psychische. Ihr Wesen besteht in ihrer Geltung, nicht in ihrer Tatsachlichkeit." 3. "Die Natur erldaren wir, das Seelenleben verstehen wir." 4. "Die Gesamtheit der Objekte, an denen allgemein anerkannte Werte haften, und die mit Riicksicht auf diese Werte gepflegt werden." Cf. also Rickert (1910), 89. 5. For Cassirer's criticism of Rickert, see e.g., Cassirer ([1942] 2000), 37; for his criticisn1 ofHeidegger, see Cassirer (1995), 219-24 [trans. 200-208]. 6. Friedman (2000), 34-37 gives a further account of Cassirer's criticisms of Rickert. 7. Hamlin and I
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Mal festgelegten Bestand und materialen 'Anschauungen' odeI' Begriffen, sondel'll als eine Funktion, die gesetzlich bestimmt ist und die daher in ihrer Richtung und Form mit sich identisch bleibt, die abel' ein Fortschritt del' Erkenntnis inhaltlich die verschiedensten Auspragungen erfahren kann. Ich wurde als apriorisch im strengsten Sinne eigentlich nur del'll Gedanken del' 'Einheit del' Natur' d.h. del' Gesetzlichkeit del' Erfahrung uberhaupt, odeI' vielleicht kurzer: del' 'Eindeutigkeit del' Zuordnung' gelten lassen: wie abel' diesel' Gedanke sich nun zu besonderen Prinzipien u. Voraussetzungen spezifiziert: das ergiebt sich auch nur erst allS dem Fortschritt del' wissenschaftlichen Erfahrung, wenngleich ich auch hier-zwar nirgends starre Schemata, wohl abel' gleichbleibende Grundmotive des Erkennens, d.h. des Forschens und Fragens zu erkennen glallbe." (AWK, inv. nr. 94) 10. "Denn noch weit engel' als es in del' fruheren Darstellung der Fall war, erscheint jetzt fur mich das logische Problem des Begriffs mit dem allgemeinen Bedeutungsproblem verknupft. Nur im Rahmen einer systematischen 'Bedelltslehre' laGt sich, wie mil' scheint, die Lehre yom Begriff zureichend begrunden und vollstandig aufbauen. Was ich jetzt scharfer und klarer zu sehen glaubte als in den Darlegllngen meiner fruheren Schrift, ist dies: daG fur eine solehe 'Bedeutllngslehre' die Mathematik und die mathematische Naturwissenschaft zwar immer ein wichtiges und untentbehrliches Paradigma bilden wird, daG sie abel' deren Gehalt in keiner Weise erschOpft" (Cassirer [1928] 2004, 130). 11. "Unter einer 'symbolischen Form' soli jede Energie des Geistes verstanden werden, durch welehe ein geistiger Bedelltungsgehalt an ein konkretes sinnliches Zeichen geknupft und diesem Zeichen innerlich zugeeignet wird" (Cassirer [1923a] 2003, 15). 12. "die Art ... , in del' ein Wahl'llehmungserlebnis, als 'sinnliches' Erlebnis, zllgleich einen bestimmten anschaulichen 'Sinn' in sich faGt und ihn zur unmittelbaren konkreten Darstellung bringt" (Cassirer [1929] 2002),234 [trans. 202]). 13. "Bezogenheit des Einzelnen, hier und jetzt gegebenen Wahl'llehmungsphanomens auf ein charakteristisches Sinn-Ganzes" (Cassirer [1929] 2002),234 [trans. 202]). 14. "echtes 'Apriori,' als wesensmaGig-Erstes" (Cassirer [1929] 2002, 235 [trans. 203]). 15. "die Ausdruckswahl'llehmllng konstituiert die Welt del' Kultur." 16. This is a fundamental phenomenological distinction for Cassirer. See "The Perception of Things and the Perception of Expression" (Cassirer 2000,34-55). 17. On the notion of Pathosformel (translated as "emotive formulas") seethe index to (Warburg 1999) and in particular "Durer and Italien Antiquity," 553-58, with the illustrations of "the arm raised in defence" formula. For a study of the link between Cassirer and Warburg, see (Krois 2002b). 18. "Die 'Objektivitat' beweist sich hier, wie in del' Natur unter anderem in del' zeitlichen Dauer, in del' Permanenz del' Form und in del' Fortwirkung del' Form (Selbst-Entfalumg del' Form). Jeder objektivierte Ausdruck zeigt diese eigentiimliche Permanenz-Hier setzt das Warbur;gJsche Problem ein, das Warburg als ein Zentralproblem alter Kulturwissenschaften erkannt hat-die Selbstandigkeit del' 'Formensprache'; ist einmal eine bestimmte 'Pathosformel' (Ausdrucksformel) geschaffen, so erhalt sie eine Art selbstandigen 'Bestand', sie geht nicht unter im Strom del' Zeit, sie entfaltet, verandert sich; abel' sie bleibt nichts destoweniger 'dieselbe'-sie wird eine kulturhistorische 'Wesenheit', die sich, wie jede
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Wesenheit, freilich nur in ihren Fortwirkungen) Wertenwirkungen) Umbildungen bekundet-" (Cassirer 2004, 126f.). 19. ''In der Metamorphose erscheint die "Permanenz," die Objektivitat-Ist einmal eine "Pathosformel" gepragt, so orientiert sich das Ausdrucks-Erlebnis von Jahrhunderten immer wieder an dieser Formel-benutzt sie als bereit-stehendes Gefiiss, in das fortschreitend immer neuer Jnhalt gefiil1t wird-und dieser neue Inhalt fiihrt, langsam und stetig, zu einer Umbildung der Form, die aber nichts destoweniger in ihren Grundziigen, in ihrer 'Struktur' beharrt-" (Cassirer 2004, 128). 20. A bibliography of the vast literature on this topic, including the relationship between astrology and algebra, is available from the Warburg Institute in London, at www.sas.ac.uk/warburgjmnemosynejorientation/astrobib/astro_biblio2.htm 21. Aristotle's logic may still be explicated in purely formal terms, as in Patzig (1968). See §25 "Aristotle and the Fourth Figure," 109-118. 22. This is explained in the study, "Concepts of Nature and Concepts of Culture" in (Cassirer [1942J 2000, 56-86). 23. See Calvert, Spence, and Stein (2004). I thank David Poppel (University of Maryland) for bringing this publication to my attention, which deals with normal perception and not just the rare phenomenon of synesthesia.
REFERENCES AWIZ (Archief Wiener lZreis). Rijksarchief in Noord-Holland, Haarlem, Netherlands. All Rights reserved. (Rights to Cassirer's letters are the property of Yale University Press). Calvert, G.A., C. Spence, and B.E. Stein (eds.) (2004). The Handbook of Multisensory Processes. Cambridge, Mass.: MIT Press. ECW (Cassirer, Ernst, Gesammelte Werke) Hamburger Ausgabe), 25 volumes. Edited by B. Recki. Hamburg: Felix Meiner, 1998ff. Cited by volume and the original pagination as given in margins. ECN (Cassirer, Ernst, Nachgelassene Manuskripte und Texte), 20 volumes. Edited by IZ.C. IZohnke, J.M. Krois and O. Schwemmer. Hamburg: Felix Meiner, 1995ff. Cited by volume and the original pagination as given in the margins. Cassirer, E. ([1907J 2001). "IZant und die moderne Mathematik" (WlitBezug auf Bertrand Russells und Louis Couturats Werke iiber die Prinzipien der Mathematik.) J(ant-Studien 12:' 1-49 (ECW 9: AuJsiitze und J(leine Schriften 1902-1921, ed. by M. Simon, 37-82). - - . ([1911J 2001). "Aristoteles und lZant." J(ant-Studien 16: 431-47 (ECW 9, 468-83). - - . ([1923aJ 2003). "Der Begriff der symbolischen Form im Aufbau der Geisteswissenschaften" (ECW 16: AuJsiitze und J(leine Schriften 1922-1926, edited by Julia Clemens, 75-104). - - . ([ 1923bJ 2001). Philosophie der symbolischen Formen) Erster Teil: Die Sprache. (ECW 11, edited by C. Rosenkranz. Hamburg: Meiner.) Translated by
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R. Manheim as The Philosophy of Symbolic Forms) Volume 1: Language. New
Haven and London: Yale University Press, 1953. _ . ([1925] 2002). Philosophie der symbolischen Formen) Zweiter Teil: Das mythische Denken. (ECW 12, edited by C. Rosenkranz. Hamburg: Meiner.) Translated by R. Manheim as The Philosophy of Symbolic Forms: Volume 1: Language. New Haven and London: Yale University Press, 1955. _ - . ([1927] 1978). "Das Symbolproblem und seine SteHung im System der Philosophie." Zeitschrift ~fiir Asthetik und allgemeine I(unstrvissenschaft 21: 295-312, 320-22 (ECW 17: Aufsiitze und I(leine Schriften 1927-1931, edited by Tobias Berben, 253-82.) Translated by J.M. IZrois as "The Problem of the Symbol and its Place in the System of Philosophy," Man and World 11: 411-28. - - . ([1928] 2004). "Zur Theorie des Bergriffs. Bemerkungen zu dem Aufsatz von G. Heymans." I(ant-Studien 33: 129-36 (ECW 17: Aufsiitze und kleine Schriften 1927-1931). - - . ([1929] 2002). Philosophie der symbolischen Formen) Dritter Teil: Phiinomenologie der Erkenntnis. (ECW 13, edited by J. Clemens. Hamburg: Meiner.) Translated by R. Manheim as The Philosophy ofSymbolic Forms) Volume 3: The Phenomenlogy of I(no wledge. New Haven and London: Yale University Press, 1957. - - . ([1932] 2003). Philosophie der Aufkliirung. (ECW Volume 15, edited by C. Rosenkranz. Hamburg: Meiner.) Translated by F.C.A. IZoelln and J.P. Pettegrove as The Philosophy ofEnlightenment. Princeton: Princeton University Press, 1951. - - . ([1939] 2005). Axel Hiigerstrom: Eine Studie zur Schwedischen Philosophie der Gegenwart. Gijtebor;gs Hogskolas Arsskrift, 45: 1-119 (ECW 21). - - . (1940). "Mathematische Mystik und mathematische Naturwissenschaft. Betrachtungen zur Entstehungsgeschichte der exakten Wissenschaft." Lychnos: 248-65. - - . (1995). Zur Metaphysik der symbolischen Formen. (ECN 1) edited by J.M. IZrois. Translated by J.M. IZrois as The Philosophy of Symbolic forms) Volume 4: The Metaphysics ofSymbolic Forms, edited by J.M. IZrois and D.P. Verene. Ne'w Haven and London: Yale University Press, 1996. - - . (1999). Ziele und Wege der Wirklichkeitserkenntnis. (ECN 2) edited by IZlaus Christian IZohnke and J.M. IZrois. - - . ([1942] 2000). The Logic of the Cultural Sciences. Translated by S.G. Lofts. New Haven and London: Yale University Press. - - . (2002). Geschichte. Mythos. (ECN 3) edited by IZ.C. IZohnke, Herbert IZopp-Oberstebrink and Rudiger IZramme. - - . (2004). I(ulturphilosophie: Vorlesungen und Vortriige 1929-1941. (ECN 5) Edited by R. I(ramme unter Mitwirkung von J. Fingerhut. - - . (in preparation). Uber symbolische Priignanz) Ausdrucksphiinomen und 'Wiener I(reis'. (ECN 4). Dilthey, W. ([1894] 1924). "Ideen tiber eine beschreibende und zergliedernde Psychologie." In Gesammelte Schriften, Volume 5, edited by G. Misch. Leipzig and Berlin: Teubner, 139-204.
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- - . ([1900] 1924). "Die Entstehung der Hermeneutik." In G. Misch (ed.) Gesammelte Schriften, Volume 5, 317-38. Leipzig and Berlin: Teubner. Friedman, M. (2000). A Parting of the Ways: Carnap) Cassire1) and Heidegger. Chicago: Open Court. Habermas, J. (1981). Theorie des kommunikativen Handlens, vol. 1: Handlungsrationalitiit und gesellschaftliche Rationalisierung. Frankfurt am Main: Suhrkamp. - - - . (1998). Faktizitiit und Geltung: Beitriige zur Diskurstheorie des Rechts und des demokratischen Rechtsstaates. Frankfurt am Main: Suhrkamp. Hamlin, C. and J.M. I
[12J Ernst Cassirer and Michael Friedman: I(antian or Hegelian Dynamics of Reason? ALAN RICHARDSON 1
It was, I believe, in 1986 that Michael Friedman agreed to do a directed reading course with me in which I first read Carnap's ([1928J 1967) Der logische Aujbau der Welt. He had by that time already published several important essays on the history of logical empiricisn1, including his seminal essay on the concept of objectivity in the Aujbau, in which he began to stress the neo-IZantian aspects of that work. Since that time Michael has become the leading figure in a growing scholarly literature on the re-evaluation of the philosophical projects of logical empiricism. I like to think that I have contributed in some measure to this literature, especially regarding the philosophical projects of Carnap's Aujbau and Carnap's early philosophy in general. In this work, I view myself to be furthering an interpretative line on Carnap's work that is very much in line with the main themes of Friedman's own interpretative work. SOlnething like this account of Carnap's early work has become a going alternative; sometimes it is even referred to as "the new received view" of Carnap interpretation. 2 While it is satisfying to hear such talk, I do not think that the neolZantian-influenced Carnap is really the new received Carnap; nor is Friedman's nuanced reading of logical empiricisn1 in general really the newly received logical empiricism. I feel a bit of dissatisfaction in reading recent literature about Carnap or about logical empiricism. One dissatisfaction comes from reading the work of senior philosophers who seem to ignore the whole literature, stretching over more the two decades now, on the re-appraisal of logical empiricism. On the basis of such work, one would think that the received view of logical empiricism had, by now, attained the status of a fixed dogma. On this received view, which we may associate especially with Quine's various accounts of the Aujbau, logical empiricism was an attempt to fulfill the promise of phenomenalist reductive empiricism through the new means of symbolic logic-and the
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Aujbau was as close to a success as such a project ever can1e. Perhaps, however, this is just the way of old grey heads. The writing of those who have entered the field of Carnap scholarship within the past two decades and have recently responded to the readings of logical en1piricism offered by Friedman is, at once, more substantial (since it engages the recent reinterpretations) and more dissatisfying. In reading much of this literature, I have a nagging sense that the important philosophical point of stressing the neo-lZantian elements of Carnap's philosophical background, the quasi-Kantian structure of his constitutional project in the Aujbau, and the lingering after-effects of these in his whole philosophical corpus has been importantly missed. Often, all the effort in recent contributions seems to be spent in finding some way of insisting that Carnap or the Aujbau is empiricist after all, as if Friedman's reading (or my own) was committed to there being no significant empiricism in the book because of its neo-lZantian elements. 3 While it may be true that Friedman and I have at various times overstressed the neo-lZantian elements of logical empiricism and of Carnap's work in particular, it has never been the case that the point of our readings is that neo- IZantianism is the whole of the story of early logical empiricism. 4 What is philosophically most rewarding, in my opinion, about Friedman's work on the history of logical empiricism is, in fact, the way in which Friedman resists telling any simple stories. Thus, I read Friedman's work on Carnap as complicating and making more interesting Carnap's place in the stories we tell of the "empiricist tradition" in analytic philosophy. Carnap's logical empiricism, on this account, relies importantly, but in con1plicated ways, on the trajectory of German neo-IZantianism. Byattending to these ways, Friedman provides resources for understanding both Carnap and empiricism in the 20th century more generally-as well as for doing philosophy of science here and now. Thus, one is certainly free to argue with Friedman about this or that passage of the Aujbau. We can even, should we wish, invent a new version of en1piricism such that the Aujbau is unproblematically empiricist (and we can do that six times before breakfast)-but we shouldn't lose sight of the key lessons of Friedman's work in the history of logical empiricist philosophy of science: the ways that the resources of formal logic are marshaled for philosophical purposes by Carnap; the methodological lessons drawn by Reichenbach, Schlick, Carnap and others regarding scientific concept formation from the development of relativistic physics; etc. These philosophical attitudes and theses among the logical empiricists stand in interesting and complicated relations to a variety of projects in epistemology and philosophy of science circa 1920. One cannot understand the systemiatic significance of the projects of logical empiricism by relying on a pris\tine empiricist tradition that links the ~~~_~~p_a~~_I-I~!J:l~,_~£I1~t_MacJl) _
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Bertrand Russell, and Carnap but excludes Immanuel IZant, Hermann Cohen, and Ernst Cassirer. I would not want to deny anyone their right to find the pure empiricist Carnap of 1928 despite all this detailed and specific historical scholarship; I would think, however, that a more interesting project than inventing a new empiricism would be to further complicate Friedman's (and my own) accounts: we should know more about the relations of early logical empiricism and the phenomenology of Edmund Husserl and Oskar Becker, the idiosyncratic positivism ofTheodor Ziehen, the historically-minded neo-IZantian Platonism of the young Alexander IZoyre and the French historians of science he influenced and was influenced by, etc. 5 Why would these things be important to know? My own answer is that such a detailed philosophical history of analytic philosophy provides a n1uch richer set of resources upon which to draw in thinking about how properly to do philosophy here and now and to explain why one has the philosophical projects one does have. If my memory serves, I learned to think this way from Friedman himself. It is certainly a way of thinking about the relations of philosophy of science and its recent history that I see on the pages of Friedman's recent work. Friedman writes, for example, in the introduction to A Parting ofthe Ways (2000) about Cassirer's "heroic" attempts to mediate the radical projects of Carnap and Heidegger, and he continues the thought by saying that attending to Cassirer "provides us with new possibilities and renewed motivation for n1aking a similar heroic effort for ourselves" (2000, xii). Friedman's own Dynamics of Reason (2001) is, of course, exactly such an heroic attempt-an attempt not directly to mediate Carnap and Heidegger, of course, but rather to use the resources of neo-IZantian thought to give an account of the objectivity of science in a way that respects the philosophical concerns and n1ethods of logical empiricism and phenomenology, even existential phenomenology. This is all fairly abstract. This abstractness reflects my sense that being asked why one ought to attend to the history of philosophy when doing philosophy an10unts, really, to being asked why one ought to understand what one is doing-I run out of things worth saying rather quickly; Rather than continue to trade in abstractions, I want to spend the rest of this essay attending to a larger issue in the history of post-IZantian philosophy that Friedman's work on the lingering influence of neo-lZantianism raises. As one might expect from issues that arise from taking IZant and neoIZantianisn1 seriously, the issue is rather complicated-even stating the issue is rather complicated. The issue concerns both the status of a priori principles in an explanation of objective knowledge and the status of transcendental explanations of the a priori in general, especially in the context of relativized or variable a priori elements. Thus, an adequate treatment of the issue would involve the interplay of several layers of philosophical
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explanation within the broadly I
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to disentangle it from the particular resources that Carnap employed in the elaboration of it. Thus, a large part of the task of Dynamics ofReason is to make plausible the idea that there are principles that playa constitutive a priori role while not tying this to a technical project in metalogic and semantics. So far, the Carnapian epistemological vision has pride of place, but the really crucial question for Friedman con1es after we have made this initial move. A variable and relativized a priori-whether given in a panoply of Carnapian formalized languages or in some other way-raises the specter of relativism. What are the reasons we move from one set of a priori principles to another? These reasons cannot, if we are dealing with the a priori in any meaningful sense, be standard empirical reasons: a priori principles cannot be falsified and new ones cannot be better able to meet the tribunal of experience. Carnap's own response to this sort ofissue was an invocation of "the pragmatic." Only when a formalized language and a confirmation theory were in hand were the genuine questions of empirical confirmation and disconfirmation even well defined; questions of whether to adopt this or that language were, thus, not questions in which confirmational status could be properly appealed to. These "external questions" answered only to issues of expediency, ease of use, and other pragmatic values. Friedman finds such an answer to relativism to be inadequate. Although Friedn1an's n10st explicit arguments for this inadequacy are within the IZuhnian, not the Carnapian, idiom, they can be easily retooled to serve as responses also to Carnap. 7 First, there are serious doubts that any list of so-called pragmatic values to govern choice of linguistic framework have been permanently endorsed or employed in the history of science. Second, the arguments the proponents of one framework would make would have to make reference to phenomena as understood and expressed in that framework-and in the Carnapian case it is provable that there is no universal meta-framework within which all such claims can be made. One cannot without remainder express what speaking in the other framework would be like while speaking in the first framework. Third, our philosophy of science is meant to explain the success of science as a set of meaningful claims about the world, but the move to purely pragmatic considerations in the account of the movement from one linguistic framework to another threatens to obviate that robustly epistemological problem and replace it with a purely practical question of the use of a technical instrument with increasing precision and accuracy. Friedman's unhappiness with the pragmatic response to the relativism inherent in the logical empiricist account of the relativized constitutive a priori is made clear in his choice of title, Dynamics ofReason. In essence, the Carnapian (and IZuhnian) account of the transition from one framework (paradigm) to the next precisely lacks a rational dynamics. There are
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transitions in the history of science but there is no account of rationality in hand to explain adequately why that transition is rational-the moments of most fundamental change in science occur without a probative logos. Faced with this, Friedman makes two moves. First, he endorses a notion of communicative rationality, borrowed with some interpretative liberty from Jurgen Habermas, to solve the problem of the instrumentalism he finds in the pragmatic response to relativism. Second, and with this we will finally return our discussion to Cassirer, Friedman endorses a move from relativized constitutive a priori principles to universal regulative a priori principles as the proper locus of principal episten1010gical interest. The key passage is this one: Since deep conceptual revolutions or paradigm-shifts are a fact of scientific life (and, I would argue, a necessity), we are never in a position to tal<:e our present constitutive principles as truly universal principles of human reason-as fixed once and for all throughout the evolution of science. We can imagine, however, that our present constitutive principles represent one stage of a convergent process, as it were, in that they can be viewed as approximations to more general and adequate constitutive principles that will only be articulated at a later stage. We can thus view our present scientific community, which has achieved ten1porary consensus based on the comrnunicative rationality erected on its present constitutive principles, as an approximation to a final, ideal community of inquiry (to use an obviously Peircean figure) that has achieved a universal' trans-historical communicative rationality on the basis of the fully general and adequate constitutive principles reached in the ideal limit of scientific progress. Indeed, we must view our present scientific comn1Unity as an approximation to such an ideal community, I suggest, for only so can the required inter-paradigm notion of comn1unicative rationality be sustained. (Friedman 2001,64-65) It is this movement from relativized constitutive principles to universal regulative principles that Friedman claims is the fundamental significance of Cassirer's "universal invariant theory of experience" (cf. Friedman, 2001, 65ft). I have already elsewhere (Richardson 2003) argued that this move importantly fails as a response to conceptual relativism. In particular, it seems to me that the limited communicative capacity across paradigms that Friedman notes has already been expressed without theganz deutsch philosophical gloss by IZuhn in this passage: The man who premises a paradigm when arguing in its defense can nonetheless provide a clear exhibit ofwhat scientific practi~~_!0-1Lh~lil~~
_
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for those who adopt the new view of nature. That exhibit can be immensely persuasive, even compellingly so. Yet, whatever its force, the status of the circular argument is only that of persuasion.... To discover how scientific revolutions are effected, we shall therefore have to examine not only the impact of nature and of logic, but also the techniques of persuasive argumentation effective within the quite special groups that constitute the community of scientists. (Kuhn [1962] 1996,94) Moreover, this passage seems to point in a more compelling direction than does Friedman's account: if we wish to understand scientific revolutions we should look at the theory ofpersuasion. Persuasion can be rational even when it cannot be captured in a formalized framework. This lesson does not require highly idealized theories of peculiarly rational communication such as we find in Habermas; Aristotle already had an articulated and informal theory of rational persuasion in his rhetoric. Nor would any plausible theory of persuasion that is meant to apply to actually existing communities of scientists require the idea of a community at the end of inquiry to serve as the ideal placeholder for the not-yet-articulated final constitutive principles of science. Friedman here seems in the grip of a highly developed but, on the face of it, quite implausible philosophical vision of the nature and requirements of human rational communication. Rather than further engage in this debate of the philosophical merits of Friedman's position, I wish to change gears and ask after the plausibility of deriving such a position from Cassirer. The key to Friedman's interpretation is the Marburg neo-Kantian slogan that the object of knowledge is not given (gegeben) but merely set as an incompletable task (aufgegeben). This view of the object of knowledge leads Cassirer to speak of the progress of knowledge in terms that recall some of the terms in which Kant separates the regulative employment of the ideas of reason from the constitutive employment of the concepts of the understanding. Thus, Kant says this of the hypothetical unity of reason: Systematic unity (as a mere idea) is only a projected unity, which one must regard not as given in itself, but only as a problem [Problem]; this unity, however, helps to find a principle for the manifold and particular uses of the understanding, thereby guiding it even in those cases that are not given and making it more coherently connected. (Kant [1787] 1998, B675) A few pages later Kant says more clearly that the hypothetical unity of reason is a necessary demand if we are even to make sense of the empirical employment of the understanding and the possibility of empirical truth:
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The law of reason to seek unity is necessary, since without it we would have no reason, and without that, no coherent use of the understanding, and, lacking that, no sufficient mark of empirical truth; thus in regard to the later we simply have to presuppose the systematic unity of nature as objectively valid and necessary. (IZant [1787] 1998, B 679) Friedman is, no doubt, correct to find echoes of such IZantian claims in Cassirer's work. It sounds very much as if Cassirer is saying that only the regulative idea of final constitutive principles that are conditions of all scientific experience (all linguistic frameworks or paradigms) secures the possibility of the historical development of knowledge and, thus, the rationality of changes in constitutive principles. Indeed, Friedman exploits something of this idea in his arguments that the new paradigms evolve "gradually" from the old ones: We can exhibit the historical evolution by which the new concepts and principles gradually emerge through successive transformations of the old concepts and principles. Practitioners of succeeding paradigms are not helpfully viewed as members of radically disconnected speech communities' as speakers of radically diverse languages, as it were, such that only intensive in1mersion in an entirely foreign culture can possibly take us from one paradign1 to another. On the contrary, successive paradign1s emerge precisely from one another, as succeeding stages in a con1mon tradition of cultural change. (Friedn1an 2001, 60) Now one n1ight object that Friedman thinks he can exhibit this only because he looks at only a small subset of the radical breaks in the history of science that IZuhn wishes to account for. It is hard to see how non-mathematical cases work for Friedman: what would be an intermediate stage between the Earth being at rest at the center of the universe and the Earth being a celestial body? But this is not the most telling worry from an interpretative point ofview. 8 We can approach that problem by asking what role the allegedly necessary ideal endpoint plays in the construction of the intermediary stages that ultimately secure for Friedman the communicative rationality of paradigm change. I must confess that I can find no such role beyond an inchoate and ultimately misbegotten desire an10ng post-revolutionary scientists to be intelligible to the holders of the former paradigm. What, rather, Friedn1an's own account of the rationality of paradigm change indicates is that revolution occurs through small steps in which particular portions of the conceptual framework of a science are called into question and rejected in favor of some other concepts. This point of view is also associated with an account of knowledge in which the objects of knowledge are given only as a task to be completed in a final end stage of
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knowledge; this view, however inchoately and abstractly specified, is Hegel's notion of specific negation: What is self-contradictory does not resolve itself into a nullity, into abstract nothingness, but essentially only into the negation of its particular content, ... , and therefore the result contains that from which it results ... Because the result, the negation, is a specific negation, it has a content. It is a fresh Notion but higher and richer than its predecessor; for it is richer by the negation or the opposite of the latter, [and] therefore contains it, but also something more, and is the unity of itself and its opposite. (Hegel [1812-1816] 1969,54) This understanding of negation is, for Hegel, "all that is necessary to achieve scientific progress" (Hegel [1812-1816] 1969,54). Hegel's system posited an Absolute in which the objects of knowledge were revealed as what they are and in which Thought reveals itself finally in its intellectual nature. Thus, there is an ideal endpoint of knowledge on this view. But, that endpoint is not the motive force in the dynamics of reason-that, rather, is given by inconsistencies or incoherencies in the content of knowledge and a general set of logical principles for overcoming such inconsistencies and incoherencies. We only can know that we have not reached the endpoint of reason because there continue to be such epistemic problems in our knowledge of nature. Indeed, one might say that for the Hegelian, the linear progress of knowledge is secured by the synthetic process by which knowledge of the world is achieved. The goal is then the ideal end state of a process of knowledge that is already guaranteed methodically to be linear, because synthesis starts from given inconsistencies in current concepts and achieves a more consistent state. Stated another way, the law of the development of knowledge guarantees a single end state to the history of knowledge. This bears comparison to Kant's own use of the regulative a priori in our knowledge of the world. The principles of pure understanding were, for Kant, already established as the constitutive a priori principles. This provides an a priori guarantee that the world of experience can be continuously unified-thus, the postulated unity of the world demanded by reason cannot be falsified and must, in the progress of knowledge, be ever more closely approximated. Given his acceptance of the variable constitutive a priori, the regulative a priori plays, for Friedman, a seemingly larger role than for Kant himself. In the absence of Kant's own regulative demand of final and ultimate constitutive principles, there may be no unity in the historical development of science. For Friedman, rational unity of historical development is not methodically guaranteed a la Hegel nor guaranteed to be approximated ever more closely via a prior and permanent
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synthetic unity uncovered by transcendental reflection. It becomes, then, for Friedman a regulative demand for communicative rationality. Now, it is difficult to determine how much of Cassirer's views are attributable to Hegelian as opposed to I
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The Marburgers may have, in the details of their philosophies, imported SOlne Hegelian elements whatever their anti-Hegelian ambitions. In the absence of a robustly transcendental perspective that allows, in the manner of IZant himself, the precise delin1itation of the a priori elen1ents of knowledge, the distinction between a properly IZantian transcendental account of reason and a properly Hegelian dynamics of reason is difficult to draw. Indeed, it might be Cassirer's place in the history of German philosophy to render the important methodological dispute between the neo- IZantians and the Hegelians otiose by offering a philosophical vocabulary that elides the key differences. It would be wise to add a bit more detail and further to specify what is at stake in this interpretative question regarding Cassirer's work. If Friedman's account of Cassirer were right, one would expect, given how well Cassirer knew the IZantian corpus, that he would draw attention to the constitutive/regulative distinction and explain the infinite task of specifying the object of knowledge in terms of the IZantian regulative a priori. Such explanations are, however, scarce in Cassirer's work. Especially striking is the paucity of such talk in Cassirer's account of the theory of relativity, since this is one of those revolutionary n10ments that sets Friedman's relativist problem. The historical remarks in Cassirer's monograph on Einstein's theory of relativity do not present Einstein's views as particularly revolutionary at all but rather as presenting sin1ply the highest stage in, so to speak, the functionalization of experience. One passage in Cassirer's monograph might seen1 to speak unequivocally and definitively in favor of Friedman's account of Cassirer's epistemology. Chapter One of the monograph considers the historical development of the concept of physical measurement. In concluding this chapter, Cassirer does speak the language of the regulative a priori: In the midst of the change of the particular theoretical instruments of measurement, the critical theory holds fast to the thought of the unity of measurement, which indeed signifies no realistic dogma but an ideal goal and a never-to-be-concluded task. Each new physical hypothesis erects, as it were, a new logical system of coordinates, to which we refer phenomena, while nevertheless the doctrine is retained as a regulative idea for investigation that all these systems converge on a definite limiting value. (Cassirer [1921] 1953, 366) Here the ideal limit seems to play some dynamical role: only through the demand for such a limiting value can steps be taken in the historical development of measurement. It is, however, not clear that this is Cassirer's considered view. Elsewhere in the monograph Cassirer speaks language that is more
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Hegelian, in which the endpoint is posited because of an independently established lawful development of the concept of form. For example, in Chapter Five he describes the mutual determination of matter and form in the critical theory of space and time. In this discussion he waxes much more Hegelian: In this way, the sensuous manifold increasingly loses its "contingent" anthropon10rphic character and assumes the imprint of thought, the imprint of the systematic unity of form. Indeed "form," just because it represents the active and shaping, the genuinely creative element, must not be conceived as rigid, but as living and moving. Thought comprehends more and more that form in its peculiar character cannot be given at one stroke, but that the existence of form is only revealed to it in the becoming of form and in the law of this becoming. In this way, the history of physics represents not the discovery of a simple series of "facts," but the discovery of new and more special means of thought. (Cassirer [1921] 1953, 421) And here the posited final unity is guaranteed by "the unity of those methodic principles" by which form is revealed in history. In this context, it is worth noting that volume three of Cassirer's magisterial Philosophy of Symbolic Forms was entitled "the phenomenology of knowledge" and that he employed the term "phenomenology" self-consciously in an Hegelian, not Husserlian or Heideggerian, meaning. In his introduction to volume three, Cassirer waxes poetically Hegelian-again with the effect that the end is guaranteed by the autonomous dynamics of reason, rather than the dynamics of reason being beholden to an already always posited end: The truth is the whole-yet this whole cannot be presented all at once but must be unfolded progressively by thought in its own autonomous movement and rhythm. It is this unfolding which constitutes the being and essence of science. The element of thought, in which science is and lives, is consequently fulfilled and made intelligible only through the movement of its becoming. (Cassirer [1929] 1957, xiv) Self-conscious knowledge as the lawful becoming of thought through the stages of human thought, Cassirer tells us, is what we must understand "if we wish to understand knowledge not so much in its result, in its mere product, as in its character of a process, in the mode and form of procedure itself" (Cassirer [1929] 1957, xv). If I am right that Friedman underplays the Hegelian elements of Cassirer's thought, then it follows that Friedman underplays the Hegelian
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elelnents of his own views. Thus, for example, while Friedman would certainly not use the language of "negation" to discuss the movement from one conceptual scheme to the next, he agrees with the general Hegelian point that the latter conceptual scheme in some sense "contains" the earlier one. This is not a point evident in Carnap or I(uhn, nor is it one that I(ant is even \vorried about. Cassirer agrees with Friedman, but offers two visions of this "containment," one more IZantian and one more Hegelian. On the first view, it is the necessary demand of reason for final constitutive principles that induces the directionality of the history of scientific con~epts and theories and, thus, of the containment relations. On the second view, it is the methodical unity of the laws of the generation of form that induce the containment and, thus, posit the final endpoint. The key difference is that in the I(antian version, the posited endpoint somehow plays a dynamical role and in the Hegelian story it does not. Friedman wishes the endpoint to playa role in the dynamics of reason. His own account of the intermediate stages of conceptual development, the stages by which one paradigm "naturally evolves" into the next, however, does not seem to bear out the claim that there is robust final causation in the realm of knowledge .11 Sorting out these admittedly rather murky issues of the relations of IZantian and Hegelian versions of the ultimate unity of knowledge will not merely help us understand and evaluate Friedman's work, however. It will help us further to understand the nature of the logical empiricist reaction to the German epistemological traditions of their time. For, whatever the ultimate disposition of Cassirer's version of the unity of reason and the unity of nature, the whole language of the demands of reason seems to have been importantly what the conventionalist maneuvers of the logical empiricism were reacting against.1 2 Neither the regulative ideal of reason in its empirical employment nor the inherent laws of the developn1ent of reason in history were endorsed by the logical empiricists. Here as elsewhere, attention to the work of Michael Friedman, both its lessons and its problems, can only help to further our understanding of the import of philosophical work of the past two hundred twenty-five or so years, our own work included.
NOTES 1. I would like to thank Mary Domski and Michael Dickson for the idea and the reality of this Festschrift for Michael Friedman and for improvements they have wrought in this essay. My thoughts in this paper have been stimulated, at times unknowingly, by conversations I have had with Abe Stone and David Sullivan. Of
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course, my biggest debt is to Michael Friedman, whose work has been the primary stimulus to my own thinking for over a quarter century. 2. Many of Friedman's essays on the history of logical empiricism are reprinted in Friedman (1999); the essay on the Aujbau referred to above is Chapter Five of that work. For recent volumes in the burgeoning literature, see, for example, Parrini, Salmon, and Salmon (2003), Hardcastle and Richardson (2003), and Awodey and I
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REFERENCES Awodey, S. and C. Klein (eds.) (2004). Carnap Brought Home: The View from Jena" Chicago: Open Court. Bonk, T (ed.) (2003). Language, Truth, and KnOJvledge: Contributions to the Philosophy ofRudolf Carnap. Dordrecht: Kluwer. Carnap, R. ([1928] 1967), The Logical Structure of the World and Pseudoproblems in Philosophy. Translated by R.A. George. Los Angeles and Berkeley: University of California Press. Cassirer, E, ([1921] 1953), Substance and Function and Einstein's Theory of Relativity. New York: Dover. - - . ([1929] 1957). The Philosophy of Symbolic Forms. Volume Three: The Phenomenology ofKnowledge. New Haven, CT: Yale University Press. Friedman, M. (1999), Reconsidering Logical Positivism, Cambridge: Cambridge University Press, - - , (2000). A Parting of the Ways. Chicago: Open Court, - - . (2001). Dynamics ofReason, Stanford: CSLI Publications. Hardcastle, G,L. and A.W Richardson (eds.) (2003). Logical Empiricism in North America. Minneapolis: University of Minnesota Press. Hegel, G.W ([1812-1816] 1969). Science of Logic. Translated by AV Miller. Atlantic Heights, NT: Humanities Press International. Hudson, R.G. (1994). "Empirical Constraints in the Aufbau." History of Philosophy Quarterly 11: 237-51. Kant,1. ([1781/1787] 1998). Critique of Pure Reason. Trans. P. Guyer and A. Wood. Cambridge: Cambridge University Press, Kuhn, TS. ([ 1962] 1996). The Structure ofScientific Revolutions. 3rd ed, Chicago: University of Chicago Press. Mancosu, P and T.A. Ryckman (2002), "Mathematics and Phenomenology: The Correspondence between 0, Becker and E. Husserl." Philosophia Mathematica 10: 130-202. Moulines, en 2001. "Die Mathematisierung der Erfahrung: Vorganger zu Carnaps Aufbau," Erkenntnis 54: 105-20. Parrini, P, WC. Salmon and M.B. Salmon (eds,) (2003). Logical Empiricism: Historical and Contemporary Perspectives. Pittsburgh: University of Pittsburgh Press. Pincock, C (2005). "A Reserved Reading of Carnap's Aufbau." Pacific Philosophical QJtarterly 86: 518-43. Richardson, A.W. (1998). Carnap's Construction of the World. Cambridge: Cambridge University Press. - - - . (2003). "Narrating the History of Reason Itself: Friedman, Kuhn, and a Constitutive A Priori for the Twenty-First Century." Perspectives on Science 11: 253-74. - - - , (2005). '''The Tenacious, Malleable, Indefatigable, and Yet, Eternally Modifiable Will': Hans Reichenbach's Knowing Subject." Proceedings of the Aristotelian Society supplementary volume 78: 73-87. - - - . (2006), "Freedom in a Scientific Society: Reading the Context of Reichenbach's Contexts." In J, Schickore and F. Steinle (eds.), 41-54.
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Roy, J-M. (2004). "Carnap's Hussedian Reading of the Aujbau." In Awodey and I
[13] From Mach to Carnap: A Tale of Confusion PAUL POJMAN
1. Introduction Michael Friedman and Alan Richardson have gone far to correct our understanding ofRudolfCarnap's Aufbau ([1928] 1967) by arguing that its primary intention is not an empiricist reduction of reality to the given, but rather a project in neo- IZantian epistemology. They have graciously allowed others to sort out the residual role of empiricism, though clearly noting that whatever such influences remain are not from the 'traditional' empiricism of the received view. It is of course Ernst Mach who has traditionally been seen as the empiricist influence. Carnap is seen as continuing the Machian program. Now, just as Carnap was misunderstood, so too has been Mach. The received view of Mach is similar to that of Carnap (that is, Mach is also supposedly attempting to reduce knowledge to experience), and is similarly incorrect. The received views of Mach and the Aufbau are thus connected. However, just as Carnap was not continuing the Machian program, so also Mach never followed the 'Machian' program. The immediate challenge then, one might think, is to get Mach right and then see what the influence of Mach on Carnap actually was. Unfortunately, the matter is even more complicated since Carnap himself apparently bought into the received view of Mach, and indeed may be partly responsible for it. The paper proceeds as follows. In Section 2, I continue Friedman and Richardson's analysis of the Aufbau by examining the remaining empiricist influences. I argue that there is ample evidence that Carnap had a second purpose in mind, one linked to Mach. I then turn to Mach in Section 3, first outlining Carnap's understanding of Mach, and then turning to Mach and arguing not only that the received view of Mach is wrong, but also
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that Carnap held it. Finally, in Section 4, I step back and investigate the actual influences of Mach on Carnap.
2. The Purposes of the Aujbau The neo-J{antian conception Friedman's two articles, "Carnap's Aujbau Reconsidered" (1987) and "Epistemology in the Aujbau" (1992), and Alan Richardson's book CarnapJs Construction of the World (1998), argue that Carnap's notion of objectivity is motivated more by neo-IZantian than traditional empiricist concerns. The standard view is that the main purpose of the Aufhau is to use the results of modern logic (from Russell and Frege) to attempt a Machian phenomenalistic reduction of reality to the given. Answering the question, "How can we have objectivity for science when only the given is certain?" a phenomenalistic reduction attempts to construct the world out of an individual's sensations, which are certain, thus pushing objectivity up. That which can be constructed out of the given is objective and potentially part of science; all else is subjective and not part of science. Friedman argues that the text of the Aujbau does not support this standard view; although Carnap certainly has empiricist influences, he is primarily concerned with a very different idea of objectivity. Most importantly, Carnap goes to great pains to make it clear that the autopsychological basis, the base beginning from private sensations, is only one possible basis for the constructional system. He also considers three other possibilities: cultural, heteropsychological, and physical. Whereas the autopsychological base consists of my private sensations, the heteropsychological includes the sensations of others. The physical base could tal(e many forms, but would loosely be the world as described by physics, and the cultural base consists of higher level cultural properties (of which one is science). These four bases appear to be in principle mutually reducible to each other, but the autopsychological, Carnap believes, holds both special difficulties (since it requires crossing the solipsistic gap), and special importance (since knowledge does begin with experience). The particular constructional systen1 outlined is intended only as an example (see, for instance, §§122, 156), and the notion of objectivity achieved is independent of it (all references to Carnap's AuJbau are to ([1928] 1967)). That is, the same notion of objectivity would have been achieved had he instead chosen a physical or heteropsychological basis. This unimportance of a phenomenalistic base choice to the project clearly contradicts the claim that the Aujbau aims to provide a phenomenalistic reduction. In fact, from the opening pages Carnap's purpose is clear. He intends to articulate an
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objeCtlVlty that is to be had through relation to a structure, and not through ostension. For science to be objective, science must concern itself with the formal relations between objects. Carnap assumes that in principle there is only 'one science', and thus one object domain. So if the logical relationships between objects in that one domain are specified in such a way that each object is uniquely defined, then objectivity is achieved (§§12, 16). Friedman argues this is much closer to neo-Kantian conceptions than empiricist ones, in that neo-Kantians also base objectivity on a form/content distinction. For neo-Kantians: Concepts do not derive their meaning 'from below'-from ostensive contact with the given. Indeed, such merely ostensive contact with the given is the very antithesis of truly objective meaning and knowledge; for objective meaning can only be derived 'from above'-from formal or structural relations within the entire system of knowledge. (Friedman 1987, 529) Richardson develops the complexities of the neo- Kantian thesis, noting also Carnap's differences with this tradition, and possible roles for empiricism. Without disputing the above work, and certainly without resurrecting the view that Carnap was primarily attempting an empiricist project, there is evidence of a secondary purpose, one identified by Carnap as an empiricist one. If we are to understand the influence that Mach did have upon Carnap, we must understand this secondary purpose.
Evidence ofa secondary) empiricist) purpose There are two main reasons for believing that Carnap intended two purposes. First, he says so, and second, for his primary purpose he need not have chosen the difficult autopsychological base. In looking further at these reasons we will also articulate what that secondary purpose was. Carnap writes in his "Intellectual Autobiography": The system of concepts was constructed on a phenomenalistic basis; the basic elements were experiences, as mentioned before. However, I indicated also the possibility of constructing a total system of concepts on a physicalistic basis. The main motivation for my choice of a phenomenalistic basis was the intention to represent not only the logical relations among the concepts but also the equally important epistemological relations. The system was intended to give, though not a description, still a rational reconstruction of the actual process of the formation of concepts. (Carnap 1963, 18; italics mine)
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Another indication occurs when he discusses epistemic primacy in the Aujbau: The system form which we want to give to our outline of the constructional system is characterized by the fact that it not only attempts to exhibit, as any system form, the order of the objects relative to their reducibility, but that it also attempts to show their order relative to epistemic primacy. (§54) And most definitively at the end of Part IV: In concluding the presentation of the constructional system, let us again emphasize what is important in this system outline and what is not. The primary purpose in the formation of a construction system was to illustrate, by way of an example, the actual content of construction theory, namely, to formulate the problems of forming such a system .... A further purpose of the outline was to show that a constructional system of all scientific objects is, in principle, possible, no matter how the details of such a system would have to be formulated. We do not only wish to assert here that it is possible in general to erect some constructional system or another; rathe~ we wish to defend the thesis that it is possible (though peJ;'haps not necessary in all points) to give the following properties to the constructional system) which are also found in the system which we have tentatively outlined . ... (§156; italics mine) The system that was "tentatively outlined" is of course one with an autopsychological basis. Besides direct statements of a secondary purpose, further evidence comes from Carnap's choice of the autopsychological basis over the heteropsychological or physical ones, a choice that entangles him (from the perspective of the primary purpose) in needless complexity. Any of the possible bases would have sufficed for his primary end; why then did he complicate matters by choosing this particularly difficult base? His answer is that the autopsychological orders objects according to their relative epistemic primacy, where "An object (or an object type) is called epistemically primary relative to another one, which we call epistemically secondary) if the second one is recognized through the n1ediation of the first and thus presupposes, for its recognition, the recognition of the first"(§54). The order of epistemic primacy proceeds "from the 'given,' i.e., from experiences themselves in their totality and undivided unity" (§67). It is quickly used as a criterion for selecting the base, choosing any possible psychological over the physical and cultural, the auto- over the heteropsychological, and individual holistic experiences over other possible aspects of the autopsychological.
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Although his empiricism is methodological rather than metaphysical (Carnap clearly distances himself from traditional empiricism and positivism (cf. §§60, 178)), the result of systematically applying epistemic primacy in choosing every aspect of the basic elements is undoubtedly where the "second purpose" of the Aujbau enters. He clearly believes that there is something significant about the autopsychological basis; and it is important that this significance has nothing to do with his primary purpose of illustrating constructional theory in general. He realized that construction theory not only brings forth objectivity on the basis of the formal structure of experience, but also has the potential to cross the epistemological gap surrounding a solipsistic basis. In §64 and §66 Carnap discusses how, although many others (including Mach) have thought that the autopsychological basis cannot form the basis for knowledge, the results of construction theory can overcome these problems: ... it seems almost impossible to proceed from an autopsychological basis to· the cognition and construction of other objects, of the heteropsychological, and of an intersubjective external world.... Since construction theory removes the obstacles, indicating and clearing the way from an autopsychological basis to the heteropsychological and to an intersubjective world (cf. §§66, 140, 145-49), there should be no reason left for adopting any other basis. (§64; see also §§2, 16, and 66.) Carnap is addressing two problems of objectivity. The first, a neo-Kantian conception of objectivity addressing the structure of logical relations, could have been solved with any basis. The second, addressing the question of how to cross the epistemological gap surrounding the solipsistic basis, required the autopsychological base. The important result is that this second problem is solved through the use of the solution of the first. In this sense, the neo-Kantian notion of objectivity is primary. To this we now turn, after which we turn to Mach.
Solutions to both problems achieved simultaneously Let us then look at the steps in the autopsychological basis, and see how Carnap arrives at objectivity, in both senses. Skipping over the preliminary constructions (quality classes, sense classes, the visual sense, visual field places, and colors), the Space-Time world is constructed in §125, and in §126 colors are embedded into it. In §128 and §129, visual things, including 'my body', are constructed, and by §131 the other senses are constructed. Finally in §132 we arrive at 'my consciousness,' on the basis of
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which the other sense qualities (including emotions and volitions) are assigned (§133). In §134 the perceptual world is constructed out of the points of tactile-visual things, and is completed through analogy of causality and substance. In §136 the world of physics is constructed by assigning "physical-state-magnitudes ... to the points of the four-dimensional number space." This level is important as it is only here, and not at the world of perception, that a consistent intersubjectivity can be had. With the world of physics completed, "it is possible to give a definite description of each individual event and each thing that belongs to the world," including humans and other biological organisms (§137). The level of physical objects is now constructed. The heteropsychological don1ain begins with the construction of the psychological content of others. We perceive other persons, and on the basis of primarily linguistic reports assign them a psychological existence analogous to our own (§§141--44). Even though others' psychological states have been constructed, the autopsychological basis is maintained: others have been constructed as a branch within 'my' system. That is, I assign psychological states to certain objects on the basis of their analogous behavior to my own. Carnap designates such a branch 'M'; although M represents another human, this branch is still within my autopsychological field (§145). M, then, is constructed within 'me' and the structure of M is analogous to mine in the sense that both contain complete structures. We will return to this later as Mach makes the same argument when leaving the autopsychological. Although both structures are complete (and thus analogous), they differ in regards to their empirical content: statements made in one system will not always agree with those in the other. "For example, if M has another hair color than I have, then we obtain two different statements about mb [my body] and mb M [M's body] respectively" (§146). In general, this analogy holds for the levels below the space-time world, but breaks down in the physical and heteropsychological domains. It is worthwhile to look closely at why this preliminary analogy holds only for levels within the autopsychological. In a strange reversal of intuition, the reason the analogy holds for 'me' and 'M' within the autopsychological is that the similarity of our structures here is not reliant on the physical content of experience, but rather on physically-uninterpreted private sensations. The empirical content of the physical world is not embedded until the physical level; before this Carnap only constructs the 'structure of experience.' Thus, within the autopsychological, the structures of ,me' and 'M' are the same, while after the autopsychological, once the content of the world (actual colors, chairs, etc.) has been placed within our structures, we will disagree in those many instances where our differing relationships to the world cause the content of our experiences to differ. This of course is the problem of solipsism.
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Canup then points out that another analogy holds where the former fails; although the former analogy between my constructional system, S, and M's constructional system, SM was incomplete from the perspective of content, an isomorphism exists be~een the spatiotemporal worlds of Sand SM. Although the construction of object might differ fi·om M to S due to their different locations with respect to 0, the structural relations within which is embedded are the same for both Sand M. This correspondence is called intersubfective correspondence. Carnap proceeds to apply this intersubfective correspondence to psychological objects; in realist language, my understanding of M's psychological states intersubjectively corresponds to my understanding of M's understanding of them. That is, within M other persons are constructed whose structures in turn intersubjectively correspond to their structures within mine. Furthermore, now that M has access to other's psychological states, intersubjectivity can be extended to the lower levels where previously it did not hold. Although there is still no intersubjective correspondence between my elementary experiences and M's, now there is a correspondence between 'my experiences' and 'my experiences as I understand M to know them' (§147). Carnap equates the set of such intersubjective objects with the domain of science (§149). But what has he achieved? First, Carnap has completed the details of one possible construction system. The domain of science is identified with those objects that are intersubjectively communicable, a result which could have been achieved with the choice of any base. Second, he has solved the problem arising from his choice of the autopsychological base, that the content of my experiences is different from others. Carnap crosses this gap by viewing the given as structure. Thus he uses construction theory in general to generate a construction from an autopsychological base. It is in this sense that the neo-Kantian aims are clearly primary.
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3. Carnap's Connection with Mach One received view of Mach is that he is an empiricist (with phenomenalistic leanings) who wishes to ground knowledge on a foundation of sensational certainty, reducing our knowledge of the physical world to private sensations. The mistake is codified in the Encyclopedia ofPhilosophy, where it is claimed that Mach sought to enable "science to be built on a foundation of certainty" (Alexander 1967, 116). Canup writes in his "Intellectual Autobiography" (1963, 50, 57) that the Vienna Circle (himself included) saw Mach this way, and indeed the text of the Aufbau (§3) suggests the same. In general, I know of no literature on Carnap that disputes Carnap's interpretation of Mach.
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Here things begin to get messy. Although Carnap sees Mach in this way, he also thinks Mach abandons a sensationalist base due to his inability to cross the solipsistic gap when starting from it. That is, in departing from empiricism, Carnap sees that he is now able to solve a problem that empiricism cannot. Carnap writes: "Mach is especially conspicuous for his non-autopsychological basis, since it does not seen1 to be in harmony with the rest of his views" (§64). Later, he offers an explanation of why others have feared the autopsychological: "If the basis of the constructional system is autopsychological, then the danger of subjectivism seems to arise. Thus, we are confronted with the problem of how we can achieve objectivity of knowledge with such a system form" (§66). And again: A system form with an autopsychological basis is acceptable only because it is recognized that science is essentially concerned with structure and that) therefore) there is a way to construct the objective by starting from the individual stream of experience. Much of the resistance to an autopsychological basis (or "methodological solipsism") can probably be traced back to an ignorance of this fact. (§66) Intersubjective knowledge had been thought to be unobtainable when basing knowledge upon a foundation of subjective experience, thus causing Mach to avoid the auto-psychological base, or at least so Carnap thought. Carnap's secondary purpose is to solve this problem that the empiricists, including Mach, were unable to solve. However, the messiness increases, as Carnap is simply wrong here. Not only is Mach not reducing knowledge to a rock bottom base of certainty, but also, he has neither fear nor need of the autopsychological. The rest of this section is devoted to textual support for the above claims about Mach. In Section 4 we will be in a position to properly evaluate Mach's influences upon Carnap. The argument will proceed as follows: (a) Although for Mach knowledge does begin with private experiences, the ego is unstable and cannot be the basis of scientific knowledge. Mach does begin at the autopsychological, but rather than believing that therein lies certainty, he argues precisely the opposite. (b) He uses the same argument from analogy that Carnap will use later to cross from the autopsychological to the intersubjective, and it is here that scientific knowledge lies. (c) Mach, like Carnap, is not committed to any particular base. This provides the best evidence that neither is attempting a phenomenalistic reduction of reality to the given. (a) In the opening pages of Analysis ofSensations ([1886] 1905), Mach offers an extensive analysis of the ego. Physiologically, the ego is a complex of sensations arising out of biological processes (both phylogenetic and
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ontogenetic), responding to the need for economical representation of sense data. There is no actual self. If there is no self, then an important philosophical problem, one that dominated discourse in Mach's era, simply dissolves; questions of the priority of the inner realm versus the outer, the psychological versus the physical, become moot. All that exist are the elements, which, although neither physical nor psychological substances, lie behind both. The elements are not to be confused with psychological elements such as sensations any more than with physical elements such as atoms (we will return to this later). Following Gustav Fechner, these elements can be seen from either of these two perspectives, but are themselves neither physical nor psychological substances. The problem of the relation of the external world to our sensations, Mach argues, only arises once the ego arises; once we understand the ego itself as a complicated sensation, there is no problem of an external world. The problem only exists once we view ourselves as a mental identity separate from physical reality ([1886] 1905, 13-14; all references to ([1886] 1905) below will omit the dates). The very concept of there being an ego apart from the world is deemed scientifically untenable by Mach. This dissolution of the problem of the external world was a central argument for a wide-ranging group of philosopher-scientists, including positivists, monists, and psycho-physicists (Mach, as with others, can be loosely included in all three groups). Fechner (the founder of experimental psychology and the psycho-physical law), Ernst Haeckel (a leading German evolutionist), Ewald Herring (a sense physiologist who propounded an evolutionary updated version of innate ideas), as well as the young William James, all held it. Not only is the ego fictitious, but it is also unstable-changing dramatically as we grow (3). In regards to the received view of Mach, there is no certainty at this level to be pushed up. True, one's experiences cannot be doubted as one's experiences (this is trivial), but they must be intersubjectivized before becoming part of science. Rather than abandon the autopsychological in order to find certainty (as Carnap suggests he does), Mach never seeks certainty but instead seeks a perspective which is immune from future refutations by new data, which is neutral between the various philosophical schools, and most importantly, which is compatible with not only physics but also the new sciences of evolutionary theory and experimental psychology. (b) Thus Mach, like Carnap later, indeed is aiming for a type of intersubjectivity. Mach crosses from the autopsychological to the heteropsychological in much the same way as Carnap: using the argument from analogy. In answering the question of how we assign psychological states to others, both Mach and Carnap assign such states to those physical objects in our field of experience which behave, etc., as we do. We saw this
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in Carnap earlier (§§142-44), and now in Mach. Mach breaks our private sensations into those sensations we label as physical objects (denoted A B C ... IZ L M). Of these physical objects, some (the IZ L M) we call human bodies and assign to them psychological states (a b c). This assignment of psychological states to certain physical objects in our sensational field is made as they behave, etc., as we do. (34) Carnap, it seems, is completely wrong in saying that Mach does not start at the autopsychological. Once the analogy between my and others' psychological states is made, the problem becomes how to coordinate 'my' experiences with those of others. As with Carnap, Mach notes that intersubjectivity arrives by comparing that which remains constant between different observers: The ascertainment of the dependence of the elements ABC on one another, IZLM being disregarded, is the task of natural science, or of physics in its broadest sense. But in reality, the ABC's are always also dependent on IZLM .... Now since many different observers KLM ... TT'L'M' ••• , TT" . Ive d ,we succee d·In e1·Iminanng . . .1.'.1.'- , L" , M" ••• are Invo the accidental influence of the variation of IZLM, etc., and we thus obtain only the element that can be stated as common property, namely the pure dependence of the ABC's on one another. In this process the KLM ..., IZ'L'M' ..., are treated like physical instruments, each with its peculiarities, its special constants, and so forth, from which the results, as finally indicated, have to be set free .... Everything then turns on the ascertainment of equality or identity of the ABC's under like circumstances,-that is to say, under like KLM's,-which comes to saying that everything turns merely on the ascertainment of spatial identities. The kind of quality of the sensations is now indifferent; it is their equality that is alone decisive. And now a single individual suffices to fix relations of dependence which are valid for all individuals. Thus from this point onwards we have obtained a safe basis for the whole field of scientific research,-a fact which inures to the advantage of psycho-physiology as well. (345; see also ([1910] 1970,39) Both Carnap and Mach, then, start from an autopsychological or solipsistic basis, both use perceptions of behaviors to assign psychological states to others, and both try to arrive at intersubjectivity (not certainty). It should be noted, however, that whereas the Aujbau is concerned with the formal construction of knowledge, Mach is describing both how we come, historically, to our understandings, and explaining why they work. (c) Perhaps the most important point to n1ake is that Mach is not committed to any particular base, and as with Carnap later, this demonstrates Mach's profound lack of commitn1ent to idealism, solipsisn1, or reductive empiricism. Although Mach undoubtedly adopts a sensationalist basis,
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such a foundation is not crucial for Mach. He recognizes the possibility of other foundations: ... many readers have found a stumbling-block in what they took, erroneously indeed, to be the general character of my conception of the universe. And, to begin with, I must say that anyone who, in spite of repeated protests from myself and fi.-om other quarters, identifies my view with that of Berkeley, is undoubtedly very far removed from a proper appreciation of my position. This misconception is no doubt partly due to the fact that my view was developed from an earlier idealistic phase, which has left on my language traces which are probably not even yet entirely obliterated. For, of all the approaches to my standpoint, the one by way of idealism seems to me the easiest and most natural. ... I feel it to be a piece of particularly good fortune that Avenarius has developed the same conception of the relation between the physical and psychical on an entirely realistic, or, if the phrase be preferred, a materialistic foundation .... (361-62) This point also comes out in another discussion of Avenarius (50-55), where Mach and Avenarius are seen as agreeing on the relation of the physical and psychical even though Avenarius "starts from the fact that naive realism, 'the natural view of the world,' stands at the beginning of all philosophizing" (51). And also: "When, therefore, speculative philosophers say 'Solipsism is the only logically consistent standpoint,' their utterance is quite intelligible in view of their struggle to reach a closed, all-inclusive, complete system of the universe. To be sure, we ought to add that materialism also is equally consistent ... This is true of all systems" (358). This point is vital. Mach is not committed to a psychological or solipsistic perspective. Mach clearly distances himself from phenomenalism and traditional empiricist reductionism. Just as Carnap's adoption of an autopsychological basis did not commit him to any epistemic position other than a search for epistemic primacy and neutrality, so also for Mach, though the fact that Mach is actively engaged in psychology causes him to write in a psychological language. Indeed the use of the term 'element' instead of 'sensations' is related to this point: "Usually these elements are called sensations. But as vestiges of a one-sided theory inhere in that term, we prefer to speak simply of elements, as we have already done" (22). Elements are meant to refer to the neutral position between idealism and materialism. Mach discusses various approaches to his standpoint. Idealism is not essential to his general program. If materialism were used as the path to the same standpoint, this would not imply that materialism were true.
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Though Mach came to a neutral standpoint through idealism, this does not mean that Mach assumes an idealistic metaphysics. In adopting the idealistic perspective, Mach simply considers this approach the "easiest and most natural." "Easiest and most natural" can be interpreted in a variety of ways, ranging from the connection with evolution and physiology (which in a sense Mach thought were primary to physics, but that is another story), to its capacity to unify the sciences due to its being the common denominator among them. We are obliged to work within a perspective; as humans we have no choice. Mach was not trying to reduce the world; he was trying to account for how our understanding of it has been, indeed still is being, built up. Mach never claims to have, or to be after, certainty. Rather his goal is psycho-environmental stability, as it is such stability that is the necessary foundation for deeper cognitive adaptation. Briefly, he viewed knowledge itself as a product of deep evolutionary processes. Humans were nature's awakening to herself, and science was the human way (and thus nature's) of bringing our earlier understandings of reality (ones evolved in pre-scientific times) in contact with the rigor of science, thus forcing our minds to re-adapt to new evidence. This process was ongoing. Epistemically, the goal of science was to literally change our cognitive structures so we would see the world differently. Included in this new way of seeing the world was an understanding of the mind-dependence offeatures of our reality such as colors, sounds, space, time, matter, etc. It is here that Mach spealcs in a sensationalist language. But he is not arguing that only sensations are real; rather, that some of what we think is real might well only be sensations produced within us by evolution for the pre-scientific purposes of survival. It is the purpose of science to re-exan1ine the very core of our knowledge systems. This may strike the current reader as metaphysics, certainly not expected fron1 a positivist; in Mach's era it seemed supported by the best available science. He saw himself as a psychologist doing a science of science. In regards to the elements, Mach thought that something like his elements could provide this stable basis for science, but he makes it quite clear that his elements are only provisional. This is emphasized even in passages that might otherwise be talcen to indicate strong endorsements of foundationalism. For instance the statement: " ... the elements in question form the real, immediate, and ultimate foundation, which it is the task of physiological-physical research to investigate," is quickly followed by "For us colors, sounds, spaces, tin1es, ... are provisionally the ultimate elements whose given connexion it is our business to investigate" (29-30; italics mine). When Mach talks about the elements as being foundations, he means they have served ontogenetically and phylogenetically as the foundations for further psychological build-up. Mach intends his references to foundational qualities of the elements or sensations to mean that these qualities are, in
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actual fact, the foundations that evolution has wired into us (we are programmed to see the world in these categories), and if scrutinized properly, can be foundations for further adaptation of cognitive structure to environment. We have no choice but to start with them. Therefore, neither Mach nor Carnap commit themselves to a sensationalist basis; the received views of both are wrong on this most significant point.
4. Conclusion Why was Mach so misunderstood? Mach himself was concerned with this question, and provides the answer, one that is amply supported by textual evidence. His point on this matter was always the same: psychologists and biologists understood him, but not philosophers and physicists. In 1913 Mach wrote: The biological theory of knowledge which has alienated some physicists from me to the degree that I neither understand their language nor they mine ... owes its origins to influences from the side of physiology. Happily, the same theory again brings me closer to biologists such as Hering and Eaberlandt, so that I may hope for a more peaceful and general understanding at a future time. (1913, 129) In his well known exchange with Max Planck, Mach again protests against being misunderstood by physicists, even going so far as offering to remove himselffrom its community ([1910] 1970,30-31,37-38). There is much more to be said here, but for our purposes, it is sufficient to establish that we should not be surprised that Carnap misunderstood Mach. So we have a mess. There exist three critical philosophical similarities between Mach and Carnap: their beginning at the autopsychological, the argument from analogy, and their non-commitment to any particular base. Yet Carnap not only does not credit Mach for these ideas, but actually criticizes him for not holding them. I take this to be evidence that Carnap did not take them from Mach, at least not consciously. Furthermore, the supposed similarity between them, that they attempted empiricist reductions of reality to the given, is doubly incorrect. However, Carnap did see Mach as doing this, and if my earlier argument is correct, Carnap sought to address Mach's apparent failure. Despite the obvious irony, we are left with this as the major influence of Mach upon the Aufbau. We have not discussed Mach's anti-metaphysical attitude, his discussions of epistemic neutrality, or his social and political stances, all of which were significant to Carnap and the Vienna Circle. These were certainly influential in a larger sense upon Carnap's thought, and it could be argued
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that they manifest themselves in the Aujbau, especially the antimetaphysical stance. Richardson raises, and argues against, the possibili ty that Carnap really got little from Mach, but mentioned him due to Mach's enormous influence upon German epistemological thought (Richardson 1998, 22). Sin1ilar questions have been raised about the origins of the Ernst Mach Verein as the predecessor of the Vienna Circle. These questions are important in that it is indeed the case that for many Mach became as much a symbol as a thinker (meaning he was discussed more than read). But there is an1ple evidence that Carnap and the Vienna Circle in general took Mach quite seriously, and presumably read him as well. The major story to tell is that even though Carnap misunderstood Mach on details, he nonetheless attempted to address a problem he thought Mach (and others) were unable to solve. Perhaps the best demonstration of the viability of his system was to solve a problem of Mach's. He did not need to do this, but certainly it is natural and proper for new projects to address current issues. In addition, Mach's stature may have even necessitated it. There is a story about the origins of twentieth-century analytic philosophy and logical positivism in which one of the icons, Rudolf Carnap, is seen as continuing an epistemological project of an earlier hero, Ernst Mach. Both parts of this story were endorsed by Carnap himself: the part about Mach in Carnap's early writings and the part about himself years later. The story is largely wrong about Carnap, and even more wrong about Mach. Of the following questions-what Carnap was doing, how Carnap understood Mach, and what the influence of Mach upon the Aujbau was-only the second has been properly understood in the last century. And although others have properly assessed Carnap's understanding of Mach, we have seen that in fact Carnap misunderstood Mach. And to make matters worse, the n10st important philosophical similarities between them are precisely the points of Carnap's misunderstandings.
REFERENCES Alexander, P. (1967). "Ernst Mach." In Edwards (ed.),Vol. 5: 115-19. Blackmore, J. (ed.) (1992). Ernst Mach-A Deeper Look: Documents and New Perspectives. Boston Studies in the Philosophy of Science, Volume 143. Dordrecht: I
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_ . (1963). "Intellectual Autobiography." In P.A. Schilpp (ed.), The Philosophy of RudolfCarnap, 3-84. Chicago: Open Court Press. Edwards, P. (ed.) (1967). Encyclopedia of Philosophy. New York: Macmillan publishing Friedman, M. (1987). "Carnap's Aujbau Reconsidered." Nous 21: 521-45. _ - . (1992). "Epistemology in the Aujbau." Synthese 93: 15-57. Mach, E. ([1886] 1905). The Analysis ofSensations and the Relation ofthe Physical to the Psychical. Translated by C.M. Williams. Originally published as Die Analyse der Empfindungen und das Verhiiltnis des Physischen zum Psychischen Gena). New York: Dover Publications. - - . ([1910] 1970). "The Guiding Principles of My Scientific Theory of Knowledge and Its Reception by My Contemporaries." In Toulmin (ed.), 28-44. - - . ([1913] 1992). "Selbstbiographie." In Blackmore (ed.), 122-32. Richardson, A. (1998). CarnapJs Construction of the World. Cambridge: Cambridge University Press. Toulmin, S. (ed.) (1970). Physical Reality. New York: Harper & Row.
[14] Quine's Objection and Carnap's Aufbau THOMAS RIClZETTS 1
In "Two Dogmas of Empiricism" (1953), Quine famously criticizes Carnap's Aujbau. According to Quine, Carnap ain1s to show how the statements "of significant discourse" (1953, 39) are translatable into a sense-datum language whose logical apparatus includes a standard set theory. Quine admiringly allows that Carnap succeeds in the first stage of the project: the definition of a variety of sensory concepts applicable to an idealized, representative cognizer. From this stage of the project, we have a temporal series of two-dimensional visual fields, two-dimensional arrays of color-qualities. The construction of the perceptual world begins with the assignment of color qualities to the spatio-temporallocations based on the distribution of color qualities over the visual fields. In this way, we construct the colored surfaces of what will become the bodies in our cognizer's environment. This assignment is fundamental for the rest of Carnap's construction. Carnap states ingenious canons to guide this assignment, which Quine epitomizes: ... qualities should be assigned to point-instants in such a way as to achieve the laziest world con1patible with our experience. (1953, 40) These canons, Quine claims, provide "no indication, not even the sketchiest, of how a statement of the form 'Quality q is at x;y;z;t' could ever be translated into Carnap's initial language of sense-data and logic" (Ibid.). Quine's surrounding discussion suggests that there will be a number of assignments of qualities to locations that satisfy Carnap's canons. Generalizing Duhem's observations about hypothesis-testing in science, Quine urges that we should not expect physical object statements to be individually translatable into a sense-datum language. He concludes that in Aujbau, Carnap does not achieve his stated goal; the rigorous character of
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Carnap's valiant attempt gives very good reason to think this goal to be an unachievable philosopher's fantasy. In his 1961 introduction to the reissue of Aufbau, Carnap himself accepts Quine's criticism of that book: One of the most important changes [in my position since I wrote Aujbau] is the realization that the reduction of higher-level concepts to lower-level ones cannot always take the form of explicit definitions; generally more liberal forms of concept introduction must be used. Actually, without clearly realizing it, I already went beyond the limits of explicit definitions in the construction of the physical world. For example, for the correlation of colors with space-time points, only general principles, but no clear operating rules were given (§127). (1961, xx; 1967, viii.)2 These retrospective remarks are puzzling. First, Carnap is clear in Aujbau on the difference between proper explicit definitions and Hilbert-style implicit definitions. He emphasizes that his definitions are to be explicit definitions. 3 Second, Carnap acknowledges in Aujbau that, after moving beyond the autopsychological realm (the sensory experiences of a representative individual) to the middle and upper levels of the construction (the perceptual, physical, and social worlds), he no longer gives explicit definitions. He notes in §122 that with the presentation of the lower levels of the constitution system the presentation of "constitutional definitions in a logical language" breaks off. The next section opens with the disclain1er: The further constitution-levels we do not put in rigorous logical-symbolic form, but only in imprecise paraphrases [andeutender Umschreibung]. (§123) Nevertheless, Carnap expresses confidence that these paraphrases give very good reason for thinking that the concepts which figure in our knowledge of the perceptual and physical worlds are explicitly definable by the application of logic to sensory concepts. He says that his sketch shows that such a construction "is in principle possible, no matter how the details of such a system would have to be formulated," and indicates that what difficulties there are here are owed more to difficulties and unclarities in the empirical sciences than to unsolved logical problems (§156). How then in 1928 and again in 1961 did Carnap evaluate the points Quine raises in his objection? To approach this question, we need a better understanding of Carnap's project than Quine gives. In particular, we need an understanding of the adequacy standards Carnap stipulates for his logical constructiQ!l~ __:w._s~~ __
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tion I, I briefly characterize Carnap's Aufbau project. I then go on to consider the underdetermination of the external world by the autopsychological realm on which Quine's objection turns. At the time of Aufbau, Carnap has fully embraced Duhem's and Poincare's basic point about the underdetermination of physical theory by observation. I argue that Carnap can and does assimilate the underdetermination of the perceptual by the autopsychological to Duhem-Poincare underdetermination. The question now is whether there is any construction of the perceptual world from the autopsychological that satisfies Carnap's adequacy standards. In section II, I suggest that Carnap's confidence in the constructability of the perceptual and physical worlds is based on a faulty understanding of higher-order quantifiers. In a word, Carnap conflates assertions of set-existence with assertions of set-definability. In section III, I note that once this error is set aside, we see an underlying continuity from the 1920s onwards in Carnap's view of the relation of observation to physical theory. I suggest that this continuity explains Carnap's retrospective remarks about Aufbau.
I Quine assimilates Aufbau to empinclst reductionism. Roughly put, according to empiricist reductionism, statements about material bodies (the external world) are equivalent in meaning to complicated statements setting forth the patterns in sensory experience that would verifY the statement. C.l. Lewis, at the height of his powers and reputation at Harvard, and Quine's teacher and colleague, was a prominent exponent of this approach. However, Michael Friedman and Alan Richardson, among others, have persuasively argued that Aufbau is not a version of empiricist reductionism, that it owes more to anti-empiricist neo-Kantian theories of knowledge. On this approach, Carnap aims in Aufbau to give a sort of logicized neo-Kantian account of the possibility of objective knowledge on the basis of subjective experience. 4 For my limited purposes, what is most important are Carnap's conception of and standards for logical construction. Carnap's constitution systems are the sites for his constructions. A constitution system begins with a formal language. Carnap takes the simple theory of types (STT) as the logical framework for his constitution systems. He assumes that, via the logicist reduction, classical mathematics is built into every constitution system. 5 This logical framework is fleshed out into a language by the specification of a domain of individuals together with primitive properties and relations of them. I will call such an extension of SIT a Carnapian formalism. In Carnap's eyes, the non-logical statements of the fleshed out framework are about these individuals. He calls the classes of individuals,
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the classes of classes of individuals, etc., that come with the logical framework quasi-objects; no statement is genuinely about them. 6 This feature of Carnap's rhetoric makes him in places sound like more of an empiricist reductionist than he in fact is. 7 A constitution system is a formalism adequate for the expression of extant scientific knowledge. To erect a constitution system, we begin with a Carnapian formalism and show how the concepts that figure in special sciences can be defined within it. The constitution system is the series of definitions that gives us a family tree (Stammbaum) of the concepts of the sciences. (See §§1-2.) Carnap himself frequently spealcs not in the formal mode, but in the material mode. He talks not of defining given vocabulary within a Carnapian formalism, but rather of constructing visual fields, the perceptual world, the physical world, and so on. I will follow Carnap's usage in my exposition. We must, however, bear in mind that this material mode rhetoric has to be redeemable in the formal mode, in terms of definitions within a Carnapian formalism. Usually, this translation poses no special problems: to effect it, we need only consider the formalization in a constitution system of material mode description of set theoretic structures which Carnap identifies with visual fields, etc. 8 By means of the definitions of the concepts (i.e., the vocabulary) of the sciences, extant knowledge can be expressed in the constitution system. Carnap believes that there are genuinely different constitution systems with different bases that provide equally legitimate arenas for the expression of extant knowledge. So Carnap's standard of definition cannot imply sameness of extension. What then do we rely on when deciding whether a group of definitions in a Carnapian formalism captures the distinctions made by some body of colloquial vocabulary? What justifies the identification of a series of definitions in a Carnapian formalism as a constitution systen1? Carnap invokes no notion of synonymy here. In §50, Carnap distinguishes the truth-value of a sentence, its logical value, from its epistemic value (Erkenntniswert). As the terminology suggests, epistemic value corresponds to Frege's sense, which Carnap in Aujbau takes to be merely psychological. Carnap says that the transformation of a colloquial statement into a constitution system statement must preserve its truth-value, but need not preserve its epistemic value. Carnap expands on this theme in §161, in which he emphasizes that it is the truth-value of statements, not the reference of their parts, that sets the standard for constitution system construction: Strictly spealcing, in science we cannot speak about the essence of an object . . . . Only in a certain improper (uneigentlich) sense does an object have an essence, an object-name a meaning (Bedeutung) .... Strictly spealcing, the question must not be phrased as "What is the
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meaning (Bedeutung) of this object-sign?" but rather as "Which sentences in which this sign can enter are true?" We can make an unambiguous assessment only of the truth or falsity of a sentence, not of the meaning ofa sign, not even an object-sign. 9 I suggest that Carnap's core standard for constitution-system construction is systematic preservation of truth-value. He assumes possession of a comprehensive, firmly established body of knowledge, including scientific laws. We can think of this body of knowledge as regimented into (first-order) quantificational notation, although this is not the way Carnap talks. Relativizing quantifiers in colloquial knowledge as necessary, the aim is to associate exu-a logico-mathematical vocabulary of colloquial knowledge with the vocabulary defined in the constitution system so that colloquial truths are associated with constitution system u-uths. This association in effect assigns interlocking type-levels to the vocabulary in the regimented colloquial knowledge statements. Constitution system consu-uction can then be viewed as a matter of demonstrating how colloquial knowledge is, in the fashion indicated, interpretable within a Carnapian formalismJo It is important for Carnap's purposes that systematic preservation of the truth-value of statements, not the reference of their parts, be the adequacy standard. Just this feature of Carnap's view allows there to be distinct constitution systems with different bases. In order both to illustrate what a constitution system is and to exhibit the insights about knowledge that constitution systems promise, Carnap outlines a particular constitution system in parts III and IV of Aufbau. This constitution system has as its basis a lifetime's stock of momentary, total sensory experiences of a representative cognizer and the relation of recollected similarity over these elementary experiences. The order of definitions in the system is to reflect a relationship of epistemic primacy. The basic idea here is that knowledge of statements involving lower-level vocabulary is the basis for knowledge of statements involving the higherlevel vocabulary. Carnap presents in considerable detail the lowest level of his constitution system, the provisional definition of vocabulary for sensory concepts. He accompanies these definitions with paraphrases in colloquial 'realistic language' of the distinctions the formal definitions are supposed to capture. To link the formal definitions with the colloquial terms, Carnap introduces the language of fictitious constructions. Carnap imagines an idealized representative cognizer first having a lifetime's worth of elementary experiences, and then ordering these experiences by the relation of recollected similarity. That is, Carnap imagines the representative cognizer, equipped with names for the lifetime's stock of elementary experiences, determining the truth-values of atomic sentences asserting that one
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elementary experience is recollected similar to another, determining, as Carnap puts it, the pair-list of the relation of recollected similarity. Carnap then describes procedures by which the representative cognizer forms classes of elementary experiences, classes of these classes, relations over these classes of classes, etc., on the basis ultimately of the pair list for recollected similarity, and this way constructs the autopsychological realm. These fictitious procedures are to help Aujbau readers appreciate how the formal definitions capture the intended distinctions as well as to present an ordering of knowledge of the autopsychological that rests everything on logic and the pair list of the relation of recollected similarity. The presentation of knowledge of the autopsychological as the result of these procedures gives us what Carnap calls a rational reconstruction of this knowledge. (See §lOO.) As I mentioned, Carnap does not give explicit definitions after the construction of the autopsychological realm. As regards the rest of the constitution system, Carnap aims only to make plausible the definability of the non-sensory vocabulary of science in the system. In moving to the construction of the perceptual world, Carnap assumes that the logic built into a Carnapian formalism permits construction of a four-dimensional manifold representing the spatia-temporal continuum. Construction of the perceptual world begins with an assignment of colors to spatio-temporal locations. The colors are to be assigned on the basis of their distribution across the temporally ordered visual fields in accordance with canons that Carnap describes in §126. The details of the canons won't concern us. In large part, Carnap's canons for the assignment are stated in mathematicallogical terms formalizable inside the constitution system. In §127, Carnap describes in colloquial 'realistic language' the aspect of perceptual cognition each formal constraint captures. It is important to note that the canons do not constitute a fictitious constructive procedure, like those specified in connection with the construction of the autopsychological realm. At several points, the canons require the exercise of judgment to trade off maximization of different desiderata. Hence, Carnap's canons as they stand do not dictate a unique assignn1ent of colors to locations, even given the visual fields. This is especially true as regards the subsequent extension of the assignment of colors to unobserved locations on the basis of analogy and testimony, and the revision of the assignment in light of scientific laws. As I observed, Carnap is fully aware that his canons fall short of a definition. Why does Carnap think that his discussion should convince his readers that an assignment of color qualities to locations on the basis of visual fields is definable in his sample constitution system? We are to imagine our representative cognizer 'painting' locations on the basis of the distribution of colors over her visual fields, exercising judgment so as to satisfy the canons of §126. Carnap's purposes require only
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that the logician be able to describe whatever assignment we imagine the cognizer to produce. There is no need for anything more general, no need, for example, for the logician to capture the cognizer's method by describing a function from series of visual fields to color assignments over locations. l l In particular, even a brute enumeration assigning colors to locations that in fact satisfies the canons of §126 would suffice as a basis for the construction of the perceptual world. I will return in section III to consider the issue of definability. I want now to consider the underdetermination of the perceptual world by the autopsychological realm. On Carnap's view, no autopsychological sentence about visual fields will be deducible from perceptual world sentences. 12 In this sense, in Carnap's outlined constitution system, physical object statements do not logically imply phenomenal 'verification conditions.' Furthermore, if there is one definable assignment of colors to locations based on a series of visual fields that satisfy Carnap's formal constraints, there will presumably be others. Nevertheless, early on in Aujbau, Carnap says of constitution systems generally: In a constitution system, all objects are constituted from certain basic objects, but in a stepwise construction. It follows from the constitution on the basis of the same basic objects, that statements about all objects are transformable into statements about these basic objects .... (§41) Carnap returns to this theme at the end of Aujbau: Logically speaking, it's like this: only when an object is constituted from the basic objects, do the previously made statements about it become in the strictest sense scientific statements, as the rule for translating statements about it into statements about the basic objects (namely about relations over elementary experiences) give such statements a verifiable sense. Verification means: testing by experiences. (§179) How is the underdetermination of the perceptual world vis-a.-vis the autopsychological realm compatible with these strong sounding reducibility claims? To address this issue, we need to consider Carnap's constitution of the physical world in §136. There Carnap advocates the constitution of the physical world "as a pure world of numbers (als einer reinen Zahlenwelt)" (§136). The pages in "Physikalische Begriffsbildung" Carnap cites at the conclusion of §136 elucidate this idea. Carnap proposes to represent General Relativistic space-time as a four-dimensional manifold, and he tentatively assumes that there are ten fundamental physical magnitudes. The
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physical world can then be identified as a class of 14-tuples. The first four numbers are the coordinates of a space-time point; the next ten numbers are the values for the ten fundamental magnitudes at that point. Physical laws are mathematical generalizations setting forth functional dependencies of values for the magnitudes at related locations. Summarizing this approach, Carnap says: The consideration of physical events as a four dimensional space-timeworld is, so to speak, the transformation of the physical consideration of an event (ein Geschehen) into the geometrical consideration of a rigid structure. The consideration of a physical event just mentioned as a set of 14-tuples carries the formalization a step further: here the physical consideration is transformed into the arithmetical consideration of a particular number system. (1926, 58£) Carnap goes on to explain that on this approach to physics proper "only numbers of a particular number system and their mathematical relations are spoken of" (1926, 59). This mathematical structure is related to the perceptual world via a many-one physico-qualitative correlation of values for fundamental physical magnitudes and sensible qualities: while each ntuple of n1agnitude-values is associated with a unique package of sensible qualities, the same sensible qualities may be associated with distinct ntuples of magnitude-values. This correlation together with a correlation of the points in the manifolds representing the space-time continua of the perceptual and physical worlds relates the physical world to the preceding levels of the constitution system. Carnap's treatment of the constitution of the physical world in §136 is very sketchy. He says little more than I have just said, and refers his readers to earlier papers, especially "Uber die Aufgabe der Physik und die Anwendung des Grundsatzes der Einfachstheit" for a fuller treatment. In this paper, Carnap follows many of his contemporaries in rejecting a pure empiricism: "Philosophy has long made known that the construction of physics cannot rest only on experimental results but that non-empirical basic principles must be brought to bear as well" (1923, 90). Carnap holds that different geometries plus physics are compatible with experience, and urges that a choice an10ng these alternatives should be made on grounds of simplicity. This appeal to simplicity to choose among empirically equivalent alternatives introduces a conventional element into the construction of physics. Indeed, Carnap calls his position "critical conventionalisn1" (1923, 94).1 3 Alan Richardson suggests that the critical conventionalism of ".A..ufgabe der Physik" stands in tension with the reductionism of Aufbau, with the Aujbau thesis that all statements are transformable into statements con-
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cerning elementary experiences (Richardson 1998, 182).1 4 I, however, see no tension here. In both pre-Aufbau papers and Aufbau itself, Carnap forswears defining fundamental physical magnitudes in terms of sensible qualities, emphasizing that the coordination of these qualities with physical magnitudes is one-many. Nevertheless, the constitution system Canup sketches appears to meet Carnap's formal requirements. The vocabulary that figures in the statement of physical theory in the constitution system is just the vocabulary of pure mathematics; and this vocabulary, Carnap believes, is definable in any constitution system. Moreover, once the vocabulary for physical magnitudes and for sensible qualities is in place, Carnap foresees no problem with expressing coordinations of these magnitudes and qualities inside the system. There is one place where the constitution system Carnap sketches does not fully conform to his advertisements for it. I noted that the order of the definitions of the autopsychological constitution system is to reflect a relationship of epistemic primacy over statements involving the defined vocabulary. The vocabulary of fundamental physics in the constitution system is logico-mathematical vocabulary. No ordering in its definition reflects the dependence of our knowledge of physics on our knowledge of the perceptual world.1 5 This dependence is instead captured in the constitution system by the coordination of physical magnitudes with sensible qualities. Let's return to the issue that prompted consideration of Carnap's construction of the physical world, namely, the underdetermination of the perceptual world by the autopsychological realm. I see no incompatibility between this underdetermination and Carnap's reductionist claims. First and most importantly, I observed how Carnap concedes an underdetermination of the physical world by the perceptual world, appealing to overall simplicity to guide the constitution of the physical world. Carnap should take the same attitude toward the decisions and trade-offs involved in applying the canons of §§126-27 to assign colors to locations. Simplicity guides our choice among assignments that otherwise satisfY Carnap's canons, so that the choice among these assignments is a matter of convention. Indeed, I see no reason not to attribute this view to him, although he does not discuss the matter. My description of Carnap's Aufbau viewpoint is a bit oversimplified. Following Friedman (1999, 161), I take Carnap to envision the prospect of revising the construction of the perceptual world in light of the construction of the physical world, including the removal of isolated contradictions generated by the coordination of sensory qualities and physical magnitudes. I suggest that in Aufbau, Carnap generalizes the application of the maxim of greatest simplicity to apply to the entire body of empirical knowledge beyond autopsychological knowledge. 16
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II So far I have assumed that it is possible in principle to state definitions that satisfy Carnap's standards for constituting the perceptual and physical worlds. But is it? Let us begin by considering the definition of the is at relation that assigns colors to locations. As I presented matters, our imagined cognizer paints the world on the basis of her visual fields by application of Carnap's canons; the logician reproduces this assignment, perhaps by brute enumeration. It is, however, difficult to see how any such assignment defined in this way will satisfy Carnap's adequacy standards for constitution systems. I justifiably take certain existential generalizations to be true, even though I take no instance to be true. So, for good reason, I take there to be red things in Tibet, even though I have via experience or testimony no knowledge of any particular red things there. A constitution system that provides expression for my perceptual knowledge will then have to assign the color red to some locations in Tibet. So some constitution system statements will be true that correspond to colloquial statements of whose truth-value I am ignorant. Nothing is amiss here. However, the danger exists that there will be constitution system truths corresponding to statements that I hold false. My rhetoric of "painting the world" insinuates that our representative cognizer assigns colors only to some restricted portion of space-time, even when testimony and analogy are taken into account. That's why the logician will be able by brute force to reproduce the assignment within the constitution system. But then the constitution system will verify a sentence corresponding to "There are no red things farther than distance D from the earth," a statement that I may for good reason take to be false, even though I have no knowledge of any particular red thing at that distance. To get around this problem, we might extend the original color assignment by some general stipulation that plops a red surface down every so often. But once again, there will be a truth within the constitution systen1 about the regular occurrence of red things at great distances that corresponds to a colloquial sentence that I for good reason reject. We have here a serious problem for Carnap's outlined autopsychologically- based constitution system. Cognizers extrapolate that unobserved regions are similar to comparable observed regions, and sometimes express their conclusions by existential generalizations. They do not, however, know how these existential generalizations are true: they lack knowledge of their instances. Such extrapolated existential generalizations must be transformed into constitution system existential generalizations about the perceptual world. The definition of the color assignment must determine how these existential generalizations are true. It is difficult to envision a definable color assignment that does this across the board without giving
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some constitution system sentence the opposite truth-value from the one that cognizers for good reason assign to its colloquial counterpart. Parallel difficulties arise for Ca.rnap's strategy for constructing the physical world. I noted how Carnap proposes to constitute the physical world as a mathematically definable structure. Suppose we represent physical magnitudes as functions over a structure representing Relativistic spacetime. Informally, we use letters "f", "g", etc. for these functions. But this won't do inside the constitution system. There we have to define in mathematical terms particular functions representing the values of the physical magnitudes over all of space-time. The defined functions must satisfY the conditions imposed by the general laws and, via the coordination of physical magnitudes and sensible qualities, by the perceptual world. It will not do, however, to identifY physical magnitudes with definable functions satisfYing these conditions, for arbitrary stipulations of values for the functions within these conditions will likely run afoul of things we have good reason to believe. The problem here is essentially the same one I raised with the constitution of the perceptual world.!7 In "Dreidimensionalitat des Raumes und Kausalitat," Canup shows himself to be aware of the underlying obstacle to the mathematical definition of the structure that is to constitute the physical world. Carnap discusses deterministic laws by which a state of the world at one time determines the state of the world at another, and says: This dependency relation (Bedingungsverhdltnis) (and not only concerning the entire infinite state of the world) is a purely logical relation that in this form cannot be applied in the practice of physical science. For in each finite region of space, there are infinitely many arguments and so infinitely many values of state-magnitudes. Even putting to one side the practical impossibility of determining these infinitely many values, it is in principle impossible (grundsi:itzlich unmoglich) to state them, since it does not concern a lawful function (da es sich ja nicht um einegesetzmaflige Funktion handelt). (Carnap 1924, 121) Carnap's point is that the infinity of values of actual physical magnitudes over any sufficiently large region will vary too irregularly to be captured by any mathematically definable function. What happens to this point in Aufbau? Carnap more or less understands consistency in terms of satisfiability, and satisfiability in terms of the truth of higher-order existential generalizations. 18 He assumes the consistency of the mathematical and empirical conditions that extant knowledge imposes on the constitution of the physical world, and so assumes the existence of a class of 14-tuples satisfYing these conditions. This existence claim is expressible by a higher-order existential
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generalization inside the constitution system. Moreover, I maintain that Carnap believes that any true existential generalization has true instances expressible in the constitution system. A definition of a class of 14-tuples satisfying all the conditions on the physical world can be extracted from any such instance. Any such definition can then be employed to constitute the physical world. Of course, we may not be able on the basis of the existential generalization actually to formulate a definition that constitutes the physical world; but the definability of the class is all Carnap claims in Aujbau. The constitution of the perceptual world, the assignment of colors to locations, can be treated similarly. On this understanding of Carnap's views, there may be distinct, incompatible autopsychologically- based constitution systems that are equally adequate rational reconstructions of extant knowledge. We may have for example an existential generalization asserting the existence of a class of 14-tuples that satisfies the conditions that extent knowledge imposes on the constitution of the physical world. There n1ay be instances of this generalization that specify distinct classes of 14-tuples, i.e., classes that represent the physical magnitudes as taking different values at the san1e location. Moreover, these differences do not reflect different applications of the principle of greatest simplicity, for this principle is applied only in arriving at the conditions formulated in our existential generalizations. However, I see no reason for Carnap to balk at this consequence. His aims require only the existence, not the uniqueness, of an autopsychologically- based constitution system. This interpretation of Carnap's understanding of issues of definability at the higher levels of his sample constitution system turns on his cloudy understanding of higher-order quantifiers. The story here is con1plicated. I will present only a sketchy and somewhat speculative account. In the 1920s, Carnap largely follows Russell in logical n1atters. In Principia Mathematica, Whitehead and Russell contextually define talk of classes in terms of existential generalizations over propositional functions. On this basis, Russell treats classes as a logical fiction, denying that they are "part of the ultimate furniture of the world" (1919, 182).Carnap follows Russell here, talci.ng propositional functions to be basic, and talk of classes to be introduced by definition in terms of propositional functions. (See §§27, 28,32, and 33. See also Carnap 1929, §8.) Carnap's calling classes "quasi-objects" reflects Russell's influence. Along with other scholars, I tal<e the propositions and propositional functions of Principia to be nonlinguistic, non-mental, non-spatial entities. 19 The contextual elimination of classes in favor of propositional functions thus represents no ontological savings; rather, it exhibits the interpretation of an ontology of extensionally individuated higher type entities in terms of an ontology of nonextensionally individuated higher type entities. Nevertheless, in Principia,
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propositional functions appear to be introduced as linguistic entities. (See Whitehead and Russell, 14 and 38). Moreover, in Introduction to Mathematical Philosophy, Russell's explanation of propositions and propositional functions is emphatically linguistic and mental: We mean by a "proposition" primarily a form of words which expresses what is either true or false. I say "primarily," because I do not wish to exclude other than verbal symbols, or even mere thoughts if they have a symbolic character . . . . A "propositional function," in fact, is an expression containing one or more undetermined constituents, such that, when values are assigned to these constituents, the expression becomes a proposition. (Russell 1919, 155£)20 I believe that Carnap instinctively and unreflectively fell into a linguistic understanding of propositions and propositional functions. I further suggest that Carnap did not think of propositional functions as designating extra-linguistic items,21 and accordingly gravitated toward an implicitly substitutional understanding of higher-order quantification. This comes out in Carnap's 1928 manuscript Untersuchungen zur allgemeinen Axiomatik. In this manuscript, Carnap defines and investigates various properties of theories formalized within the framework ofSTT. As Awodey and Carus (2001) explain, the metalogical concepts Carnap deploys are not our standard ones that figure in Godel's completeness and incompleteness theorems. 22 For Carnap, SIT is itself the background theory in which the investigation is conducted as well as the language in which theories are formalized. There is then no distinction between object-language and metalanguage. Furthermore, as Carnap uses high-order quantification to frame his metalogical notions, there is also no distinction between syntax and semantics. Canup defines a concept of a model of a theory, and of isomorphism of interpretations using higher-order quantification. In his alleged proof of the Gabelbarkeitssatz, he assumes that if a theory has nonisomorphic models, then non-isomorphic models are specifiable for it in STT. In the context Carnap has established, this assumption is a special case of the assumption that any true higher-order existential generalization has a true instance. 23 Although Carnap follows Russell in the 1920s in adhering to a type theoretic approach to logic, he rejects Russell's ramified theory oftypes in favor of the simple theory of types, there by gaining the power of impredicative higher-order quantification and avoiding the need for the axiom of reducibility for the logicist reduction. (See Carnap 1929, §9, 21.) However, the admission of impredicative higher-order quantifiers makes a substitutional construal of those quantifiers viciously circular. The strains in Carnap's hazy conception of higher-order quantification catch up with
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him in 1932. By then Carnap, working on what would become The Logical Syntax of Language, was trying to define "analytic" for a language with higher-order quantifiers suitable for classical mathematics. Carnap's attempted definition-in effect a truth definition for the logico-mathematical parts of the language-treats the first-order quantifiers over natural numbers substitutionally. He tries to treat second-order quantifiers substitutionally as well. In correspondence with Carnap, Godel points out that impredicativity renders Carnap's definition viciously circular, and that imposition of a requirement of predicativity renders the language too weak for Carnap's purposes. Carnap's initial response to Godel exhibits the difficulty Carnap has in conceiving of undefinable subclasses of numbers as within the range of his second-order quantifiers. 24 In all this, Carnap is not simply careless and confused. He does not approach matters with the array of metalogical concepts that emerge with Godel's completeness and incompleteness results. 25 Carnap's writings show how these concepts may fail to be salient to someone approaching logic from the vantage point of Frege's and Russell's work, even to someone like Carnap, who was by the mid-1920s well informed about developments in the field. 26
III I have maintained that Carnap has in AuJbau a sophisticated appreciation of the empirical underdetermination of physical theories emphasized by Poincare and Duhem. Carnap's response to this underdetermination is his critical conventionalism. I suggested that in AuJbau, Carnap, in his attitude toward the constitution of the perceptual world, extends his critical conventionalism from physical theory to everyday knowledge. This point about everyday knowledge sounds Quinean, but Quine himself freely acknowledges Carnap's priority.27 I argued that Carnap's critical conventionalism is compatible with the enterprise of constitution system construction-with the interpretation of extant knowledge within a Carnapian formalism via series of definitions for the vocabulary of the sciences. I offered an explanation of why Carnap thought that constituting definitions for the perceptual and physical worlds were available in his outlined autopsychologically- based constitution system. In conclusion, let's return to Carnap's 1961 introduction to the republished AuJbau. I noted that Carnap's remark that he unwittingly went beyond the limits of strict definition in the discussion of the constitution of the perceptual world does not fit the text of AuJbau. I think we can, nevertheless, make good sense of Carnap's retrospective discussion here of the AuJbau constitution of the perceptual and physical worlds.
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Carnap says nothing in "Aufgabe der Physik" about the form that coordinations of sensible qualities and physical magnitudes take in the constitution system. He begins to become interested in this issue in his Physilzalische Begriffsbildung. There Carnap is especially concerned with the transition from sensible qualities to sensible quantities, the transition from a qualitative vocabulary for sensible qualities ("warm", and "warmer than") to the quantitative temperature scale. Canup appears to envision operational definitions for terms like "as warm as" that figure in the definition of quantitative concepts. He still thinks of laws for fundamental physical magnitudes as purely mathematical statements that gain empirical significance only through many-one coordinations of fundamental magnitudes with sensible magnitudes. Operational definitions go by the board with the realization in "Testability and Meaning" that sensible magnitudes like temperature are not operationally definable in Carnap's extensional framework. And by Logical Syntax, having come to a clear, objectual understanding of higher-order quantification as well as a grasp of the distinction between object-language and metalanguage together with that between syntax and semantics,28 Carnap realizes that assignments of physical magnitudes to space-time locations are not likely to be definable in mathematical terms. 29 He is well on the way to the views of "Methodological Character of Theoretical Terms" with its correspondence-rules or bridge principles linking theoretical concepts with dispositional concepts introduced into a formal language for science by reduction sentences linking them to an observation language. I think it is this line of development that Carnap has in mind in his 1961 introduction to the reissued Aujbau. There Carnap retrospectively assimilates the Aujbau procedure for assigning color-qualities to spacetime points to "the introduction of 'theoretical concepts' through theoretical postulates and correspondence rules." The earlier quotation from the 1961 introduction continues: Actually, without clearly realizing it, I already went beyond the limits of explicit definitions in the construction of the physical world. For example, for the correlation of colors with space-time points, only general principles, but no clear operating rules were given (§127). This procedure is related to the method of introducing concepts through postulates to which I shall return later. The positivist thesis of the reducibility of thing concepts to autopsychological concepts remains valid, but the assertion that the former can be defined in terms of the latter must now be given up .... (p. viii) Friedman comments:
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In the Aujbau, however, the new mathematical logic of Principia Mathematica provides Carnap with all the philosophical concepts and distinctions he needs. Carnap thereby achieves a standpoint that is both non-psychological and truly metaphysically neutral, and at the same time he transforms the neo-IZantian tradition into something essentially new: 'logical-analytic' philosophy. (1999, 141) This transforn1ing process does not stop with Aujbau. Among the several factors that drive it forward is Carnap's increasing sophistication about the technical tools he is using, about what they can and cannot accomplish. As regards Quine's objection, we should not think of Carnap as surrendering a dream of the reduction of science to sense-data, but rather as coming to a better appreciation of what can and cannot be defined in the formal languages in which he is worldng. 3o When we refract Carnap's Aufbau-era philosophy of science through his post-Godel grasp of the logical resources deployed in Aujbau, the gulf between Aufbau and his mature views shrinks. It is these continuities that Carnap acknowledges in the 1961 introduction.
NOTES 1. In writing this paper, I have benefited from comments from Steve Awodey, Andre Carus, Michael Friedman, Peter Hylton, and especially Greg Frost-Arnold and Warren Goldfarb. 2. I cite Aujbau by section numbers. Page citations to Carnap's 1961 introduction are to the German edition and the George translation. My translations from Aujbau are based on the George translation. 3. See §§lS and 38-39. In §39 Carnap's terminological distinction between "explicit definitions" and "definitions in use" can mislead. As the example at the end of §39 shows, Carnap's definitions in use are a kind of explicit definition. In §lS Carnap cites Carnap (1927), in which the distinction between explicit definitions and Hilbertian implicit definitions is discussed at length. In this paper Carnap cites Frege's criticisms of Hilbert. Carnap attended Frege's 1914 lectures in which Frege discussed the point. Frege (2004) includes a translation of these lecture notes. 4. See especially Friedman (1999) and Richardson (1998). S. In Aujbau, Carnap does not discuss the difficulties with logicism, for example the status of the axiom of infinity. Although he speaks in §107 of logic as grounded in stipulations for the use of certain signs, there is no worked out philosophy oflogic/mathematics in Aujbau. By the end of the 1920s, Carnap is thinking about how to maintain logicism, a topic that is at the forefront of his interests through the completion of Carnap ([1934] 1937). For a discussion of -the development of Carnap's views in this period, see Goldfarb (1996) and Awodey and Carus (2001).
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6. See §§27 and 33. In §27, Carnap emphasizes, "The objects of science are almost without exception quasi-objects." We will see below that the concept of reference is problematical for Carnap in Aufbau, and officially plays no role in his understanding of constitution systems. 7. This rhetoric leads Carnap to say that the statements of a constitution system are all about the basic individuals ofthe system, thus obscuring the free-wheeling use Carnap makes of the set theory built into Carnapian formalisms to satisfy his adequacy standards for constitution systems. See §1l9, the final paragraph. Indeed, Carnap's remarks in §§179-80, especially the penultimate paragraph of §180, suggest that he thinks that the extension of the basic relation of recollected similarity over the elementary experiences algorithmically fixes the truth-value of all the statements of the constitution system. On the interpretation I shall present, this suggestion is mistaken. 8. Friedman emphasizes the importance of the formal character of Carnap's conception of construction in Friedman (1999), especially pp. 138-40. 9. Years ago Alan Richardson called my attention to this passage and its significance. He also observed that in connection with a discussion of the semantical paradoxes, Carnap speaks of "the pseudo-relation of meaning (Bedeutens), which generally does not occur in a purified language" (Carnap 1929,21). 10. This brief discussion of Carnap's adequacy standards for constitution systems requires elaboration in order to be convincing. For example, I need to argue that the notion of coextensiveness Carnap invokes in §§35, 47, and 50 in his discussion of reducibility of predicate-functions (Aussagefunktion) needs to be understood in terms of systematic preservation of truth-value. I also need to take up some the points about Aufbau that Goodman raises in chapter one of (1951). 11. Friedman suggests that Carnap proposes to define the assignment by a formalization of the canons of §126 as a recursive definition over the types of STT (Friedman 1999, 160-61). I find this suggestion implausible. First, none of Carnap's remarks suggest that he has such a definition is mind. Second, Carnap in the 1920s limits STT to finite types. Friedman's recursion over the types will not be formulable within STT thus limited. 12. Some of Carnap's rhetoric suggests otherwise. See especially §179, the end of the third paragraph. See the discussion in note 7 above. I think that Carnap's project of epistemic constitution system construction is very different from verificationist programs. This difference manifests the influence of· neoKantian ideas on Aufbau. I don't think Carnap fully appreciates the difference, when he falls in with verificationism after his move to Vienna in 1926. 13. Carnap's central goal in this paper is to distinguish his conventionalism from Poincare's and Dingler's. On Carnap's view, these thinkers apply the principle of simplicity just to the body of geometrical and physical laws. On this basis, Dingler urges retention of Euclidean geometry and Newtonian physics in the face of General Relativity. Carnap urges that this approach gains simplicity of laws only at the price of complicating the application of those laws to empirical phenomenon, complicating the coordination of physical magnitudes with sensible qualities. He urges that the principle of simplicity be applied to the abstract laws together with the application of these laws to the description of empirical phenomena. On this basis, he opts for General Relativity.
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14. In my discussion of Carnap's views on the constitution of the physical world, I am generally indebted to chapter 7 of Richardson (1998). 15. As Greg Frost-Arnold pointed out to me, something similar happens with the construction of the space-time manifold for the perceptual world. Carnap points out in §125 that the two dimensional character of the visual fields and his empirical constraints for assigning colors to locations require a manifold with at least three spatial dimensions. The actual construction of this manifold is a purely logical matter, so that the construction itself does not reflect these en1pirical constraints. 16. In "Aufgabe der Physik," Carnap discusses only basic physical laws, including geometry, and the coordination of physical magnitudes with sensible qualities. He does not discuss how the latter are to be assigned to space-time locations, i.e., he does not discuss the construction of the perceptual world. Aujbau fills in this gap in the earlier account. 17. As noted above, Carnap (1926) describes the constitution of the physical world somewhat differently, as a class of 14-tuples, but the same difficulty arise for this approach. The problen1 is to define in purely mathematical terms the class of 14-tuples that constitutes the physical world. 18. See Carnap's discussion of consistency and satisfiability in Carnap (2000, 96-101). For a discussion of Carnap's views here, see Awodey and Carus (2001, 158). I think that it is reasonable to read Carnap's 1928 understanding of consistency and satisfiability back into Aufbau. 19. See Hylton (1990) Chap. 7 and Goldfarb (1989). Matters here are complicated by Russell's adoption in 1910, as volume one of Principia is being con1pleted, of the multiple relation theory of judgment. This theory of judgment brings with it the rejection of Russell's earlier metaphysics of propositions and propositional functions in favor of a metaphysics of facts. It is unclear how to reconcile the logic of Principia with the metaphysics required by the multiple relation theory. 20. Russell's rhetoric here bespeaks a turning in his philosophical orientation toward a naturalist approach to language and content during his 1918 prison sentence, when he read extensively in psychology. See Russell (1986), Appendix iii. Russell's new outlook is adumbrated in Russell (1919b) and in Russell (1921). Russell briefly discusses this shift in (1956, 108). For a treatment of it, see Dreben (1996), 45-48. 21. In AuJbau §28, Carnap both calls unary propositional functions "properties" or "property-concepts," and speaks of them as representing (darstellen) properties. Parallel usage occurs in Carnap (1929) §2. I do not think that Carriap in either place conceives of properties platonistically. This fits with his eschewal of reference (Bedeutung) in favor of truth and falsity in discussions of constitution system, discussed above. 22. Awodey and Carus note that there is good reason to think that Carnap's efforts spurred Godel to develop his far more fruitful approach to n1etalogic. For further discussion of this point, see Goldfarb (2005). 23. Carnap's alleged proof of the Gabelbarkeitssatz is in Carnap (2000), 133-39. Awodey and Carus (2001), 156f discusses Carnap's argument. Awodey and Reck (2002), 26 presents the point I highlight. 24. Selections from the Carnap-Godel correspondence are contained in Godel (2003),335-59. The relevant letters are Godel to Carnap, 11.9.1932 and Carnap
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to Godel, 25.9.32. Carnap discusses the matter in ([1934] 1937) §34d, the final paragraph. 25. For a discussion of the gradual, fitful emergence of these distinctions, especially the distinction between derivability and semantic consequence, see Awodey and Reck (2002). 26. For a review ofCarnap's work in logic in the 1920s, see Reck (2004). 27. See Quine (1960), 19 and Quine (1995), 13. 28. In Logical Syntax, the distinction between syntax and semantics corresponds to Carnap's distinction between metalogical concepts that are either definite (decidable, recursive) or based on decidable concepts (semidecidable, recursively enumerable) and those that are not, the higWy indefinite ones. See §34a. 29. See Logical Syntax §50 with its discussion of the distinction between logical and descriptive expressions, especially the example of the fundamental tensor determining the metrical structure of space-time. Here Carnap observes that if space-time has constant curvature, the fundamental tensor is logically definable term. If space-time has variable curvature "depending on the distribution of matter," the sign for fundamental tensor will be descriptive, not logical. 30. My discussion of Carnap's 1961 introduction is indebted to and builds on Friedman's discussion in §iv of Friedman (1999).
REFERENCES Arrington, R.L. and H. Glock (eds.) (1996). Wittgenstein and Quine. London: Routledge. Awodey, S. and A. Carus (2001). "Carnap, Completeness, and Categoricity: the Gabelbarkeitssatz of 1928." Erkenntnis 54: 145-72. Awodey, S. and E.Reck (2002). "Completeness and Categoricity. Part I: Nineteenth-century Axiomatics to Twentieth-Century Metalogic." History and Philosophy ofLogic 23: 1-30. Awodey, S. and C. Klein (eds.) (2004). Carnap Brought Home: The View from Jena. Chicago: Open Court. Carnap, R. (1923). "Uber die Aufgabe del' Physik und die Anwendung des Grundsatzes del' Einfachstheit." Kant-Studien 28: 90-107. - - - . (1924). "Dreidimensionalitat des Raumes und Kausalitat: Eine Untersuchung tiber den logischen Zusammenhang zweier Fiktionen." Annalen der Philosophie und Philosophischen Kritik 4: 105-30. - - . (1926). Physikalische Begriffsbildung. Karlsruhe: Verlag G. Braun. - - - . (1927). "Eigentliche und Uneigentliche Begriffe." In Symposion: Philosophische Zeitschrift fur Forschung und Aussprache vol. 1, 355-74. - - . ([1928] 1961). Der logische Aufbau der Welt. Hamburg: Felix Meiner. Original publication (Hamburg: Felix Meiner). Translated by R. George (2003) as The Logical Structure of the World and Pseudoproblems in Philosophy. Chicago: Open Court.
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(1929). Abriss der Logistik, mit besonderer Beriicksichtigung der Relationstheorie und ihrer Anwendungen. Vienna: Verlag von Julius Springer. - - . ([1934] 1937). The Logical Syntax of Language. Trans. by A Smeaton, London: Kegan Paul. Originally published as Die logische Syntax der Sprache, Vienna: Verlag von Julius Springer. - - - . (1956). "The Methodological Character of Theoretical Terms." In Feigl and Scriven (eds.), 38-76. - - - . (2000). Untersuchungen zur allgemeinen Axiomatik. Edited by T. Bonk and J. Mosterin, Darmstadt: Wissenschaftliche Buchgesellschaft. (This posthumously published manuscript was written in 1928.) Dreben, B. (1996). "Quine and Wittgenstein: the Odd Couple." In Arrington and Glock (eds.), 39-61. Feigl, H. and M. Scriven (eds.) (1956). The Foundations ofScience and the Concepts of Psychology and Psychoanalysis, Minnesota Studies in the Philosophy of Science, vol. 1. Minneapolis: University of Minnesota Press. Frege, G. (2004). Frege)s Lectures on Logic: Carnap)s Student Notes, 1910-1914. Translated and edited by E.H. Reck and S. Awodey. Chicago: Open Court. Friedman, M. (1999). "Epistemology in the Aujbau." In Reconsidering Logical Positivism. Cambridge: Cambridge University Press, 114-62. Giere, R.N. and AW Richardson (eds.) (1996). Origins of Logical Empiricism, Minnesota Studies in the Philosophy of Science, vol. 16. Minneapolis: University of Minnesota Press. Gbdel, K. (2003). Collected Works, Vol. IV (Correspondence A-G), S. Feferman, et al. (eds.). Oxford: Oxford University Press. Goldfarb, W. (1989). "Russell's Reasons for Ramification." In Savage and Anderson (eds.), 24-40. - - - . (1996). "The Philosophy of Mathematics in Early Positivism." In R.N. Giere and AW Richardson (eds.), 213-30. - - . (2005). "On Gbdel's Way In: The Influence of Rudolf Carnap." The Bulletin of Symbolic Logic 11: 185-93. Goodman, N. (1951). The Structure of Appearance. Indianapolis: The BobbsMerrill Co. Hylton, P. (1990). Russell, Idealism) and the Emergence of Analytic Philosophy. Oxford: Oxford University Press. Quine, Wv. (1953). "Two Dogmas of Empiricism." In W.v. Quine, From a Logical Point of View. Cambridge, Mass: Harvard University Press, 20-46. Originally published in Philosophical Review 60 (1951): 525-34. - - . (1960). Word and Object. Cambridge, MA: MIT Press. - - - . (1995). From Stimulus to Science. Cambridge, MA: Harvard University Press. Reck, E. (2004). "From Frege and Russell to Carnap: Logic and Logicism in the 1920s." In S. Awodey and C. Klein (eds.), 151-80. Richardson, A. (1998). Carnap)s Construction of the World. Cambridge: Cambridge University Press. Russell, B. (1919). Introduction to Mathematical Philosophy. London: George Allen and Unwin.
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_ . ([1919] 1986). "On Propositions: What They Are and How They Mean." In Russell (1986), 276-306. _ . (1921). Analysis ofMind. London: George Allen and Unwin. _ . (1959). My Philosophical Development. London: Routledge. _ . (1986). Collected Papers, Vol. 8 .. Edited by J.G. Slater. London: George Allen and Unwin. Savage, c.w. and c.A. Anderson (eds.) (1989). Rereading Russell: Essays on Bertrand RusseWs Metaphysics and Epistemology, Minnesota Studies in the Philosophy ofScience, Vol. 12. Minneapolis: University of Minnesota Press. Whitehead, A.N. and Russell, B. ([1910] 1962). Principia Mathematica to *56. Cambridge: Cambridge University Press.
[15J "Let me briefly indicate why I do not find this standpoint natural." Einstein, General Relativity, and the Contingent A Priori DON HOWARD
Introduction In his recent Dynamics of Reason (Friedman 2001) and other writings (see, especially, Friedman 2002a, 2002b), Michael Friedman urges the reintroduction of a robust notion of the a priori within the philosophy of science as·a bulwark against the relativism that many see as the most worrisome consequence of Thon1as IZuhn's critique of neo-positivist fantasies about a theory-choice algorithm (1962), a bulwark not afforded by either the linguistic conventionalism of Rudolf Carnap (1934, 1950) or the semantic and theory holisn1 ofW.V.O. Quine (1951, 1960). Taking inspiration from early work by Hans Reichenbach (1920) and Ernst Cassirer (1921), Friedman argues that the history of mechanics and the theory of gravitation from Newton to Einstein reveals an enduring structure wherein selected elements of theory, such as Newton's laws in mechanics and the principle of equivalence in general relativity, playa contingent a priori role relative to the specific theories within which they live. They are a priori in the sense that their assumption is necessary in order to invest other elements of the theory in question with empirical content. In the Newtonian case, it is the law of universal gravitation that becomes, thereby, an empirical law; in Einstein's case, it is the field equations of general relativity. These a priori elements are contingent in the sense that, with the progress of theory, they can be revised or jettisoned, but this is held always to happen in such a way that both the successes and the failures of a predecessor theory can be explained from the perspective of the successor theory, as when Newtonian dynamics is viewed as a low-velocity limit of special relativity, even if, from the point of view of the predecessor theory, the successor theory appears incommensurable with the predecessor. Friedman dubs this diachronic relationship one of 333
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"prospective irrationality" and "retrospective rationality" (Friedn1an 2001,93-103). That a theory such as general relativity can be so parsed is indisputable, and Friedman's perspective offers helpful insight into the systematic role that certain principles can play in the larger body of physical theory. Open to dispute, however, is the question whether this is the correct view of the history of mechanics and gravitation theory or the most helpful view of the manner in which theories acquire empirical content. It is, surely, a noteworthy historical fact that the author of the general theory of relativity, Albert Einstein, pondered precisely this view in its earlier forn1s associated with Reichenbach and Cassirer, and found it wanting. Einstein recognized that any theory can be reconstructed in such a way that some elements of the theory serve as that via which other elements acquire their empirical content, and he argued that some such parsing is always necessary if one is to exhibit clearly the manner of a theory's attachment to the empirical. What he disputed was the necessity of anyone such parsing of theory, arguing that other parsings are always possible, and that it is the arbitrariness in our choice among parsings that is the main reason for not viewing the elements privileged in anyone such parsing as a priori. Who is right? Friedman or Einstein? That is one question that will concern us in what follows. A more fundamental question is this: Why, in his reconstruction of the history of mechanics and gravitation theory, does Friedman not tell the story of Einstein's rejection of more or less exactly the same view of the contingent a priori that Friedman defends? This is a question about method in arguments for the contingent a priori. We are given no transcendental argun1ent-in the strict I
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relation to the empirical was the product of long, careful, philosophically sophisticated reflection on all that is at stake in the choice? If Einstein was wrong, then we need an argument showing how and why he was wrong. My own view is that Einstein's argument against the contingent a priori is a compelling one. 2
Inertial Motion, the Equivalence Principle, and the Contingent A Priori Friedman (2001) distinguishes three (or four) strata of scientific knowledge that are held to be discernible at least in the history of mechanics and gravitation theory from Newton to Einstein: 1. Empirical laws ofnature, such as Newton's law of universal gravitation or Einstein's field equations for general relativity.
2. Constitutively a priori principles. These are contingent and revisable, but not "empirical in the same sense" as the empirical laws of nature, because they are presuppositions for the possibility of the empirical testing of empirical laws. They function in an a priori manner only relative to a specific theory. They include: 2a. Mathematical and geometrical principles, such as the calculus for Newton and differential geometry for Einstein. They are a priori in the sense that they function as the language in which basic mechanical principles and empirical laws of nature are formulated. 2b. Basic principles of mechanics, such as Newton's laws of motion and Einstein's principle of equivalence. These are a priori in the sense that they are necessary conditions for the possibility of the empirical laws' possessing empirical content.
3. Philosophical meta-paradigms or meta-frameworks, such as Scholastic natural philosophy or the mechanical world view. According to Friedman, a similar structure is expected to be found in theories generally, but a detailed exposition of this structure is provided only for the case of mechanics and gravitation theory from the seventeenth through the twentieth centuries. Our concern here is with the relationship between basic principles and empirical laws. The general pattern is evinced already in Newtonian gravitation theory, where the laws of motion "are presuppositions of the properly empirical laws," their function being "precisely to mediate between
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abstract mathematical representations and the concrete empirical phenomena these abstract mathematical representations are intended to describe" (Friedman 2001, 77). They do this via their work in picking out the center-of-mass frame of, first, the planetary system and, then, the galaxy as a whole (the universe as Newton knew it), thereby approximating absolute space as the frame of reference with respect to which unforced motion is properly viewed as inertial and, thus, the frame of reference with respect to which a force law such as the law of universal gravitation claims empirical validity. 3 An analogous relationship is held to obtain between Einstein's field equations for general relativity and the principle of equivalence. Of the space-time structure determined by the field equations, Friedman writes: Such a variably curved space-time structure would have no empirical meaning or application, however, if we had not first singled out some empirically given phenomena as counterparts of its fundamental geometrical notions-here the notion of geodesic or straightest possible path. The principle of equivalence does precisely this, however, and without this principle the intricate space-time geometry described by Einstein's field equations would not even be empirically false, but rather an en1pty mathematical formalism with no empirical application at all. (Friedman 2001, 38) There are, of course, differences between the Newtonian and Einsteinian cases. Inertial motion in general relativity corresponds to free fall in a gravitational field, whereas inertial motion in Newtonian mechanics is completely unforced and gravity plays no role in structuring space-time itself. Still, the analogy runs deep: The principle of equivalence depicts the space-time trajectories of bodies affected only by gravitation as geodesics or straightest possible paths in a variably curved space-time geometry, just as the Newtonian laws of motion, when viewed from the space-time perspective, depict the trajectories of bodies affected by no forces at all as geodesics or straightest possible paths in a flat or Euclidean space-time geometry. (Friedman 2001,38) Though functioning in the indicated a priori fashion with respect to basic empirical laws, elements of theory such as the principle of equivalence cannot-as contingent principles-be wholly devoid of connections to the world of experience. Friedman acknowledges that they must at least have empirical "counterparts" and empirical motivations (Friedn1an 2001, 86-92). But they cannot possess empirical content in the same_~~y
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as empirical laws, for it is such principles that ground the possibility of the laws' having empirical content. Contingent a priori principles thus playa role similar to that of coordinating principles in the orthodox logical empiricist story about empirical content, but with the difference that they are not mere conventions. 4 For Friedman, they are distinguished £l'om empirical laws not as that which may be freely varied, as with stipulative, coordinating definitions of rods and clocks, but structurally and functionally as that without which the rest of a theory would lack content. That there must be some such functional distinction between empirical laws and a priori principles is a central premise behind Friedman's larger philosophical ambitions for the contingent a priori. He argues that it is only by recognizing the existence of such asymmetries between the roles played by different components of theory, and by viewing the principles that effect the coordination with experience as more than mere conventions, that we can block the slide into epistemological relativism that is, on Friedman's analysis, the fate of Carnap, Kuhn, and Quine. Thanks to their investigations into the crucial framing work performed by languages and paradigms, Carnap and Kuhn were at least working in the right direction, according to Friedman, even if they lacked the gumption to embrace fully the a priori status of those frames. Quine's holism, on the other hand, fails utterly as an antidote to relativism thanks to its turning any principled, functional difference in kind among components of theory into at best a difference in degree of centrality to the web of belief and, consequently, in degree of resistance to revision. It is, moreover, simply historically false as a story about how principles like Newton's laws and the principle of equivalence actually function (Friedman 2001,33-46). Much is as stake, therefore, in assessing Friedman's argument for the contingent a priori way of regarding the status of such principles.
Duhem's Critique of Poincare and Le Roy Einstein's doubts about the contingent a priori are, as announced above, the main issue to be explored in assessing Friedman's way of regarding the principle of equivalence. Let us begin, however, with Pierre Duhem. That Duhem was Quine's patron saint is one excuse for so beginning. More pertinent is the fact that Duhem's theory holism was-so I have long argued (see Howard 1990)-among the most important influences shaping Einstein's philosophy of science. Most pertinent, however, is the following ironic fact, namely, that one ofDuhem's own most important deployments of his theory holism against those who would regard some physical principies as insulated against easy empirical refutation thanks to their role, qua
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definitions, in securing the empirical status of laws, concerned precisely the example of the law of free fall. The immediate target of Duhem's criticism was the conventionalism of Henri Poincare (1901) and Edouard Le Roy (1901). I quote Duhem at length so as to make clear the almost uncanny relevance of his discussion of the issue: Foremost among the assertions in which we shall aim at eliminating the appearance of paradox, we shall place one that has recently been formulated and discussed .... It has been developed at length and forcefully by H. Poincare with regard to principles of mechanics. Edouard Le Roy has also formulated it with great clarity. That assertion is as follows: Certain fundamental hypotheses of physical theory cannot be contradicted by any experiment, because they constitute in reality definitions, and because certain expressions in the physicist's usage take their meaning only through them. Let us tal(e one of the examples cited by Le Roy: When a heavy body falls freely, the acceleration of its fall is constant. Can such a law be contradicted by experiment? No, for it constitutes the very definition of what is meant by "falling freely." If while studying the fall of a heavy body we found that this body does not fall with uniform acceleration, we should conclude not that the stated law is false, but that the body does not fall freely, that some cause obstructs its motion, and that the deviations of the observed facts from the law as stated would serve to discover this cause and to analyze its effects. Thus, M. Le Roy concludes: "laws ... constitute the very criterion by which we judge appearances as well as the methods that it would be necessary to utilize in order to submit them to an inquiry whose precision is capable of exceeding any assignable limits." (Duhem 1906, 208-209) What Le Roy terms "laws" are what are presently under discussion under the heading ofputatively contingent a priori principles or coordinating principles, namely, those elements of a theory whose role is to secure the empirical import of the other elements of a theory. Is Le Roy correct? What if experiment seemed to contradict our expectation that a body would fall freely? It follows from Duhem's holism that we are not constrained to postulate heretofore unsuspected forces influencing the body's fall. No, as always, we have a choice. We could postulate new forces or we could change the "law," that is, change the definition of "free fall." Duhem remarks: We have here nothing more than a particular application of the principle set down above in Section 2 of this chapter. A disagreement
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between the concrete facts constituting an experiment and the symbolic representation which theory substitutes for this experiment proves that some part of this symbol is to be rejected. But which part? This the experiment does not tell us; it leaves to our sagacity the burden of guessing (Duhem. 1906,211) A second major example adduced by Duhem is the principle of inertia, about which Duhem's antagonists-this time Poincare, himself-had made claims more or less identical to that made on behalf of the law of free fall. Can we regard the principle of inertia as being susceptible to empirical test? Duhem writes: One thing is indeed certain, namely, that whatever the motion of a material point is, when seen from a first frame of reference, we can always and in infinite ways choose a second frame of reference such that seen from the latter our material point appears to move in a straight line with uniform motion. We cannot, therefore, attempt an experimental verification of the principle of inertia; false when we refer the motions to one frame of reference, it will become true when selection is made of another term of comparison, and we shall always be free to choose the latter.... Therefore, it would be absurd to wish to subject certain principles of mechanics to direct experimental test .... Does it follow that these hypotheses placed beyond the reach of direct experimental refutation have nothing more to fear from experiment? That they are guaranteed to remain immutable no matter what discoveries observation has in store for us? To pretend so would be a senous error. Taken in isolation these different hypotheses have no experimental meaning; there can be no question of either confirming or contradicting them by experiment. But these hypotheses enter as essential foundations into the construction of certain theories of rational mechanics .... In truth, hypotheses which by themselves have no physical meaning undergo experimental testing in exactly the same manner as other hypotheses. Whatever the natute of the hypothesis is, we have seen at the beginning of this chapter that it is never in isolation contradicted by experiment; experimental contradiction always bears as a whole on the entire group constituting a theory without any possibility of designating which proposition in this group should be rejected. (Duhem
1906,213-15) Duhem discusses the law of free fall and the principle of inertia in chapter six, "Physical Theory and Experiment," of La Theorie physique) son objet
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et sa structure (Duhem 1906), the chapter presenting the core argument for theory holisnl. The evidence strongly suggests that Einstein read La Theorie physique in 1909 or 1910 (see Howard 1990, 1994). He had clearly assimilated the lesson of Duhem's holism by the winter semester of 1910-1911, when, in lectures on electricity and nlagnetism at the University of Zurich, he explained to his students how one can introduce the concept of charge in the interior of a solid charged body, even though one cannot employ here the fiction of a test charge:
We have seen how experience led to the introduction of the concept of electrical charge. It was defined with the help of forces that electrified bodies exert on each other. But now we extend the application of the concept to cases in which the definition finds no direct application as soon as we conceive electrical forces as forces that are exerted not on material particles but on electricity. We establish a conceptual system whose individual parts do not correspond immediately to experiential facts. Only a certain totality of theoretical materials corresponds again to a certain totality of experimental facts. We find that such an el[ ectrical] continuum is always applicable only for representing relations inside ponderable bodies. Here again we define the vector o[ f] el[ ectrical] field strength as the vector of the mech[anical] force that is exerted on a unit of pos[itive] electr[ical] charge inside a ponderable body. But the force thus defined is no longer immediately accessible to exp[ eriment]. It is a part of a theoretical construction that is true or false, i.e., corresponding or not corresponding to experience, only as a whole. (Einstein 1910-1911, 325) A few years later, Einstein's sympathy for theory holism was reinforced when he encountered in the writings of Schlick (1915, 1917, 1918) a version of theory holism owing nlore to David Hilbert than to Duhem,5 and both semantic and theory holism became a cornerstone of Einstein's mature philosophy of science (see Howard 1984, 1994,2004, forthcoming). How did this holism shape Einstein's reaction to lZant and to the notion of the contingent a priori known to him mainly through Reichenbach's work?
Einstein's Critique of the A Priori Like many of his generation, Einstein had read and studied lZant from an early age. He apparently read all three Critiques for the first time in his teens. He elected to enroll in August Stadler's lectures on lZant while a student at the ETH in the 1890s. On many later occasions he returned to Kant, as when in 1918 he read the Prolegomena. He had personal connec-
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tions to both the Marburg critical idealist community, centered around Hermann Cohen and Paul Natorp, and the Berlin critical realist community, centered around Alois Riehl. And he carried on sustained and sophisticated discussions with thinkers like Reichenbach and Ernst Cassirer, who were, in the 1920s, renegotiating the relationship between the Kantian tradition and the new empiricisms flowering in Berlin and Vienna. 6 The general tendency of Einstein's mature reaction to Kant is clear at least from July of 1918, when, while on vacation, he wrote to Max Born: I am reading Kant's Prolegomena here, among other things, and am beginning to comprehend the enormous suggestive power that emanated from the fellow and still does. Once you concede to him merely the existence of synthetic a priori judgments, you are trapped. I have to water down the "a priori" to "conventional," so as not to have to contradict him, but even then the details do not fit. Anyway it is very nice to read, even if it is not as good as his predecessor Hume's work. Hume also had a far sounder instinct. (Born 1969, 25-26) On more than one later occasion he made essentially the same point about preferring to designate as conventional those elements of theory whose characterization as a priori is the mark of one's allegiance to the Kantian tradition. Why did he prefer to regard as conventional what the followers of Kant deemed a priori? One reason was Einstein's skepticism about the Kantian doctrine of intuition, a skepticism amusingly expressed in a letter of 15 September 1919 to Ilse Schneider: I have received the mentioned dissertation by S. [Ewald Sellien] (Epistemology and Relativity Theory) .... Kant's celebrated view on time reminds me of Andersen's tale about the emperor's new clothes, except that instead of the emperor's clothes, it concerns the form of intuition. (Rosenthal-Schneider 1980, 83-84; see Sellien 1919) But the more fundamental reason for Einstein's doubts about the a priori was rooted in his commitment to theory holism of the Duhem and (early) Schlick varieties. In the early 1920s, especially, Einstein's correspondence and writings are dense with discussions of precisely why and how he does not find the nOtion of the a priori a helpful one. His reasons are not those of a crude "tabula rasa" empiricist who ignores the knower's active contribution to cognition. On that point he agrees with the Kantian. What then is the reason? Here is how he expressed his reservations to Cassirer in a letter of 5 June 1920 after reading in manuscript Cassirer's Zur Einsteinschen Relativitdtstheorie (Cassirer 1921):
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I can understand your idealistic way of thinking about space and time, and I even believe that one can thus achieve a consistent point of view. To me, as a non-philosopher, philosophical contrarieties appear more contrarieties of en1phasis than contrarieties of a principled kind. What Mach calls connections, are for you ideal names, which experience first makes possible. But you emphasize this side of knowledge, whereas Mach wants to make them appear as insignificant as possible. I acknowledge that one must approach the experiences with some sort of conceptual functions, in order for science to be possible; but I do not believe that we are placed under any constraint in the choice of these functions by virtue of the nature of our intellect. Conceptual systen1s appear empty to me, if the manner in which they are to be referred to experience is not established. This appears most essential to me, even if, to our advantage, we often isolate in thought the purely conceptual relations, in order to permit the logically secure connections to emerge more purely. (As quoted in Howard (1994), 53-54) What is objectionable in the doctrine of the a priori is the idea that anyone feature of our representation of nature is fixed. Einstein elaborates on this point in a letter to Eberhard Zschimmer of27 September 1922: That which you work out in your essay seems correct to me, at least from the physical side, which is all that I can judge with certainty. In my opinion, though, the important question for the opposition of relativity theory and I
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cific theories, an a priori relative to a specific theoretical or phenomenal domain? Cassirer (1921) had suggested one such perspective on the a priori in arguing that an a priori spatio-temporal form-one mathematically weaker than the Euclidean spatial form that Kant regarded as a priori-was picked out by various invariant topological, conformal, and other mathematical facts about space-time in general relativity. Cassirer here effected a link between invariance and the a priori that was also at the center of Carnap's analysis of the space problem in his doctoral dissertation (Carnap 1921). Contingency is not highlighted as a property of an a priori distinguished by its invariance properties, the signature of the metric, for example, bidding fair to be a henceforth abiding, if newly discovered fact about spacetime. A more explicitly contingent a priori was championed by Reichenbach (1920), his a priori principles of coordination being singled out not as the invariant but as that which secures our space-time theory's linkage with the empirical realm. Reichenbach argued that the received notion of the a priori involved both apodicticity and a constitutive function, whereas the lesson of relativity theory was that the two aspects of the a priori should be disentangled. No necessity attaches to what we take as a priori. That changes with the forward march of science. But at any stage in the development of theory some elements must play the fundamental constitutive role of providing empirical content for theoretical primitives via a priori principles of coordination. There is no evidence that Einstein knew of Carnap's doctoral dissertation. He knew well, however, the projects of Cassirer and Reichenbach. Yet his own thinking about an a priori grounded in something other than claims about the nature of the intellect seems to have been stimulated more by a seemingly very different project, namely, Hermann Weyl's proposal of a unified theory of electricity and gravitation (WeyI1918a, 1918b, 1918c). In a Weyl space-time, length is defined only locally and transforms according to a linear differential form whose components become tl1e components of the electromagnetic four-potential. Weyl's philosophical motivation for restricting the fundamental geometrical structures of a space-time theory to those definable locally, a kind of structure he termed a "purely infinitesimal geometry," came from the transcendental phenomenology of Edmund Hussed, the idea being that one has epistemic access only to such structure as is definable in the space of intuition of an ideal observer? What focused Einstein's thinking about elements of theory that might be regarded as a priori for reasons other than those having to do with the nature of the human intellect was not so much Weyt's transcendental phenomenological argument for the purely local perspective as it was Einstein's own reaction to the Weyl theory.
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A notorious consequence of the restriction to the purely local is that length is path dependent. Even before Weyl's first publication appeared, Einstein had objected to this feature ofWeyl's theory, arguing that it was refuted by such facts as the stability of spectral lines-shouldn't atoms with different histories radiate at different frequencies?-and that it made impossible the empirical interpretation of the theory because it implied that there could be no structure in nature corresponding to an ideal measuring rod (or an ideal clock), a freely transportable, practically rigid instrument for checking metrical relations as predicted by the theory. A measuring rod that gives different results depending on its history is no good measuring rod at all (see Einstein 1918). There is irony here in the circumstance that Einstein, the epistemological holist, assumes in his critique ofWeyl a view antithetical to that holism in arguing that selected theoretical primitives, such as the infinitesimal metrical interval, have to be given direct empirical interpretations independently of the theory in which those primitives are embedded. In effect, Einstein is arguing for a privileged role-he would not, himself, use the term "a priori"-for coordinating definitions of basic metrical notions. Einstein recognized the tension with his elsewhere asserted holistic fornl of conventionalism, whose correctness in principle he never denied, but he persisted in the denland for a direct empirical interpretation of basic metrical notions. Why? Einstein's best known discussion of why one must demand direct empirical interpretations of basic metrical notions is contained in his widely-read and influential 1921 lecture, "Geometrie und Erfahrung." The focal question is whether one must assume a coordination between practically rigid bodies in the world of experience and fundamental geometrical primitives: Why is the equivalence of the practically-rigid body and the body of geometry-which suggests itself so readily-rejected by Poincare and other investigators? Simply because under closer inspection the real solid bodies in nature are not rigid, because their geometrical behavior, that is, their possibilities of relative disposition, depend upon temperature, external forces, etc. Thus, the original, immediate relation between geometry and physical reality appears destroyed, and we feel impelled toward the following more general view, which characterizes Poincare's standpoint. Geometry (G) predicates nothing about the behavior of real things, but only geometry together with the totality (P) of physical laws can do so. Using synlbols, we may say that only the sum of (G) + (P) is subject to experimental verification. Thus (G) may be chosen arbitrarily, and also parts of (P); all these laws are conventions. All that is necessary to avoid contradictions is to choose the
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ren1ainder of (P) so that (G) and the whole of (P) are together in accord with experience. Envisaged in this way, axiomatic geometry and the part of natural law which has been given a conventional status appear as epistemologically equivalent. Sub specie aeterni Poincare, in my opinion, is right. The idea of the measuring rod and the idea of the clock coordinated with it in the theory of relativity do not find their exact correspondence in the real world. It is also clear that the solid body and the clock do not in the conceptual edifice of physics play the part of irreducible elements, but that of composite structures, which must not play any independent part in theoretical physics. But it is my conviction that in the present stage of development of theoretical physics these concepts must still be elTIployed as independent concepts; for we are still far from possessing such certain knowledge of the theoretical principles of atomic structure as to be able to construct solid bodies and clocks theoretically from elementary concepts. (Einstein 1921, 7-8) Holism is, in principle, correct. It is only the immaturity of current physical theory that forces us to demand direct empirical interpretations of geometrical primitives. But what, if anything, are the implications for whether we are to regard as a priori the coordinations that provide those interpretations? That is not immediately obvious. Einstein returned to this cluster of questions when, in 1924, he reviewed a couple of books on relativity theory by the minor neo-IZantians JosefWinternitz (1923) and Alfred Elsbach (1924). In the first of these reviews, Einstein again asserts his basic view according to which what others regard as the a priori is more helpfully seen as the conventional: Thus Winternitz asserts with IZant that science is a mental construction on the basis of a priori principles. That the edifice of our science rests and must rest on principles that are not themselves derived from experience, will be acknowledged without doubt. For me, doubt only arises if one asks about the dignity of those principles, that is, about their irreplaceability. Are those principles at least in part so constituted that their modification would be incompatible with science, or are they collectively mere conventions, like the ordering principle of the words in a lexicon? W. inclines toward the former view, I to the latter. (Einstein 1924a, 21-22) But in the second review, Einstein provides his most careful and lucid statement both about what he finds uncongenial in the notion of the a pri0ri and about why one should still privilege in some fashion (as conventions) coordinating definitions of geometrical primitives. First about the a priori in general:
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This does not, at first, preclude one's holding at least to the Kantian problematic, as, e.g., Cassirer has done. I am even of the opinion that this standpoint can be rigorously refuted by no development of natural science. For one will always be able to say that critical philosophers have until now erred in the establishment of the a priori elements, and one will always be able to establish a system of a priori elements that does not contradict a given physical system. Let me briefly indicate why I do not find this standpoint natural. A physical theory consists of the parts (elements) A, B, C, D, that together constitute a logical whole that correctly connects the pertinent experiments (sense experiences). Then it tends to be the case that the aggregate of fewer than all four elements, e.g., A, B, D, without C, no longer says anything about these experiences, and just as well A, B, C without D. One is then free to regard the aggregate of three of these elements, e.g., A, B, C as a priori, and only D as empirically conditioned. But what remains unsatisfactory in this is always the arbitrariness in the choice of those elements that one designates as a priori, entirely apart from the fact that the theory could one day be replaced by another that replaces certain of these elements (or all four) by others. (Einstein 1924b, 1688-89) Connecting theory to the empirical realm requires our fixing some elen1ents of theory. What is not "natural" is the privileging of anyone such fixing, the choice of what to hold fixed being, in principle, entirely arbitrary. A few pages later, Einstein relates this general perspective on the a pri0ri to the question of choosing between the holist's view from eternity and the here-and-now demand for direct empirical interpretations of geometrical prin1itives. The locus is a critique of Elsbach's defense of Paul Natorp's claims that apparent deviations from Euclidean spatial structure can be accommodated by changes in physical law and that the spatial metric is not empirically determinable because it is ideal, not real (Natorp 1910): The position that one takes on these claims depends on whether one grants reality to the practically-rigid body. If yes, then the concept of the interval corresponds to something experiential. Geometry then contains assertions about possible experiments; it is a physical science that is directly underpinned by experimental testing (standpoint A). If the practically-rigid n1easuring body is accorded no reality, then geometry alone contains no assertions about experiences (experiments), but instead only geometry with physical sciences taken together (standpoint B). Until now physics has always availed itself of the simpler standpoint A and, for the most part, is indebted to it for its fruitfulness; physics employs it in all of its measurements. Viewed from this
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standpoint, all of Natorp's assertions are incorrect . . . . But if one adopts standpoint B, which seems overly cautious at the present stage of the development of physics, then geometry alone is not experimentally testable. There are then no geometrical measurements whatsoever. But one must not, for that reason, speak of the "ideality of space." "Ideality" pertains to all concepts, those referring to space and time no less and no more than all others. Only a complete scientific conceptual system comes to be univocally coordinated with sensory experience. On my view, Kant has influenced the development of our thinking in an unfavorable way, in that he has ascribed a special status to spatio-temporal concepts and their relations in contrast to other concepts. Viewed from standpoint B, the choice of geometrical concepts and relations is, indeed, determined only on the grounds of simplicity and instrumental utility.... Concerning the metrical determination of space, nothing can then be made out empirically, but not "because space is not real," but because, on this choice of a standpoint, geometry is not a complete physical conceptual system, but only a part of one such. (Einstein 1924b, 1690-91) Though standpoint A is the one we have standardly employed, for good reason, and though adopting standpoint B now would be "overly cautious'" standpoint B-the holist standpoint-is, nevertheless, the right one from a first-principles point of view and is the right perspective from which to assess claims for an a priori component in scientific theory. Einstein's basic attitude toward the a priori remained unchanged to the end of his life. The knower contributes something to scientific cognition, but nothing about the nature of the intellect fixes that contribution. More specifically, our putting empirical content into theories requires our fixing some coordinations between theory and world, but how we fix those coordinations is, in principle, arbitrary. He said this in a 1928 review of Emile Meyerson's La Deduction Relativiste: There is no feature, no characteristic, of the system we are seeking, about which we can know a priori that it must necessarily belong to this system due to the nature of our thought. This also holds for the forms oflogic and causality. We can only ask how the system of science (in its states of development thus far) is composed, but not how it must be composed. The logical foundations of the system as well as its structure are thus (from a logical point of view) conventional; their only justification lies in the performance of the system vis-a-vis the facts, in its unified character, and in the small number of its premises. (Einstein 1928,
253)
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He said it again in his 1936 essay, "Physics and Reality": Concerning the manner in which we are to build and connect concepts, and the n1anner in which we are to coordinate them with sense experiences, not the least can be said a priori, in my view. Only success in regard to establishing an order among sense experiences is decisive. The rules for connecting concepts must be stipulated only in general, for otherwise knowledge of the kind for which we strive would be impossible. One has con1pared these rules with the rules of a game, rules that are in themselves arbitrary but whose determinateness first makes the game possible. However, this stipulation can never be a final one, rather they can only claim validity for an intended domain of application (i.e., there are no ultimate categories in I
Regulative Principle, Empirical Law, or Contingent A Priori? Einstein's general attitude toward the a priori is one thing. His specific attitude toward the methodological role of the principle of equivalence in general relativity is another. The heuristic work done by the principle of equivalence in guiding Einstein to general relativity was something upon which he remarked, and he was usually careful to explain his precise understanding of the content of the principle as asserting only the strictly local equivalence of inertia and a homogenous gravitational field (see Norton 1989). How did he view the principle's systematic, methodological role in general relativity? Perhaps Einstein's most original contribution to our understanding of scientific method is his distinction between "principle theories" and "con~tructive theories." Though this distinction informed Einstein's way of \
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doing physics from as early as 1908,8 it found its first longish public discussion in a 1919 article in the Times of London. The "principles" that make up a principle theory like special relativity or macroscopic thermodynamics are empirically well-grounded generalizations that serve a heuristic role by constraining the search for the deeper constructive theories that provide ultimate explanations of phenomena in terms of models constructed from an ontology of systems, states, and interactions. Special relativity comprises two such principles: the principle of special relativity, itself, and the light principle. The first and second laws of thermodynamics are the principles in macroscopic thermodynamics. In general relativity we have again two principles: the principle of general relativity and the equivalence principle. Noteworthy in Einstein's discussion of the latter in 1919 is his introducing it, after the fashion of all such "principles," by stressing its empirical origins and credentials, specifically the "long known experience" of the equivalence of gravitational and inertial mass (Einstein 1919, 13). Far from the principle's being set in a category different in kind from ordinary empirical laws, the principle of equivalence is accorded its special role precisely because of its being a generalization from experience. Nor does it work by way of effecting coordinations between theory and world (Friedman 2001, 90). It works, instead, by constraining our musings about constructive foundations to respect the wealth of experience encapsulated in it. Privileged it is, but as an empirical generalization, not an a priori principle. Viewed thus, the principle of equivalence has only as much security as any battle-tested empirical generalization, which means that it could well turn out to be wrong when we extend our experience into new domains. That's what happened to the second law of thermodynamics when we pressed into regimes where microscopic fluctuations became manifest. And the same has been the fate of the equivalence principle-if it is taken to imply that, in the absence of other forces, massive bodies must follow geodesic trajectories-for particles with spin do not necessarily follow geodesic trajectories. How are we helped by one's characterizing as constitutively, if contingently and relatively a priori, a principle that is just plain false? If we restrict our ken to systems without spin, must we still view the principle of equivalence as an empirical law? Einstein did so regard it, but then he was not at home with modern axiomatizations of general relativity a la Ehlers, Pirani, and Schild (1972), which take as primitive the notions of freely-falling test bodies and photon trajectories. In the EPS framework it is natural to regard the principle of equivalence as making possible the empirical interpretation of the primitive notion of free fall by picking out the inertial motions. But the naturalness of the perspective does not compel us to respond to possible deviations from free fall by positing new forces, any more than Duhem felt himself so compelled, and
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there is, by no means, a modern consensus on regarding the principle of equivalence as anything other than a testable empirical proposition (see Ciufolini and Wheeler (1995) for a view contrary to EPS and to Friedman). Einstein was more familiar with axiomatizations of relativity theory of the kind to be found in Reichenbach's 1924 Axiomatik der relativistischen Raum-Zeit-Lehre, wherein the rods and clocks that provide direct empirical interpretations of basic metrical notions are taken as primitive (Reichenbach's "matter axioms") and the extension of the framework to general relativity explicitly renders the principle of equivalence empirically testable. Had he been familiar with the EPS framework, would he have found it more helpful? My guess is that, for the kinds of reasons asserted in "Geometrie und Erfahrung" (Einstein 1921) and the review of Elsbach (Einstein 1924b), he would still have preferred a rods-and-clocks axiomatization. But one would expect him also to point to the very possibility of two such very different ways of axiomatizing general relativity as illustrating the more general point that he made in the review of Elsbach, the point about the inherent arbitrariness of anyone way of parsing a theory into empirical laws and coordinating principles.
Conclusion: So What? Who cares what Einstein thought? Is the foregoing anything more than a long-winded argument from authority? As such fallacies go, and given the subject matter, arguing from the authority of Einstein is more respectable than arguing from the authority of many others would be. Einstein made his mistakes, but as recent experience with problems such as that of the "hole" and "point-coincidence" arguments in general relativity has shown, much can still be learned from our seeking carefully and in detail to excavate Einstein's views on fundamental conceptual, interpretive, and methodological problen1s in the physics with which he grappled. Still, advancing an argument from authority is not the point. The important point is the one mentioned at the outset. Given that the primary evidence for claims about the contingent a priori will have to be historical evidence, why privilege contemporary rational reconstructions over the self-understandings of the historical actors then1selves? To be sure, the historical actors cannot be counted upon to understand themselves, their actions, and their creations better than we do today, and historical distance often brings deeper insight thanks partly to the very detachment that such distance brings. But historical distance can distort as well as clarify. Ears grow deaf to the vocabularies of the past. Period problematic settings are obscured by our tendencies to whigishness and presentism.
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How then weigh Einstein's views on the a priori in physical theory against the reconstruction proffered by Friedman? The main reason for taking Einstein's view of the matter seriously is that he was at least as good a physicist and philosopher of science as are those involved in the current debate over the contingent a priori, he thought long and hard about the very issues under dispute, and he defended and refined his views through decades of close critical debate with the best minds of his own day. Does it not make sense to invite such a thinker to be part of our contemporary conversation? Does it not make sense to weigh what is, in effect, his own reconstruction of his scientific achievements against later reconstructions? Does he not deserve a place at the table?
NOTES
1. The analysis ofWeyl's transcendental idealism in Ryckman (2005) perhaps affords a kind of transcendental argument for an a priori element in space-time theory in the form of the demand that only such structures be admitted as fundamental in physical theory as are epistemically accessible in the space of intuition of the ideal observer. But that some specific structure is thus epistemically accessible is, presumably, a universal and eternal, if previously unknown, fact. That circumstance seriously mitigates the degree of contingency ascribable to such elements of theory. A general dilemma suggests itself: transcendental arguments do not yield contingent conclusions. So historical evidence could well be the only possible evidence for the contingent a priori. 2. Some aspects of Einstein's philosophy of geometry are discussed in Friedman (2002a). In particular, Friedman there provides a helpful discussion of Einstein's views on how geometry is related to experience, simating Einstein's position in a longer history reaching back to Poincare and Helmholtz. All the more surprising, therefore, that Einstein's arguments against granting a role for the a priori in the empirical interpretation of physical theories is not discussed in those of Friedman's writings bearing directly on the purportedly contingent a priori role of the principle of equivalence in general relativity. 3. For a detailed reconstruction of this way of understanding the relationship between the laws of motion and the law of universal gravitation, see Friedman (1992), 141-77. 4. Reichenbach (1928) is the locus classicus for this notion of coordinating definitions, though the idea first makes its appearance several years earlier in Reichenbach's work (see Reichenbach 1924) as the now merely conventional descendent of the robust notion of the constimtive a priori introduced in Reichenbach (1920). It was Moritz Schlick who urged upon Reichenbach the reconcepmalization of the constimtively a priori principles of coordination as merely conventional coordinating definitions. For details, see Howard (1994), Friedman (1994), and Ryckman (2005).
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5. Note that this is the pre-Vienna Circle Schlick, whose epistemology in the 1910s differed markedly from the more orthodox logical empiricism that he famously defended in the 1920s and 1930s. Foremost among the differences are the theory holism and the blunt defense of realism-in opposition to Mach-in the 1910s. For more on the development of Schlick's thinking see Howard (1994). 6. For more on Einstein's early exposure to Iunt, see Beller (2000) anc;i Howard (1994) and (forthcoming). 7. For a fuller discussion ofWeyl's program as a species of transcendental idealism, see Ryckn1an (2005) and the critical discussion of it in Howard (2005b). 8. See Howard (2005a) for an argument that the distinction is already implicit in Einstein's 1905 photon hypothesis paper as well as a more general discussion of the way the distinction works and possible historical sources and antecedents.
REFERENCES Beller, M. (2000). "Kant's Impact on Einstein's Thought." In Howard and Stachel (eds.) (2000), 83-106. Born, Max (ed.) (1969). Albert Einstein-Hedwig und Max Born. Briefwechsel) 1916-1955. Munich: Nymphenburger. Carnap, R. (1921). Der Raum. Ein Beitrag zur Wissenschaftslehre. InauguralDissertation zur Erlangung der Doktorwiirde der hohen philosophischen Fakultat der Universitat Jena. Gottingen: Dieterich'schen Univ.-Buchdruckerei, W. Fr. Iuestner. Reprinted as "I
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_ . (1918). "Nachtrag." Koniglich Preuflische Akademie der Wissenschaften (Berlin). SitzungsbeYichte. Physikalisch-mathematische IOasse. 478. _ . (1919). "Time, Space, and Gravitation." Times (London). 28 November 1919,13-14. _ . (1921). Geometrie und E1fahrung. El'weiterte Fassung des FestJJortrages gehalten an del' Preussischen Akademie der Wissenschaften zu Berlin am 27. Januar 1921. Berlin: Julius Springer. _ . (1924a). Review of Winternitz (1923). Deutsche Literaturzeitung 45: 20-22 _ - . (1924b). Review of Elsbach (1924). Deutsche Literaturzeitung 45: 1685-92. _ - . (1928). Review of Meyerson (1925). Revue philosophique de la France et de l'#ranger 105: 161-66. Page numbers and quotations from the translation published as an appendix to E. Meyerson (1985), 252-56. - - - . (1936). "Physik und Realitiit." Journal of The Franklin Institute 221: 313-47. - - - . (1949). "Remarks Concerning the Essays Brought together in this Cooperative Volume." In P.A. Schilpp (ed.), 665-88. Elsbach, A. (1924). Kant und Einstein. Untersuchungen iiber das Verhiiltnis der modernen Erkenntnistheorie zur Relativitiitstheorie. Berlin and Leipzig: Walter de Gruyter. Friedman, M. (1992). Kant and the Exact Sciences. Cambridge, MA: Harvard University Press. - - - . (1994). "Geometry, Convention, and the Relativized A Priori: Reichenbach, Schlick, and Carnap." In Salmon and Wolters (eds.), 21-34. Reprinted in Friedman (1999), 59-70. - - - . (1999). Reconsidering Logical Positivism. Cambridge: Cambridge University Press. - - . (2001). Dynamics of Reason: The 1999 Kant Lectures at Stanford University. Stanford, CA: CSLI Publications. - - - . (2002a). "Geometry as a Branch of Physics: Background and Context for Einstein's 'Geometry and Experience.'" In Malament (ed.),l93-229. - - . (2002b). "Kant, Kuhn, and the Rationality of Science." Philosophy of Science 69: 171-90. Howard, D. (1984). "Realism and Conventionalism in Einstein's Philosophy of Science: The Einstein-Schlick Correspondence." Philosophia Naturalis 21: 618-29. - - . (1990). "Einstein and Duhem." Synthese 83: 363-84. - - - . (1994). "Einstein, Kant, and the Origins of Logical Empiricism." In Wesley Salmon and Gereon Wolters (eds.), 45-105. - - - . (2004). "Einstein's Philosophy of Science." The Stanford Encyclopedia of Philosophy (Spring 2004 Edition), Edward N. Zalta (ed.), URL = . - - - . (2005b). "'And I Shall Not Mingle Conjectures with Certainties': Einstein on the Principle Theories-Constructive Theories Distinction." Invited lecture delivered at the conference "2005: The Centenary of Einstein's Annus Mirabilis," The British Academy, London, March 4-5, 2005.
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- - . (2005b). "'No Crude Surfeit': A Critical Appreciation of The Reign of Relativity." Paper delivered at the Annual Meeting of the Pacific Division of the American Philosophical Association, San Francisco, CA, March 26, 2005. - - - . (forthcoming). "Einstein and the Development of Twentieth-Century Philosophy of Science." In M. Janssen and C. Lehner (eds.) (forthcoming). Howard, D. and J. Stachel (eds.) (1989). Einstein and the History of General Relativity. Boston: Birkhauser Boston. - - - . (2000). Einstein: The Formative Years) 1879-1909. Boston: Birkhauser. Janssen, M. and C. Lehner (eds.) (forthcoming) The Cambridge Companion to Einstein. New York: Cambridge University Press. IZuhn, T.S. (1962). The Structure of Scientific Revolutions. Chicago: University of Chicago Press. Le Roy, E. (1901). "Un positivisme nouveau." Revue de Metaphysique et de Morale 9: 138-53. Malament, D. (ed.) (2002). Reading Natural Philosophy: Essays in the History and Philosophy ofScience and Mathematics. Chicago: Open Court. Meyerson, E. (1925). La Deduction Relativiste. Paris: Payot. - - - . (1985). The Relativistic Deduction: Epistemological Implications of the Theory of Relativity. Translated by D .A. Sipfle and M. Sipfle. Dordrecht, Boston, and Lancaster: D. Reidel. Natorp, P. (1910). Die logischen Grundlagen der exakten Wissenschaften. Leipzig and Berlin: B. G. Teubner. Norton, J. (1989). "What Was Einstein's Principle of Equivalence?" In Howard and Stachel (eds.) (1989), 5-47. O'Raifeartaigh, L. (ed.) (1972). General Relativity: Papers in Honor ofJ.L. Synge. Oxford: Clarendon. Poincare, H. (1901). "Sur les Principes de la Mecanique." Bibliotheque du Congres Internationale de Philosophie. Sec. 3, Logique et Histoire des Sciences. Paris: A. Colin. Reprinted as: "La Mecanique classique." In La Science et FHypothese. Paris: Flammarion, 1902, 110-34. Quine, W.V.O. (1951). "Two Dogmas of Empiricism." Philosophical Review 60: 20-43. Reprinted in W.V.O. Quine (1953), 20-46. - - - . (1953). From a Logical Point ofView. Cambridge, MA: Harvard University Press. - - . (1960). Word & Object. Cambridge: MIT Press. Reichenbach, H. (1920). Relativitiitstheorie und Erkenntnis Apriori. Berlin: Julius Springer. - - - . (1924). Axiomatik der relativistischen Raum-Zeit-Lehre. Braunschweig: Friedrich Vieweg und Sohn. - - - . (1928). Philosophie der Raum-Zeit-Lehre. Berlin and Leipzig: Walter de Gruyter. Rosenthal-Schneider, 1. (1980). Reality and Scientific Truth: Discussions with Einstein) von Laue) and Planck. T. Braun (ed.). Detroit: Wayne State University Press. Ryckman, T. (2005). The Reign ofRelativity: Philosophy in Physics 1915-1925. New York: Oxford University Press. Salmon, W. and G. Wolters (eds.) (1994). Language) Logic) and the Structure of
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Scientific Theories. Pittsburgh: University of Pittsburgh Press; Konstanz: Universitatsverlag. ScWick, M. (1915). "Die philosophische Bedeutung des Relativitatsprinzips." Zeitschrift fur Philosophie und philosophische Kl'itik 159: 129-75. _ . (1917). Raum und Zeit in der gegemviirtigen Physik. Zur Einfuhrung in das Verstiindnis der allgemeinen Relativitiitstheorie. Berlin: Julius Springer. _ . (1918). Allgemeine Erkenntnislehre. Berlin: Julius Springer. Sellien, E. (1919). Die erkenntnistheoretische Bedeutung der Relativitiitstheorie. "Kant-Studien" Erganzungshefte, no. 48. Berlin: Reuther & Reichard. Schilpp, P.A. (1949). Alben Einstein: Philosopher-Scientist. Evanston, IL: The Library of Living Philosophers. Weyl, H. (1918a). "Gravitation und Elektrizitat." Kdniglich Preuflische Akademie der Wissenschaften (Berlin). Sitzungsberichte. Physikalisch-mathematische Klasse. 465-80. (1918 b ). Raum, Zeit, Materie. Vorlesungen uber allgemeinen Relativitiitstheorie. Berlin: Julius Springer. - - - . (1918c). "Reine Infinitesimalgeometrie." Mathematische Zeitschrift 2: 384-411. Winternitz, J. (1923). Relativitiitstheorie und Erkenntnislehre. Eine Untersuchung uber die erkenntnistheoretischen Grundlagen der Einsteinschen Theorie und die Bedeutung ihrer E~ebnisse fur die allgemeinen Probleme des Naturerkennens. Wissenschaft und Hypothese, Vol. 23. Leipzig and Berlin: B.G. Teubner. Zschimmer, E. (1923). "Die anschauliche Welt und die Invarianz der Naturgesetze nach A. Einstein." Beitriige zur Philosophie des deutschen Idealismus 3: 22-37.
PART IV History and Philosophy of Physics
[16] How Hume and Mach Helped Einstein Find Special Relativity JOHN D. NORTON 1
1. Introduction In 1946, at the age of sixty-seven, Einstein sat down to record his autobiographical reminiscences of a life in science. Einstein's scientific work had then already become a revered source of stimulation for a new generation of philosophers who sought philosophical enlightenment in Einstein's physical theorizing. Einstein too had long made clear that there was a reverse influence: he in turn drew stimulation from the philosophical literature. From as early as 1912, one could read in his publications in physics that his work on general relativity had been n10tivated by the writings of Ernst Mach, especially through what Einstein had come to call "Mach's Principle." In his autobiographical reminiscences, Einstein now affirmed a similar debt for his special theory of relativity. Though this was not the first time he had made remarks of this type, their prominence in an otherwise brief and authoritative account of the discovery of special relativity could leave no doubt of their importance. He wrote of the decisive moment in which he abandoned the absoluteness of sin1ultaneity and thereby discovered special relativity: Today everyone knows, of course, that all attempts to clarifY this paradox [of light that leads to special relativity] satisfactorily were condemned to failure as long as the axiom of the absolute character of time, or of simultaneity, was rooted unrecognized in the unconscious. To recognize clearly this axiom and its arbitrary character already implies the essentials of the solution of the problem. The type of critical reasoning required for the discovery of this central point was decisively furthered, in my case, especially by the reading of David Hume's and Ernst Mach's philosophical writings. (Einstein 1949, 51) 359
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An earlier remark in a letter of Decen1ber 14, 1915, to Moritz Schlick makes the relative importance of HUll1e and Mach clear:
Your exposition is also quite right that positivism suggested reI. theory, without requiring it. Also you have correctly seen that this line of thought was of great influence on my efforts and indeed E. Mach and still much more Hume, whose treatise on understanding I studied with eagerness and admiration shortly before finding relativity theory. (Papers, A, Vol. SA, Doc. 165) It was Hume more than Mach. Einstein's avowal of intellectual debts to Hume and Mach have long been recognized and exan1ined. 2 My purpose in this paper is to present a more detailed account of what, I believe, Einstein intended with these remarks, illuminating the account with recent work in the history of Einstein's discovery of special relativity. I will suggest that what Einstein learned from Hume and Mach is not quite what one might initially expect. Einstein's discovery is concerned with space and, more essentially, time; and Hume and especially Mach's works are known for their critical analyses of the notions of space and time. Yet the match is not so perfect. Hume's and Mach's analyses of space and time address many aspects of the notions of space and time. But they pass over the specific aspect that was the entirety of Einstein's conceptual breakthrough of 1905: an analysis of the simultaneity of distant events that shows that observers in relative n10tion need not agree on which events are simultaneous. That is not to be found in Hume's and Mach's writings. What is to be found, however, is an account of the nature of concepts in general: concepts are dependent entirely on our sense impressions or sensations; they are inapplicable as representations of reality, that is, fictional, in so far as they extend beyond our sense experience. Neither Hume nor Mach saw this fictional character as a tool that could be used in theory construction; fictional concepts were false representations to be elin1inated from one's account of nature or at best tolerated if, as Hume held of causation, the elimination was unachievable. Here Einstein differed. One does not have to eliminate a fictional concept. Its presence indicated an arbitrariness in our physical theorizing. It could be retained as long as its arbitrary character was recognized and it was accommodated in such a way as to preclude unwitting introduction of false presumptions. At the decisive moment in his discovery of special relativity, Einstein did just this. He recognized that the traditional concept of the simultaneity of distant events was not fixed by experience; and that its use had tacitly committed us to a false presumption, the absoluteness of simultaneity - its independence from the state of motion of the observer. So he
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replaced it with a new concept of simultaneity. It was introduced by a freely chosen definition that exploited the arbitrariness of the concept. That definition brought no tacit commitment to the absoluteness of simultaneity. In the context of the postulates of his new theory, it led to the relativity of simultaneity, the dependence of judgments of simultaneity of distant events on the state of motion of the observer. This paper will tell the story of Einstein's discovery and its debt to the writings of Burne and Mach. In Section 2, I will review the problems in electrodynamics that occupied Einstein for over seven years. Their recalcitrance finally led Einstein to seek a radical solution outside electrodynamics as a last desperate measure, a reformation of our notion of time and simultaneity. In Section 3, I will describe how Einstein justified this extraordinary departure by means of an austere account of the nature of concepts in scientific theories. In Section 4, I will seek to show how this account drew essentially on the writings of Mach and Burne. Section 4.3 contains some speculation over why Einstein singled out Burne over Mach. I will suggest that Mach's writings may have been less important since Einstein regarded them as denying the freedom of concept formation Einstein needed in 1905 to introduce his new definition of distant simultaneity. Finally in Section 5, by way of a conclusion, I will reflect on how Einstein's use of Burne's and Mach's philosophical writings was highly selective. Bis goals were as much to understand Burne's and Mach's thought as to find in them ideas that may be useful in his creative work as a physicist. Understandably this latter goal induced Einstein to be undeterred by systematic problems in Burne's and Mach's writings and to ignore consequences that did not suit his purpose of creating new physical theories. To preclude confusion, let me stress here that I consider just the question of how Burne's and Mach's work figured in Einstein's discovery of special relativity in 1905. I shall not consider the large influence of Mach especially on Einstein's later work in general relativity.
2. Einstein's Electrodynamical Pathway to Special Relativity Einstein (1952) recalled the long years of intense effort that preceded his 1905 discovery of special relativity-"the seven and more years that the development of the Special Theory of Relativity had been my entire life." Bere I will review what we know of the struggles of these years, emphasizing that they were devoted almost entirely to grappling with problems in electrodynamics. That the real issue lay in a reconceptualization of the notion of simultaneity entered only in the final five to six weeks of his
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seven or more years of work. This review will help us to appreciate just what, on my best account, lay behind his reconceptualization of simultaneity and his drawing on the philosophical work of Hume and Mach. It was not an impulsive experiment in speculative philosophy. It was the culmination of years of labor, the adoption of a new approach to an old problem to which he was compelled by the failure of all other avenues. We might well suppose that the courage to take this step was supplied as much by his desperation to solve a stubbornly recalcitrant problem as by the persuasiveness of Hume's and Mach's writings. While the review of this section gives a specific context for Einstein's reconceptualization of simultaneity in 1905, the account of later sections does not depend on the details of how Einstein came to identify simultaneity as the stumbling block. Perhaps all that really matters is that it came after years of failure with ordinary solutions, so that Einstein was willing to entertain something extraordinary.
2.1 The magnet and conductor thought experiment Our documentation of Einstein's interest in the problems that led to special relativity begins with events as early as the sumn1er of 1895, when a sixteen year old Einstein wrote an essay proposing experimental investigation of the electromagnetic ether, the medium proposed by 19th-century theories as the carrier of electromagnetic fields and light (Papers, Vol. 1, Doc. 5). This ether supplied a preferred state of rest for the universe, but a long tradition of experiments in the 19th century had failed to detect this state of rest. In the introductory section of the paper in which Einstein unveiled special relativity, "On the Electrodynamics of Moving Bodies," Einstein (1905) pointed to these experiments as grounds for doubting that there is an ether state of rest and as evidence for the principle of relativity, the assertion of the equivalence of all inertial states of motion. Traditional textbook accounts give pride of place to one of these experiments, the Michelson-Morley experiment. We now know that the experiment played only a n1inor role in Einstein's thoughts (see Holton 1969, Shankland 1963/73, and Stachel 1987). Einstein did know of the experiment prior to 1905 (as was finally revealed by Einstein's correspondence of 1899 see Papers, Vol. 1, Doc. 57). However, it appeared to Einstein to do little more than support the idea that physics must conform to the principle of relativity; it did not establish the constancy of the speed of light, as later textbook accounts commonly assert. These 19th-century experin1ents played some role in Einstein's thought. Einstein (1920) made clear, however, that another reflection provided the real impetus: "The phenomenon of magneto-electric induction
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compelled me to postulate the (special) principle of relativity." Here Einstein referred to the magnet and conductor thought experiment that Einstein laid out in the opening sentences of his "On the Electrodynamics of Moving Bodies." While 19th-century experiments had revealed no ether state of rest, the theoretical structure of Maxwell's electrodynamics seemed to depend upon it. When a magnet is at rest in the ether, it is surrounded by a magnetic field. If the magnet is set in motion, however, the magnetic field changes in strength with time, as the magnet moves past some fixed point in the ether. That change, according to Maxwell's theory, causes a new entity to appear, an induced electric field as shown in Figure 1. (Einstein noted that the induced electric field has "a definite energy value," as if to emphasize its reality.) induced electric field
magnetic
~
yfield
~"f9=r=l~ ??C~magnetat rest -l~ ~
in the ether ~
~ !f2);:: =~[]====I~
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FIGURE 1. Magnets at rest and moving in the ether
One might suppose that this induced electric field would be an unequivocal indicator of whether the magnet is at rest in the ether or not, giving a direct observational means for distinguishing motion from rest in the ether. What Einstein realized was that this induced electric field could not be used as such an indicator because of an almost conspiratorial coincidence. To take a slightly simpler case than the one Einstein considered in 1905, imagine that an observer located on the magnet tries to test whether the induced electric field is present by looking for the current it engenders in a conductor: •
If the magnet is at rest in the ether, there would be no current simply because there is no induced electric field.
•
If the magnet is moving in the ether, there would be an induced electric field and that field would generate a current in the conductor. However, a second effect, the motion of the conductor in the magnet's own magnetic field, would also induce a second current of exactly equal magnitude but opposite direction. The two currents would cancel and there would no net electric current in the conductor.
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The electrodynamics of Einstein's time treated motion and rest of a magnet in the ether as very different cases. But as far as observables were concerned-the measurable current in a conductor-the two cases were the same. Once again experiment fails to reveal motion with respect to the ether state of rest. Einstein saw this as a strong indication that electrodynamics must somehow be modified so as to eradicate its dependence on this elusive state of rest; that is, it must be conformed to the principle of relativity. The magnet and conductor thought experiment did a great deal more than just give Einstein the impetus to this conclusion. It also gave him a theoretical device that would later form an essential part of his completed theory. Classically, an electric field is an absolute quantity. Either it is present or not; and all observers, whatever their states of motion, would agree on its presence or absence. This is contrasted with relative quantities, such as the kinetic energy of a body. These quantities vary with the state of motion of the observer. A roadside observer will assign a large kinetic energy to a speeding car. The car occupants, however, will assign the car no kinetic energy, for the car is at rest with respect to them. Einstein's early deliberations on the magnet and conductor thought experiment revealed to him that an electric field had a similar relative existence. If one were at rest relative to the magnet, the field surrounding the magnet would manifest as a pure magnetic field. If one were in motion relative to the magnet, that same field would n1anifest as a magnetic field with an electric field. That is, the state of motion of the observer would determine whether or not the one field would appear to have an electric field within it or not. The importance of this insight into field transformations was that it provided Einstein a theoretical device with which to implement the principle of relativity in electrodynamics, as has been pointed out by Earman et al. (1983) and Rynasiewicz (2000). The induced electric field of the magnet and conductor need no longer be regarded as revealing the absolute motion of the magnet. It now merely revealed the relative motion of magnet and observer-an effect fully in accord with the principle of relativity. All that was required was to find the general transformation law that would work in all cases and eliminate the need for an ether state of rest in the theory. One might hope that this device could be used to implement the principle of relativity within Maxwell's electrodynamics and it is natural to suppose that Einstein explored the possibility. As I have shown in Norton (2004, Section 2), 3 the exploration would be encouraging, initially. Maxwell's electrodynamics is based on four field equations. Two only are needed to analyze the magnet and conductor thought experiment. As long as we consider them alone, it proves quite easy to find a field transformation law that allows an account of the magnet and conductor thought
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experiment, fully in accord with the principle of relativity. Further exploration rapidly generates disastrous consequences, however, and the most important is the following. The other two of Maxwell's equations can also be treated in a way that is in accord with the principle of relativity. However, one must use a different field transformation law for these other twO equations. So Maxwell's theory cleaves into two parts, each of which can be made to conform to the principle of relativity, but not when they are joined. While Einstein would have had every reason to think that there was something very right in this notion of field transformations, he would also have had to see that something more was needed if electrodynamics was to be rendered compatible with the principle of relativity.
2.2 An emission theory oflight Using his device of field transformations, Einstein could bring conformity with the principle of relativity to one part of Maxwell's theory or to the other, but not to both together. It takes only a little reflection to see that this circumstance is inevitable. A theory that implements the principle of relativity in the context of ordinary Newtonian space and time must treat velocities in a quite particular way. Assume an observer watches a gun fire a bullet. The bullet velocity, as determined by the observer, must be given by the velocity of the bullet with respect to the gun added vectorially to the velocity of the gun with respect to the observer. Now MaA'Well's electrodynamics is also a theory of light; according to it, light consists merely of waves that propagate in the electromagnetic field. In the Newtonian context, the velocity of the propagation of light must be treated just as the velocity of everything else. By direct analogy with the bullet and gun, to determine the observed velocity of light we must add vectorially the velocity of propagation of light with respect to its emitter to the velocity of the emitter with respect to the observer. This rule is the characteristic property of a so-called "emission" theory oflight. Any theory oflight that implements the principle of relativity in the Newtonian context-that is, any "Galilean covariant" theory-must be an emission theory (but not conversely). The difficulty Einstein faced is that Maxwell's electrodynamics cannot be an emission theory. One of the most important consequences of the theory is that light in a vacuum always propagates at the same speed, c=186,OOO miles per second, with respect to the ether. Its velocity of propagation is unaffected by the velocity of the emitter. So Maxwell's theory cannot conform to the principle of relativity in a Newtonian context. This might not be apparent ifwe only look at each of the two parts of Maxwell's theory mentioned above individually. For neither part alone is sufficiently
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rich to determine a velocity of propagation for light. But the t\Vo parts combined are able to do this. Einstein's response, as John Stachel (1982) emphasized, was to seek to modify Maxwell's theory in such a way as to convert it into an emission theory. Einstein left us no direct record of his explorations. The strongest clue of their contents lies in his repeated remarks that they were like those of Walter Ritz (1909a, 1909b), who, subsequently to the advent of special relativity, sought to construct a Galilean covariant, emission theory of electrodynamics and light. In Norton (2004, Sections 3-4; forthcoming, Section 4.3), I have put considerable effort into reconstructing the sort of theory that Einstein explored. It turns out that there is a single quite plausible theory that exploits Ritz's key theoretical maneuver while at the same time leaving unchanged that part of Maxwell's theory that treats the magnet and conductor, so that Einstein's device of field transformations could still be employed. While initially promising, this and all other Galilean covariant theories of electrodynamics prove unsatisfactory. Combing through Einstein's later correspondence and writings we find a plethora of reasons given for the failure of an emission theory of light. The theory must fail, Einstein asserted, because the physical state of a light ray must be determined completely by its intensity, color and polarization; and an emission theory cannot be formulated as a field theory governed by differential equations; and an emission theory would allow light to back up on itself, in the sense that light emitted earlier by accelerating sources could be overtaken by light emitted later; and there would be problems with shadow formation. It is not at all straightforward to see how the objections work. I have urged (Norton 2004, Sections 5-6; Norton forthcoming, Section 4.3) that they can be put into cogent and compelling form if we embed them in Einstein's famous chasing a light beam thought experin1ents of his Autobiographical Notes (1949, 49-50). Indeed, I suggest, if we don't, it is hard to understand the importance Einstein accords to the thought experiment in his recollections. For our purposes, what n1atters most is that these explorations, however they may have proceeded, required years of effort that brought4 Einstein mounting frustration and a willingness to entertain a radical solution.
2.3 ((The StepJJS: The relativity ofsimultaneity The breakthrough came some five to six weeks prior to completion of special relativity. Einstein faced an impasse in t\Vo incompatible demands. He felt compelled to conform Maxwell's electrodynamics (in the form given by Lorentz) to the principle of relativity, but that seemed impossible since Maxwell's electrodynamics required a constant speed for light. Yet all his
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efforts to modifY Maxwell's electrodynamics to an emission theory had failed. The tension could be reduced to the incompatibility of two requirements: the principle of relativity and the constancy of the speed of light. 6 Einstein suddenly realized these two requirements were not incompatible after all. The circumstances of this realization have become part of the heroic lore of discovery. Einstein recounted the story in a lecture in Kyoto on December 14, 1922:7 Why are these two things inconsistent with each other? 1 felt that 1 was facing an extremely difficult problem. 1 suspected that Lorentz's ideas had to be modified somehow, but spent almost a year on fruitless thoughts. And 1 felt that was a puzzle not to be easily solved. But a friend of mine living in living in Bern (Switzerland) [Michele Besso] helped me by chance. One beautiful day, 1 visited him and said to him: '1 presently have a problem that 1 have been totally unable to solve. Today 1 have brought this "struggle" with me.' We then had extensive discussions, and suddenly 1 realized the solution. The very next day, 1 visited him again and immediately said to him: 'Thanks to you, 1 have completely solved my problem." My solution actually concerned the concept of time. Namely, time cannot be absolutely defined by itself, and there is an unbreakable connection between time and signal velocity. Using this idea, 1 could now resolve the great difficulty that I previously felt. After 1 had this inspiration, it took only five weeks to complete what is now known as the special theory of relativity. What Einstein alluded to was his recognition, laid out in detail in Section 1 of his 1905 relativity paper, that the principle of relativity and the constancy of the speed of light could be rendered compatible if one was willing to allow that observers in relative motion might disagree on which spatially distant events are simultaneous. He argued that the simultaneity of spatially distant events could not be directly experienced. So we had to specifY by a definition which spatially distant events were simultaneous. 8 Einstein's definition, in slightly modified form, is shown in Figure 2. An observer at the midpoint of a platform will judge the emission of light signals at clocks A and B at either end to be simultaneous, if those light signals arrive simultaneously at the observer.
FIGURE 2. Einstein's defmition of simultaneity
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Innocuous as the definition may seem, Einstein proceeded to show that it yielded a startling conclusion if we also adhered to the principle of relativity and the constancy of the speed of light. Consider that same synchronization procedure, as it would appear to an observer who moves uniformly to the left. What that observer would see is shown in Figure 3.
A-------------+)0-+(------- B
FIGURE 3. Einstein's procedure as seen by a moving observer
The new observer would judge the platform observer to be moving away from the signal emitted by clock A and towards the signal emitted by clock B. So, the fact that the signals arrive simultaneously at the platform observer shows that the two emission events were not simultaneous. The emission event at clock A must have happened earlier to give the light signal time to catch up with the fleeing platform observer; and the emission event at clock B must have happened later to compensate for the approach of the platform observer. In Figure 3, the signal from A must cover the greater distance AO and the signal from B must cover the lesser distance BO. Thus the platform and resting observer disagree on whether the two emission events are simultaneous, an illustration of the relativity of simultaneity. This inference requires the constancy of the speed of light; the moving observer must also assign the same speed c to light in a vacuum. Thus a signal traversing the greater distance AO requires more time than one traversing the lesser BO. The deduction would fail if we assumed an emission theory since the moving observer would assign unequal speeds to the two signals. It also invokes the principle of relativity in so far as both observers use the same definition of simultaneity. What Einstein's analysis shows is that the inconsistency of the principle of relativity and the constancy of the speed of light is only apparent. They can co-exist if we give up the notion that simultaneity is absolute, that is, the notion that all observers must agree on whether two events are simultaneous. That recognition also answers the obvious objection to attempts to retain both the principle of relativity and the constancy of the speed of light. If we conjoin them, we are to believe that all inertial observers will measure the same speed for light. But how can that be possible? If an observer chases after light at great speed, would not the moving observer measure a speed for light less than that of a resting observer? We can now see why that slowing need not happen. All judgments of the speed of light
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depend upon measurements of time that use synchronized clocks. We might see how much time a light pulse takes to traverse a platform, such as shown in Figures 2 and 3, by comparing the times read by clocks A and B as the light pulse passes. If the two observers are in relative motion, they will disagree on the simultaneity of distant events. As a result, they will synchronize their clocks differently. If they both use Einstein's procedure of Figure 2 to synchronize their clocks, it is easy to see that each has adjusted the synchrony of their clocks in just the right way to ensure that each will measure the same constant c for the speed oflight. 9 Since judgments of simultaneity arise throughout kinematics, Einstein now needed to ascertain how our traditional notions of space and time must be modified to accommodate this new result of the relativity of simultaneity. That accommodation is the working out of the special theory of relativity, a new theory of space and time. The new theory solves Einstein's original problem of conforming Maxwell's electrodynamics to the principle of relativity. As Einstein showed in his 1905 paper, it turns out that, within the new theory of space and time, Maxwell's electrodynamics already conforms to the principle of relativity; all that was needed was the selection of the appropriate transformation rules for electric and magnetic fields. Indeed, as I show in a simple thought experiment in Norton (forthcoming, Section 4.2), the field transformation laws Einstein considered necessitate modifications to notions of simultaneity if they are used within Maxwell's theory.
3. Einstein's New View of Concepts 3.1 Concepts must begrounded in experience What licensed Einstein's taking of "the step" is a new approach he explicitly decided to adopt towards concepts such as simultaneity in physical theory. The view is quite simple: a concept can properly represent something physically real only in so far as the concept is grounded in experience. That adopting this view enabled "the step" is expressed quite clearly in a 1924 recollection: 10 After seven years of reflection in vain (1898-1905), the solution came to me suddenly with the thought that our concepts and laws of space and time can only claim validity insofar as they stand in a clear relation to experiences; and that experience could very well lead to the alteration of the concepts and laws. By a revision of the concept of simultaneity into a more malleable form, I thus arrived at the special theory of relativity.
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And Einstein elaborated the view in his 1916 interview with Max Wertheimer who reported: l l ... an illustration which Einstein offered in discussion. Suppose somebody uses the word 'hunchback.' If this concept is to have any clear meaning, there must be some way of finding out whether or not a man has a hunched back. If I could conceive of no possibility of reaching such a decision, the word would have no real meaning for me. 'Similarly,' Einstein continued, 'with the concept of simultaneity. The concept really exists for the physicist only when in a concrete case there is some possibility of deciding whether the concept is or is not applicable. Such a definition of simultaneity is required, therefore, as would provide a method for deciding. As long as this requirement is not fulfilled, I am deluding myself as physicist (to be sure, as nonphysicist too!) if I believe that the assertion of simultaneity has real meaning.' This view about the meaning of concepts can be found in Einstein's more general writings. His popular text on relativity theory (Einstein 1917, §8) asserts at the relevant moment: "The concept [of simultaneity] does not exist for the physicist until he has the possibility of discovering whether or not it is fulfilled in an actual case."
3.2 The pur;ging of the a priori from concepts Einstein's principal concern was the danger accompanying the use of concepts not properly grounded in experience. Use of a concept in a physical theory typically requires some sort of factual presun1ption. The danger was that use of a concept in a physical theory might inadvertently commit us to false physical presumptions, which we would introduce unwittingly as a kind of a priori knowledge, since it entered our theorizing merely through our choice of concepts and not through en1pirical investigation of the presumption. For example, before relativity theory, simultaneity was taken to be a two place relation between events. Events A and B could be simultaneous simpliciter; after relativity theory, it was recognized that events A and B can be sin1ultaneous only with respect to an observer or frame of reference. So use of the older concept had required the tacit presumption that judgments of simultaneity are independent of observer or frame of reference. Here is how Einstein recounted our failure to recognize the inadequate grounding in experience of distant simultaneity prior to relativity theory:
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The illusion which prevailed prior to the enunciation of the theory of relativity-that, from the point of view of experience the meaning of simultaneity in relation to spatially distant events and, consequently, that the n1eaning of physical time is a priori clear-this illusion had its origin in the fact that in our everyday experience we can neglect the time of propagation of light. We are accustoll1ed on this account to fail to differentiate betvveen "simultaneously seen" and "simultaneously happening"; and, as a result, the difference between time and local time is blurred. The lack of definiteness which, from the point of view of its empirical significance, adheres to the notion of time in classical mechanics was veiled by the axiomatic representation of space and time as given independently of our sense experiences. Such a use of notions-independent of the empirical basis to which they owe their existence-does not necessarily damage science. One may, however, easily be led into the error of believing that these notions, whose origin is forgotten, are logically necessary and therefore unalterable, and this error may constitute a serious danger to the progress of science. (Einstein 1936, 299) Einstein urged that we preclude such unwitting introduction of a priori presumptions by proper attention to the experiences needed to warrant the application of the concept. Where no such experiences are possible, the concept is fictional or arbitrary. In his analysis in "On the Electrodynamics of Moving Bodies," Einstein had no difficulty identifying the experiences needed to warrant application of the concept of simultaneity for events at the same place. However, he could not identify experiences sufficient to warrant application of the concept of simultaneity for events at different places, so that concept is fictional or arbitrary. At this point, someone of a Machian bent would call for the purging of the concept from the theory as idle metaphysics with no grounding in experience. Einstein, however, was willing to retain the concept as long as its arbitrary character was recognized and in a way that no longer allowed the unwitting introduction of a priori presumptions. In the case of distant simultaneity, Einstein achieved this by introducing distant simultaneity through a definition-a freely chosen stipulation-carefully designed to minimize the danger of introducing false physical presumptions. In the context of Einstein's two postulates, his definition of simultaneity had the consequence that judgments of simultaneity of distant events would vary with changes of the state of motion of the observer. Einstein's procedure had purged kinematics of the false presumptions about simultaneity that permeated the Newtonian view of space and time.
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4. The Debt to Mach and Hume We have seen that Einstein expressed a debt to the philosophical writings of Hume and Mach in his discovery of special relativity. We are now in a position to see what that debt was. For the demand that concepts must be properly grounded in experience permeates both of their writings. Indeed much of their philosophical critiques amounts to the purging of a priori elements from concepts that do not meet this demand. Because of the prominence of this view of concepts in Einstein's recollections of the breakthrough of 1905 and also in Hume and Mach's writings, I believe this view was the debt Einstein acknowledged to Hume and Mach, rather than any particular analysis by Hume and Mach of the notion of tin1e. Of course, it is not so easy to disentangle the particular analyses of the notion of time from the general view of concepts. Mach's (1960, Ch. 2, §VII) celebrated critique of Newton's Absolute Space is simply a direct application of the general view. Newton's concept, Mach complains, is a fiction not given in the facts of experience. With only a little more effort, one finds Hume also applying his view of concepts to time in ways we could imagine impressing a young Einstein. 12 Since none of these analyses explicitly treat simultaneity, even if they were the ones that impressed him, Einstein would still need to abstract their general approach to concepts and then apply that abstracted approach to distant simultaneity in order to make the breakthrough of 1905.
4.1 Mach While Einstein attributed greater influence to Hume, it is easier for us to see that Einstein found this view of concepts in Mach's writings. The reason is an appreciation Einstein wrote of Mach's work as Einstein's 1916 obituary for Mach. We need not read Mach's writings and try to ascertain how Einstein may have read or misread them and what in them attracted him. The obituary tells us directly how he read Mach's work and what he valued in it. What looms large in his appreciation is Mach's treatment of concepts just along the lines sketched above. Einstein (1916) described Mach's general orientation towards concepts, which contains the basic doctrine that concepts have meaning only in so far as they are empirically grounded: Science is, according to Mach, nothing but the comparison and orderly arrangement of factually given contents of consciousness, in accord with certain gradually acquired points of view and methods.... . . . concepts have meaning only in so far as they can be found in things,
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just as they are the points of view according to which these things are organized. (Analysis of concepts) He proceeded then to find in Mach the warning against the use of concepts disconnected from their experiential grounding: Concepts that have proven useful in ordering things can easily gain authority over us such that we forget their worldly origin and take them as immutably given. They are then rather rubber-stamped as a 'necessity of thought' and an 'a priori given,' etc. Such errors often make the path of scientific progress impassable for a long time. And then Einstein found in Mach the license to purge our theories of concepts that extend beyond their experiential grounding or to alter them to bring them into accord with that grounding: Therefore, it is not at all idle play if we are trained to analyze long familiar concepts, and to point out upon which circumstances their justification and usefulness depends; and how they evolved in particular from the givens of experience. Thereby their all too powerful authority is broken. They are removed, if they cannot properly be legitimized; they are corrected if their association with given things was too sloppy; they are replaced by others if a new system can be established that, for various reasons, we prefer. These remarks refer to Mach's historico-critical approach to understanding our present scientific concepts through an account of their historical development, a central theme in Mach's critical writings on physics, whose importance Einstein recognized. Einstein then proceeded to illustrate how Mach applied this view in his well-known critiques of Newton's absolute space and time and mechanics (Mach 1960, Ch. 2. VII Newton's Views of Time, Space and Motion). The illustrations, quoted at length, are too well known to bear repetition here. Judgments of time are revealed to be really just expressions of dependence of one thing on another, such as the oscillations of a pendulum or the position of the earth; Newton's bucket experiment reveals only what happens when there is relative rotation between the water and the rest of the universe; it does not reveal an absolute motion, which has no grounding in sense experience. Einstein here attributes to Mach the view of concepts that Einstein found decisive in his discovery of special relativity: concepts must be properly grounded in experience and there are great dangers in using concepts that fail to be so grounded. Einstein clearly made this same
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connection, for he proceeded to aver that Mach had just the critical apparatus needed to discover special relativity, if only he had worked in another time: It is not improbable that Mach would have hit upon relativity theory if, in the time that he was of young and fresh spirit, physicists would already have been moved by the question of the meaning of the constancy of the speed of light. In this absence of this stimulation, which follows from Maxwell-Lorentz electrodynamics, even ~Aach's critical urge did not suffice to arouse a feeling for the necessity of a definition of simultaneity for spatially distant events. We must recall the context of Einstein's remarks, an obituary written to honor Mach. So we might well understand that Einstein here overlooked Mach's opposition to introducing arbitrary concepts into theories. Einstein was n10re forthright about the lack of fertility of Mach's approach the following May when he wrote to his friend Besso, "I do not inveigh against Mach's little horse; but you know what I think about it. It cannot give birth to anything living, it can only stamp out harmful vermin.,,14 And again he averred in a lecture in Paris of 6 April 1922, ". . . in fact what Mach has done is to make a catalog, not a system.,,15
4.2 Hume We saw above that Einstein informed Schlick in 1915 that Hume had "still much more" influence than Mach. However there is son1e uncertainty over just the work to which Einstein referred. He informed Schlick that he studied Hume's "treatise on understanding," "shortly before finding relativity theory." But does that betoken Hume's A Treatise ofHuman Nature? Or is it Hume's later An Enquiry concerning Human Understanding? I will follow the editors of Papers, Vol. 2, who decide the former is intended. They note (pp. xxiv-xxv) that the first part of Hume's Treatise ofHuman Nature was then available in a recently published German translation; that Einstein recalled reading Hume in German translation; and that it was known to belong to the reading list of the Olympia Academy, the small reading group formed by' Einstein and his friends Conrad Habicht and Maurice Solovine in 1902. Ifwe turn to Hume's Treatise, we find that Hume's entire approach is based on a view of concepts that agrees with that of Einstein and Mach in the essential observation that concepts ("ideas") are grounded in sense experience ("impressions"). Hume concluded his introductory section (Book 1, Part 1, Section 1) with the synoptic assertion:
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... all our simple ideas proceed either mediately or immediately, from their correspondent impressions. This then is the first principle I establish in the science of human nature .... Hume then unleashed this basic principle upon the ideas of metaphysics, demanding that they all derive from impressions. Hence, he noted, the idea of time depends upon our having changing impressions (Book 1, Part II, Section III): As 'tis from the disposition of visible and tangible objects we receive the idea of space, so from the succession of ideas and impressions we form the idea of time, nor is it possible for time alone ever to make its appearance, or be taken notice of by the mind. . . . time cannot make its appearance to the mind either alone, or attended with a steady unchangeable object, but is always discover'd by some perceivable succession of changeable objects.
What of ideas that extend beyond their grounding in impressions? These are denounced as falsehoods, as, for example, in the case of those who try to apply the notion of duration in time to perfectly unchangeable objects (Book 1, Part II, Section III): I know there are some who pretend, that the idea of duration is applicable in a proper sense to objects, which are perfectly unchangeable.... But to be convinced of its falsehood we need but reflect on the foregoing conclusion, that the idea of duration is always deriv'd from a succession of changeable objects, and can never be convey'd to the mind by any thing stedfast and unchangeable. For it inevitably follows from thence, that since the idea of duration cannot be deriv'd from such an object, it can never in any propriety or exactness be apply'd to it, nor can any thing unchangeable be ever said to have duration. Ideas always represent the objects or impressions from which they are deriv'd, and can never without a fiction represent or be appl'd to any other.... This mode of analysis-illustrated here in its application to the notion of duration-is repeated throughout Hume's Treatise. What of the idea of substance, a substratum that carries properties? "We have therefore," Hume ( Book 1, Part 1, Section VII) assures us, "no idea of substance, distinct from that of a collection of particular qualities, nor have we any other meaning when either talk or reason concerning it." Most famously this is the basis of Hume's celebrated critique of causation. There is no necessary connection relating cause to effect beyond the habit of mind that assigns it to the sorts of constant conjunction in sense experience that we call causal (Book 1, Part III, Sections XIV, XV).
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4.3 Why Hume more than Mach? Why was it that Hume influenced Einstein "still much more" than Mach in Einstein's discovery of relativity?16 It might just have been happenstance. Perhaps Einstein read Hume at just the lnoment in his work on electrodynamics that reflections on the origin of our concepts were apposite. Or perhaps there is a relevant, systematic difference between Hume's and Mach's writings that might explain why Einstein found more inspiration in Hume. There is such a difference. At the decisive nl0ment, Einstein realized the fictional character of the concept of the sinlultaneity of distant events. He exercised the freedom implicit in this fictional character to assert through definition which distant events are simultaneous. The exercising of this freedom in turn calls to mind Einstein's frequent characterization of the concepts of scientific theories as "free creations of the human spirit." (For discussion, see Howard 2004, Section 2.) While Einstein seems to have found this freedom to conform with Hume's writings and perhaps even to be encouraged by them, he found it directly contradicted in Mach's writings. Let us take Einstein's view of Mach first. As we saw in Section 4.1, Einstein may well have felt it inappropriate to dwell on Mach's philosophical weal(ness in writing Mach's obituary. However Einstein felt no such reservation years later in private correspondence with his old friend Besso. He wrote to Besso on 6. Jan. 1948: [Mach] took convincingly the position that these conceptions, even the most fundamental ones, obtained their warrant only out of empirical knowledge, that they are in no way logically necessary. . . . I see his weal(ness in this, that he more or less believed science to consist in a mere "ordering" of empirical "material"; that is to say, he did not recognize the freely constructive element in the formation of concepts. In a way he thought that theories arose through discoveries and not through inventions. He even went so far that he regarded "sensations" not only as material which has to be investigated, but, as it were, as the building blocks of the real world; thereby, he believed, he· could overcome the difference between psychology and physics. If he had drawn the full consequences, he would have had to reject not only atomism but also the idea of a physical reality. (Speziali 1972, Doc. 153; translation Holton 1968,231) Mach's view of science as the mere ordering of sensations left Einstein no room for invention, the freedom of construction of concepts invoked in his 1905 definition of distant simultaneity.
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Bume is mentioned in one other place in Einstein's Autobiographical Notes. Einstein noted (p.l3): Bume saw clearly that certain concepts, as for example that of causality, cannot be deduced from the material of experience by logical methods. A familiar aspect of Hume's view (c.g. Hume 1978, Book 1, Part III, Section IV, 165-66, 170) is that we proceed from sense impressions to the concept of cause by custom or habit. That aspect might well suggest to Einstein the sort of freedom in concept formation that Einstein invoked in his 1905 definition of simultaneity. At least Hume's view would seem compatible with this freedom, for Hume does not call upon us to eradicate the notion of causation as necessary connection, but merely to recognize its true origin and nature. However, Einstein read Mach's strict antimetaphysical attitude as directly contradicting this freedom.
4.4
Taking the step
We saw above that Einstein reported to Schlick that he had studied Hume's writing "shortly before finding relativity theory." So we can well imagine how the decisive moment came. After exploration upon exploration had led to nothing, Einstein finally saw that Maxwell's electrodynamics already conformed to the principle of relativity. That it appeared not to, he now saw, was an illusion fostered by the false presumption of absolute simultaneity. With the support of his readings of Hume and Mach, he determined that the concept of absolute simultaneity was fictional and he found the courage to discard it. In its place, he introduced a new definition of distant simultaneity, exercising the freedom to modifY concepts that may have been encouraged by his reading ofHume, but not of Mach. We have essentially no further clues as to the nature and even order of his deliberations at this moment. Since Einstein's later reporting of the relativity of simultaneity is routinely given in terms of clocks and how light signals may be used to synchronize them, it is easy to presume that Einstein discovered the relativity of simultaneity by reflection on clocks and light signals. I7 Elsewhere I have urged caution in making this presumption and explored other possibilities (Norton 2004, Sections 7-8; Norton forthcoming, Sections 4.5, 6). None of the deliberations Einstein reported prior to the step involve clocks and light signals. They pertain to light as a waveform extended in space, rather than as a signal, a spatially localized pulse. IS There are other pathways Einstein could follow that did
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not require contemplation of clocks and synchronization by light signals. Lorentz's (1895) Versuch, which Einstein had read, used Lorentz's "theorem of corresponding states" to establish that optical experiments would not reveal the motion of the earth as long as it was slow. 19 Lorentz's result depended essentially on his use of "local time," which mathematically served precisely the same function as the relativity of simultaneity in special relativity. All that was needed was to give a different physical interpretation to Lorentz's mathematics. Might the step have been taken by Einstein, en1boldened by Burne and Mach, recognizing that Lorentz's local tin1e was just time and not a n1athematical contrivance? There is even an observational pathway to this conclusion. Einstein recalled two experiments as having been important in guiding him to special relativity: Fizeau's experiment on the speed of light in moving water and the observation of stellar aberration, the deflection of starlight due to the relative motion of the earth and the star. What is distinctive about the observed effects in both experiments is that they arise entirely from Lorentz's local time. That means that, in the context of special relativity, they arise entirely from differences in judgments of simultaneity by observers in relative motion. In the case of stellar aberration, for example, a wave front changes its direction of propagation as we transform between observers in different states of motion. The change of direction arises entirely because the two observers use different judgments of simultaneity in assembling the local fragments of the wave into a spatially extended wavefront at one instant. If Einstein noticed this, he would have found an observational basis for the relativity of simultaneity, largely independent of the details of Maxwell's electrodynamics and requiring no thought of clocks and their synchronization by light signals. My conjecture is that just such a recognition explains Einstein's attribution of in1portance to these two experiments.
5. Conclusion: Einstein as Physicist and Philosopher When Einstein expressed a debt to the philosophical writings ofHume and Mach in his discovery of special relativity, he did not refer to a particular doctrine about space and time in their writings. While both cast their critical eyes on the notions of space and time, neither gave the sort of analysis of the concept of simultaneity that Einstein needed. Rather, what Einstein found helpful in their writings was a view about concepts. They must be grounded in experience and, in so far as they extended beyond that grounding, they are fictional or arbitrary. Rather than merely abjuring these fictional concepts (Mach) or at best tolerating them (Burne), Einstein saw them as revealing an arbitrariness in our physical theorizing.
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The fictional concept could still be introduced through a freely chosen definition, as long as the definition did not tacitly commit us to false presumptions. That view of concepts enabled Einstein to abandon the notion of absolute simultaneity when he finally realized that this notion was all that obstructed his conforming of Maxwell's electrodynamics to the principle of relativity. Be replaced it by a new notion of simultaneity introduced through a definition that did not commit him to the absoluteness of simultaneity. In his "Reply to Critics" in the volume Albert Einstein: PhilosopherScientist (1949a)) Einstein mused that a scientist "must appear to the systematic epistemologist as a type of unscrupulous opportunist," drawing selectively on the views of the realist, idealist, positivist and Platonist as it suited the scientist's purposes. As Boward (2004) has noted, this selectivity may not betoken an incoherent position rather than one that agrees partially with each of the views listed. Nonetheless Einstein's selectivity is surely evident in the present episode in Einstein's use of the work of Burne and Mach. Be talces those parts that serve his purpose in physical theory. But he ignores other less helpful consequences of their critiques and leaves the systematic elaboration of the philosophical doctrines to others. And we can hardly blame him since his goal is to develop new physical theories, not new philosophies. To see the selectivity, consider a problem that Einstein did not address and which arises directly from the severity of Burne's critique. The application of his view of concepts led Bume, as we saw above, to the view that it is meaningless to apply the notion of duration in time to unchanging bodies. Einstein never seemed to adopt the full consequences of that conclusion; and ifhe had, it would have crippled his subsequent theorizing. It contradicts directly his famous cosmology of 1917, which models the universe as a uniform distribution of matter in space, both of which remain perfectly changeless over an infinity of time (Einstein 1917a). Indeed Einstein introduced this model exactly because he felt it met the demands of Mach's critique of inertia. There is a deeper systematic difficulty. In formulating the view as requiring concepts to be 'grounded' in sense experience, I have chosen a somewhat vague locution. Just what does this grounding amount to? In the narrowest view, the meaning of the concept simply is the actual sense experience that grounds it. If that is the view, we must ask whether and how we can arrive at concepts that are universally meaningful when all our experiences are, it would seem, specific. Take two people who have sense experience of what we would ordinarily say are different instantiations of the same concept. In the narrowest view just mentioned, these two people would form different concepts on the basis of their sense experience. Yet if we reject this narrowest view, what sorts of
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extensions beyond the grounding experiences are admissible and why are they admissible? This problem is tackled by Hume with some energy. He maintained that a concept can meaningfully extend beyond the specific impressions that ground it. While we cannot experience directly everyone of the infinitely many shades of blue, Hume urged in the Treatise (Book 1, Part I, Section I) that we can readily conceive of a missing shade in a gradation of shades of blue presented to us, even though we may never have experienced this particular shade directly. It is not just that we can generalize from many instances of the same shade to the concept of that shade; we can also proceed to different shades. Mach inclines towards a very lean reading of the grounding in experience. Science, he urged, was merely economical description of experience. He wrote (1882, 197, 207) that "Physics is experience, arranged in economical order," and "The goal which it [physical science] has set itself is the simplest and most economical abstract expression of facts." Thus he did deny the applicability of physical science beyond the actual facts at hand, this being the foundation of his critique of Newton's ideas of absolute space and time. Mach's celebrated analysis of Newton's bucket experiment offers a clear statement of his resistance even to small extensions beyond actual fact, just as it shows that this austerity was too radical for Einstein. The relevant fact in the experiment is that, when a bucket of water rotates with respect to the remainder of the universe, there is a concavity formed in the surface of the water. Might this concavity be due to some sort of interaction between the water in the bucket and the distant stars? Would such an interaction mean that the walls of the bucket, if made leagues thick, would drag the water with it through this interaction when the walls were rotated? Mach recognized the temptation to guess at the possibility. However, in his celebrated remark on Newton's bucket, Mach denounced such theorizing. Or that, I believe, is the only reading that conforms to Mach's other pronouncements on science as an economical summary of experience. Mach wrote: Newton's experiment with the rotating vessel of water simply inforn1s us, that the relative rotation of the water with respect to the sides of the vessel produces no noticeable centrifugal forces, but that such forces are produced by its relative rotation with respect to the mass of the earth and the other celestial bodies. No one is competent to say how the experiment would turn out if the sides of the vessel increased in thickness and mass till they were ultin1ately several leagues thiclc The one experiment lies before us, and our business is, to bring it into accord with the other facts known to us, and not with the arbitrary fictions of our imagination. (Mach 1960, 284)
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As I have discussed in detail elsewhere (Norton 1993), Einstein's response to this same problem is revealing. He chose to endorse exactly the speculation that Mach disavowed. Einstein inferred that the walls of a very massive bucket, if rotating, would drag the water, attributing this same conclusion to Mach. That conclusion became his "relativity of inertia" and later "Mach's principle," the notion that drove Einstein through years of theorizing on general relativity, before he abandoned it. Einstein chose fertility for a new physical theory over philosophical cogency. So how did Einstein explain that concepts can supply us meaningful content beyond the experiences that ground them? He wrote: The very fact that the totality of our sense experiences is such that by means of thinking (operations with concepts, and the creation and use of definite functional relations between them, and the coordination of sense experiences to these concepts) it can be put in order, this fact is one which leaves us in awe, but which we shall never understand. One may say "the eternal mystery of the world is its comprehensibility." ... In speaking here of "comprehensibility," the expression is used in its most modest sense. It implies: the production of some sort of order among sense impressions, this order being produced by the creation of general concepts, relations between these concepts, and by definite relations of some kind between the concepts and sense experience. It is in this sense that the world of our sense experiences is comprehensible. The fact that it is comprehensible is a miracle. (Einstein 1936,292) His explanation-that it is a wondrous miracle beyond our comprehension-would surely be the last resort of a desperate philosopher. But it is a comfortable resting point for a physicist whose real concern lies in physical theory and who wants to call up philosophical analysis only when it suits his physical ends. 2o
NOTES 1. It is a pleasure to present this paper in honor of my colleague and friend, Michael Friedman, whose work on the entanglements of science and philosophy has enlightened and energized us all. I thank Dan Steel for pointing out to me years ago the connection between Einstein's work of 1905 and Hume's critique of concepts and causality. I am grateful for helpful discussion to Stephen Engstrom, Don Howard, Gerald Massey, Paul Pojman and the participants in the conference "Synthesis and the Growth of Knowledge" (University of South Carolina, October 1-3, 2004); to its tireless organizers, Michael Dickson and Mary Domski; and especially to Robert Rynasiewicz for his unflinching resolve in bringing criticisms
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of an earlier version of this paper to my notice, some of which proved to be correct and important. 2. This literature is enormous. For an entry into it, see Holton (1968), Stachel, et al. (1989a), and, most recently, Howard (2004). 3. For a simplified account, see Norton (forthcoming, Section 4). 4. As a footnote to Einstein's (1920) recollection of the magnet and conductor thought experinlent he Vlrote, "The difficulty to be overcome lay in the constancy of the velocity of light in a vacuum, which I first believed had to be given up. Only after years of [JahrelangJ groping did I notice that the difficulty lay in the arbitrariness of basic kinematical concepts." Wertheimer (1959, 216) also reports on the strength ofinterviews with Einstein in 1916 that Einstein was occupied with the problem "for years." 5. As Pais (1982, 163) reports Einstein called the breakthrough. 6. Knowing what is about to ensue, it is hard for modern readers to do anything but applaud Einstein's stubborn insistence that we elinlinate the ether state of rest from physics in favor of the principle of relativity. But the situation was not so straightforward in 1905. Physicists had learned to accon1modate principles whose truth was contradicted by the basic entities of the science. For example, the second law of thermodynanlics, Carnot's principle, assured us of an inexorable rise in entropy with time. The atonlic theory of matter, however, asserted that this rise was only very probable, with that assertion resting on the existence of atoms that were by supposition so small as to elude direct detection. 7. The story quoted is from notes taken by Jun Ishiwara. Multiple translations of uncertain quality can be found. The one given is from Stachel (2002, 185). 8. In this sense, Einstein deemed judgments of the simultaneity of distant events to be conventional. To what extent did his views conform to the thesis of the "conventionality of simultaneity," as developed by Reichenbach and many others? (See Janis 2002.) One nlight think that Einstein would have to subscribe to this thesis were he only to pursue the logical consequences of his assertion that distant simultaneity can be introduced by a definition. He would then be comn1itted to many of the fanliliar consequences of the thesis, such as the adnlissibility of nonstandard £-coordinate systems and the division of all physical quantities into factual two-way quantities and conventional one-way quantities. Yet outside of the first part of his 1905 paper, Einstein mal<:es no systematic acknowledgen1ent of this distinction of one-way and two-way quantities; and his treatment of coordinate systems is sufficiently different from our modern approach as to mal<:e it unlikely that he would have regarded a non-standard £-coordinate system as licit in special relativity. (See Norton 1989 and 1992.) 9. Does this mean that the constancy of the speed of light is a conventional artifact of how we set our clocks? It does not. Einstein's definition of simultaneity requires certain facts to obtain. It would fail if an emission theory of light were true, since such a theory allows light to have many speeds. And it would fail in an ether theory, with light propagating at c with respect to the ether. For observers moving at c would find light frozen and be unable to use it to determine the simultaneity of events. 10. This is a translation in Papers Vol. 2, p. 264, of a voice recording, transcribed and presented in the German in Herneck (1966).
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11. While Wertheimer presents the remarks as a quotation, they are more likely recreations from notes. 12. Having asserted that we have no means to ascertain exact equality of measure for geometrical figures, Hume urged that the idea of a perfect correction of imperfect, real measures "is a mere fiction of the mind, and useless as well as incomprehensible" (Hume 1978, Book 1, Part II, Section IV). He then extended the critique to the notion of the duration of time: This appears very conspiCUOUSly with regard to time; where tho' 'tis evident we have no exact method of determining the proportions of parts, not even so exact as in extension, yet the various corrections of our measures, and their different degrees of exactness, have given us an obscure and implicit notion of a perfect and entire equality. I am grateful to Gerald Massey for drawing my attention to this discussion. 13. The translations that follow are based loosely on Engel (1997). 14. 13 May 1917; Papers, Vol. 8a, Doc. 339. Translation from Holton (1968, 240). 15. As quoted in Holton (1968,239). 16. The remark is repeated in a somewhat vaguer context in a letter by Einstein to his old friend Besso of 6 Jan. 1948 (Speziali, 1972, Doc. 153). Responding to a suggestion by Besso of Mach's influence, Einstein responded: Now, as far as Mach's influence on my development is concerned, it was certainly great.... How far [Mach's writings] influenced my own work is, to be honest, not clear to me. In so far as I can be aware, the immediate influence of D. Hume on me was greater. I read him with Konrad Habicht and Solovine in Bern. However, as I said, I am not in a position to analyze what is anchored in unconscious thought. 17. In this regard, we may well wonder if Einstein was informed by Poincare's earlier analysis of Lorentz's local time in terms of light signals and clocks. More generally, we may wonder whether Einstein drew on Poincare's conventionalist views when Einstein invoked the freedom to fix the concept of simultaneity through chosen definitions. If Einstein hit upon the relativity of simultaneity without reflecting on clocks and light signals, a possibility we cannot rule out, then there would be no significant role for Poincare's analysis of local time. Moreover, if a reading of Poincare's work was important to Einstein's taking of the step, we should ask why he acknowledges help only from Hume and Mach in stating the philosophical debts incurred in the discovery of special relativity. Einstein did elsewhere acknowledge a debt to Poincare. For example, writing to Besso on 6 March 1952 (Speziali 1972, Doc. 182) of his early readings with Habicht and Solovine, he noted: "These readings [of Hume] were of considerable influence on my development-along with Poincare and Mach." But nothing in this acknowledgement specifically suggests the discovery of special relativity. Poincare's name is not mentioned elsewhere in Autobiographical Notes. Poincare does figure prominently in the appended "Reply to Criticisms" (1949a). But Poincare is discussed only for his contribution to the discussion of the conventionality of geometry. For further discussion, see Darrigol1996, 302; Darrigol2004.
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18. Einstein recalled in several places that the insight came to him suddenly, suggesting that he may not have needed to develop a new framework of clocks and synchronization procedures to see it. See Einstein's 1922 recollections in IZyoto above " ... and suddenly I realized the solution ..." and in the 1924 recording " ... the solution came to me suddenly with the thought that our concepts ...." 19. More precisely, there would be no observable effect in quantities of the first order in v/ c, where v is the speed of the earth. 20. This is not to say that Einstein's selectivity was inadvertent. Indeed, as we saw in Einstein's remarks to Besso of 6 Jan 1948 (Section 4.3 above), Einstein was fully aware of the austerity of Mach's system and properly complained that a full development of Mach's views would bring not just skepticism about atoms but about physical reality itself.
REFERENCES Asquith, P.D. and T. Nickles (eds.) (1983). PSA 1982. East Lansing: Philosophy of Science Association. Barbour, J. and H. Pfister (eds.) (1993). Mach)s Principle: From Newton)s Bucket to QJtantum Gravity. Einstein Studies. Boston: Birkhauser. Darrigol, O. (1996). "The Electrodynamic Origins of Relativity Theory." Historical Studies in the Physical Sciences 26: 241-312. - - - . (2004). "The Mystery of the Einstein-Poincare Connection." Isis 95: 614-26. Earman, J., C. Glymour, and R. Rynasiewicz (1983). "On Writing the History of Special Relativity." In Asquith and Nickles (eds.), Volume 2,403-16. Einstein, A. (1905). "Zur Electrodynamik bewegter Knrper." Annalen der Physik 17: 891-921. Translated as "On the Electrodynamics of Moving Bodies," in Lorentz et al. (1952), 37-65. - - - . (1916). "Ernst Mach." Physikalische Zeitschrift 17: 101-104; Papers, Vol. 6, Doc. 29. - - - . (1917). Uber die spezielle and die allgemeine RelativitCitstheorie (Gemeinverstandlich). Braunschweig: Friedr. Vieweg & Sohn; Papers, Vol. 6, Doc. 42. 15th expanded edition. Translated by R.W. Lawson (1954), Relativity: the Special and the General Theory. London: Methuen. - - - . (1917a). "IZosmologische Betrachtungen zur allgemeinen Relativitatstheorie." In Preussische Akademie der Wissenschaften) Sitzungs-berichte, 142-52; translated as "Cosmological Considerations on the General Theory of Relativity," in Lorentz et al. (1952), 175-88. - - - . (1920). "Fundamental Ideas and Methods of the theory of Relativity, Presented in Their Development." Papers, Vol. 7, Doc. 31. - - . (1936). "Physics and Reality." In Ideas and Opinions, 290-323. New York: Bonanza, n.d. - - . (1949). Autobiographical Notes. P.A. Schilpp (ed.) La Salle and Chicago: Open Court. Reprinted and corrected from P.A. Schilpp (1944).
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_ . (1949a). "Remarks Concerning the Essays Brought Together in this Cooperative Volume" (Running Heading: "Reply to Criticisms"). In P.A. Schilpp (ed.) (1949), 1-94. ___. (1952). "In Memory of Albert A. Michelson on his 100th Birthday." Einstein Archive 1-168.00; facsimile, Einstein Archives Online, URL = http://www.alberteinstein.info Eisenstaedt, J. and A. Kox (eds.) (1992). Studies in the History of General Relativity: Einstein Studies. Vol. III Boston: Birkhauser. Engel, A. (1997). The Collected Papers of Albert Einstein. Volume 6. The Berlin Years: Writings, 1914-1917. English Translation ofSelected Texts. E. Schucking, consultant. Princeton: Princeton University Press. Herneck, F. (1966). "Zwei Tondokumente Einsteins zur Relativitatstheorie." Forschungen und Fortschritte 40: 133-35. Holton, G.J. (1968). "Einstein, Mach, and the Search for Reality." In G.J. Holton (1973),219-59. - - - . (1969). "Einstein, Michelson and the 'Crucial' Experiment." Isis 60: 133-69; Ch. 9 in G.J. Holton (1973). - - - . (1973). Thematic Origins of Scientific Thought: Kepler to Einstein. Cambridge, MA: Harvard University Press. Howard, D.A. and J. Stachel (eds.) (2000). Einstein, The Formative Years, 1879-1909: Einstein Studies, Volume 8. Boston: Birkhauser. Howard, D.A. (2004). "Einstein's Philosophy of Science." The Stanford Encyclopedia ofPhilosophy (Spring 2004 Edition), Edward N. Zalta (ed.), URL = plato.stanford.edu/archives/spr2004/entries/einstein-philscience. Hume, D. (1978). A Treatise ofHuman Nature. 2nd ed. Oxford: Clarendon. - - _ . (1988). An Enquiry concerning Human Understanding. Chicago, IL: Open Court. Janis, A. (2002). "Conventionality of Simultaneity", The Stanford Encyclopedia of Philosophy (Fall 2002 Edition), Edward N. Zalta (ed.), URL = plato.stanford. edu/archives/ fa1l2002/ entries/ space- time-convensimul. Janssen, M. et al. (2002). The Collected Papers of Albert Einstein: Volume 7: The Berlin Years: Writings, 1918-1921. Princeton: Princeton University Press. ("Papers, Vol. 7.") Janssen, M. and C. Lehner (eds.) (forthcoming). Cambridge Companion to Einstein. Cambridge: Cambridge University Press. Kox, A.J., et al. (eds.) (1996). The Collected Papers of Albert Einstein: Volume 6: The Berlin Years: Writings, 1914-1917. Princeton: Princeton University Press. ("Papers, Vol. 6.") Lorentz, H.A. (1895). Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Kdrpern. Leiden: E.J. Brill. Lorentz, H.A., et al. (1952). The Principle ofRelativity: A Collection of Original Memoirs on the Special and General Theory of Relativity. With notes by A. Sommerfeld; translated by W. Perrett and G.B. Jeffery. New York: Dover. Mach, E. (1882). "The Economical Nature of Physical Inquiry." In Mach (1943), 186-213. - - - . (1943). Popular Scientific Lectures. 5th ed.. Translated by T.J. McCormach. LaSalle, Illinois: Open Court.
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- - - . (1960). The Science ofMechanics: A Critical and Historical . A.ccount of Its Development. 6th ed. Translated by T.J. McCormach from the ninth German edition. La Salle, IL: Open Court. Norton, J.D. (1989). "Coordinates and Covariance: Einstein's View of Spacetime and the Modern View." Foundations ofPhysics 19: 1215-63. - - - . (1992). "The Physical Content of General Covariance." In Eisenstaedt and IZox (eds.) (1992), 281-315. - - - . (1993). "Mach's Principle before Einstein." In Barbour and Pfister (eds.) (1993) Vol. 6., 9-57. - - - . (2004). "Einstein's Investigations of Galilean Covariant Electrodynamics prior to 1905." Archive for History ofExact Sciences 59: 45-105. - - - . (forthcoming). "Einstein's Special Theory of Relativity and the Problems in the Electrodynamics of Moving Bodies that Led him to it." In Janssen and Lehner (eds.). Pais, A. (1982). Subtle is the Lord: The Science and the Life of Albert Einstein. Oxford: Clarendon. Ritz, W. (1909a). "Uber die Grundlagen der Elektrodynamik und die Theorie der Schwarzen Strahlung." Physikalische Zeitschrift 9: 903-907; reprinted as Ritz (1911),493-502. - - - . (1909b). "Zum gegenwartigen Stand des Strahlungsproblems: Erwiderung auf den Aufsatz des Herrn A. Einstein." Physikalische Zeitschrift 10: 224-25. - - - . (1911). Gesammelte Werke. Oeuvres. Paris: Gauthier-Villars. Rynasiewicz, R. (2000). "The Construction of the Special Theory: Some Queries and Considerations." In Howard and Stachel (eds.), 159-201. Shankland, R.S. (1963/73). "Conversations with Einstein." American Journal of Physics 31(1963): 47-57; 41(1973): 895-901. Schilpp, P.A. (ed.) (1949). Albert Einstein: Philosopher-Scientist. Evanston, IL: Library of Living Philosophers. Schulmann, R., A.J. Kox, M. Janssen, and J. Illy (eds.) (1998). The Collected Papers ofAlbert Einstein. Volume 8. The Berlin Years: Correspondence) 1914-1918. Part A: 1914-1917. Princeton: Princeton Univ. Press. ("Papers, Vol. 8A"). Speziali, P. (ed.) (1972). Albert Einstein Michele Besso: Correspondence. Paris: Hermann. Stachel, J. (1982). "Einstein and Michelson: the Context of Discovery and Context of Justification." Astronomische Nachrichten 303: 47-53; reprinted in Stachel (2002), 177-90. - - - . (1987). "Einstein and Ether Drift Experiments." Physics Today 40: 45-47; reprinted in Stachel (2002), 171-76. - - - . (2002). Einstein from 13) to 'Z.): Einstein Studies, VOlume 9. Boston: Birkhauser. Stachel, J., et al. (eds.) (1987). The Collected Papers ofAlbert Einstein: VOlume 1: The Early Years: 1879-1902. Princeton: Princeton University Press. ("Papers, Vol. I") - - . (eds.) (1989). The Collected Papers ofAlbert Einstein: Volume 2: The Swiss Years: Writing, 1900-1902. Princeton: Princeton University Press. ("Papers, Vol. 2."). - - - . (1989a). "Einstein on the Theory of Relativity." Headnote in Papers, Vol. 2,253-74. Wertheimer, M. (1959). Productive Thinking. New York: Harper & Bros.
[17] The Paracletes of Quantum Gravity JAMES MATTINGLY 1
1. Background for Quantum Gravity The physics community is embroiled in a profound and far-reaching revolution, a revolution that will overturn the bedrock theories of matter, space-time, and gravity on which almost all of the physical theories of the twentieth century depend. This revolution is pressing and urgent and now is the time to choose up sides. At stake is no less than the theory of quantum gravity! However, since the revolution is now in at least its fifty-fifth year, it may be time to take stock and make some suggestions about what might be slowing it down, and about what strategic moves might advance the cause. In the twentieth century physicists developed two of the most successful descriptions of nature ever produced, general relativity and quantum mechanics. General relativity, Einstein's theory of gravitation, rejects the notion of gravitational force and replaces it with a conception of gravitational interaction carried out by changes in the actual geometry of space and time, now space-time; these changes in geometry reflect the distorting effect of matter and energy in space-time. The story is complicated in its details, but simple in its conception: the presence of matter distorts the geometry of space-time where the matter is located, and these changes are propagated at the speed of light throughout space-time; on the other hand, matter not subject to outside forces travels on geodesics, the straightest paths in space-time. This mutual interaction between spacetime and the matter-energy fields therein gives rise to behavior that, in the limit of small masses (planets, moons), reproduces the appearances of Newton's theory of gravity but that deviates from the predictions of that theory in the case of larger masses (suns, galaxies). As is well known, the 387
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predictions of general relativity have been born-out by observation, and it has superseded Newtonian gravitation theory. Quantum mechanics is at the heart of our current best theories of matter. According to quantum mechanics, matter is significantly different than our classical intuitions suggest. Classically we expect that matter (whether it is ultimately grainy or continuous) behaves deterministically, and that given the state of one small bit of matter at one time (its mass, its velocity, its position, the forces acting on it) we can predict, at least in principle, its state at any future time. In practical tern1s, carrying out this operation proves to be quite complicated, even in the classical picture, since attending to all the forces acting on the bit of matter requires knowledge of the state of many other particles, and since our observations will inevitably be less than perfect. But still it is generally taken to be the case that in principle, once all these niceties are taken into account, the behavior of matter is deterministic classically for generally well-behaved systems. (See Earman 1986 for discussion and important qualifications.) In the quantum theory things are quite different. Even given a specification of the entire state of a quantum system, it is not possible, even in principle, to specify the future values of all variables that can be associated with some bit of matter in that system. Views of how to articulate the situation differ, but n10st will agree that there is more that one might want to know about a system than is contained in the most complete physical description of that system. For exampIe, we tend to consider it inappropriate to apply at the same time properties like momentum and position with arbitrary precision to the same bit of matter. The predictions of this theory have also been born-out, to an unprecedented degree of precision, and it has superseded classical conceptions of matter. One additional feature is shared by both quantum mechanics and general relativity: they tal(e their proper domain of validity to be all of nature. There are regimes in nature where the importance of one or the other for making predictions is minimal, but even in those regimes the theories are supposed to be applicable. It is then important to determine whether the two theories are con1patible. For if not, at least one must be wrong. As typically stated, the theories do appear to be incompatible. For quantum field theory-the version of quantum mechanics developed for use with special relativity2-is usually taken to require a fixed, non-dynamical space-tin1e and so it is incompatible with general relativity; and general relativity is a theory about classical space-time and is usually understood to apply to classical matter fields and so it is incompatible with quantum field theory. One might then conclude that what is called for is a theory that captures the best insights of both theories, a theory that recognizes the indeterminacy of matter and the dynamical character of space-time, a theory in which the gravita-
rOnal field is also a quantum field. That theory would be qU~~~~~~~
_
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There is, in fact, a theory that does incorporate the best features of quantum field theory and general relativity-the semiclassical theory of gravity. The semiclassical theory is really a cluster of theoretical approaches that preserve the classical nature of the gravitational field while showing how to do quantum field theory on a dynamical space-time. There is near universal agreement in the physics community that the semiclassical theory is untenable as a final theory of gravity. However, despite this widespread agreement, there are very few actual arguments that are offered as support for that agreement. (I will rehearse some of these in what follows.) Moreover, none of those arguments is any good. "Any good" really is the right expression here: of the arguments that aren't obviously false on their face, each is obviously question begging, or requires a minimal grasp of the physics of general relativity and quantum mechanics to refute, so that it is hard to believe that anyone is persuaded by any or all of them. Indeed I have argued elsewhere (Mattingly 2005, 2006) that it is best to think of these arguments as excuses for not investigating the semiclassical theory rather than as attempts to establish its non-viability. We can understand the community best, I believe, by taking the non-viability of semiclassical gravity as a primitive insight rather than a hard won conclusion. But we can also see that so taking semiclassical gravity, even if it is non-viable, obscures any real flaws the theory might have, and thereby obscures the real question that a fully quantized theory of gravity might be an answer to. Since certain field theorists on the cutting-edge of attempts to quantize gravity have now recognized that they may be asking the wrong questions, and since they have invited input from philosophers of science and begun to engage with their projects (see, e.g., Isham 1993, Rovelli 2001), what follows here may be seen as an attempt to show how we might go about finding the right question. I do not know the right question that when answered will produce a theory of gravity for quantum mechanical matter, and certainly I don't know the right answer. What I offer here is a general suggestion for how the practice of physics can itself provide the right question. Attempts to produce, ab novo, a quantum theory of gravity are not necessarily misguided, and no argument I make here should be taken as an attempt to establish that they are. Rather, I want to suggest that there is a method that has historically produced great results, a method that begins with our current practice. The method aims to extend the domain of that practice as widely as possible, aims to elaborate the claims of that practice as precisely as possible, and in pursuit of those aims, finds the basic flaw in the current practice and brings it into view. The method, which I outline below, shows how to transform that flaw into the right question, whose answer is the transformation in our practice that results in a theory that properly unites gravitation theory with quantum mechanics. I call this method the "Lorentzian strategy"; I will elaborate it below.
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2. Cassirer and Scientific Theory Change For analysts of scientific communities and the roles such communities play in the development of science, quantun1 gravity provides an important and ongoing field of study. Similarly, theorists of the development of science concerned with questions of the rationality of scientific transitions should be very interested in developments here; the issue ofwhether and how well practitioners of the various stages, theories, and general approaches are able to communicate with and understand each other is particularly pressing given the importance in the philosophy of science of IZuhn's work (and reactions to that work) detailing what he identified as an important incommensurability between distinct theoretical paradigms. Here I will explore one account of the development of science, the construction of quantum gravity. The account is a partial reconstruction of Ernst Cassirer's philosophy of scientific theory change. There is, I think, some confusion about how best to understand Cassirer's picture of the world and of man's place in it and I therefore do not wish to engage here with his full project, nor with standard interpretations of his contribution to philosophy. Instead I will outline an interpretation of the philosophy of science in his Einstein)s Theory of Relativity) and claim that it contains the prescription for a robust methodology of scientific theory change. While much work remains to be done to explicate fully this prescription and fit it into the larger context of Cassirer's philosophy, I restrict my attention to an application of the prescription to the case of quantun1 gravity. I argue that some key features of a revolutionary break with prior theory are absent in the current state of attempts to integrate quantun1 mechanics with general relativity, and I suggest that pursuing aggressively a "Lorentzian" strategy (pursuing the semiclassical theory of gravity) is likely to yield more fruit than the alternative strategy of "quantizing the gravitational field." Michael Friedman (2000) has suggested that we return our attention to Cassirer, in part to bridge the divide between analytic and continental approaches to philosophy that has grown ever wider in the last few decades. Cassirer's pivotal position in philosophy at the beginning of that divide by itself makes it likely that a nuanced understanding of his mature philosophy will provide foundations for that bridge. But in returning to Cassirer we are rewarded in many other ways as well. Something one notices immediately is that what we have in Cassirer's account of the development of science may be seen as the implementation of an early version of a dynamical conception of I
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standing of the role of philosophy in providing guidance through the conceptual spaces of science that are enlarged by scientific revolutions, and so I won't address that issue here. Cassirer as well suggests that we see our current scientific conceptions as embeddable in a more comprehensive framework as a special case: Newtonian mechanics as the limit of special relativity, special relativity as itself the limit of general relativity. But Friedman and Cassirer will differ on their view of whether there is more than one way to effect this embedding properly in any given case: Friedman will claim that considerations internal to the scientific debate cannot settle, in general, which way to extend a given theory and thus we require the separate discipline of philosophy to guide our choice; Cassirer will claim that there is always a best scientific decision to be made at a given juncture and that the critical philosophy illuminates the epistemological imperative behind this decision. I begin with Cassirer's account of the transition from Lorentzian electrodynamics to Einstein's special theory of relativity. I then extract from that account a general prescription for fundamental theory change in the sciences-the Lorentzian strategy. Note that for my purposes it suffices to assume that this fundamental change proceeds in roughly the way that Kuhnian revolutions in science generally proceed. However, fundamental differences in the Kuhnian and Cassirerean accounts emerge that allow one to characterize the revolutions as progressive and rational when described according to Cassirer's prescription. I follow, in Section 4, with an account of the semiclassical theory of gravity, and in particular, I address premature reports of its death. While there is an important analogy between the semiclassical theory's relation to quantum gravity and Lorentzian electrodynamic's relation to special relativity, there is an important disanalogy between the approach the physics community is taking to the transition to quantum gravity and its approach to the transition to special relativity. One important aim here is to redirect the physics community to the latter approach. We will see that it was important for the transition to special relativity that Lorentzian electrodynamics had gone as far as it had. This general point of Kuhn's is worth emphasizing: revolutions happen under the pressure of anomalies that eventually precipitate a crisis in the earlier framework. And it is important that the revolutionary developments proceed in reaction to those anomalies. Section 4 shows in an abbreviated manner that such burdens are simply not present in the case of semiclassical gravity. Absent these, it isn't clear what a theory of quantum gravity should be fixing. The Sections 3 and 4 provide, I think, a strong analogical argument that semiclassical gravity is worth pursuing in our quest for a fully quantized gravity theory. I turn finally in Section 5 to canvassing some current physics research for possible routes to quantum gravity that start with the
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semiclassical theory. There is such work, and it rewards our attention: analysis of the regime where quantum effects (fluctuations, superpositions, entangled states) are of roughly the same n1agnitude as gravitational effects (tidal forces, massive curvature irregularities, significant gravitational radiation) is ongoing and very interesting. The programs of Barcelo, Liberati and Visser, Verdaguer, Hu, Calzetta, and Martin, and Gunzig and Saa, for example, are, in my estimation, applications of the Lorentzian strategy in quantum gravity. Let me note here that the situation in quantum gravity has changed significantly in the last decade. It is no longer the case that no approach to quantum gravity is showing progress. And yet the problems seem as severe as they ever were-the new approaches are, at this point, merely n1athematical exercises. The "data" appealed to by the various groups of quantum gravitation theorists are very old, and all programs are responsive to it. Thus the data alone cannot help us to choose among approaches. What is needed is either some other way to choose among basic approaches or a method of producing new approaches. I think the Lorentzian strategy can perforn1 both tasks: it provides an impetus to new experiment, and a clear expression of the limitations in the semiclassical theory that will have to be overcome in a quantum theory.
3. The Lorentzian Strategy The Lorentzian strategy is a conservative approach to theory change that requires that we work stepwise with minimal mutilation of the current theory. Given certain resources and presented with certain data, Lorentz set out to produce a fusion of Maxwell's electrodynamics and Newtonian mechanics. In particular he attempted to produce a viable theory of moving charged bodies. He had steady success over the years but was finally presented with empirical and theoretical contradictions. 3 Einstein is able to see through this maze of technical apparatus and can finally understand all at once the deep conflict between Newtonian mechanics and Maxwellian electrodynamics that makes then1 so hard to unify. How did this all happen, and is the story we tell about it generalizable? The honest answer is "I don't know," but my answer here will be "the Lorentzian strategy, and I hope so." In addition to his native genius Einstein had a deep awareness and appreciation of efforts to unite the mechanical and electrodynamical theories. These efforts were precisely the carrying out of the Lorentzian strategy: extending as widely as possible the domain to which the theory is applied, and applying it there with as much precision as possible. This strategy (in1plicit or explicit) made manifest apparent contradictions between basic principles: one, the constancy of the
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speed of light for all observers, the other the ~rinciple of.relativ~ty i.ts~lf. Einstein showed how to resolve these contradlCtIons, WhIle mamtammg the apparently conflicting principles. An investigation of the transition to relativity illuminates another aspect of the philosophy of science that is amenable to a Cassirerean analysis: the nature of scientific theories. It is beyond my purpose here to layout a full presentation of that view. I have elsewhere begun to address the relation between Kuhn's incommensurability thesis and the theory of scientific theories he appealed to (Mattingly 2004), and while much more remains to be done, the basic insight is this: much of Kuhn's thesis of incommensurability between scientific frameworks relies on a statement view of theories that bears strong resemblance to the logical empiricists' syntactic conception. Ironically, one can carry out much the same analysis and rephrase Kuhn's argument for incommensurability using a model theoretic view of theories along the lines of the semantic conception. But one can show that a hybrid model that is committed to principles as bearers of scientific meaning can both respect Kuhn's historiographical insights into the revolutionary changes in science that punctuate its history, and reject his thesis that science does not progress according to scientific principles. Here, though, I will focus on just one component ofCassirer's dynamical version of the critical philosophy: the role of what he calls "paracletes of thought" in the development of our scientific theories. In Cassirer's account the paracletes, or awakeners, of thought are apparent contradictions in the older theory that are jointly asserted as axioms for the new theory and which thereby are seen not to conflict after all. In the case of special relativity he finds two pairs, one from the observational side-the Fizeau and Michelson-Morely experiments-and one from the theoretical side-the constancy of the speed of light and the relativity of all motion. In Fizeau's experiment testing light transmission through liquids, we find that the velocity W of light in a moving medium is W = w + v (1- ~ ) where w is the speed of light in the medium at rest, v the speed of the medium, and n the index of refraction of the medium. In the MichelsonMorely experiment, on the other hand, we have the famous null result that the speed of light is unchanged with the direction of motion of the earth through its supposed containing medium. Since we have found that light travels at constant speed regardless of the speed of its source, or of the observer, we advance that constancy as a fundamental theoretical principle. We also assume that any inertial frame is sufficient for the description of natural phenomena, and articulate that assumption in the Galilean transformations between coordinate systems. Here on the theoretical side we are not presented with such a clear apparent contradiction, nor is the resolution simply to maintain the truth of both principles. It is necessary to reinterpret the Galilean principle as a spe-
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cial version, in context, of the general prescription of the relativity principIe. While the actual equations of transformation must be given up and replaced with their Lorentzian counterparts, we still assert what is regarded as the core content of the principle: that all (inertial) fran1es are sufficient for the framing of physical law. Cassirer's discussion of the empirical and conceptual foundations of the theory of relativity makes it quite clear what is at stake. We have taken the theory of Maxwell on the one hand, and the theory of Newton on the other, as far as we can. We have found both an empirical contradictionthe Fizeau versus Michelson-Morely experiments-and an apparent conflict of basic principles-the constancy of the speed of light, and the relativity principle of Galileo. These contradictions, and especially the conflict of basic principles are what Cassirer calls the "paracletes of thought." When we are presented with this basic conflict, we are finally forced to chose between abandoning one of our cherished principles, and abandoning a story about the nature of substance that we had been using to produce the basic conflict. In such an instance, Cassirer claims, the choice is clear, and it is also a scientific choice. Science is in service of the unifying, systematizing drive, and as such, proceeds only according to principles. Thus, when we are finally brought to the point of actual contradiction, we must discover a means to modify the formulation of our basic principlesat whatever cost to our intuitions about the nature of substance, or the essence of matter-in order to bring them once again into harmony. It is Cassirer's view that only thus can our basic understanding of nature be extended. But if this is so, we must show how we can go about finding these points of crisis, a method for producing the awakeners of thought. Whether or not we endorse Cassirer's radical claim-that only by identifying apparent conflicts between basic principles, and resolving the contradictions by re-articulating these principles and maintaining them as basic postulates of the new theory-we can see the potential advantages of finding such apparent conflicts. The Lorentzian strategy develops hybrid and apparently transitional theories with an eye toward driving out the kinds of contradiction that were resolved in the move to special relativity. It took careful attention to the twin facts of the constancy of the speed of light and the relativity of motion before we were able to see both the manifest contradiction between the two, as well as the way to resolve that contradiction. In a similar way I suggest, in particular, that it will take sustained attention to the facts of dynamical space-time and a quantum mechanical stress energy tensor before we can see the actual contradiction between quantum field theory and general relativity (or generalizations thereof). Note that the answer in the case of special relativity was very unlike the apparent question that Lorentz had to deal with. There was a !_~~__g~~~!i_
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establish a matter theory compatible with the apparent special frame picked out by the ether; a radical relativization of simultaneity was not obviously in the future. So the real question answered by the special theory of relativity was not apparent in advance of Lorentz's efforts. I want to suggest that sustained attention to efforts to extract as much physics as possible from hybrid theories is worthwhile as an important possible source of insight into the true problem that needs to be solved by fullfledged theories. A partial answer to the question of how Einstein's insight was possible is that the Lorentzian strategy highlighted a conflict between the formulations of basic principles-relativity and the constancy of the speed of light-that was resolved in properly formulating them. Einstein realizes that the actual rule of transformation must depend on other principles, that relativity is not given by the transformation function of Galileo (or any other particular function) but that the relativity principle (in concert with the light principle) dictates the proper transformation function between inertial frames. In partial answer to the question of whether the strategy is generalizable, I offer the example of semiclassical gravity. We can compare Cassirer's view of Einstein with that of one of the chief architects of loop quantum gravity, itself a chief candidate for the correct theory of quantum gravity, Carlo Rovelli. Rovelli was for years intimately connected with philosophers and historians of science in his position at Pittsburgh University and through his presence on dissertation committees in the Department of History and Philosophy of Science there. According to Rovelli, one key to understanding Einstein's development of relativity is the latter's twin appreciation of Maxwell's theory and Galileo's principle of relativity. He says: Einstein understood that Galileo's great intuition-that the notion of velocity is only relative-could not be wrong. I am convinced that in Einstein's faith in the core of the great Galilean discovery there is very much to learn, for philosophers of science, as well as for contemporary theoretical physicists. So, Einstein believed the two theories, Maxwell and Galileo: he assumed that they would hold far beyond the regime in which they had been tested. Moreover, he assumed that Galileo had grasped something about the physical world, which was, simply, correct. And so had Maxwell. Of course details had to be adjusted. (Rovelli 2001,116) The significance for the whole question of how to think about the development of science is contained in this passage. For while the historical impression that Rovelli has of Einstein's theory of science and his development of relativity theory may lack the subtlety of modern views, that
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impression appears to form part of the core ofRovelli's own theory of science. We see here an important way in which the character of the practice of science develops partly in light of, in indifference to, and in reaction against the work of historians and philosophers of science. Physicists such as Rovelli, despite their familiarity with philosophical analyses of science, and despite their willingness to engage with philosophers4 and despite their appreciation of the importance of those historical, philosophical, and sociological studies that undermine simplistic notions of the relentless progress of science toward the absolute truth, are still convinced that they are making progress, that they are learning about the world and advancing beyond the understanding of the world had by their teachers and historical role models. They are right.
4. Semiclassical Gravity Is quantum gravity really necessary? Can't we get along without it? Is there truly some crucial reason why all physical fields must be quantum mechanical in nature? I will use the term "semiclassical gravity" completely generally to apply to any theory that (1) couples a continuous, Lorentzian signature metric to the expectation value (or otherwise suitably averaged and "classicalized") stress-energy quantity (which need not be a tensor field itself) and (2) reduces to general relativity in the classical limit h ~ O. Here I can do no more than sketch the standard objections to semiclassical gravity.s More extensive treatments can be found in (Mattingly 2005, 200~; Callender and Huggett 2001, Introduction) and references therein; what follows is a summary of (Mattingly 2005, 2006). The paucity of evidence and the weakness of what little there is suggest a need to look much harder at the issue of how, and why, and whether semiclassical gravity fails. Arguments that semiclassical gravity must fail divide nicely into three classes, two of which are based in physics and one in philosophy. There are objections of an experimental nature, of a theoretical nature, and of a metaphysical nature. Let us consider them in turn.
1. Experiment There are only two experiments that are appealed to in arguments against the semiclassical theory. The first is Page and Geilker's. To test the semiclassical model of gravity they contrive a situation where the position of a large mass is determined by the outcome of a quantum experiment. They then test the gravitational field and show that it is affected only by the loca-
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tion of the mass-not partly by the location of the mass and the location of where the mass might have been located had the quantum experiment turned out differently. But since in the semiclassical theory the gravitational field is determined by the expectation value of the mass density, they conclude that the theory is empirically inadequate. Their analysis is in two parts. It first claims that only a model of quantum mechanics that does not involve wave-function collapse can be used for semiclassical gravity, because any collapse model must violate the semiclassical Einstein equations. The claim is non-trivial and they do not argue for it except by noting that they don't see how to avoid it. But the burden is on them here in advancing their claim. Moreover, even if their claim is U'ue, it shows only that a modification of their specific model of semiclassical gravity is required, and not that the gravitational field must be quantized. The second claim is then that any no-collapse model requires that expectation values never change. There is one version of how to understand measurement in quantum mechanics that might support that claim, but they make no argument that we should adopt that version, and most who study this issue agree that we should not. Nor do they show exactly how this interpretation of measurement in quantum mechanics would support their claim. The second experiment suffers from serious flaws of analysis as well. Moreover it hasn't been performed. Eppley and Hannah propose a solar system sized gravity wave detector. They would use it to show that classical wave behavior for gravity is incompatible with the quantum character of matter. I leave aside their argument here except to note that they appeal crucially to an understanding of Heisenberg's uncertainty principle on which it is impossible for a particle to be localized sharply simultaneously in both momentum and position space. This view is rarely defended today by serious students of quantum mechanics since there are empirically adequate interpretations of quantum mechanics for which these relations reflect our ignorance and are not fundamental features of the world. On the de Broglie-Bohm theory, for example, position and mechanical momentum are always well defined. It is not inconceivable on such an interpretation that one could find a way to measure these quantities simultaneously with arbitrary accuracy; one simply cannot do so using purely quantum mechanical means. Yet the de Broglie-Bohm interpretation is empirically equivalent to the standard, Copenhagen interpretation. Similarly Everett's relative state formulation of quantum mechanics is empirically identical to the Copenhagen interpretation, and particles always have well-defined position and momentum in that formulation. Thus we cannot assume that sharply localizing a particle simultaneously in both momentum and position space is inconsistent with the uncertainty principle. But even granted their argument, their device fails because it is
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so massive that by the time it was built, it would have fallen into its own black hole. Even by the standards of a thought experiment-one which would be impossible in practice, but conceivable in principle-Eppley and Hannah's proposal fails (Mattingly 2006).
2. Theory The following remarks are far too hasty, but will have to serve here. General theoretical arguments in the physics literature typically refer either to the incoherence of the semiclassical theory or to its mathematical intractability. But there are satisfiable axioms for sen1iclassical gravity (see, e.g., Wald 1994). These are not axioms for a full version of the theory, but they are sufficient to refute any argument that claims that a theory coupling a classical field to a quantum expectation value is inconsistent. So semiclassical gravity is mathematically consistent. The fact that a classical field can be coupled consistently to a quantum mechanical expectation value is not a guarantee that the semiclassical theory can serve as a final model of gravity, and there are some arguments suggesting that the mathematical problems of constructing such a theory are insurmountable. None of these arguments, however, does more than establish that there are hard mathematical problen1s that would have to be solved. But after a half century of attempts to construct a full quantum gravity, the fact of mathematical difficulties should perhaps be seen as a challenge to overcome, rather than an excuse not to try.
3. Metaphysics Sometimes one hears that the unity of nature demands that all physical fields be of one metaphysical type. There are indeed, in the literature on quantum gravity, arguments that attempt to turn quite vague claims about unity into specific arguments against hybrid models of gravity. None of these is very striking. But what is striking is that physicists should take them at all seriously. Without assigning some specific content to the notion of unity it is difficult to see how metaphysical arguments for unity are relevant. Rovelli has offered what seems to be the best attempt at a general argument in favor of quantizing the gravitational field. He tells us that we have learned in the course of the twentieth century both that gravity is a dynamical field, and that all dynamical fields are quantum mechanical. Stated this baldly though, the argument is clearly questionbegging. But no plausible seeming argument for the general claim has been offered.
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It simply has not been established that a semiclassical model of gravity is doomed to failure. Granting even the soundness of arguments prompted by quantum mechanics, which urge a modification of general relativity, would not suffice to establish that the gravitational field must therefore be quantized. Indeed, semiclassical gravity is a program that grants the former without subscribing to the latter. Some of the most compelling arguments for quantizing the gravitational field are motivated by puzzles that are solved already in the semiclassical regime. For example, there are theorems establishing that general relativity necessarily displays singular behavior. This singular behavior is often taken to show that the theory cannot be complete, and that the gravitational field must be quantum mechanical. But each of these theorems fails in semiclassical theories because of peculiar facts about quantum mechanical matter. That is, coupling the gravitational field of general relativity to quantum mechanical matter fields results in a theory that is not inherently singular (Mattingly 2001). We should recognize that semiclassical gravity is not best understood as a monstrous hybrid that should be eradicated as quickly as possible. Hybrid it may be, but it has a status similar to that of the combination of electrodynamics and mechanics in the latter part of the nineteenth century-it is the going theory and should be taken seriously. If it turns out to have insuperable flaws, these should be discovered in the course of investigating the theory itself. The payoff of continuing aggressively to pursue the theory is at least two-fold: on the one hand, we learn more about the theory and how it can be used as a tool of empirical investigation; on the other hand, we may learn something crucial about how actually to find a theory of quantum gravity.
5. Lorentzian Strategy for Quantum Gravity Here I will merely gesture at important work in the semiclassical theory. This work is interesting both for its own sake as work in the semiclassical theory, and for what it promises to show us about the weaknesses and limitations of that theory. Many of the objections raised above against the theory are seen to be irrelevant, or easily overcome in this work. And while the theory has not yet broken down, a better sense is emerging for where one should push it to make it reveal the paracletes of quantum gravity. A very interesting direction is taken by Barcelo, Liberati and Visser (2003). They propose the possibility of testing semiclassical gravity using phase perturbations (roughly, distortions in the speed of shock waves) in a physical model of the whole universe-a Bose-Einstein condensate. Their idea is to tune the interactions of the various components of the model (the elements of the condensate) in order to simulate a Friedman-
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Robertson-Walker space-time. Such space-times are expected to be very good models of our own universe since, most importantly, they correspond to universes with matter that are expanding in accordance with Hubble's law-where the expansion rate between any two points is proportional to their separation. Their experiment shows that their model universe naturally divides into two distinct kinds of system. They have, on the one hand, a system that looks like an expanding universe much like our own, and, on the other hand, a remainder that looks like particle creation associated with this expansion n1uch like that in the early stages of our universe. While their condensate does not expand, they are able to define a system of coordinates for it such that the radial coordinate is increasing in accord with the Hubble paran1eter of an expanding space-time. And while there is no real particle creation in the condensate, there are coherent sound waves (phonons) that behave much like quantum particles. The punchline, of course, is that they calculate that there is a significant opportunity to observe analogues of these effects in our own universe, and test thereby various theories of semiclassical gravity-and that these opportunities are much greater than those associated with other sources of semiclassical scale effects. So what we see here is a physical model of a semiclassical universe that can help us to constrain the range of regimes in our own universe that are appropriately described by the semiclassical theory. Of particular interest is that such work will help to model what breakdowns on semiclassical gravity look like as quantum gravitational effects become important. One might expect to see which parts of the semiclassical model are in conflict with each other. In addition to the analogue effects described by Barcelo et al., there are a number of ongoing experiments that are expected to provide sufficient constraints on matter models to eliminate certain of these models for the universe as a whole. The results are very interesting and important in their own right, and suggest that soon candidate models of quantum gravity will be subject to experimental test. But equally interesting for present purposes is how the experiments highlight once again. the role the semiclassical theory plays in settling "facts" of cosmological significance. Gunzig and Saa (2004) show that certain n10dels of quintessence (these are theories of exotic matter that is not part of the standard model of particle physics), which are apparently ruled out by singular effects at the transition between attractive and repulsive gravitational regimes, are not ruled out in certain models of semiclassical gravity. One crucial point is that the classical action is modified by the inclusion of a total divergence tern1. While such terms have no bearing in any classical case, here the term, if necessary to reproduce experimental results, would potentially force a n10dification of the
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action considered in loop quantum gravity, and hence change its target "reduction regime." That is, changes in the action necessitated by semiclassical considerations produce changes in possible models of one of the most promising candidates for a full quantum gravity. Finally I mention other work meant to alter the semiclassical theory itself-preserving its basic form while insulating the approach from the apparent problems of the original proposal. Hu and Verdaguer (2002), Calzetta and Hu (2003), and Martin and Verdaguer (2004) have been exploring stochastic semiclassical gravity. In the stochastic program a term is added to the matter-energy side of the equation-a term describing the fluctuations in the quantum matter fields. The standard semiclassical theory doesn't include the effects of fluctuations and so is unlikely to be correct in regimes where, for example, fluctuations in matter density are comparable in magnitude to the expected matter density itself. So the addition of an explicit fluctuation term is intended to extend the range over which the semiclassical theory is valid. Without considering specific details I do want to point out that there is a great deal of freedom in the form that such a fluctuation term may take. There is thus a great deal of freedom available now for extending the semiclassical theory to find potential deviations from experiment, and to modifY it in the light of those deviations. Consideration of these three classes of work on the semiclassical theory malces it plausible to suggest that pursuing the semiclassical theory is a viable strategy of the kind undertaken in pre-relativistic electrodynamics. There were attempts to build mechanical models of ether systems dating back to Ma:~.'well's day, attempts to test the theories by consideration of boundary effects, and importantly, efforts to extend the theory by incorporating new classes of interaction into the theory (ether contraction effects, e.g.). These all have analogues here in the semiclassical theory. I am interested in how the Lorentzian analogy plays out here. And so I want to advance the idea that we may see, on the one hand, an apparently ad hoc maneuver meant to save semiclassical gravity and show how to extend its domain of applicability by a suitable augmentation of the right-hand side of the basic equation for semiclassical general relativity. This is the explicit introduction of a stochastic fluctuation term in the stochastic program ofCalzetta, Hu, Martin, and Verdaguer addressed above. 6 On the other hand, we may see an attempt to track down the observational defects of the entire class of semiclassical gravitation theories in the observation of quantum effects in the metric either directly or analogically. This is seen in Gunzig and Saa's (2004) and in Barcelo, Liberati, and Visser's (2003) work, respectively. We see here the theoretical and observational components of the strategy that we saw in the case of special relativity. However, I am sensitive to the danger of forcing the analogy and so want
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to make clear that the interest of semiclassical approaches for philosophy of science transcends the analogy. Perhaps it is useful to phrase the approach this way: the reappraisal I suggest of Cassirer's analysis of the transition to special relativity should occasion a closer look by philosophers of physics into the semiclassical regime of gravitation. But neither should we regard acceptance of Cassirer's view as required in order to make the semiclassical theory interesting to philosophers, nor should we take his view as naively prescriptive of what progress in quantum gravity must look like. Still, if radical new physics ultimately is required for quantum gravity to advance to a complete theory, it is not at all unreasonable, prima facie, to suppose that we will uncover that radical physics in pursuit of the semiclassical progran1.
6. Conclusion Let me close by discussing two important missing components of the view that I am trying to advance. These components are not, I think, crucial for the basic outline that I have presented, but some attention to then1 would make for a more complete picture. First, there has been no mention of the quantum mechanical measurement problem. On the face of it, such an omission would seem surprising. After all, the issue of what to do about the apparent fact that the universe is in a definite state despite the absence of an outside observer is behind many of the important approaches to quantum cosmology, and these in turn are behind many of the programs in quantum gravity as well as setting many of the desiderata for such programs. Moreover some, in particular Roger Penrose (1989), have suggested that gravity may be the way quantum states are collapsed and that to understand the measurement problem one will have to understand the connection between gravity and quantum mechanics. More attention should be devoted to the issue of measurement. However I think that the issue is irrelevant to the question of how we should go about quantizing gravity (or even whether we should). How the collapse of the wave function takes place (if it does at all) should, I think, be discovered as we go along. My suggestions toward a philosophy of science are, I think, clearly directed more toward the development of the sciences than toward straightforward, traditional problems of interpretation of scientific theories themselves. It would be nice to have a solution to the measurement problem. I don't see that such a solution is required for progress in quantum gravity. Second, a more complete account of Cassirer's scientific methodology would be of considerable interest here. I have only offered the sketchiest of outlines of part of his view. Producing a more detailed analysis of the
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relation between Cassirer's views and contemporary physics, and contemporary debates about progress in physics and science more generally, are tasks for another occasion. However, the parts of Cassirer's views that I have singled out seem to provide a solid foundation for beginning to understand both the development of quantum gravity, and the sense in which we are seeing true scientific progress in that development.
NOTES 1. I am grateful to helpful comments from the attendees of the Synthesis and the Growth of Knowledge conference. I would like to thank Thomas Ryckman especially for helpful clarifying remarks about Cassirer's basic views. 2. This is yet a third fundamental development of the twentieth century. The theory claims that all inertial systems are equivalent for the description of nature, and that the speed of light, regardless of the speed of the source of that light, has the same value in all of these frames. It follows from these claims that simultaneity, time intervals, and space intervals are all relative to the inertial frame in which they are measured. Special relativity is related to general relativity in that in very small regions of space-time, general relativity reduces to special relativity. What to say beyond that is subject to considerable debate. 3. Incidentally, it is instructive to see that Lorentz himself provides a counterexample to the radical version of Kuhn's incommensurability thesis. Lorentz is quite capable of understanding and rationally evaluating his program against that of Einstein. While he sees certain advantages for his program, and indeed thinks that it is possible that his program will be revived in the future, he also can see that Einstein's theory has scientific advantages over his own. (See Lorentz 1939.) 4. This is a welcome change both from Feynman's "shut up and calculate" injunction and the blind adherence to naive falsificationism of a few decades past. 5. Note that even should they succeed, they do not amount thereby to any sort of argument that we should quantize the gravitational field, for any number of other options remain. These options include such a radical proposal as deriving both quantum mechanics and relativity from some deeper theory to which they are both only approximations, for example. 6. Of course, they don't see their work in this light, but neither did Lorentz apparently. Instead I believe both "groups" are engaged precisely in the effort to extend their respective theories as far as possible. If the theories never fail, so be it. But if they do, then we have ready guides to their replacement.
REFERENCES Barcelo, C., S. Liberati, and M. Visser (2003). "Probing Semiclassical Analogue Gravity in Bose-Einstein Condensates with Widely Tunable Interactions."
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Physical Review A68: 053613. Version cited here at: arXiv:condmat/0307491v2. Callender, C. and N. Huggett (eds.) (2001). Physics Meets Philosophy at the Planck Scale. Cambridge: Cambridge U. Press. Calzetta, E. and B.L. Hu (2003). "Noise and Fluctuation in Semiclassical Gravity." arXiv:gr-qc/9312036v1. Cassirer, E. ([1921] 1953). Substance and Function and EinsteinJs Theory of Relativity. New York: Dover. Earman, J. (1986). A Primer on Determinism. Dordrecht: Reidel Eisenstaedt, J. and A. I(ox (eds.) (2005). The Universe of General Relativity: Einstein Studies, Volume 11. Boston: Birkhauser. Fokker, A. and P. Zeen1an (eds.) (1939). Collected Papers. The Hague: Nijhoff. Friedn1an, M. (2000). A Parting of the Ways: Carnap) Cassire1) and Heidegger. La Salle, IL: Open Court. Gunzig, E. and A. Saa (2004). "Quantum effects and superquintessence in the new age of precision cosmology." International Journal of Modern Physics D 13: 2255-60. Version cited here at: arXiv:gr-qc/0406068v1. Hu, B.L. and E. Verdaguer (2001). "Recent Advances in Stochastic Gravity: Theory and Issues." Classical and Quantum Gravity 20: Rl-R42. Version cited here at: arXiv:gr-qc/0211090v1. - - - . (2002). "Stochastic Gravity: A Primer with Applications." arXiv:grqc/0211090v1. Isham, C. (1993). "Prima Facie Questions in Quantum Gravity." arXiv:grqc/9310031. Lorentz, H. (1939). "De zwaartekracht en het licht een bevestiging van Einstein's gravitatietheorie." In A. Foklcer and P. Zeeman (eds.), Vol. IX, 264-74. Martin, R. and E. Verdaguer (2004). "On the Semiclassical Einstein-Langevin equation." Physics Letters B465: 113-18. Version cited here at: arXiv:grqc/9811070v2. Mattingly, J. (2001). "Singularities and Scalar Fields. Matter Theory and General Relativity." Philosophy of Science 68: S395-S406. - - - . (2004). "Overcoming Metaphysics (Again!): Cassirer, Weyl, and Carnap on the Ontology of Spacetime." Presented at the International Conference on the Ontology of Spacetime, Montreal, Quebec, May 11-14,2004. - - - . (2005). "Is Quantum Gravity Necessary?" In Eisenstaedt and I(ox (2005), 325-38. - - - . (2006). "Why Eppley and Hannah's Thought Experiment Fails." Physical Review, D73, 681 6402547. Penrose, R. (1989). The EmperorJs New Mind. Oxford: Oxford University Press. Rovelli, C.(2001). "Quantum spacetime: what do we know?" In Callender and Huggett (2001),101-22. Wald, R. (1994). QJtantum Field Theory in Curved Spacetime and Black Hole Thermodynamics.
[18] Beauty Doth of Itself Persuade: Dirac on Quantization, Mathematical Beauty, and Theoretical Understanding MICHAEL DICKSON 1
"Beauty itself doth of itself persuade The eyes of men without an orator" -SHAKESPEARE, The Rape ofLucrece
1. Dirac on Mathematical Beauty A5 many physicists have, Dirac insisted on the importance of mathematical beauty for the practice of theoretical physics. His view on this point seems not to have changed through his career. For example, in (1939) he writes: A5 time goes on, it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen. And in (1963) he writes: I think it's a peculiarity of myself that I like to play about with equations, just looking for beautiful mathematical relations which maybe don't have any physical meaning at all. Sometimes they do. And not long before he died, he described mathematical physics thus (1982): It is simply a search for pretty mathematics. It may turn out later that
the work does have an application. Then one has good luck.
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However, in addition to these and similar generalities, Dirac also made some more suggestive, concrete pronouncements about mathematical beauty, which begin to point one towards something like an account. While mathematical beauty surely encompasses a variety of qualities, one finds in Dirac an emphasis on two: simplicity and generality.2 I will briefly consider each in turn. Dirac had a decidedly physical understanding of mathematical simplicity, as, in essence, the minimization of the number of 'boundary conditions', understood somewhat loosely. For example, writing about a sense in which he found quantum mechanics to be 'ugly', Dirac (1939) objects to the fact that one must impose the 'collapse' of the wavefunction from outside the theory: The increase thus arising in the non-mathematical part of the description of the universe provides a philosophical objection to quantum mechanics .... The objection does show ... that the foundations of physics are still far from their final form. This quotation is striking because Dirac assumes that a 'philosophical' objection indicates a problem with the foundations of physics, the objection in question being the fact that the mathematical expression of the theory is not 'simple'. Later in life Dirac attempted to derive various boundary conditions or physical constants from others, mostly fallaciously or implausibly, but tirelessly nonetheless. Perhaps the most infamous example involves the ratio of the electrostatic force to gravitational force between the proton and electron in a hydrogen atom-this ratio is approximately 10 39 . As Dirac observed, the ratio of the time it takes light to traverse the diameter of a classical electron and the estimated current age of the universe is also approximately 10 39 . This coincidence led Dirac to speculate that some theory links these two ratios (so that, in particular, the ratio of electrostatic to gravitational force must get smaller over time). If Dirac were correct (but he was not 3 ), then the total number of fundamental constants would, so the argument goes, be reduced, to the credit of the beauty of the theory.
2. Mathematical Beauty and Understanding Dirac's insistence on the importance of mathematical beauty in theoretical physics-indeed, his insistence that less than fully beautiful theories are ipso facto somehow incomplete or in need of revision, or otherwise unsatisfactory-may strike one as odd at best. For what does mathematics, much less
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mathematical beauty, have to do with the world, and with our understanding of the world? On Dirac's behalf,4 and despite the excesses of some of Dirac's own speculations (see the end of section 1), I claim that mathematical beauty is connected, in an important way, with our theoretical understanding of the world. A contrary view was famously at issue amongst other founders of quantum theory. This contrary view emphasizes the notion of theoretical understanding as some form of mechanical visualization, or, to use the catch-word at the time, Anschaulichkeit, roughly translated as 'visualizability'. For example, Schrodinger (1926) writes: My theory was inspired by L. de Broglie and by brief but infinitely far-seeing remarks of A. Einstein . . . . I naturally knew about [Heisenberg's] theory, but because of the to me very difficult-appearing methods of transcendental algebra and the lack of Anschaulichkeit, I felt deterred by it, if not to say repelled. The point, of course, is that Schrodinger's wave mechanics was supposed to be somehow 'visualizable', because it describes a wave propagating in space. Whether Schrodinger's theory is really visualizable is extremely debatable. The wave takes on complex, not real, values, and it propagates not in 3-space but in configuration space. Indeed, Heisenberg seems to have understood this point quite well, in a letter to Pauli: The more I think about the physical portion of the Schrodinger theory, the more abscheulich [abominable] I find it . . . . What Schrodinger writes about the Anschaulichkeit of his theory 'is probably not quite right', in order words, it's crap. (Pauli 1979, letter 136) 5 Heisenberg, further, attempted to make the case that his own theory (matrix mechanics) was no worse off than Schrodinger's: The prediction that, say, the velocity in the x-direction is 'in reality' not a number but the diagonal term of a matrix, is perhaps no more abstract and no more 'unvisualizable' than the statement that the electric field strengths are 'in reality' the time part of an antisymmetric tensor located in spacetime. (Heisenberg 1927) Below, I will suggest that Heisenberg made a strategic mistake here by apparently buying in to the general view that theoretical understanding must involve visualization (and then attempting to argue, quite unpersuasively, that his own theory is indeed visualizable).
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This view of understanding has some philosophical pedigree. For some, it is perhaps a conceptual truth that understanding involves mechanical 'pictures'; for example, Salmon (1984) argues that [Explanations] show us how particular occurrences come about; they explain particular phenomena in terms of collections ofparticular causal processes and interactions-or, perhaps, in terms of noncausal mechanisms, if there are such things. For others, the point is perhaps more contingent, but true nonetheless. For example, Cushing (1991) writes: My basic position in this paper is that, not an a priori, but rather an historicist argument shows that picturable models or explanations are necessary for producing in us a sense of understanding. This necessity is rooted in the way we think about and understand physical phenomena. Indeed, it has been suggested that even our understanding of n1athematical physics involves n1echanism. Speaking of n1echanistic metaphors used in the service of understanding Maxwellian electrodynamics, Cat (2001) writes: Scientific metaphors organize electric and magnetic phenomena for investigation and provide a geometrical, mechanical and dynamical (energy-based) vocabulary through which to understand the phenomena as well as the relevance of the mathematical formalism. They provide a vocabulary in which to conduct the investigation. Metaphors that play these roles surely enhance scientific understanding. Without wishing to call into question the importance of 'mechanistic' understanding (and without implying that the above authors believe it to be the only sort of understanding), I would like to call attention to a dif ferent philosophical tradition, one in which, I will argue, mathematical beauty can be seen as contributing to theoretical understanding. This tradition has a solid philosophical pedigree of its own: You would loudly proclaim that you know of no other way in which any given object can come into being except by participation in the reality peculiar to its appropriate universal .... If anyone should fasten upon the hypothesis itself, you would disregard him and refuse to answer until you could consider whether its consequences were mutually consistent or not. And when you had to substantiate the hypothesis itself, you would proceed in the same way, assuming whatever more ultimate hypothesis commended itself to you. (Plato, Phaedo 101c-e
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The general idea, of course, is that understanding is achieved by seeing the thing to be understood as an exalTIple of, or member of, a more general class of things. (And described at this level of generality, one could ascribe the notion to Aristotle just as well as to Plato.) Return, now, to Dirac's view of mathematical beauty, according to which beauty is (in part at least) constituted by simplicity and generality. Hovv might beauty qua beauty contribute to understanding? The case of simplicity is clear. Increasing simplicity, in the sense of reducing the number of boundary conditions, increases the 'mathematical part' (as Dirac says) of a theory. In other words, phenomena that previously were described in part by mathematical theory and in part 'by hand' (boundary conditions, which are given, or at any rate motivated, not by a theory, but by 'the world,6) come to be increasingly described by theory alone. Hence, almost by definition, we have a better theoretical understanding of the phenomena in question. Moreover, if one accepts the PlatonicAristotelian principle that bringing a phenomenon under a general concept constitutes a form of understanding, then theoretical understanding is itself a genuine form of understanding. Of course, if one accepts that principle, then increasing generality is also a form of understanding, and in this case, in a twofold sense. Suppose that we begin with some theory, T, that describes some phenomena, P. Theoretical work produces a new theory, T*, which is more general than T. In virtue of T*, we have a new kind of understanding of P, according to which P comes under not only concepts from T but also concepts from T*. More important, however, we now have an understanding of Titself, in virtue of seeing it as a special case of the more general theory, T*. In this paper, I am more interested in theoretical understanding in this latter sense ('understanding of a theory' as opposed to 'using theory to understand some phenomenon'). I believe that Dirac was also, generally, more concerned with this form of understanding.. Indeed, his being so would explain why sometimes he takes simplicity to be distinct from beauty, and why he prefers the latter of over the former: The research worker, in his efforts to express the fundamental laws ... should strive mainly for mathematical beauty. He should take simplicity into consideration in a subordinate way to beauty... It often happens that the requirements of sin1plicity and beauty are the same, but where they clash, the latter must take precedence. (Dirac 1939) I turn, now, to an example, from Dirac himself, to illustrate the way in which beauty (primarily as generality) contributes to theoretical understanding (in the sense of 'understanding theory').
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3. Dirac's Argument About Quantization One of Dirac's major contributions to the foundations of quantum mechanics involved 'quantization', the analysis of the relationship between classical mechanics and quantum mechanics, in a way that would make our deep understanding of the former of some use in the latter. Dirac (1925) puts it thus: It is well known that the experimental facts of aton1ic physics necessitate a departure from the classical theory of electrodynamics in the description of atomic phenomena. This departure takes the form, in Bohr's theory, of the special assumptions of the existence of stationary states of an atom, in which it does not radiate, and of certain rules, called quantum conditions .... In a recent paper Heisenberg puts forward a new theory, which suggests that it is not the equations of classical mechanics that are in any way at fault, but that the mathematical operations by which physical results are deduced from them require modification. All of the information supplied by the classical theory can thus be made use of in the new theory. (Dirac was being generous. Heisenberg was far from describing a complete mathematical theory to replace the theory of classical physics.) The program, then, is to understand Heisenberg's equation of motion as somehow analogous to the classical equations of motion, replacing 'classical' operations with 'quantum' ones. Part of this replacement involves replacing classical real-valued variables with N X N matrices (where typically N = 00). One of the key unsolved problems faced by this program is to find an appropriate notion of 'taking a derivative' within quantum theory. In particular, how ought one tal(e the derivative of a matrix? Dirac (1925) writes (with slight change of notation): We shall now determine the form of the most general quantun1 operation djdv that satisfies the laws djdv(x+y) djdv(xy)
=
=
dxjdv + dyjdv [Linearity]
(dxjdv) y + x (dyjdv) [Leibniz Rule].
(One of the fundamental problems here is that matrix multiplication is non-commutative. Note that the order of x and y is preserved in the last equation.) Dirac shows that the answer is that for any N X N matrices x and v, there must be some N X N matrix a that defines the derivative of x with respect to v by
Beauty Doth of Itself Persuade dx/dv
=
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xa - ax
where xa is the matrix product of x and a, and similarly for ax? This result is highly suggestive, for as Dirac noticed (and he was apparently the first to do so), this 'quantum derivative' has the form of a Lie Bracket, which means that algebraically it looks like the Poisson bracket of classical mechanics, which is used to express the time-derivative of observables. 8 Hence, Dirac argued, the time-derivative of a quantum observable (represented by a self-adjoint matrix) F is i[F, H], where H is the (quantummechanical) Hamiltonian. For Dirac, this result was part of a consistency proof for quantum mechanics, in much the same spirit as some early consistency proofs for non-Euclidean geometries, which proceeded by defining an appropriate map between the non-Euclidean geometry in question and some fragment of Euclidean geometry (for example, the surface of a sphere). The consistency of the latter then guarantees the consistency of the formerY The problem faced by Heisenberg's theory was, in part at least, the question whether it forms a consistent mathematical theory. Dirac's result suggests a general program for verifYing the consistency of Heisenberg's theory: Find a map (homomorphism) from the aigebra 10 of classical observables to the algebra of quantum observables, a map that talces Poisson brackets to 'quantum Lie brackets' given by ih[· , . ]. Indeed, 'quantization' (roughly, 'translating' the classical description of a system into a corresponding, and presumably more accurate, quantum description) proceeds along more or less the lines envisioned by Dirac. Alas, however, Dirac's program in full generality has failed-there is, provably, no homomorphism between the classical and quantum algebras of observables with the requisite properties. That is, there is no fully general and rigorous scheme for quantization. l l Nonetheless, Dirac's result is quite important, for it in a sense shows one where Heisenberg's equation comes from-the only reasonable definition for a time-derivative of matrices is (1). And keep in mind that as far as quantum theory is concerned, the mathematical framework of N X N matrices is completely general. However, Dirac apparently felt that this framework was not general enough, for in his textbook he made a much more general argument, in the very general context of non-commutative algebras. He writes (1930): We assume the quantum FB. ['quantum Poisson bracket', i.e., now, a Lie bracket derivation on an arbitrary noncommutative algebra] to satisfY all the conditions (2) to (6) [i.e., just the conditions required to be a Lie Bracket derivation], it being now necessary that the order of the factors ... be preserved through the equation .... These conditions
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are already sufficient to determine the form of the quantun1 P.B. uniquely, as may be seen from the following argument. Letting [. , . ] be the commutator and [[. , . ]] be the sought-for 'quantum Poisson bracket', Dirac points out that for any two non-commutative pairs, f,g and a,b, [[', .]] can be evaluated in two ways (because ofnon-commutativity). Equating them yields
[[ h g ]] [ a, b]
=
[f, g] [[ a, b]]
(2)
Dirac then makes a familiar move (see note 7):12 Since this condition holds with f and g quite independently of a and b, we must have
[[ f,g]]
=
ik [f, g]
(3)
where k must not depend on f and g, nor on a and b, and must also commute with [[ a,b]]. It follows that k must simply be a number. (Dirac 1930, notation slightly changed) A further argument (ibid.) based on the analogy with the classical Poisson bracket establishes that k = i.
4. The Significance of Dirac's Argument To review, here is the story thus far. In 1925 Dirac proved his result for the algebra of N X N matrices. He probably quickly realized that the result also holds for the proper quantum-mechanical case, namely, hermitian N XN n1atrices (including N = 00). In his textbook (the first edition of which appeared in 1930), Dirac 'generalized' the argument to the case of arbitrary non-commutative algebras. (This form of the argument is not valid as it stands-see note 12.) The general form of the argument seems to have been quite important to Dirac, for it was never removed from his textbook, despite major revisions elsewhere in the book at each new edition. Moreover, Dirac mentions the (conclusion of the) argument in numerous articles and (even more often) public lectures. However, one wonders why. Apart from the fact that Dirac may well have eventually realized that the argument is in fact invalid,13 there is the obvious point that as far as quantum mechanics is concerned, the original (and valid!) 1925 argument is enough to derive the form of the quantum Poisson bracket. Why bother with the more general argument?
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The immediate answer is that the more general argument is more beautiful, i.e., more general. And beauty, as I claimed above, generates understanding: In this case, it allows one to see quantum theory as an example of a more general class of theories (whose algebras of observables are noncommutative, but not necessarily isomorphic to the algebra of N XN Hermitian matrices). Let us consider this point in slightly more detail. How might Dirac's argument be patched up? There are (at least) two reasonable strategies. The first would be to characterize algebras in which the argument fails, and argue that they are somehow physically uninteresting. For example, consider a proposed 'quantum Poisson bracket' of the form
[[f,g]]
=
z[f,g]
(4)
where z is in the center of the algebra (but not a multiple of the identity).1 4 Is such an algebra interesting, physically? Arguably not. Note that for any Hamiltonian, H: dz/dt
=
z[z, H]
=
O.
(5)
In other words, z never changes its value no matter what interactions the system might undergo. (Hneed not be a 'free Hamiltonian' but could, for example, be the effective Hamiltonian for a system that is experiencing external forces. Or evolving under the influence of external potentials.) But then why is z in the algebra in the first place? In what sense is it an 'observable'?15 The important point about this strategy is that, if successful, it would show why Dirac's argument works in the context of quantum theory. For example, if it were true that the only cases where the argument failed were cases where the algebra of observables has a non-trivial center, then we would be in a position to make a connection between 'being a member of a class of theories whose algebras of observables have a trivial center' and 'having a Poisson bracket of the form (3)'.1 6 Such a result would, I claim, amount to 'theoretical understanding' of the sort described in section 2. The second strategy for patching up Dirac's argument is to characterize directly the algebras in which the argument does work. For example, let F x,y be the free polynomial algebra over K generated by x and y (K = R or C) .17 Then the only Lie bracket derivations on F x, yare of the form [[f,g]] = ik [f,g] with k EK.1 8 What is interesting about the free algebra over two things? Dirac may well have had this algebra (or something akin to it) in mind at times, taking the 'two things' to be position and momentum. The free algebra over position and momentum, is, in a sense, a 'very general' theory of position
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and momentum, in which one makes no assumptions about the relationship between position and momentum or any polynomials involving them, apart from whatever follows from their explicit polynomial structure. (In particular one does not assume commutativity.l9) Again, if some result along these lines were true, we would be in a position to understand why quantum theory has the Poissonian structure that it does, seeing it as a member of a class of theories, all with the same structure. Suppose that we managed to follow through one of these strategies successfully. It is clear that the result would be, by Dirac's lights, a potentially beautiful result, for it would be quite general, and include quantum theory as a special case. And I have already claimed that it would constitute a form of theoretical understanding, simply in virtue of the fact that we would have understood an important feature of quantum theory (its dynan1ical structure) as following from quantum theory's membership in some broader class of theories, characterized in some perspicuous way. But what further evidence can be given that subsun1ption of this sort really is a form of theoretical understanding? "By their fruits ye shall know them." One reason for pursuing these highly abstract projects is that with such results in hand, we would be in a better position to make further theoretical progress. Dirac (1931) affirms this view in quite a strong, and striking, way: There are, at present, fundamental problems in theoretical physics ... the solution of which ... will presumably require a more drastic revision of our fundamental concepts than any that have gone before. Quite likely, these changes will be so great that it will be beyond the power of human intelligence to get the necessary new ideas by direct attempts to formulate the experimental data in mathematical terms. [Here Dirac must have Heisenberg's matrix mechanics in mind.] The theoretical worker in the future will, therefore, have to proceed in a more direct way. The most powerful method of advance that can be suggested at present is to employ all the resources of pure mathematics in attempts to perfect and generalize the mathematical formalism that forms the existing basis of theoretical physics. Notice that Dirac refers to purely mathematical generalization as 'more direct'. In what sense could it be 'more direct' than the path that Heisenberg (allegedly) took, namely, the path of directly formulating experimental data (in this case, about spectra) in mathematical tern1S? The answer must be that the n1athematical formulation of experimental data is, for Dirac, a step on the way towards true scientific understanding, which comes by way of a highly general, and therefore beautiful, theory that has as a special case the phenomena in question. 2o Such a theory provides
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explanations in terms of general physical concepts which, precisely because of their generality, explain, in part, by unification (because diverse phenomena all fall under the highly general concept). Consider, for example, the mathematical concept of linearity. Many otherwise diverse theories are 'linear', and precisely because of this shared property, they share other properties. Moreover, seeing various phenomena as governed by linear theories allows one to understand, in part, why they behave the way that they do. For example, linearity arguably describes phenomena in which causal influences act independently to produce an effect. One finds such phenomena (or, approximately such phenomena) in diverse areas, from economics to population genetics to fundamental particle physics. Hence generalization is itself a form of theoretical progress. Moreover, it leads to further progress by pointing the way towards new, and better, theories. (These new theories are the 'fruits' of having the generalized theories in hand.) For example, understanding how linearity is connected with various physical features of the world will allow one to see when it might be appropriate to give up on linearity. Or, to take the case at hand, one might hope that a more complete understanding of the conditions under which Dirac's argument works would lead to a more complete understanding of the origins of the dynamical structure of quantum theory, and thence, quite possibly, to the development of an improved theory, either a theory of quantization, or even a successor to quantum theory itself.
5. 'Unreasonable Effectiveness' Plenty ofphysicists (and others) have remarked, as Dirac did, on the apparent role of mathematical beauty in theoretical physics. The most famous such remark comes from Wigner (1960): The mathematical formulation of the physicist's often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena. This shows that the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language. The usual reading ofWigner is that he is arguing some sort of Pythagorean claim, one where beauty plays also an important role: The observation which comes closest to an explanation for the mathematical concepts' cropping up in physics which I know is Einstein's
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statement that the only physical theories which we are willing to accept are the beautiful ones. (Ibid.) Hence the mystery is, supposedly, quite deep indeed. Why should 'the world' care about mathematical beauty? Well, for Wigner, we do not know that it does! He writes: However, Einstein's observation can at best explain properties of theories which we are willing to believe and has no reference to the intrinsic accuracy of the theory. (Ibid.) In Wigner's view, we 'do not know why our theories work so well', not because we do not understand how Nature 'knows' about mathen1atical beauty, but because we don't know why it has appeared thus so far. As far as Wigner is concerned, there is no reasonable answer to this question. Our attitude should sin1ply be to "be grateful for it and hope that it will remain" (ibid.). Wigner is not alone in his wonder at the success of mathematics in physical sciences. A more recent, and more extended, version of the argument is due to Steiner (1998, 46):21 1. Mathematical concepts arise from the aesthetic impulse in humans.
2. It is unreasonable to expect that what arises from the aesthetic impulse in humans should be significantly effective in physics. 3. Nevertheless, a significant number of these concept are significantly effective in physics. 4. Hence, mathematical concepts are unreasonably effective in physics. Thus argues Steiner, but there are some obvious (and, in some cases, wellworn) objections to this sort of argument. Regarding 1, it has been pointed out that the mathematical concepts themselves (or at any rate those that are applied) arise from the investigation of the physical.world. Linearity, discussed above, is an oft-cited example. Regarding 2, one might suppose that effectiveness in physics is itself determined by human criteria, so there is no surprise. Regarding 3, some (e.g., Wilson 2000) have argued that the number is not at all significant; moreover, the concepts are not significantly effective, perhaps even ineffective. And regarding 4, Wigner himself pointed out that strictly speaking all that we know (if we know this much) is that mathen1atical concepts have been effective, and in this case, nothing of interest follows except to be grateful and hope (see above). We need to distinguish two 'mysteries'. First, why should mathematics be applicable in the first place-why should nature be mathematizable?
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Second, granted that mathematics is applicable, why should mathematical beauty be a guide to how to pursue applied mathematics-why should nature 'care' about beauty? Wigner and Steiner elide the first mystery into the second, but it seems that they are quite different issues, if only because the second presupposes that mathematics is indeed applicable. Moreover, the objections above seem to be, if not conclusive, quite persuasive with regard to the first mystery, while the analogous objections to the mysteriousness of the guiding role of mathematical beauty in theoretical physics (and perhaps elsewhere) are not nearly so convincing. Let us briefly consider replies to those analogues. Regarding 1, while one might claim that 'the world', or the way that we understand it, has properties (e.g., independent action of forces) that naturally give rise to mathematical concepts (e.g., linearity), it is far less plausible to claim that 'the world' is beautiful in the relevant sense. (What would it mean to say that 'the world is general'?) On the other hand, one might claim that what gives rise to the value of beauty is not some feature of the world, but some feature of our understanding, or theorizing about the world. However, as I shall emphasize below, to admit this much is to concede the point. Regarding 2, it might be that theoretical progress is defined by us, but it is not, generally, defined in mathematical (and specifically aesthetical) terms, so the question remains about the connection between how it is defined and mathematical beauty. Why does mathematical beautification lead to, or count as, progress in the physical sciences (ifit does)? Regarding 3, the issue is not whether all (or most, or many) cases of theoretical understanding or progress are achieved by the pursuit of beauty, but why beautification counts as understanding or progress. In other words, the issue of whether there are cases of theoretical progress that are not achieved by appeal to or pursuit of mathematical beauty is not in question-surely there are. The issue is why achievement of mathematical beauty is, in at least some cases, for that reason (among others 22 ) an achievement of theoretical understanding. And regarding 4, the claim that beautification counts as understanding does not depend on an 'induction' from past cases where beautification led to understanding. Rather, it depends on a particular conception of understanding, or, more forcefully, on the nature of human understanding. Hence, prima facie, the standard (and convincing) resolutions of the first mystery, the mystery of why nature is after all mathematizable, do not carryover neatly to (convincing) resolutions of the second mystery, the mystery ofwhy beautification counts towards increased understanding. On the other hand, I am certainly not claiming that the role of mathematical beauty in science is, after all, a great mystery.23
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Indeed, the discussion, above, of two different sorts of theoretical understanding was meant already to provide a kind of 'deflationary' account of the mystery, by pointing to a context, a particular account of theoretical understanding, in which generalization in particular, but simplification also, just are cases of understanding. I would then suggest, with Dirac, that one ought to view mathematical beauty (in part!) as generality, and that mathen1atical generality does have an important role to play in science, both at the level of contributing to a scientific understanding of physical phenomena and at the level of contributing to our understanding of theories themselves (and thence to our ability to construct better theories). The very applicability of mathematics to the world ought, perhaps, to continue to inspire wonder, but about the importance of mathematical beauty there is no real mystery. Beauty should of itself persuade.
NOTES 1. Thanks to an audience at Stanford University for very helpful comments. Special thanks to Diarmuid Crowley, who was very helpful with some of the mathematical issues mentioned in this paper. 2. One feature, in addition to simplicity and generality, that probably should be counted in favor of beauty is 'power', roughly, the capacity to produce many and varied theorems and explanations. Indeed, power in the presence of simplicity and generality is probably an important type of beauty in mathematics. If my memory is correct, I have Bas van Fraassen to thank for making this point to me. Dirac does not, to nlY knowledge, ever speak explicitly of mathematical power, but I believe that one can read his statements about 'physical meaning' and 'application' as statements about physical power. As for purely mathematical power, it does seem that the only cases where simplicity and generality are interesting in the first place is when they are accompanied by mathematical power. 3. Assuming that the basic forms of the laws of gravitation and electrostatic attraction are correct, Dirac's speculation apparently requires that the gravitational constant, G, appearing in Newton's law of gravity decrease in time. There is data from at least the 1970s that rules out anything even close to the rate of decrease required by Dirac's speculation. 4. Perhaps I should say, 'motivated by Dirac', for while I do think that the account to follow is a plausible reading of Dirac, one must admit that there simply is not sufficient textual evidence to make a very strong claim that the account to follow was intended by Dirac. 5. Dirac's theory did not fare much better in Heisenberg's eyes. In a subsequent letter, he referred to it as "learned crap" (Pauli 1979, letter 147). 6. The scare-quotes are to indicate that the implied distinction between theory and world is, of course, problematic. Nonetheless, I would insist that within a theoretical context (i.e., in the context of any given physical theory), there is a spectrum from the more theoretically justified claims to the more obser~~~~I1~J!y
_
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justified claims, and that boundary conditions and claimed values of fundamental physical constants are often at the latter end. 7. Using linearity and the Liebniz rule, written in terms of mau'ix multiplication: Lnlm1k
allm;lllH'~' x n'1z Ybn' = Lnlklk
ank;nlll' xn'k' Ylun + Lklmlk x nk akm;k1m l Yk'm,!
The linear operator a ('differentiation by v') is independent of x and y: 0kk,a nm ;n 1n omm~anll;n'kl + 0nn,akm;krm l
l
1'
::::;
The matrix a in Dirac's result turns out to be related to the operator a by a nm
=
abn;kn'
which is independent of k given the constraints. 8. Given two (infinitely differentiable) functions (observables), fand g, on phase space, their Poisson bracket is given by
If,g) := (of/oX)(Op/dq) - (oflop)(opjox) The time-derivative of a classical observable is given by If, H), where H is the Hamiltonian (total energy) observable for the system in question. 9. Of course, the map must be well defined and 'consistency-preserving'. At least the former condition will prove to be a problem for Dirac. 10. More precisely, the program is this. First, recall that an algebra is a vector space, A, over a field, K, together with a bilinear operation that maps A XA to A. A derivation on A is an endomorphism, : A ~A, where
(ab)
=
(a)b + a (b)
for a,b EA. Finally, a Lie bracket on A is a bilinear, anti-symmetric, operation satisfying the Jacobi identity:
(a, (b,
ell
+ Ib, (e,
all
+ Ie, la, b)l
=
0
where a,b,c E A and la,b) is the Lie bracket of a and b. A Lie Bracket is a derivation just in case:
la,
be)
=
la, b}e
+
b(a, el.
Both classical and quantum theory are examples of algebras with a Lie Bracket derivation. Dirac's program, then, is to find a homomorphism from the one to the other. 11. The original result is due to Groenewold (1946), subsequendy generalized by van Hove (1951). See Gotay et al. (1996) for a contemporary discussion and review. 12. As it turns out, Dirac's argument is invalid. This point came clear to me during lengthy discussions with Diarmuid Crowley. We also derived some partial results about the conditions under which the argument is valid, but they do not appear to be particularly illuminating. Nonetheless, I am much clearer about the situation thanks to discussions with Crowley, and especially thanks to his mathematical expertise. 13. The language that Dirac uses to describe the argument changes noticeably over time. Compare, for example, his statement (1930) that "These conditions are already sufficient to determine the form of the quantum P.B. uniquely" with a much later statement d1at "It turns out that this bracket expression correspondsis very closely analogous to-i[a,b]" (1978, taken from a 1975 lecture). This shift
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reflects an apparently general shift in Dirac's language (describing this argument) from words like 'determine' and 'implies' to phrases like 'is analogous to'. (Of course, there could be other explanations for the shift, or it may be an accident.) 14. The center, Z(A), of an algebra, A, is the set of all z A that commute with every a E A, i.e., za = az. In standard quantum theory (e.g., matrix mechanics), the center of the quantum algebra of observables generally contains only multiples of the identity. It is fair to assume that in fact Dirac was not considering algebras with non-trivial centers. Note, finally, that (4) escapes Dirac's argument. Plugging into (2), one gets
z[h g][a, b] = [h g] z[a, b], which is indeed an equality under the condition that z commutes with everything. (Indeed, this example shows that it is enough for z to commute with all commutators, which is in general a weaker condition.) 15. There may well be answers to this question. Indeed, superselection rules in quantum theory are sometimes exactly of the form 'observable F is in the center of the algebra of observables'. An example is the purported charge superselection rule of 'local relativistic quantun1 field theory' (see Haag 1997). On the other hand, there is an important sense in which, in these cases, the quantity in question plays more the role of a 'parameter' than an 'observable'. 16. Well, clearly we must either generalize (3) a bit, or restrict attention to algebras that, as vector spaces, are over the field of complex numbers. 17. The notion of a free algebra is best defined category-theoretically. Here I'll offer just an intuitive characterization, as the closure under 'all formal linear combinations and products' of {x, y}. 18. This result, the proof of which I omit here, comes from joint work with Diarmuid Crowley. Crowley has further surmised that the set of all algebras in which Dirac's argument works is a category. 19. Of course, the quantum algebra of observables is not a free algebra. Hence we are still quite far from our goal, which would be to find some class of algebras that includes both the quantum algebra and F x,y, and all of which have the same Poissonian structure. 20. Note that direct formulation of experin1ental data could easily fail the test of simplicity as well, in Dirac's sense of 'simplicity'. 21. Steiner seems to read Wigner as clainung that the mystery of the applicability of n1atl1ematics is in fact in need of explanation. However, as I noted above, Wigner seemed to thinl( that there is no explanation (or at least not one that is available to us), that the best we can do is be glad of it. 22. The claim here is not tl1at beautification is by itselfsufficient for the achievement of understanding, but that there are cases where the beautification is itself one among otller causes of tlle achievement of understanding. When I say, following the title of this essay, that beauty should 'ofitself' persuade, then, I understand 'ofitself' not to mean 'by itself, alone', but 'essentially, rather than accidentally', i.e., because of what it is, and not because of some accidental connection with understanding. 23. My explanation of the importance of mathematical beauty is closest in spirit to objection 2, but in the end it is not the same, for it does not depend on the idea that we define \vhat counts as theoretical progress-I do not thinl( that we
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do. Rather, it depends only on the identification of one aspect of what does, in fact, count as theoretical progress.
REFERENCES Cat, J. (2001) "On Understanding: Maxwell on the Methods of Illustration and Scientific Metaphor." Studies in History and Philosophy of Science Part R: Studies in History and Philosophy of Modern Physics 32: 395-441. Cushing, J. (1991) "Quantum Theory and Explanatory Discourse: Endgame for Understanding?" Philosophy ofScience 58: 337-58. Dirac, P. (1925) "The Fundamental Equations of Quantum Mechanics." Proceedings of the Royal Society ofLondon A 109: 642-53. - - . (1930) The Principles ofQuantum Mechanics. Oxford: Clarendon Press. - - - . (1931) "Quelques Problemes de Mecanique Quantique." Annales de rInstitut Henri Poincare 1: 357-400. - - - . (1939) "The Relation between Mathematics and Physics." Proceedings of the Royal Society (Edinburgh) 59: 122-29. - - - . (1963) "The Evolution of the Physicist's Piculre of Nature." Scientific American 208: 45-53. - - - . (1978) "The Development of Quantum Mechanics." In Directions in Physics, ed. H. Hora and J. Shepanski. New York: John Wiley and Sons, 1-20. - - - . (1982) "Pretty Mathematics." International Journal of Theoretical Physics 21: 603-605. Gotay, M., H. Grundling, and G. Tuynman (1996) "Obstruction Results in Quantization Theory." Journal of Nonlinear Science 6: 469-98. Groenewold, H. (1946) "On the Principles of Elementary Quantum Mechanics." Physica 12: 405-60. Haag, R. (1997) Local Quantum Physics. Berlin: Springer-Verlag. Heisenberg, W. (1927) "Ueber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik." ZeitschriftfurPhysik43: 172-98. Pauli, W. (1979) Scientific Correspondence, Vol. I., ed. A. Hermann. New York: Springer-Verlag. Salmon, W. (1984) Scientific Explanation and the Causal Structure of the World. Princeton: Princeton University Press. Schrodinger, E. (1926) "Uber das Verhaltnis der Heisenberg-Born-Jordan'schen Quantenmechanik zu der meinen." Annalen der Physik 79: 734-56. Steiner, M. (1998) The Application of Mathematics as a Philosophical Problem. Cambridge, MA: Harvard University Press. van Hove, L. (1951) "Sur Certaines Representations Unitaires d'un Groupe Infini de Transformations." Memoires / Academie Royale de Belgique, Classe des Sciences 26: 1-102. Wigner, E. (1960) "The Unreasonable Effectiveness ofMathematics in the Natural Sciences." Communications in Pure and Applied Mathematics 13: 1-14. Wilson, M. (2000) "The Unreasonable Uncooperativeness of Mathematics in the Natural Sciences." The Monist 83: 297-315.
[19J Theory, Coordination, and Empirical Meaning in Modern Physics SCOTT TANaNA 1
1. The Relativized A Priori, Coordinating Principles, and Conceptual FrameworkB Michael Friedman's conception of the relativized a priori is inspired by the early logical positivists, who, according to Friedman, were primarily concerned with the place and meaning of the IZantian a priori in the face of the apparent failure of Euclidean geometry to describe space and time (see Friedman 1983, 1999). As readers of this volun1e probably know, the early logical positivists distinguished two aspects of the a priori in IZant's philosophy. The first is that the a priori is immutable; for exan1ple, the form of our intuition of space and time is necessarily Euclidean. The second is that it is constitutive of our knowledge; that is, the a priori is a precondition for knowledge and provides the form without which our experiences would not count as knowledge and perhaps would not even be experiences. Friedman claims that the early logical positivists (in particular, Reichenbach) rejected the first aspect of the a priori but accepted something of the second (Friedman 2002,174). Taking motivation from these aspects of early logical positivisn1, Friedman argues that a theoretical or conceptual framework is a prerequisite for the application, description, and even conceptualization of a physical theory. For example, Newton's second law says that forces are associated with corresponding accelerations, but the notion of acceleration assumes an established notion of free motion, and that in turn assumes an established geon1etry. In general, there are a number of specific foundational prerequisites for any physical theory. A framework for a theory consists of a structured set of the prerequisite abstract mathematics and physical principles that establish both the context within which and the language with which the theory is specified: a kinematics presupposes a
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geometry, a dynamics presupposes a kinematics, and so on. A theory simply is not well defined without a specification of a particular framework, and so the principles of the framework are a priori in the second sense. But while some set of a priori principles is required to provide a framework in which the theories can be specified, no particular, given set is required; we have a choice regarding the form of that framework. Similar phenomena can be described by theories with different frameworks. For example, consider a phenomenon in which all objects in the absence of specific forces move along certain curved lines. Such a phenomenon might be described by a theory with a flat geometry and universal forces, or by a theory with a curved geometry, where what would have been forced motion in the first framework is now considered to be free motion. For Friedman, the choice to adopt a new theoretical framework may be based on a variety of factors, including well-articulated meta-principles and desiderata for a theory as well as empirical considerations. However, I believe that Friedman has a difficult time accounting for the role that empirical phenomena play in such a choice, because while the phenomena are supposed to be concrete and independent of the abstract framework, they must also be general and well defined, which is possible only in the context of such a framework. According to Friedman, the abstract mathematical formalisms of modern physical theories have no empirical content on their own. An abstract mathematical apparatus only makes claims about empirical phenomena once one indicates how the abstract structure is to be applied to empirical phenomena. One indicates this connection via what Friedman calls (following Reichenbach) "coordinating principles." These principles coordinate "mathematical representations with experience." That is, they connect elements of the abstract structure to empirical phenomena, and thereby give the abstract theory empirical content: [Coordinating principles] serve as general rules for setting up a coordination or correspondence between the abstract mathematical representations lying at the basis of [the physics] . . . and concrete empirical phenomena to which these representations are intended to apply .... And without such general rules of coordination we simply have no idea what it means for concrete empirical phenomena to be described by the mathematical representations in question-either correctly or incorrectly. (Friedman 2001, 76-77) Friedman's paradigm of a coordinating principle is the light principle in special relativity, the principle that the speed of light is constant in any inertial reference frame. 2 Minkowski space-time by itself is merely an abstract mathematical forn1alism. The light principl~~~~~_cj~~~~_~~_~952~4ig~!~~_what----
Theory) Coordination) and Empirical Meaning in Modern Physics 425 are otherwise just abstract elements of Minkowski space-time with concrete empirical phenomena. In particular, the light principle coordinates the null straight lines of the former with possible paths of light rays in the latter, and the angle of those lines in a chosen space-time basis with the speed of light in a reference frame. When null straight lines are taken to correspond to possible paths of light, and coordinatizations 3 of the Minkowski formalism are taken to correspond to the coordinate systems of inertial reference frames, then the conformal structure of Minkowski space-time indicates that the speed of light will be constant and independent of inertial reference frame. That coordination then tells us in general how to apply the Minkowski structure to the empirical phenomena of observed distances and times in reference frames, or to translate the observed phenomena in a reference frame into Minkowski space-time terms. Until we do this, Minkowski space-time is not even a space-time and does not indicate anything about any phenomena, never mind specific phenomena concerning light, length contraction, or time dilation. 4 In the old Kantian picture, such coordination is not needed, as our experiences immediately and irrevocably occur in a particular a priori framework. In particular, Euclidean geometry is the only possible spatial form of our experience for Kant, and our experience of the world cannot at all be isolated from that form. Friedman rejects the necessity of Kant's particular Newtonian-Euclidean framework, but like the early logical positivists, he maintains that some choice is required, and he retains the idea that the a priori, once chosen, is constitutive of the objects described by theories (Friedman 2001, 2002). That is, the characteristics of the theory depend on the principles themselves, and so a conceptual framework establishes the form and nature of those objects in the framework. 5 Because of this constitutiveness, the objects of the framework, as objects ofthe framework, cannot be isolated from the form of the framework-theoretical objects are deeply embedded within a particular framework and inherit the structure of the framework of their embedding. That is, the things described by a theory, as theoretical objects conceived of and described by the theory, have structures and properties that depend on the framework involved. This constitutiveness thus rules out a naive reading of what it means to be "described by a theory" that would allow that one and the same object can be described differently by different theories. The objects and events of a classical space-time are different from those of a Minkowski space-time because the frameworks are different. 6 However, this constitutiveness cannot be all encompassing, for on Friedman's view we can choose among different coordination procedures that would coordinate the empirical phenomena of experiences to different frameworks. If we have a choice of the form of the scientific characterization of our experiences, then apparently there is something about
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those experiences that is independent of the form prior to our choice. So apparently we, unlike IZant, must reject the aspect of the a priori that makes the form and content of our experiences inseparable. 7 Coordinating principles (at least implicitly) must refer both to an abstract structure and to the "concrete empirical phenomena" of our experiences, as they provide the abstract structure empirical content by indicating how they are related. But if this is so, the empirical phenomena must be specified independently of that framework. If the phenomena were already constituted by the framework, the coordinating principles would not be necessary. If they were not able to be specified independent of the framework, the principles would be empty. The separability of form and empirical content suggests a difficulty not well handled on Friedman's picture. In the early discussions between Schlick, Reichenbach, and Cassirer on the idea of coordination (Zuordnung), there was much disagreement about what it is to which a priori structures are coordinated (see, e.g., Ryckman 2005, ch. 2 or Coffa 1991, ch. 10). In general, the idea was that the conceptual is coordinated with the physical world. But what is this world? Is it a pre-existing reality which becomes designated by our signs, some set of elements from our experience which becomes conceptualized through the framework, or some other sort of "undefined" which is only defined through coordination? Friedman's interest, like Reichenbach's, is not in the individual nature of cognition but rather in more general features of scientific epistemology. Friedman's idea may seem to be that the concrete empirical phenomena are something like our direct observations or perceptual experiences. For example, in one place he describes the coordinated as "concrete sensible experience" (Friedman 2001, 76). However, his actual examples are much more broad and intersubjective than direct personal sense experiences. One example, of course, is the phenomenon of the constancy of the speed of light; another concerns "the observable relative motions in the solar system" (Friedman 2001, 76-77). Both are indicative of Friedman's broader claim that the coordination of the relativized a priori concerns the non-personal, intersubjective facts of "concrete empirical phenomena." Like Reichenbach, Friedman therefore needs to account for the independence of the empirical phenomena in non-personal, intersubjective terms. However, this is a difficult task on Friedman's conception of the relativized a priori alone, and one I believe he has not yet addressed. I suggest that modifying his conception will help in this task. I will argue below that abstract theories gain empirical content through coordination to phenomena defined in other frameworks. To develop this view I will focus on special relativity, the empirical phenomena of which, I will argue, are based in a classical framework. I will then apply this perspective to quantum mechanics and suggest that the use of classical concepts in quantum
Theol'Y, Coordination, and Empirical Meaning in Modern Physics 427 mechanics is required and natural, even though the justification for their use is not as well developed as it is for relativity.
2. Theory-Ladenness and Other-Theory-Ladenness What, then, are the empirical phenomena to which abstract theories are coordinated? In particular, what are the phenomena of special relativity? One sometimes hears that the empirical or observational content of relativity is specified via definitions of space and time in terms of operations with rods and clocks. But operationalism is far too limited a picture to account for the empirical content of theoretical terms. Our concepts of distance and time play many roles in a wide range of experimental and theoretical settings. Corresponding to these many roles are many ways of measuring distance, time, and speed. For instance, we can measure the speed oflight by using rods and clocks, or (as we shall see below) through interference effects, or through any variety of other means. We can measure distance in a variety of ways that do not involve rods. And we can measure time with pendula, digital clocks, cesium atoms, quasars, or a variety of regular periodic phenomena. The empirical claims of relativity are independent of one's method of measurement: time dilation is not merely a slowing of clocks, but is a slowing of all time-dependent phenomena. The problem is that there does not appear to be a good operational justification for why we take all of these different ways of measuring distance and time to be measuring the same things (respectively), as it is not clear that they are at all operationally intertranslatable. But it is clear that many different operations are understood as measuring the same thing. Duhem expresses this idea very well: Suppose the following sentence is pronounced to a physicist: "if we increase the pressure by so many atmospheres, we increase the electromotive force of a battery by so many volts." It is indeed true that the initiated person who knows the theories of physics can translate this statement into facts and can do the experiment whose result is thus expressed, but the noteworthy point is that he can do it in an infinity of different ways. (Duhem [1906] 1991, 148-49) Duhem continues to say that one could, for instance, measure the change in the electromagnetic force through any of a variety of galvanometers, voltmeters, and so on. It is our scientific theories that tell us that these different operations measure the same things. Neither operationalism nor any pure observation language can provide the empirical content of a theory, as the empirical
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content of a theory is not restricted either to particular operations or to particular, direct, perceptual observations. Part of the reason for the fall of logical positivism was the growing consensus that observations in science are theory-laden and that there is no clear observational/theoretical distinction. So while there are many facts about the operations one can perform on light, and there are many different observations one could make in the process of measuring the speed of light, there is but one fact about the speed of light, and it is the latter fact that embodies the empirical content of special relativity. Now, theory-Iadenness often is thought of as a claim about the psychology of perception. As such, it is the claim that when we observe some phenomenon, what we actually see and how we literally perceive it depends on background theories and beliefs. One special class of cases are optical illusions, where two people (or the same person at different times) may receive exactly the same sensory input from a purposely ambiguous line drawing but see different things (e.g., an old woman and a young woman). The classical example of this sort of theory-ladenness of phenomena from IZuhn is the deck of cards that has a red six of spades and a black four of hearts (IZuhn [1962] 1996, 62). When shown these cards quickly in succession with other cards from the deck, people tend not to notice anything wrong because, the argument goes, we unconsciously and immediately interpret ambiguous sensory input in accordance with expectations determined by background "theories" about decks of cards. These examples and others like them are meant to show that our perceptual processes can be affected by many things, including our expectations about a particular setting. IZuhn went so far to say in Structure that a scientist used to seeing cloud chamber tracks "sees" a particle where the untrained eye sees only tracks ([1962] 1996, 195). However, there is a broader but related notion of theory-ladenness that concerns a broader notion of "observation" in a loose sense. 8 This notion may also be present in IZuhn's Structure, but it is not related to the actual psychological process of perception. It rather concerns the empirical aspect of theory investigation in the sense in which I interpreted Duhem's quotation above. In this sense, a scientist may make many "observations" in a lab that are only indirectly related to her perceptual systems and which involve much explicit interpretation of the workings of her instruments, inferences from other "background" theories, and so on. This sense of observation is meant to be loose specifically because it suggests that the en1pirical investigation of a phenomenon, even if for the explicit purpose of testing a theory, itself involves the conscious and deliberate use of much theory, so that the analogue of observation in this conception of empirical investigation is fairly disconnected from the actual specifics of any perceptual process.
Theory) Coordination) and Empirical Meaning in Modern Physics 429 For example, if a theory predicts that the current in a wire will have a particular value under certain circumstances, we can test the theory by "observing" the current. Of course, a scientist does not directly see the current, and she need not even see the ammeter being used to read off the current to get the result (perhaps she is read the values by a computer-generated voice). Independently of how she learns the value of the current, her "observations" of the current are being used to develop evidence for the theory. There are reasons for preferring this notion of "observation" in science, precisely because observation in the scientific setting ought to be intersubjective-something all potential participants would agree to-and direct perceptual sensation is just not amenable to that. Thus the term "observation" loosely refers to the recording of empirical facts thought to be relevant for a particular theory. The idea of theory-ladenness of observations in this sense of "observation" is the idea that the description and even conceptualization of the intersubjective results of an empirical investigation involve a range of background theories which we often talce to be unproblematic for the purposes of the experiment or measurement at hand but which nevertheless are not neutral. Theory-ladenness may pose a problem if the theory being invoked in the measurement of a phenomenon is the same one being investigated as an explanation of the phenomenon. Indeed, this is the alleged issue in paradigm non-commensurability. However, this problem is avoided if the theories involved in "observation" of phenomena need not be the theories being used to explain or predict those phenomena. Circularity problems can be avoided if observations are other-theory-laden. How are these issues relevant to Friedman's conception? This second sense of theory-ladenness is analogous to the idea that a conceptual framework is required for the conceptualization and description of empirical phenomena. In a footnote discussing the problems with operational definitions, Friedman indicates how coordinating principles, instead of operational definitions, can be used to define the basic concepts concerning measurements of empirical phenomena: [I]f we define the uniform passage of time by stipulating that some actual periodic process (such as the diurnal rotation of the earth) is uniform, this would be an operational definition of "equal times." Defining "equal times" by the laws of motion, by contrast, explicitly provides for the possibility of correcting and refining any and all such concrete coordinations without limit. (Friedman 2001, 77n6) Friedman suggests that there are benefits to using the more general definition via coordinating principles. The point for us is that the concept of
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equal time is here defined theoretically via the a priori principles of a framework; we might therefore say that the observations are not merely theoryladen, but are framework-laden. The potential problem with such framework-ladenness is that the abstract framework cannot be both the framework constituting the framework-laden phenomena and the abstract formalism that only has empirical content through coordination with the independent empirical phenomena. Consider the light principle. Friedman claims that this principle is an empirical fact, something we can discover about the empirical world, and that in special relativity we elevate this fact to a coordinating principle. As a fact we learn prior to the establishment of relativity, it is discoverable through empirical means independent of the formalism of special relativity. Because it claims that the speed of light is the same in any inertial reference frame, the concepts of speed and light must be understood independently of the new formalism. Reichenbach himself recognized the possible problem posed by theory-ladenness and realized that if his concept of coordination was to work, he needed an independent theoretical grounding of empirical observations. Reichenbach's method of successive approximations (Reichenbach [1920] 1965, [1924] 1969) suggested that actual observations in relativity are laden with pre-relativistic physics but that they are legitimized as approximations of the more accurate relativistic physics. The version of Reichenbach's views most relevant here occurs in his 1924 Axiomatization of the theory afrelativity, where he suggests that the thing coordinated is neither an immediate perception nor direct reality but rather itself a conceptual construct: The difficulty concerning coordinative definitions is similar to that concerning elementary facts: the physical thing that is coordinated is not an immediate perceptual experience but must be constructed from such experience by means of an interpretation. If I establish the coordinative definition "a light ray is a straight line," then the coordinated physical thing, the light ray, is a construction going beyond perception. (Reichenbach [1924] 1969,8) We coordinate not direct perceptual experience, but rather theory-laden constructions. These constructions go beyond experience by interpreting perceptions in light of a particular theory. This interpretation is an expression of the theory-ladenness of measurement, of which Reichenbach gives a nice brief characterization: We shall have to make use of the scientific theory itself in order to interpret the indications of our measuring instruments. Thus we shall not
Theory, Coordination, and Empirical Meaning in Modern Physics 431 say 'a pointer is moving' but 'the electric current is increasing.' (Reichenbach [1924] 1969,5) This interpretation requirement is important because it generates precisely our difficulty: how can the concrete empirical phenomena be constructs within a theoretical fi-amework and also be independent from the theoretical framework? Reichenbach's answer to this difficulty is that the theories or fi-ameworks involved in the two parts of the question are not the same; we use the old pre-relativistic framework as an approximation upon which we can base, through successive approximations, the new theory: All axioms of our presentation have been chosen in such a way that they can be derived from the experiments by means of pre-relativistic optics and mechanics. All are facts that can be tested without the use of the theory of relativity. In particular, they are all formulated without the use of the concept of simultaneity at distant points ... The particular factual statements of the theory of relativity can all be grasped by means of pre-relativistic conceptions; only their combination within the conceptual system of a theory is new. (Reichenbach [1924] 1969, 6-7, emphasis original) Such a perspective would indeed address our problem, and I suggest that it is in part correct. However, the view I develop below differs significantly from Reichenbach's in a number of ways, primarily regarding the questions of whether such use of pre-relativistic physics might be avoidable and how it ought to affect our interpretation of the theory. 9
3. Layers of Frameworks One of the most interesting aspects of Friedman's relativized a priori is the idea that intersubjective and precise descriptions in physical theories can only be given in the context of a presumed conceptual fi-amework. However, I suggested above that if the concrete empirical phenomena themselves play such a significant role in specifYing the empirical meaning of our modern theories' abstract mathematical structures, then the phenomena themselves must be intersubjective and independent of the theories and their chosen fi-ameworks. My suggestion is that concrete phenomena do require their own conceptual fi-amework; the "same" phenomenon can be measured and specified in different ways, so it is not quite "concrete," but rather is conceptualized. However, the concrete phenomena can depend on a fi-amework different fi-om the one to which they are coordinated. Coordination would
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then be a process of establishing a relationship between two different frameworks. I suggest that by amending Friedman's picture in this way we can both accept his fundamental insights and account for the independence that I have suggested is needed for coordination. These suggestions bear similarities to Bogen and Woodward's suggestion that theories make empirical predictions not about particular data points but rather about theory-laden phenomena such as the melting point of lead, or the existence of neutral currents (Bogen and Woodward 1988). On their view, a theory does not explain particular data points of an experiment (which are due to contingent and varied facts of the particulars of the situation) but only explains the experimental phenomenon. While I believe there may be some issues with their account (e.g., whether or not the theory can thereby also explain particular data points depends on one's idea of explanation), my account is not about explanation, and I do believe that often the best way to think about the empirical content of theories is in terms of their relation to entire phenomena. It is therefore worth considering how Bogen and Woodward's view applies to the account I am proposing. Bogen and Woodward's view is most easily understood in one of two ways. One way is to suggest that phenomena just are patterns in data, in which case they may be just as much a feature of the data points of a particular experiment as are the data points themselves.l° The other is to suggest that they are real features of the world, independent of whatever is done in a particular experiment or measurement. I suggest a third view: phenomena are theoretical constructs that can be specified only within a conceptual framework. Different frameworks can construe the "same phenomena" in quite different mathematical and theoretical ways. Here I borrow again from Duhem, who says that "the result of an experiment in physics is an abstract and symbolic judgment": What is the value of the volume occupied by the gas, what is the value of the pressure it supports, what is the degree of temperature to which it is brought? Are they three concrete objects? No, they are three abstract symbols which only physical theory connects to the facts really observed. (Duhem [1906] 1991, 146) The idea here is also what is dictated by Friedman's retention of the constitutiveness of the a priori-the objects of a theory are not concrete physical objects but framework-dependent representations of them. This applies just as well to the phenomena explained by a theory, as the objects of the phenomena are objects of some theory or other. This third view suggests that at least for certain physical theories, the en1pirical content comes through the theory's connection not just to a par-
Theory, Coordination, and Empirical Meaning in Modern Physics 433 ticular phenomenon, but rather to the entire conceptual framework of that phenomenon. l l As I will argue below, special relativity has empirical content through the connection between elements of Minkowski space-time and the classical space and time of an empirical inertial reference frame. Predictions about a particular experiment or measurement are then determined both by how the theory coordinates its formalism to the framework of the phenomenon as well as by what the theories of the framework of the phenomenon say about the particular situation. For example, special relativity dictates that the speed of light will be constant in any direction; classical, non-relativistic theories can then tell you how this will be evident in the particulars of, say, a Michelson-Morley setting, by the lack of changes in interference fringes (more on this below). Another example worth considering is the way Bohr's model of the atom fl.·om early quantum theory accounted for the atomic spectra.l 2 The model accounted for the Balmer series, which is a phenomenon concerning the patterns in the frequencies oflight emitted from excited hydrogen atoms. Because frequency is a wave concept, the predictions of Bohr's model therefore concerned properties of light according to a particular theoretical conception of light-the wave theory-and the conceptual framework necessary for the formulation of that theory. The model's empirical content depended on the way transitions between stationary states were associated with emitted frequencies of light. On this view, the Balmer series and other such series are the concrete empirical phenomena for the model. These phenomena are not simply directly observable but themselves are theoretical in nature. They are wave phenomena; they are regular patterns in the frequencies of light emitted by excited hydrogen atoms. They are phenomena only when conceptualized that way; if conceptualized another way, there may not be any such patterns at all (recall the choice between universal forces or a curved geometry). That is, their description and our understanding of them make sense only in a particular conceptual framework concerning the wave theory oflight. Because the model's empirical content concerns emitted frequencies of light, it assumes that framework for the phenomena. Thus any method of measuring frequency within that framework is an appropriate measurement of the atomic spectra accounted for by the model. In suggesting these ideas I am not merely describing the role background theories and auxiliary hypotheses play in deriving observations. The fact that one needs additional background theories to make predictions is consistent with the idea that the observations could be described in a pure observational language, which I deny. The point I am making is that the place where the background theories come in is important. On this view the observations are themselves theoretical in nature, and so one cannot make claims about these observations without invoking
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the theories and foundations of those phenomena. In other words, the idea is not merely that we need auxiliary hypotheses and background theory to help us derive statements about observations. Rather, the background theory constitutes the empirical phenomena and establishes the theoretical language that describe them, and it is an assumption of the new theory that the observations are objects of the "background" theory. A new, abstract theory has empirical content via a coordination to this background theory and framework; it can thus make claims about the phenomena of that framework via the chosen coordinating principles. Now, even though coordination is between frameworks, the relationship is not symmetric. Rather, one framework is taken as a given for the other. The former is assumed to appropriately capture the aspects of the phenomena used as the empirical foundation for the other. Of course, that framework itselfwill not account for all the aspects of the phenomena (e.g., the Balmer series, or time dilation), but it does define the things involved in the phenomena (e.g., frequencies of light, distance and time) and thus serves as the theoretical space in which the empirical phenomena find their home. On this view some such conceptual frameworks are used to define the empirical content of others. What makes a framework empirical, then? I suggest that there is no real difference in kind between the two and that whether or not a framework is empirical is relative. However, ideally this use of one framework as the empirical foundation for another is justified in some way, perhaps because the one is somehow closer to "observable" in the perceptual sense. I will not try to give any real account of what makes a certain framework empirical for another here, although I will consider the particulars of the case for special relativity below. Now, the conceptual frameworks of this discussion, even the "empirical" ones, all involve abstract structures of geometry and laws of motion as well as dynan1ical theories. But these are merely abstract theoretical formalisms if they are not grounded to empirical phenomena. An empirical framework is just a conceptual framework being used to define thestructure of the phenomena for some other theoretical framework, so independent of this other one, the story will be the same as what is told for any-e.g., it requires coordination principles itself. That is why being empirical is relative. I call this view a framework -layered (or theory-layered) conception of empirical content: Connecting an abstract theory to actual observations involves a series of layers of theoretical frameworks. The relationship between theory and observations is mediated by other theories such that there are layers of theories or frameworks involved in connecting a theory to observations.1 3
The01')', Coordination, and Empirical Meaning in Modern Physics 435 I believe this conception of framework-layering to be potentially fruitful in physical theories in general. However, here I will focus mainly on its application to special relativity and then, below, to quantum mechanics.
4. Non-Relativistic Concepts in Special Relativity I argue that for special relativity to have empirical meaning, its phenomena must be understood within the context of their own, independent conceptual framework. Without an independent framework to indicate the empirical (and not the abstract structural-theoretic) meanings of light, distance, and time, the light principle would not be able to provide empirical content to relativistic space-time.l 4 The relativized a priori of special relativity is not all-encompassing and is not constitutive of either the empirical phenomena or our experiences of them. Both of these occur from the perspective of a particular frame of reference, and within that perspective we have an independent notion of space and time, what it means to make a measurement of the speed of light, and so on, independent of relativity. I therefore suggest that the form of Minkowski space-time does not concern the form and content of our experience and observations within a reference frame. From the perspective of special relativity, the objects of physics (the theoretical and mathematical abstract objects of the theory) exist in the framework of relativistic space and time; the objects of "experience," which constitute our observations (relatively speaking, given the above caveats), exist in a pre-defined, pre-relativistic framework. What is this independent framework? Consider first the way we established the empirical fact that the speed of light is constant. Although the Michelson-Morley experiment originally was meant to measure the speed of earth relative to the ether, we now treat it as having established the constancy of the speed oflight. The speed oflight was thought to be constant in the ether frame, and the earth had to be moving relative to the ether in one direction or another. To try to measure the speed of the ether wind relative to the earth, Michelson and Morley set up an apparatus for measuring differences in the speed of light in two different directions. The apparatus split a beam of light and sent it along two perpendicular arms with mirrors at each end, so that the beams were reflected back and then recombined. Because of the presumed motion of the earth relative to the ether, there would be some orientation of the apparatus (which could rotate) in which light along one arm would be traveling with and then against the ether "wind" while the other would be traveling perpendicular to that wind. In this orientation, light traveling along the first arm would be expected to take longer to travel the round-trip than would light on the second, and the difference in the time it would take for the light to travel
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the distances along the two directions would show up as a phase difference measurable via interferometry. If the apparatus were rotated, then at some orientation the apparatus would be in this state of maximum difference between the time to make the one round-trip and the time to make the other. At some other orientation there would be no difference between the times. Michelson and Morley looked for changes in the interference pattern as the apparatus rotated, and found no such change. This lack of change in the interference pattern indicated that the speed of light along the two arms did not change when the orientation changed, and so indicated that the speed of light is the same in any direction in the lab fran1e. 15 This empirical phenomenon poses a problem for the ether theory if one uses the classical Galilean transformation rules that translate the velocity in the ether frame to the apparent velocity in the reference fran1e moving relative to the ether by adding or subtracting velocities. On the basis of these rules, the velocity of light relative to the lab frame ought to equal its velocity in the ether minus the velocity of the earth relative to the ether. Maxwellian ether theory says that the speed of light is c within the Maxwellian ether rest frame. Under the assumption that the Earth is moving relative to the ether in some direction, the speeds as measured in our reference fran1e ought not be the same in most orientations of the apparatus in the lab frame, and there would be some difference in the interference pattern as the apparatus turned. But the empirical phenomenon is that the speeds are the same. Special relativity accounts for this phenomenon by dropping the Galilean velocity addition rules and introducing the Lorentz transformations for calculating frame-dependent distances and time in different inertial frames. This change establishes by stipulation the validity of Maxwell's equations within any frame (not merely the ether frame of rest). It also drastically alters our conceptions of space and time outside our reference frame. The Lorentz transformations indicate that the distance and time between two events as measured in one inertial reference frame will in general differ from the distance and time measured in a different one. According to special relativity, there is no global time and no global space because there is no global simultaneity relation. Instead, we have a global four-dimensional manifold of space-time events upon which our referenceframe-dependent simultaneity relations are projected. There is a welldefined and invariant space-time interval between any two events, and it is, for example, an invariant feature ofspace-time whether lines between those events are time-like, space-like, or null. But the distance or time between two events is reference-frame-dependent. As a consequence, certain features of relativistic space-time become difficult to interpret; in particular, they are difficult to understand in terms of common intuitive ideas about
Theory) Coordination) and Empirical Meaning in Modern Physics 437 space and time (for instance, those concerning whether the future is undetennined) because space and time have no well-defined global meaning. Because special relativity tells us that the world is not really as we thought, one might question the degree to which the independent concepts of space and time apply at all to the world according to relativity. However, despite the fact that the concepts of space and time do not apply globally, they remain valid concepts within an inertial reference frame. Space and time have not been dropped in relativity. There remain robust notions of "proper" time for and "proper" length of an object that correspond to the time and space of the inertial reference frame in which it is at rest; indeed, the proper length can in some respects be understood as the "real" length of an object. I argue not only that the empirical content of special relativity relies on our having independent conceptions of space and time that apply within an empirical reference frame but also that special relativity takes them for granted and does not explain what they are or why they apply.16 One of the most important things we hold onto in these empirical descriptions is the distinction between space and time. Indeed, anyempirical phenomenon in a reference frame is a phenomenon in Euclidean threedimensional space plus time. Of course, special relativity tells us that this space and time is only a particular decomposition of space-time into individual space and time components. Space-time itself has no global simultaneity that would allow us to apply this particular perspective globally. Nevertheless, the phenomena of the reference frame do occur with a particular such perspective. Moreover, it seems we cannot avoid this situation. First, we arguably have no way of measuring the space-time interval between two events without first measuring the distance and time between those events with respect to a particular reference frame. All our knowledge of the relationships between space-time events comes via a particular space-time perspective. Second, the nature of relativity is that its empirical content (its predictions, etc.) concerns precisely the individual space and time perspectives of different reference frames. Indeed, that is exactly the point of Lorentz transformations. Special relativity assumes that all the objects at rest in an inertial reference frame share the same common perspective of space and time. It can justify this assumption to some degree, as the Lorentz transformations return the same values of distance and time for two objects at rest with respect to one another. Yet it is also part of the fundamental assumptions of a reference frame that this is true. Moreover, there is more to a reference frame than just the objects at rest with respect to one another. The novelty of relativity concerns what it tells us about the application of the reference-frame-dependent perspective of space and time to moving
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objects. An object moving relative to that reference frame "experiences" a different perspective of space and time than an object at rest in that reference frame. But the space and time of the objects at rest in that frame also apply to the n10ving object with respect to that frame, and the Lorentz transformations tell us how observers in motion with respect to one another disagree on simultaneity, elapsed times between events, and the lengths of objects both at rest and in motion with respect to those frames. However, relativity does not tell us why the space and time of the objects at rest apply to objects moving in that frame as well-it just assumes that they do. Part of what it means to be a reference frame (inertial or not) is that it establishes distinct space and time dimensions that are applied to all objects, moving or not, from the perspective of that frame. Relativity just tells us how the distinct space and time perspectives of different inertial reference frames are related.
5. Non-Relativistic Framework$ in Reference Frames What does it mean for there to be a distinct space and time of a reference frame but not of the global world? That is one of the interesting questions to be asked in the interpretation of relativity, and one without a straightforward answer. While relativity gives a mathematical account of these relations, it is less clear what ontological account we can give of them. Central to the topic is the nature of inertial reference frames. In Minkowski spacetime, an inertial reference frame is associated with a coordinatization of all space-time, so at least in that abstract representation, it is infinitely extended. But reference fran1es are also concrete and empirical, picked out by actual, concrete objects that serve as reference points. As such they are tied to something much more local in nature-not necessarily a set of rods and clocks, but something like an actual laboratory, a train or rocket ship, or even an entire planet. We therefore need to distinguish a reference fran1e itself from both the concrete objects with respect to which that frame is established and the framework of that frame (and, in particular, the coordinate system of that frame). Moreover, we need to distinguish all of these from a particular coordinatization of Minkowski space-time which we may take to correspond to them. I am suggesting that the abstract structure of Minkowski space-tin1e really only applies to empirical space and time via the coordination between it and the independent fran1ework of empirical phenomena. This occurs through the establishment of a correspondence between substructures of Minkowski space-time and reference frames. Note that these reference frames are not merely the coordinatizations in Minkowski space-time, as it
Theory, Coordination, and Empirical Meaning in Modern Physics 439 is the latter to which the former are coordinated. So what are the former? The reference frame is a particular point of view, picked out by a particular object. In that sense, it may be concrete and unconceptualized, but in its role of providing for the measurement of position and time, it becomes an independent conceptualization of the wayan object establishes a perspective of space and time with respect to that object, and includes therefore not only the geometrical conception of classical space and time of a frame (i.e., relative to an object) but also the physics required for establishing measurements of those things. A coordinatization in Minkowski space-time does not provide all these. Special relativity does not define what space and time are in a reference frame. The light principle may on some views provide the canonical measure of distance and time of distant events-we can send a signal out and back, assume the speed is the same in both directions so that the event of hitting the other object occurs halfWay through the process, and determine the distance to the object at that time based on the speed oflight. 17 Yet it can serve this role only with an already presumed notion of time. What it means to say that the light principle coordinates abstract structure with empirical phenomena is that that measure defined through this procedure is a measure of distance, so that all our other ways of measuring distance, etc., correspond to that one. Thus the light principle tells us how to correctly apply (or translate) our independent notions of space and time, especially with respect to moving bodies. The space and time of a reference frame are then associated with the theory when the coordinates of the reference frame are put into correspondence with a particular coordinatization of Minkowski space-time. The difficulty with taking special relativity to define a new notion of distance can be seen by considering the question of how we fix a notion of inertial reference frames. One way of viewing relativity is that inertial frames are now defined by the light principle. Since the light principle dictates that the speed of light is constant in any inertial reference frame, it tells us that any reference frame in which the speed of light is not constant is not inertial. But the constancy of the speed of light can only fix a notion of inertial reference frame if we have prior notions of space and time, and, in particular, a prior understanding of reference frames, so we must have an independent conception of and ability to apply space-time notions. IS Note then that if the light principle is used to define distance, it can no longer define inertial frames. On this view, the light principle tells us which reference frames are inertial, and then we can associate each of their spaceplus-time frameworks with a coordinatization of Minkowski space-time. I suggest that this prior, independent account of space and time in a reference frame is carried forward from our pre-relativistic conceptions. Consider the theories involved in the Michelson-Morley experiment and
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the experiment's interpreted "null" result. While Michelson and Morley certainly had direct perceptual observation experiences while performing their experiment, their results did not concern those particular, direct experiences. Rather, those direct, phenomenal experiences were interpreted as "observations" concerning the fringes in the interference pattern and ultimately to the "observation"-the empirical phenomenon-that the speed of light is the same in any direction in the reference frame of the lab, stationary on the earth. I suggest that it is these "observations" that we think of as the experimental result, and the generalization of these observations that the speed of light is independent of an observer's motion is what is elevated to a coordinating principle on Friedman's account. Michelson and Morley assumed a three-dimensional Euclidean space and a distinct tin1e dimension, as well as good chunks of wave theory and plenty of other classical physical conceptions. These theoretical tools together form a framework in Friedman's sense: there is a chosen geometry of space and time, a Newtonian kinematics on top of the geometry, particular theories of light propagation based on these other foundations, and so on. Michelson and Morley's result that the speed of light is constant cannot be inferred and does not even have its intended meaning outside the conceptual framework in which speed, light, etc., are all understood in classical and wave-theoretic terms. To work as a coordinating principle, the light principle must retain the pre-relativistic framework for the description of empirical phenomena. I suggest, then, that even after the development of relativity, we cannot abandon the pre-relativistic framework. One could replicate the Michelson-Morley experiment today and use entirely the same theoretical inferences to interpret the experiment. Of course, the point of the experiment would not be the same, since with relativity the speed of light is postulated to be the same. If the experiment turned out otherwise, one would conclude either that there was some mistal<:e in the execution or interpretation of the experiment, or that the lab was not moving inertially. While relativity introduces many changes to space-time physics, it does not change the framework of our practice in the lab. It does not even really change what we assume in order to measure the duration of a process (e.g., the duration of the life of a particle) from the perspective of one's reference frame. It does not even really change how we measure lengths of objects, even moving ones. Normal measurements of length for either a moving or stationary object will, if taken correctly, produce the appropriate length for that reference frame. Relativity does not alter these measurements but rather reconceives them and restricts their range of applicability; it allows us to translate the measurements of an empirical phenomenon into what is, according to the theory, the appropriate physics for the phenomenon in another frame (e.g., in order to predict a moving particle's
Theory, Coordination, and Empirical Meaning in Modern Physics 441 half life, we might need to translate between the observed time in one reference frame and the proper time in the particle's rest frame). Here we have to be careful. I am suggesting that our descriptions and conceptualizations of empirical phenomena (which are both abstract and close to our experiences) are embedded in a pre-relativistic framework of three-dimensional space plus time. However, I am suggesting that that framework accounts only for the structure of the phenomena, not the phenomena themselves. That is, I am not at all suggesting that the old framework can explain the constancy of the speed of light, time dilation, length contraction, or the velocity-dependence of mass. These features of the phenomena of our reference frames are accounted for by the way the structure of the phenomena is coordinated with the structure of the theory of special relativity, but the fact that it is only the new theory that accounts for the phenomena does not change the fact that the phenomena remain phenomena in space and time. Even the most famous example of special relativity discussions-the twin paradox-does not pose a challenge to the fundamental use of prerelativistic physics within a reference frame. 19 If two watches are synchro- . nized and one is subsequently sent on a roundtrip ride to Jupiter at speeds near the speed of light, the watches will no longer be synchronized. While there are some issues with treating this phenomenon in the context of special relativity because of the acceleration involved, it (or, rather, its real-life analogue with supersonic jets, etc.) is still seen as evidence for special relativity. Now, it may seem that this example shows us that we do in fact have to reject classical notions of time and space. Mter all, if we are doing experiments with objects that are moving, or even if any part of our apparatus is moving, it seems that we do have to change the way we measure duration, because we have to worry about relativistic effects. But all we can measure is the reference frame perspective. The watch on the rocket ship provides a measure of the time of the ship's reference frame, not of the time in our reference frame. There are various equally good measures of the duration of the flight; one measure of the duration is from the perspective of the reference frame on earth, another is from the perspective of the other frame. Of course, the numbers on the watch in the end are observed from our reference frame on earth; however, those numbers are not a record of the time elapsed in our reference frame. Indeed, this example shows the danger of thinking of time as being defined by clocks. The physical processes governing the watch in the rocket ship can be specified in either frame, and the time of either frame is central to either specification. To account for what the watch will display, one needs to ensure that one determines the watch's proper time either by specifYing those processes in the watch's reference frame or by converting the specification of those processes with the Lorentz transformation.
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Although there are frame-independent objects in Minkowski spacetime, any specification of a physical process involves a reference frame. All special relativity has to do in order to make empirical claims, and arguably all it does do, is to indicate how the space and time of a reference frame is related to the space and time of others by saying how it relates to Minkowski space-time. Any predictions about the phenomena in a reference frame's space and time can then be inferred on the basis of the reference frame framework. It does not matter how distances or time are measured, as long as the measurement itself does not involve inferences from information from other reference frames. Even special relativistic effects evident in these distances and times do not depend on how they were measured; the Lorentz transformations do not explain only certain ways of measuring distance or time (e.g., with rods and clocks). Any measure of time (according to the classical theories) will show evidence of time dilation. The relationship is similar to the one described by Duhem-a new theory may account for the current in a wire, and that account will be the same regardless of how the current is measured, as long as those measurements share the same theoretical framework for understanding current. I conclude that while special relativity does of course predict phenomena that are very different from the predictions of classical theory, those phenomena are still phenomena of three-dimensional space and time that are understood and measured, within a reference frame, in ways continuous with our understanding and measurements before relativity. Of course that understanding, and the results of those measurements, are limited in applicability and restricted to a reference frame. But within a reference frame, they remain perfectly good, and so it seems appropriate to call the framework of empirical phenomena pre-relativistic or classical.
6. Reichenbach and Approximations On this view, the use of pre-relativistic physics to give independent descriptions for coordination is not merely a temporary measure that can be overcome after the establishment of the new theory. That is, I am not merely claiming that classical physics works fine as an approximation of the new, correct description of the empirical phenomena, which we could use at least in theory, and which we ought to use if we really want to be more precise. By contrast, Reichenbach's method of successive approximation, which is similar in some respects to this view, seems to imply that while we do use approximate, non-relativistic descriptions of phenomena as the things to which we coordinate relativistic structure, we can still say that the correct account of the phenomena is a purely relativistic one. Reichenbach says that we can eliminate the problem of initially not having independent
Theory, Coordination, and Empirical Meaning in Modern Physics 443 relata for the coordination by using "coordinative definitions whose degree is not important and which, in particular, do not make use of relativistic definitions" (Reichenbach [1924] 1969,8). The idea is that while the classical theories with which we describe empirical phenomena are strictly false, they are good enough for the purposes of coordination. His view seems to be that using the old theory will not hurt: Let us assume that the theory by means of which we explain a certain fact is false and is to be replaced by a different one. It is nevertheless possible that the new theory, when used for the interpretation of this one fact, makes hardly any difference quantitatively, whereas it leads to considerable changes with respect to other assertions. For example, whatever assumptions may be made about the motion oflight in strong gravitational fields, its motion within the telescope will in practice be assumed to be always the same. (Reichenbach [1924] 1969,6) The differences between Reichenbach's view and the one I am proposing here consist mainly in the following four points. First, Reichenbach's justification seems to be that we could use the new theory to describe the empirical phenomena without relying on the old theory. But special relativity has no empirical content without those prerelativistic descriptions, so to say that we could do without them is just wrong. Second, Reichenbach's theory-ladenness is rather thin in a number of respects. 20 First, he considers the only required independent theories to be pre-relativistic optics and mechanics. As I note above, to capture the full generality of our practice we need to admit more into our empirical descriptions, including wave optics in pre-relativistic Euclidean space and time (constrained to the reference frame). I also recognize that the pre-relativistic theoretical structures themselves are abstract and need to be coordinated with something. So not only is the empirical layer thick with the structure of Friedman's conceptual framework, but it requires its own empirical phenomena, which are presumably closer to direct experience. Third, while Reichenbach draws attention to the theory-ladenness of observations, he nevertheless continues to treat the content of observations as relatively unproblematic. The meaning and import of the use of pre-relativistic theories is not particularly relevant to the meaning of relativity. In this respect, the theoretical baggage of our observations is quite ancillary, and we could imagine the new theory with or without the use of that previous theory. I want to argue, on the contrary, that the empirical content of our theory depends not only on the coordination principles but also on the form of the observational or empirical framework. Indeed, there is a sense in which the empirical meaning of the theory is derivative
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of the theoretical conceptions in the empirical framework, as the empirical content of the theory concerns how the theory relates to these other theoretical concepts and the phenomena they describe. Last, Reichenbach strongly contends that the objects described by relativity and the observation-theories are one and the san1e, that the light of relativity and the light of pre-relativistic ray optics are identical. I am less interested in starting with an analysis of the identity of types of objects and would rather focus on our theoretical descriptions of them. If one follows the neo-IZantian line that foundational principles are constitutive of objects, then coordinating principles in a sense connect different theoretical worlds. I believe that our first interpretive task is to analyze the status of and the relationship between the two theoretical frameworks and only then ask whether there is a way we can say that the concepts in each refer to the same objects or the same properties of the same objects. In the final analysis, Reichenbach's view does not account for the significantly different epistemological status of the two relata of the coordination relation and does not fully account for the necessity of the independent empirical description. On the perspective I have been suggesting, every relativistic description of a phenomenon has empirical content just in virtue of the pre-relativistic framework of referenceframe-dependent space and time. Although the pre-relativistic framework of the empirical phenomena in a reference frame is abstract, its concepts can arguably be made to line up with something along the lines of our actual experience of distance and time in a way that Minkowski space-time in the abstract cannot. In general, one would hope the one framework is "closer" in some sense to the actual, concrete, empirical foundations of science, whether they be experiences or not. But it is also possible that this dependence of one framework on another is merely contingent-due perhaps only to the fact that the one was simply developed prior in time and thus became well accepted so as to remain the basic empirical framework. However, even without a general account for why one framework can count as empirical relative to another, it remains that a framework requires another for coordination, and that an abstract framework, including the one for relativity, has empirical meaning only through coordination with the framework assumed for some set of empirical phenomena.
7 . Understanding Relativity with Layers This perspective suggests that in interpreting special relativity we cannot ignore the role classical space and time, as conceived of and used in classical physics, play in providing the empirical content of relativistic spacetime. Classical Newtonian space-time (Galiliean tran_s_~~~ll:"l~~9}}~ __ witb.
_
Theory, Coordination, and Empirical Meaning in Modern Physics 445 absolute space )21 allowed one to think of the space of a reference frame as intimately tied to the absolute space, as each inertial reference frame is isomorphic to absolute space and they all share the same simultaneity relation. Therefore, one could think of the space of the inertial reference frame as embedded in the absolute space, and the time of the inertial frame as the same time as the time of the absolute space. While absolute space might not play any role in direct observations, it is an important feature of this space-time that the structure of the space of each reference frame is somehow "the same" as the structure of the absolute structure. Even with Galilean space-time, which keeps the Galiliean transformations but drops absolute space, whatever interpretation ofspace and time one wants to give can be applied equally well to all frames, and there is no other object besides the frames requiring interpretation. But with special relativity, the separate space and time of a reference frame are associated with a particular coordinatization of Minkowski space-time. While the space and time coordinates of a reference frame can be put into correspondence with the latter, the empirical space and time within which we make measurements is structurally quite different from relativistic space-time. So if Minkowski space-time is a reflection of the way things fundamentally are, our empirical experiences do not reflect fundamental reality. But since relativity relies on the classical notions of space and time as the framework for the phenomena, it would be wrong to say that those experiences do not reflect an aspect of reality; they reflect features of the empirical phenomena in reference frame, which, even if not fundamental, nevertheless are real and in fact are brute features of the theory. In a sense then, relativity is really a theory about the space and time that, while itself abstract, is closely tied to the space and time of our experiences of the old-fashioned classical space-pIus-time kind. If the empirical content of relativity is explicitly concerned with pre-relativistic notions of space and time, then this is an important feature of the theory which we need to consider when we interpret what it says about space and time. 22 On the view I am presenting, the new theory relies on our pre-relativistic conceptions as part of the framework for the empirical phenomena. Within the applicability of the framework, there is continuity in both their theoretical definitions and in their use. In particular, relativity does not explain what space and time are but rather relies on the pre-relativistic theoretical conceptions that allow us to measure distance, duration, and speed in a variety of ways. However, relativity also restricts the applicability of the framework. By restricting the use of these concepts to an inertial reference frame, which provides only one perspective of space-time, it also reconceives them. Pre-relativistic space and time are now something akin to secondary properties. They are the way space-time appears to us in reference
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frames; they are the features of the reference-frame-dependent phenomena, but they are not as the world really is. 23
8. Quantum Mechanics I believe that the perspective I have been suggesting is not peculiar to special relativity but applies more generally, at least in physics. Here I will explore what this perspective might imply for the interpretation of quantum mechanics, and I suggest that from this perspective, certain problems of interpretation seem less problematic than is often assumed. Now, while the case for quantum mechanics is more complicated than the one for relativity, I will argue that both rely on classical concepts and that the difference between them is related to the nature and the justification of our use of those concepts. I will only sketch the outline of an analysis here, but I think it will suffice to show that the perspective is potentially fruitful. I believe that some of the difficulties in the interpretation of quantum mechanics have been complicated by treating too naively the relationship between operators and empirical phenomena (see Tanona 2004b, cf. Daumer et al. 1996). By thinking that operators non-problematically represent observable properties of systems, "ve miss the opportunity to understand how the theoretical structures gain empirical content by analyzing how they are coordinated with empirical phenomena. I suggest that applying the view above to quantum n1echanics might provide motivation for adopting an updated Copenhagen interpretation along the following lines: 24 An independent and basically classical framework is needed for the description of measurement results, because without such an independent framework for the phenomena, quantum mechanics is just an empty forn1alism. We have independent pre-quantum mechanical concepts that we use for describing measurements, but their use is restricted to a measurement context, and we can coordinate quantum mechanical operators (thought to represent something, but not necessarily classical properties) with these measurement-context-dependent pre-quantum mechanical concepts. "Having" a precise property such as position means that the pre-quantum framework of the position measurement context can be precisely coordinated with some substructures of the quantum formalism; in particular, the now measurement-context-dependent concept of position can be precisely coordinated with an appropriate quantum operator. Non-commutativity indicates that this coordination can only be done with some subset of operators at a time, and which ones (e.g., position vs. momentum) can be coordinated will depend on the measuren1ent context. The formalism allows us to translate our measured results to inferences about other properties only if the corresponding operators comn1ute.
Theory) Coordination) and Empirical Meaning in Modern Physics 447 According to this view, we might say that the classically-based conceptions of properties we use during measurement merely reflect appearances-the phenomena-and not fundamental quantum mechanical properties. On this view, properties like position are real, but they are real only as phenomena in measurement contexts and are only related to or coordinated with the fundamentally real quantum structures behind the phenomena. It is a mistake to attribute that concept of position to underlying reality, but it is also a mistake to think we can do without it. Thus the fact that we can assign a precise position at one time and not another is not an indication that a system sometimes does and sometimes does not have a precise quantum property of position, but rather is an indication of the way quantum mechanical features are reflected in our mostly classical empirical perspective of them. I suggest that this view is not very different from the one described above for special relativity. Just as we make a distinction in special relativity between reference-frame-dependent space and what we might take to be the "real," relativistic space-time, we ought to make a distinction in quantum theory between measurement-context-dependent notions and the "real" quantum mechanical features of a system. As special relativity relates framework-laden observations in a reference frame to the Minkowski space-time structure "behind" the phenomena, quantum mechanics relates framework-laden observations in a measurement context to the quantum mechanical structure "behind" the phenomena. I suggest that momentum, position, and the other "observables" of a quantum system all have other-theory-laden empirical meanings in their frameworks similar to the ones they had before the development of quantum mechanics. That is, quantum mechanics does not replace those concepts but rather coordinates its new structure with them, and because the classical frameworks of these concepts are retained as the framework of the phenomena, the quantum mechanical formalism gains empirical content through its coordination with these frameworks. This coordination tells us what it means empirically for a quantum mechanical operator to correspond to a particular value. Like the pre-relativistic concepts in relativity, the prequantum concepts are reconceived and restricted in applicability, and we can explain new phenomena of position, momentum, etc., that one could not have even predicted previously (e.g., the uncertainty relations ).25 The above applies not only to properties like position, but even to the quintessential quantum property of spin. The paradigm measurement of spin is made with Stern-Gerlach magnets, which allow us to measure a spin magnetic moment the same way we would any classical magnetic moment. Our equations for the measurement are structurally similar to the classical ones. We first determine the properties of the instrumentation used, and, by characterizing the effects of the magnets in terms of a classical magnetic
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field gradient, we correlate the change in n10mentum of the particle wit its magnetic moment. Then the results are interpreted in an essentially classical way: the force of the deflection is correlated with the magnetic moment because the change in momentum depends on the magnetic mOlnent, the gradient of the magnetic field, and the angle between them. The actual spatial angle of the deflection depends on the mass and velocity of the particle, and the deviation fron1 center will depend on the time elapsed after passing through the magnets. That our account is then quantized accounts for the new fact that there are only a limited selection of possible spin magnetic moments, but it does not change our description of the results in terms of the particle's having one of those magnetic moments (at least in the measurement context), and does not change what it is to measure magnetic moment. 26 To indicate how spin is measured, one must say how it is to be connected to actual measurement possibilities. One might simply claim that, e.g., a measurement correlates the position of a pointer on some instrument with the spin, but that story obscures the fact that our characterization of the experiment is that it is a measurement of magnetic moment, in its pre-quantum sense. So the empirical content related to the spin operator comes through its coordination with magnetic moment. Without such a coordination with an independent classical context, the abstract quantum mechanical descriptions (which allow us to describe superpositions and entanglement) are empirically meaningless. Quantum mechanics strictly applies only to closed systems, but to malce a measurement we must provide an external, independent phenomenon to which we can coordinate the "observable" being measured, and this indicates why we turn to pre-quantum concepts to provide the boundary, external conditions for measurement. Understanding that we can reconceive our concepts and use them this way forestalls the questions that lead to interpretational difficulties-questions such as how a system can be in a superposition of observable properties or how it can possess a fundan1ental property at one time and not another. By accepting the role of classical concepts as descriptions of empirical phenomena, we more or less accept without debate one of the fundamental characterizations of the measurement problem: despite the changes in our notions of properties which quantum mechanics introduces, our experiences-and the basis of the empirical phenomenaremain essentially classical. Instead of seeming deeply problematic, this situation now appears somewhat natural. Of course, we will want an account of the relationship between these different frameworks, and there certainly are serious issues involved here, especially concerning entanglement. But we ought not let these issues move us to dismiss an important aspect of the theory, and we ought not dismiss the role of pre-quantum
Theory) Coordination) and Empirical Meaning in Modern Physics 449 mechanical theories in the interpretation of quantum mechanics. We unproblematically retain classical notions in relativity for our accounts of phenomena, and I suggest that quantum mechanics ought not be different in this respect. That is, classical concepts need not be problematic merely because we use them. I suggest that these theories of modern physics are similar in that they both involve abstract formalisms which need to be coordinated with phenomena in theory-laden, neo-classical frameworks in order to have empirical content. They differ, however, in the status of the principles that coordinate the frameworks and in the status of the restriction and reconception of the theories within those frameworks. A too-facile comparison to relativity would be the following: special relativity informs us of the relativity of inertial reference frames and quantum mechanics informs us of the "relativity" of "measurement contexts." The problem is that in special relativity there is a general principle that associates an entire class of structures within Minkowski space-time with the reference frames of the empirical domain. In quantum mechanics there is no such easily identifiable class corresponding to measurement contexts, and no such general principle. At best there would seem to be a different coordination for each measuring context, such that we can associate values of the fundamentals of the theory with the empirical phenomena only of a particular measurement (e.g., the coordination between spin states and empirically-measured magnetic moment). In relativity, this coordination is general. This comparison suggests other disanalogies which highlight the interpretational problems of quantum mechanics. First, we famously have no good account of measurement context in quantum mechanics, whereas we do have good accounts of inertial frames in relativity. Without a good characterization of the domain of the empirical theories and what it is they are describing, we cannot do more than vaguely reconceive the theories and indicate what the empirical layer is about. Second, we have no good justification for the retention of aspects of classical physics for those domains. With special relativity, we can justifY our grouping of objects at rest with respect to one another in the same reference frame because relativistic effects only occur when considering objects in motion. The quantum analogue to the relativistic justification would probably be based on the correspondence principle. However, the correspondence principle is quite difficult to state precisely, and it is unclear how we ought to interpret the limit condition usually used (11 ~ 0). Among other things, this limit condition does not commute with t ~ 0, it may not be physically realizable (e.g., be equivalent to a limit of high quantum number), and it may not even get us the limit value we want, anyway.27
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Last, we have no clearly stated coordinating principles and have no clear justification for the possible candidates for coordinating principles. For special relativity, the light principle is associated with the idea that the laws of physics ought to be the same for all observers. This principle grounds the relationship between relativistic space-time and our pre-relativistic experience of space and time within a frame. We do not need a further (e.g., causal) account of what space and time in a reference frame are and how they are related to relativistic space-time, as we have what some argue is a good independent justification for the coordinating principle that relates them. We do not currently have such a principle for quantum mechanics, though it is not in1possible that we could find one. 28 These are not new issues, of course. However, I believe that this perspective sheds new light on the issues and indicates a way in which we n1ight want to view these common problems and possibly address them. For instance, on this view we do not need to explain the apparent classicality of quantum systems because quantum mechanics requires and relies on that apparent classicality-indeed, in a sense, the theory is about what happens in that classical world. On this view, then, the characterization of collapse as a separate physical process is misguided because the phenomenon which collapse is supposed to address concerns not an actual process within quantum mechanical theory but rather the coordination between en1pirical measurements and representations of quantum systems. Until we first get clear on this relationship, it is premature to propose new processes to account for features of that relationship. Of course, we will need to indicate how we can justify and reconceive the empirical theories as descriptions of quantum mechanical phenomena but not underlying structure. This is no trivial task. However, because we do not need to fully explain the role of classical framework in the theory, it is a different project from many typical attempts to address the measurement problen1.
9. COl1.clusion I believe that Friedman's conception of the relative a priori provides a quite fruitful picture of the structure and foundations of physical theories. However, it requires that there be external and rich empirical phenomena with which the abstract structures of the theories can be coordinated, and yet it does not really give an account of how that is possible. I have suggested that for any particular theory, these external empirical phenomena are other-theory-Iaden and require frameworks with their own internal structure of the same sort that Friedman argues are required by any physical theories. Abstract physical theories gain empirical content through
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coordination with other frameworks assumed to be appropriate for the en1pirical phenomena; thus, empirical content is gained through the coordination of layers of frameworks. The choice of a new framework then does not necessarily require the rejection of the old; indeed, it may be impossible to completely reject the old, as the old framework sometimes is the empirical foundation upon which the new theory is built. I have argued that the empirical framework for both special relativity and quantum mechanics is provided by a constrained and reconceived classical physics. Both assume and rely on the classical physics for the conceptualization of empirical phenomena, and in a sense neither explain those conceptions. The difference between the interpretive status of the two theories is not to be found in their reliance on classical concepts to describe empirical results, but rather in the status and foundation of the coordinating principles, the accounts of what the empirical layers are and how they relate to the structure of the theories, and the justification within the theory for the use of these other theories for constitutively describing the empirical phenomena.
NOTES 1. I thank Mary Domski for initiating this project, for all her effort on it, and for her very helpful early suggestions concerning the topic of this paper, which really helped me frame my approach. My thanks also go to Michael Dickson for his work on this project and for his insightful comments on this paper, which, as always, forced me to clarify and improve my views in more ways than I can specifically acknowledge in the paper itself. Many thanks, of course, go to Michael Friedman, whose helpful comments and suggestions on the topic of this paper are only the latest in a long list of things for which he has my gratitude. I am very pleased to be able to contribute in whatever small way I can to scholarship on the topics he has shaped and defined for the field. I also thanl( the audiences who provided many helpful comments on talks related to this paper. I have tried to interpret and incorporate suggestions appropriately; any ren1aining issues are of course my responsibility. 2. There are of course different ways of stating the light principle. This one will work for us. 3. A coordinatization is an application of separate space and time coordinates to Minkowski space-time points. A particular coordinatization may be picked out through the process of coordination between Minkowski space-time and empirical phenomena in reference frames, but the former is conceptually distinct from the latter. See below. 4. One note about the use of the word "theory": A theory involves both a framework and coordinating principles. While one might isolate a particular formalism as the abstract representation of a theory, the formalism requires certain other foundational structures and principles for its proper interpretation. Moreover,
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one needs coordinating principles to specify how it is to be applied. Thus, while the light principle is a coordinating principle on Friedman's view, that does not mean that it is necessarily to be thought of as being "outside" the theory. Because the formalism in the absence of the coordinating principles is only abstract and without empirical meaning, it seems difficult to think of just the formalism as a theory. Rather, it seems appropriate to think of the theories Friedman discusses as consisting of a formalism in addition to the framework in which that formalism is specified and the rules that indicate how that formalism is to be applied to empirical phenomena. 5. Note that while the theories of physics that are the focus of Friedman's attention all have a geometry of space or space-time as the foundation of their frameworks, this need not be a general requirement. 6. Of course, one may want to argue that these different conceptualizations are meant to refer (and can be thought of as referring) to the same real objects in the real world independent of conceptualization. Nothing I say here need be inconsistent with that interpretation, although it would require an additional argument that is out of the scope of this paper. It remains that the objects, as objects of the theories, are quite different in the above cases. 7. Note that one may want to distinguish our experiences from concrete empirical phenomena. See below. 8. My thanlcs to Michael Dickson for pointing out the importance of the distinction between this and the narrower and perhaps more widely known psychological sense. 9. I will have more to sayan this below. 10. My thanlcs to Bruce Glymour for suggesting this point, as well the related point above concerning when a theory explains data points. 11. This claim suggests another distinction between the view I am proposing and Bogen and Woodward's, as their idea that theories explain empirical phenomenon (and not data) might be interpreted as a claim that empirical content of a theory comes via the connection between theory and phenomenon. 12. For details, see Tanona (2002). 13. While this layering raises the question of how the layers can be grounded at bottom, that is a question I will not address here, where I will limit my attention to the relationship between two frameworks. 14. If the meaning of the terms are just the structural- theoretic ones of the theory (e.g., light rays are defined as null straight lines), then the light principle reduces to a tautology. 15. The experiment actually measured only the two-way "there and back" speed, not the one-way speed. Note, too, that the empirical phenomenon supports the gen~ralization that the speed of light is the same in all inertial frames only by abstracting away from the experiment. 16. However, special relativity does say how they apply. 17. Note that this application of the principle does not by itself tell us when two objects are at rest with respect to one another-to do that requires at least two such procedures together with assumptions that the apparent speed of light does not change between them. 18. For discussion of numerous issues relevant to this topic, see Torretti (1983)_._
Theory) Coordination) and Empirical Meaning in Modern Physics 453 19. My thanks to Michael Dickson for suggesting this example for consideration. 20. My thanks to Michael Friedman for suggesting this description of Reichenbach's views on these matters. Of course, any mischaracterization of Reichenbach's views here is due to me. 21. Here I am borrowing Earman's (1989) terminology. But I also follow Friedman's (1983, 62-64) characterization of different senses of "absolute," each with a different contrast (e.g., relative, relational, dynamical). Here I am using "absolute" as a contrast to relative, as there is a fi-ame-independent space. Newton's absolute space was also non-relational, probably substantival, and certainly non-dynamical, but the important thing here is that it is non-relative (although it is relevant for the interpretation of the theory that this space is often thought of as a non-relational "container" for objects). 22. I have been using language that may make space seem substantival. However, I believe the story is really neutral with regard to this question, and it should apply equally as well whether the space of a reference frame is thought to be substantival or merely relational. Either way, it is the form of that space (and the fact that it applies only within a reference frame) that is important. 23. Unlike secondary qualities such as color, however, we have no way of measuring the space-time "behind" the space-plus-time appearances in a reference frame except through that space and time. 24. "Copenhagen interpretation" here refers to something like Bohr's original views, not the standard interpretation with von-Neumann collapse that sometimes goes by that name. 25. Of course, there are plenty of features of quantum mechanical systems that are not classical, just as there are plenty of features of relativistic space-time that are not classical. Moreover, there is no law that says only small systems behave quantum mechanically. Indeed, there can be quantum behaviors at large scales. This view is rather that while we do not use classical physics to describe the causes of measurement results, we do assume a classical framework as the perspective for describing those results. 26. I provide a similar argument in Tanona (2004b). 27. Makowski and G6rska (2002) provide a good summary and analysis of these issues. 28. For some very interesting work related to this, see Michael Dickson's attempt to formulate such general principles based on the idea of quantum reference frames (Dickson 2004 and 2006).
REFERENCES Bogen, J. and J. Woodward (1988). "Saving the Phenomena." Philosophical Review 97: 302-52. Coffa, A. (1991). The Semantic Tradition from Kant to Carnap: To the Vienna Station. Cambridge: Cambridge University Press.
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Daumer, M., D. Durr, S. Goldstein, and N. Zanghi (1996). "Naive Realism about Operators." Erkenntnis 45: 379-97. Dickson, M. (2004). "A View from Nowhere: Quantum Reference Frames and Uncertainty." Studies in History and Philosophy ofModern Physics 35: 195-220. - - - . (2006) "Non-relativistic Quantun1 Mechanics." Philosophy of Physics. Handbook of the Philosophy of Science. 2 Volumes. J. Butterfield, J. Earman, D.M. Gabbay, P. Thagard, and J. Woods (eds.). Dordrecht: Kluwer Academic Press, 275-416. Duhem, P. ([1906] 1991). The Aim and Structure of Physical Theory. Princeton: Princeton University Press. Reprint edition. Earman, J. (1989). World Enough and Spacetime: Absolute and Relational Theories ofMotion. Boston: M.I.T. Press. Friedman, M. (1983). Foundations of Space-Time Theories: Relativistic Physics and Philosophy ofScience. Princeton: Princeton University Press. - - - . (1999). Reconsidering Logical Positivism. Cambridge: Cambridge University Press. - - . (2001). The Dynamics ofReason. Stanford: CSLI Publications. - - - . (2002). "Kant, IZuhn, and the Rationality of Science." Philosophy ofScience 69: 171-90. IZuhn, T. ([1962] 1996). The Structure of Scientific Revolutions. Chicago: University of Chicago Press. 3rd edition. Makowski, A. and IZ. G6rska (2002). "Bohr's Correspondence Principle: The Cases for which it is Exact." Physical Review A 66( 6): 062103. Reichenbach, H. (1965). The Theory of Relativity and A Priori I(nowledge. Berkeley: University of California Press. Original German edition published in 1920. - - - . (1969). Axiomatization ofthe Theory ofRelativity. Berkeley: University of California Press. Original German edition published in 1924. Ryckman, T. (2005). The Reign of Relativity: Philosophy in Physics 1915-1925. Oxford: Oxford University Press. Tanona, S. (2002). From Correspondence to Complementarity: The Emer;gence of Bohr)s Copenhagen Interpretation ofQJtantum Mechanics. Dissertation, History and Philosophy of Science. Bloomington, IN, Indiana University. - - - . (2004a). "Uncertainty in Bohr's Response to the Heisenberg Microscope." Studies in History and Philosophy ofModern Physics 35: 483-507. - - - . (2004b). "Idealization and Formalism in Bohr's Approach to· Quantum Theory." Philosophy of Science 71: 683-695. Torretti, R. (1983). Relativity and Geometry. Oxford, Pergamon Press.
[20] The "Relativized A Priori": An Appreciation and a Critique THOMAS RYCKMAN 1
In a number of recent writings, Michael Friedman has revivified, and in so doing, considerably enriched, Hans Reichenbach's early view that a priori principles have a theory-relative and fallible, but constitutive significance for physical theory. Recall that according to Reichenbach in 1920, i.e., in the immediate aftermath of general relativity, cognition of the objects of physical theory is understood in fundamentally Kantian terms. However, the Kantian problematic of "constitution" has been considerably streamlined and updated: it is now to be understood as a "problem of coordination" whereby (parts of) purely formal axiomatized mathematical theories are "coordinated" to empirically attested phenomena, constituting the latter as objects of physical theory. But mere coordination (Zuordnung) itself is not sufficient for constitution of the object of physical knowledge; rather, it must be mediated by so-called "principles of coordination" that constrain the assignment of certain formal mathematical structures to concrete observable phenomena. In this way, a mathematized physical theory acquires empirical content. For the theory of general relativity, such principles include those of "general relativity" (i.e., general covariance, understood as encompassing full space-time coordinate generality), the local validity of the inertial frames of special relativity, and the principle of causality. Unlike Kant's static picture of Newtonian mechanics as the culmination of physical science, Reichenbach expressly allowed for the continued growth of physical cognition, and so for the fact that coordinating principles themselves are fallible and theoryrelative. Yet simply positing coordination principles as constitutive but theory-relative is not the endpoint of Reichenbach's account of the global changes in the character of physical cognition in the wake of the general theory of relativity. Rather, any adequate epistemology of science must now allow for a generalized solution to the coordination problem in
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physics, i.e., of finding a consistent set of coordination principles for any empirically successful new physical theory that (as did general relativity) contains those of the preceding theory as a limiting approximation, and in turn will be contained by a successor set. This inter-theoretic process of modifying or generalizing the set of coordination principles across theorychange exemplifies a method that Reichenbach termed "successive approximation," in essential agreement with what we today recognize as the convergentist logic of approxin1ate truth of scientific realism. 2 As is well known, Reichenbach's subsequent philosophical developn1ent amply illustrates how his growing realist and empiricist tendencies completely effaced even this relativized transcendental approach to the constitution of physical objects through principles of coordination. Therein lies a philosophical moral that merits closer examination. Highly sympathetic to Friedman's goal of showing that the constitutive office of transcendental philosophy has lost none of its relevance for philosophical understanding of modern science, I nonetheless have reservations regarding his attempt to refashion a contemporary transcendental idealism on the basis of Reichenbachian coordination principles. Thus, I shall project a somewhat different genealogy and role for the "relative a priori)) in physical theory. To begin my critical remarks, I'd like to focus on Friedman's appraisal that it is precisely the appearance of the n10dern axiomatic conception of mathematical theory by the turn of the century, primarily associated with Hilbert) that makes the Reichenbach conception of coordination principles appear so compelling in the guise of the "relative a priori)) in physics. On this new view of mathematics there is thus more need than ever for principles of coordination to mediate between abstract mathematical structures and concrete physical phenomena. So it is in no way accidental that coordination as a philosophical problem was first articulated by scientific philosophers deliberately trying to come to tern1S with Einstein's theory of general relativity. Indeed, Reichenbach in 1920, together with Moritz Schlick in virtually contemporaneous· work, were the first thinkers explicitly to pose and to attempt to solve this philosophical problem. And the solution at which they both arrived is that there is a special class of non-empirical physical principies-variously called coordinating principles or constitutive principles by Reichenbach, conventions in the sense of Henri Poincare by Schlick-whose function is precisely to establish and secure the required connection between abstract mathematical structures and concrete sensory experience. (Friedman 2001, 78-79) However termed, the principal intent of this solution of Schlick and Reichenbach was to completely revamp the traditional I
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constitution of cognition in physics, jettisoning any reliance of mathematical-physical cognition on intuition. 3 As is conspicuous in the ensuing logical empiricist method of "rational reconstruction" of physical theories, this move was pivotal in arriving at the "received view" ca. 1960 according to which there is no transcendental constitutive function at all, whereas coordinating principles (now "bridge principles") certainly do playa necessary role in providing the mathematical machinery of physical theory with an empirical interpretation. Although arguably the object-constituting role of mathematics in physics is neither exhausted by, nor to be identified with, the coordination of mathematical conceptions to empirical phenomena, the subsequent tradition of logical empiricist philosophy of science collapsed the former issue into the latter, ironically setting the stage for the Quinean naturalism that would eventually supplant it. My dissent from this chain of events begins with a critique of the (admittedly Kantian-inspired) logical empiricist conception of cognition as a coordination of two independent faculties or sources of cognition, an account immensely reinforced by the supposedly Hilbertian conception of modern mathematics, as purely formal, axiomatized theories. So here is a brief synopsis, and a critique, of how it happened that the raison dJetre of transcendental philosophy of science-the problem of constitution of objects of physical cognition-came to be regarded, in Reichenbach and Schlick, as the problem of coordinating abstract mathematical theory to concrete empirical phenomena, subsequently leading to the canonical logical empiricist thesis that empirical cognition in physics first requires "coordinative definitions" linking bits of mathematical theory (e.g., the metric of space-time) to observable physical objects ("rigid rods" and "perfect clocks"). It is easy to see that this account of physical cognition reflects, at not too distant a remove, Kant's emphasis that knowledge requires two independent sources, the "passive" faculty of sensibility, and the "active" faculty of understanding. However, for reasons having to do with the demise of the Kantian account of pure intuition, primarily prompted by developments in mathematics and physical science in the second half of the nineteenth century, the doctrine of cognition as a coordination purposefully lacked altogether the intricate Kantian machinery of the "Transcendental Schematism" for relating the content of cognition given within the faculty of sensibility (with its necessary Euclidean and Newtonian mathematical structure) to the otherwise purely formal, and empty, categories of the understanding. 4 Instead, drawing upon the nineteenth-century development of the modern notion of a function, the relation between the two sources of cognition is regarded as established via a mapping of one into the other. Indeed, at the turn of the twentieth century, epistemologies employing the set theoretic notion of a mapping or coordination were quite common, almost a paradigm, and not merely
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among neo-IZantians. 5 However, complementing the modern notion of a function, the conception of cognition as coordination also drew support from other period developn1ents in logic and in mathematics. First and perhaps foremost was the modern formal or "axiomatic" conception of mathematics, associated primarily with Hilbert and his influential axiomatization of Euclidean geometry. It is well known that the logical empiricists cited Hilbert as the patron saint of this "formalist" view of mathematics, while Schlick made heavy epistemological weather of the method of implicit definitions, taken from Hilbert's exemplary formal axiomatization of Euclidean geometry (Hilbert 1899, and Schlick 1918, §7). In actual fact, Hilbert's views of the nature of mathematics were rather different and considerably more nuanced. To be sure, what one attempts to achieve in axiomatization, or in Hilbert's term, via the "axiomatic method," is envisaged as a pure conceptual structure severed from any and all particular physical or visualizable interpretation. But this represents an advanced stage of mathematical thinking only rarely attained in physics. In any event, the axiomatic method is principally directed towards metan1athematical issues of completeness, consistency and independence of the axioms, without any particular concern for any intended empirical interpretation, even in the case of physical axioms. This approach is amply demonstrated in Hilbert's long involvement with the axiomatization of physical theories. 6 On the other hand, Hilbert maintained that mathematical knowledge, as all human cognition, originates in intuition and experience, as the logical empiricists might have learned had they heeded the epigram to Hilbert's Grundlagen der Geometrie (1899), which is just IZant's most concise statement of how cognition results from the relation of intuitions, concepts, and ideas. So fiingt denn aIle menschliche Erkenntnis mit Anschauung an) geht von da zu Begriffen und endigt mit Ideen. (A702jB730)7 I-lilbert's prominent placement of IZant's declaration at the beginning of his own text was not a momentary infatuation or a heavy-handed attempt at irony, as some have thought, given Hilbert's reputation as a formalist. Rather, it is a message given special emphasis over many years in Hilbert's several lecture courses on "the axiomatic method."8 It recurs again in his writings regarding his own axiomatization, in Noven1ber 1915, of Einstein's relativistic theory of gravity coupled to a hypothetical "theory of matter," a generally invariant generalization of Gustav Mie's theory of electrodynamics. As Hilbert stressed in this case, axioms (in particular, the axiom of general invariance) are Ideas, in precisely the IZantian sense. 9 But above all, Hilbert consistently maintained that intuition is the fundamental source of guidance of mathematical construction. Indeed, the image of
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Hilbert as a "dogmatic formalist," a view widely held in the debates over foundations of mathematics in the 1920s and even until recently, is not at all accurate but a figment of the historical imagination.l° In support of this claim, I need cite only Hilbert's own critique in 1919-1920 ofwhat would become the logical empiricist view that "the propositions of mathematics are analytic propositions": Were this viewpoint correct, mathematics should then appear as nothing more than a series oflogical arguments heaped one upon the other. One would find nothing but an arbitrary series of conclusions driven by the power of logic alone. But in reality, nothing ofthe kind exists. Indeed, the conceptual structure of mathematics is constantly led by intuition (Anschauung) and experience so that mathematics for the most part represents a closed structure free of arbitrariness. 11 I am persuaded that Hilbert's affirmation-"the conceptual structure of mathematics is constantly led by intuition"-is both surely right, and highly suggestive for the problem of application of mathematics in physics. There can be little doubt that the conceptual structure of mathematics grows, and often is led, by generalizing upon results obtained from something like the Kantian conception of the construction of mathematical concepts in pure intuition, i.e., on a basis of primitive operations visualizable within a quasi spatio-temporal intuitive continuum. Here I shall give just two examples: in finitary arithmetic the intuition of the successive concatenation of occurrences of the numeral" 1" (or simple "strokes"), and in differential geometry, the bounded intuition of an immediately surveyable relation of comparison of direction of vectors at neighboring points, underlying the analytical operation of parallel transport of a vector. But there is still another development in logic implicitly assumed by the logical empiricist conception of cognition as a coordination mediated by "definitions," namely, Frege's attack on psychologism in logic, his utter hostility to any invocation of mental processes underlying logical objects and laws, and the resulting logocentrism, the idea that logic neither has nor requires an extra-logical foundation since it is itself constitutive of rational thought. 12 From Frege, the logical empiricists, and in particular Carnap, inherited a conception of logic as both constitutive of objectivity and as neither requiring nor permitting an extra-logical foundation in intuition-mere shorthand, according to Schlick, for psychology. As Carnap indeed emphasized in the Aujbau) a necessary condition for being an object of empirical cognition in the sciences is that such an object possess a univocal definition in the purely structural terms afforded by the logic of classes and relations of Principia Mathematica. From the perspective of this paper, what is most striking about Aujbau is the Fregean logocentrism
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that enabled Carnap to outline a constructive solution to how such structures obtain their particular empirical content via chains of extensional reductive definitions, ultimately to "elementary experiences," all without any reliance on intuition. Befitting its self-described task as a "theory of constitution" (J(onstitutionstheorie)) Aujbau signally departs from the epistenl0logical assumption guiding Schlick, Reichenbach, and the subsequent rational reconstructionist tradition in philosophy of science, that empirical cognition first requires "coordinative definitions" in order that the autonomous form or structure of theory be provided a content by senseexperience. Yet Aujbau crucially retains the noli mi tangere presumption of Schlick and Reichenbach that logical and nlathematical structure is the autononl0US locus of objectivity in physics, explicable only in its own terms. While the logical empiricists, and mathematical logic generally, would leave behind Frege's own understanding of a universal logic, even Carnap's later Principle of Tolerance assumes that all inquiry and scientific discourse presupposes one or another logical-linguistic framework, which of course may be changed. After all, to declare that there are no morals in logic is just a backhanded way of affirming that justification of a logico-linguistic framework can only be made on pragmatic, not epistemic, grounds. Though the matter cannot be argued here, Frege's vastly more influential brief against psychologism in logic (especially within analytic philosophy) may be instructively contrasted with Russerl's, where, in excluding empirical laws of psychology from logic, the view is retained that pure logical laws possess an ideality that can be evidentially brought forth in "immediate insight." But if one proceeds from Husserl's, rather than Frege's, critique of psychologism in logic, then logic and mathematics are not to be acccepted "at face value" in the constitution of physical objects. Rather, each is a species of objectification that, as all such species, is an achievement of what Husserl termed, somewhat mysteriously, "transcendental subjectivity," the locus of the formation of all objective meaning. As I have learned from Husserl via Hermann Weyl, the end result is not only the rejection of any dualistic account of cognition as a coordination of two independent sources of knowledge, but as well a possibility of understanding the application of mathematics in fundamental physical theory that avoids the Scylla and Charybdis of scientific realism and instrumentalism (or, indeed, "IZuhnian" irrationalism). Perhaps a bit more might be said here, though obviously the claim requires far nlore articulation to appear at all promising. 13 While the laws of logic are a pure conceptual order, nonetheless even logical and mathematical "objectivities" are subject to intentional-analytic investigation through what Husserl termed "essential analysis" (Wesensanalyse) (Husserl 1929). The ternl is unfortunate but then the underlying thought is not at ---------------------------
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all familiar within analytic philosophy. For Hussed, each kind of object (phenomenologically considered, all objects are objects of intentional acts) has its own characteristic immediacy of givenness, the way in which the object can be characterized relative to consciousness as "itself there," "there in the flesh." Hussed famously claimed that phenomenology was a method for the intuition of essences, the stock of non-temporal general or universal predicables that must belong to an object, making it the thing it necessarily is. In this way, an object, whether possible or actual, "constitutes itself," in the sense that its various necessary and contingent aspects are given in consciousness as interconnected in such a way as to be aspects of one object. But the ideal objects of mathematics and logic are themselves essences, and indeed, Hussed regarded mathematics as the paramount example of an eidetic science, wherein the ostensibly apodictic mode of evidence is that given in "essential insight" (Wesensschau).l4 Essential analysis is then a step-by-step phenomenological reflection upon the intentional acts through which logical and mathematical objects are "constituted" as categorical complexes, each essence having its place in a hierarchy according to generality and specificity, ranging from highest genus to eidetic singularities. Employing reflective acts of "eidetic intuition," essential analysis reveals the subjective conditions of adequate evidence wherein a given meaning-intending act regarding a particular object can be seen (or not seen) to be "fulfilled" or attained. Such phenomenological experiences of adequate evidence (which are possible ad infinitum) comprise a pro tem (not absolute) justification for the objects or forms that first enable a subject "to perform all the sorts of acts in which theoretical knowledge is made real" (Hussed1900, §65, Eng!. trans. 233). Thus on Hussed's view, logic (and mathematics) are not to be accepted "at face value" but like all objectivities (objects of knowledge or subjects of rational propositions) are to be considered as founded on subjective acts of thought, ultimately rooted in the analysis of pre-predicative intentional experience. It should be pointed out that such phenomenological analyses are to be regarded, neither as a metaphysics nor an empirical psychology, but as purely descriptive of the intentional contents and structures of cognizing consciousness. To this point, I have sought to identifY sources of motivation for, and objections to, the logical empiricist conception of cognition as a coordination. Now I want to turn to the role of principles of coordination that Reichenbach and Friedman both see as essential in first enabling the coordination that allows cognition to be carried out. Initially, in accord with the distinction Anderson (1967) had drawn between covariance and invariance groups, Friedman's reconstruction of Reichenbach's 1920 position held that the distinction between the constitutive and empirical is that between the absolute and the dynamical objects of a theory. This is to say that the constitutive function of the "relatively a
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priori)) coordination principles is to establish and define the absolute (i.e., non-dynamical) spatio-temporal framework within which the particular dynamical laws operate, a necessary condition for the empirical testing and confirmation of these laws.
The idea is as follows. Each of the theories in question (Newtonian physics, special relativity, general relativity) is associated with an invariance group of transformations that presents us with a range of possible descriptions of nature-a range of admissible reference frames or coordinate systems-that are equivalent according to the theory. The choice of one such system over another is therefore arbitrary, and Reichenbach's thought is that those elements left invariant by the transformations in question ... are precisely the constitutive elements of the theory. Thus, for example, in Newtonian physics the relevant group of transformations is the Galilean group, and so ... the underlying structure of Newtonian space-time is constitutively a priori; particular fields defined within this structure, such as the gravitational field, the distribution of mass, and so on, do not then count as constitutive. Similarly, in special relativity the relevant group of transformations is the Lorentz group, and so ... the underlying structure of space-time of Minkowski space-tin1e is constitutively a priori; particular fields defined within this structure ( ... ) do not count as constitutive. Finally, in general relativity the relevant group includes all one-one bidifferentiable transformations (diffeomorphisms), and so only the underlying topology and manifold structure remain constitutively a priori. (Friedman 1994, 66) Now I an1 in sympathetic agreement with one crucial aspect of this account, clearly identified already by Cassirer (1921) and highlighted recently in Nozick (2001), tracking the progressive development of the concept of physical objectivity in terms of those objects remaining invariant under wider and wider groups of admissible transformations. In the Anderson-Friedman sense, this process of generalizing the invariance group of a theory serves to replace absolute with dynamical structures. There are, I think, problems in setting out the distinction between constitutive and empirical elen1ents of a theory along the lines of absolute and dynamical objects.l 5 But there is no need to go into details, for in more recent publications, Friedman's position has changed somewhat, eliminating all reference to absolute and dynamical objects. As with Reichenbach, the task of relative a priori principles is still to coordinate two independent contributions to physical cognition, analytically reconstructed as distinct components of empirically successful physical theories. But Friedman now emphasizes the role such principles play in defining a framework or "space
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of intellectual possibilities" within which physical laws first find application and en1pirical meaning. On the one side of the coordination are purely abstract conceptual (mathematical) structures; on the other are the concrete empirical phenomena to which these representations are intended to apply.l6 In the case of general relativity, the coordination is attained through theory-specific "principles of coordination": the light principle (the constancy and source-independence of light velocity) and the principle of equivalence. As these examples make clear, the role of such principl~s is to provide empirical meaning to the mathematically expressed precise laws of nature, furnishing "the necessary framework within which the testing of properly empirical laws is then possible." Without these two "mathematical-physical presuppositions" in the case of general relativity, Einstein's field equations lack all empirical content. In this example, the applicability of the Riemannian theory of manifolds to space-time physics appears as the prime instance of such a constitutive coordination. Of course, Friedlnan's presentation reflects the modern understanding that derives not from Reichenbach but from the sophisticated n1athen1atical and conceptual treatment of Weyl, who first developed the notion of a manifold with affine connection and the physical interpretation of affine geodesics as the paths of freely falling test bodies.l 7 According to the (weak) principle of equivalence, free fall is not locally distinguishable from uniform motion. This enables a non-flat affine structure to be associated with the attestable local behavior of freely falling bodies. In addition, the conformal (causal) structure of space-time geometry can be physically represented by the paths of light rays. As Weyl first showed in 1921, thus physically represented, the affine (or projective) and conformal structures together allow construction of the metrical structure of space-tin1e (up to a local factor of scale), together with its 3+1 signature, obviating any need for a "coordinative definition" of the metric via rods and clocks in the absence of "universal forces." It should be observed that these geometrical notions are represented by coordinate-independent physical processes; of course, this is not the case with measurements made with scales and clocks. In any case, the physical representation of these and other geometrical concepts is not at all "arbitrary" or "conventional," but abide the reasonable requirement of consistency that "the physical interpretation of the mathematical notions occurring in a physical theory must be con1patible with the equations of the theory" (Trautman 1980,4). In so many words, the physical representation of aspects of geometrical structure is guided by the dynamical character of the metric and by Einstein's requirement of general covariance. From the onset of general relativity up to a current school of quantum gravity, these constraints-in one formulation, prohibiting any geometrical structures other than the metric tensor from representations of space-time itself (Wald 1984, 57)-have governed
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the conception of possible object in physics. Historically, this regulative claim was made independently and in various formulations by Hilbert, Weyl, Eddington, Cassirer, and by Einstein himself. I8 Our discussion thus far shows only that in providing the geometrical theory with an en1pirical interpretation, certain particular physical processes transpiring in space-time and governed by empirically established physical laws, can be naturally represented by fundamental concepts of differential geometry. In our account, the work of associating the latter with the former is accomplished through an intentional relation of representation rather than the notion of coordination that, in its constitutive employment, retains the odor of epistemological dualism. But it is via this relation of representation that the deductive structure and computational resources of Riemannian geometry can be brought to bear on the further representation of physical quantities, and of the relations between them as expressed in dynamical law, in particular Einstein's field equations. For when we accordingly say that that the universe has a geometrical structure (in fact, the structure of Einstein's equation relating space-time curvature to mass-energy sources), what actually are we asserting? Is this representation to be regarded as a faithful (or approximately faithful) picture of reality? Are we to be realists about space-time structure? Here the logical empiricists, invoking the arbitrariness of coordinative definitions, balked. Carnap interpreted the question as pertaining to a pragmatic preference for a "thing" language, whereas Reichenbach became a realist about the causal structure of space-time, and a scientific realist more generally.l9 Friedman, by contrast, rightly lays emphasis on the theory-relative nonconventionality of the light principle and the equivalence principle, that is, on the fact that Einstein's conceptual analysis elevating these quasi-empirical principles to postulates was required in order that the mathematical structures of (semi-) Rien1annian geometry could be applied to empirical phenomena. But without saying something further about the possible meaning of, e.g., the statement that paths of freely falling bodies are intentionally represented in general relativity as affine geodesics, I do not see that he has a viable response to the claim of the approximate truth of our fundamental physical theories beloved by scientific realism. In short, drawing attention to how parts of the formalism of Riemannian geometry acquires empirical content in general relativity is one thing, and a valuable service to the foundations of physics. But this accomplishment remains within the "natural attitude," the purview of practicing science,20 whereas the problem of transcendental constitution arises only when we reflect on such "coordinations" and attend to the intentional relation of representation that makes them at all possible. In this regard, transcendental constitution is a specifically philosophical, indeed a phenomenological and not a physical problem.
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What sort of response then can be given to scientific realism, itself a specifically philosophical problem? On the account sketched here it begins with the realization that representation is inherently intentional, and that it involves taking a conception of an object (its sense or noema, according to Husser!) as going proxy for the object itself, i.e., the object as it is /mown or experienced for the object an sich. This would encompass the Duhem-Cartwright point that symbolic representation in physics, no matter how supple, always involves idealizations and simplifications that lead away from the facts purportedly represented. But there is a larger issue concerning the very meaning mathematical representation in physics can have. I believe that any satisfactory answer to scientific realism lies in attacking its "epistemic optimism" that such representation is a better and better approximation to the fundamental structure of nature, because this metaphor omits altogether the fundamental reflective insight that even in physical theory, cognition is basically founded upon the intentionality of consciousness, and that all constitution is a matter of what Husser! called "sense-bestowal." This is in fact how Weyl understood the problem of constitution, in the very book that is the basis of the modern understanding of general relativity. The given to consciousness (Bewufltseins-Gegebene) is the starting point in which we must place ourselves in order to comprehend the sense and the justification of the positing of actuality. (Weyl 1918, "Introduction" ) For Weyl, undertaking such a task enjoined that the mathematical physics of general relativity must itself be set on the grounds of intuitive evidence (through, as Weyl termed it, a "new kind ofaxiomatization") in order to understand how objects of field theory are "constituted" within the mathematics of differential geometry. In particular, Weyl reconstructed the Riemannian theory of manifolds on the new basis of purely local relations of comparison (affine connection, metrical connection), giving these relations the visualizable meaning of a comparison of vectors or tensors at immediately adjacent points. From such primitive relations, the entire calculus of tensors can be developed, and thus the full range of mathematical structures of space-time theory. Then in principle such structures as appear in physical theory representing objective four-dimensional physical quantities are to be understood as having purely a symbolic and representative function, not a designating one; they are symbols for physical reality fashioned ultimately from the cloth of human intuition, not abstract names depicting, or mirroring, that reality without any relation to mind or consciousness. The relation of consciousness to transcendent reality is not a matter of our representations reaching out to this reality,
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depicting its essential contours, but is given. evidentially in phenomenological-reflective consideration upon the meaning of objects presumed within the "natural attitude" of the working scientist to have this transcendent reality (Ryckman 2005, Chapters 5 and 6). As I understand the animus guiding Friedman's revival of Reichenbach's Neo-IZantian account of relative a priori principles of coordination, Friedman aims thereby to provide a critical alternative to both Quinean holism and IZuhnian-inspired narratives (alas, to IZuhn's dismay) of scientific revolutions as inherently irrational transitions between incomn1ensurable paradigms. Against Quine, Friedman maintains that his "relatively a priori)) constitutive principles are not simply empirical statements relatively deeply embedded within our overall system of belief. Rather, by first making possible the empirical knowledge structured and framed by such principles, they have a unique and necessary "meta-en1pirical" standing that Quinean holism cannot recognize without surrendering its fulcrum point, the attack on the analytic/synthetic distinction. With regard to alleged IZuhnian relativism, Friedman maintains that in the development of modern science the relative a priori object-constituting principles comprise a converging series, successively refined in the direction of greater generality and empirical adequacy. As mentioned above, modern mathematical reformulations may be required to make such convergence salient. In the particular instance of the transition from Newtonian gravity to general relativity, the generalization emerges n1uch more perspicu0usly through a generally covariant formulation of Newtonian gravitational theory in the frame of four-dimensional space-time, first broached by Cartan and Friedrichs in the 1920s, as Friedman showed in his wonderful book on Foundations of Space-Time Theories (1983). But contrary to relativism's account of scientific change as discontinuous and arbitrary, governed by no context-free or super paradigmatic norms, such reforn1ulations enable, at least in hindsight, fundan1ental conceptual shifts in science to be seen as a rational process, evolving through continuous change, the later paradigm or framework arising by "natural transformation" from an earlier one. 21 I am entirely in agreement with the spirit of these counters to both Quine and relativism. Only for the reasons given above, I think that the articulation of transcendental philosophy's critique must take on board something like the intentional-analytic framework of phenomenological idealism, whose "only task and accomplishment is to clarify the sense (Sinn) of [the real (real)] world, exactly that sense in which everyone regards it as actually existing and as actually valid. "22 Otherwise we run the risk that by uprooting holism and relativism, naturalism and scientific realism slips in through the back door.
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NOTES 1. Special thanks to Mary Domski and Michael Dickson for their Herculean efforts in making this workshop such an enjoyable as well as intellectually rigorous event. I wish to thank Bill Demopoulos, Don Howard, and Tom Ricketts for their comments. 2. It is instructive to observe that the four-dimensional, generally covariant formulations of Newtonian gravity by Cartan (1923) and Friedrichs (1927) appeared just a few years later, for these, as Friedman has argued, provide needed grist otherwise lacking for Reichenbach's mill of successive approximation. 3. In two papers, (1994) and (1995), Friedman first surveyed the contorted steps of tllis transformation, pointing out how inappropriate to general relativity is Schlick's appropriation of Poincare's geometric conventionalism. 4. In particular, the goal of the Schematism (A137-147/B 176-87) is to show how appearances are determined through application of particular categories. Kant termed the conditions of application of the pure categories of understanding "schemata," rules (of the imagination) for recognition of the universal in the particular. In Husser!'s transcendental phenomenology, such recognition is attained through the method of "eidetic variation" witllin the broader framework of "essential analysis" (Wesensanalyse). See the discussion in Ryckman (2005), chapter 5. 5. Indeed, Schlick regarded the mere coordinating (Zuordnen) of one object with another as "a fundamental, irreducible act of consciousness, a simple ultimate tllat can only be stated, towards which every epistemologist must ultimately advance" (Schlick 1918, 326). It is debatable whether, as in Dedekind, such acts for Schlick nlight be located within a kind of "transcendental psychology." For discussion, see Ryckman (1991), 81-82. 6. See Corry (2004). 7. "Thus all human cognition begins with intuitions, goes from there to concepts, and ends with ideas." 8. For example, Hilbert (1991). 9. Hilbert (1915) and (1923). For discussion, see Brading and Ryckman (2008). 10. For a corrective to the view that Hilbert was a "dogmatic formalist," see Sieg (1999). 11. Hilbert 1991, 5. 12. See e.g., Ricketts (1994). 13. See Ryckman (2007b). 14. "Ostensibly" is the key word, for even such evidence can, through later experience (in Husser!'s sense of intentionality), be revealed to be deceptive; see Husser! (1929), §58. 15. See Ryckman (2005), chapter 2, §2.2.1 and pp. 245-49. 16. See Friedman (2001), 45, 79, 83; (2000), 382; (2002), 195-196, and 213. 17. Indeed, Reichenbach's "constructive axiomatization" of the theory of general relativity, based on rigid rods and ideal clocks, expressly aimed to show that Weyl's more sophisticated treatment manifested no epistemological advantage; for discussion, see Ryckman (2005), chapter 4.
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18. See Ryckman (2005), and Brading and Ryckman (2008). 19. See Ryckman (2007a). 20. The term "natural attitude" is, of course, Husserl's designation for that (phenomenologically) unreflective mode of thought ingrained in the practice of both science and everyday life that is left "in place" by phenomenological investigation. 21. E.g., Friedman (2001), 63. 22. Husserl (1931),21; German original in Husser! (1952),152-53.
REFERENCES Anderson, J.L. (1967). Principles ofRelativity Physics. New York: Academic Press. Bergmann, P.G. and V. DeSabbata (eds.) (1980). Cosmology and Gravitation: Spin) Torsion) Rotation) and Super;gravity. New York and London: Plenum Press. Boghossian, P. and C. Peacocke (eds.) (2000). New Essays on the A Priori. Oxford: Clarendon Press. Brading, IZ. and T. Ryckman (2008). "Hilbert's 'Foundations of Physics': Gravitation and Electromagnetism within the Axiomatic Method." Studies in the History and Philosophy ofModern Physics, 39: 102-153. Cartan, E. (1923). "Sur les varietes a connexion affine et la theorie de la relativite generalisee." Annales de PEcole Normale Superieure 40: 352-412. Translated by A. Magnon and A. Ashtekar (1986) as On Manifolds with an Affine Connection and the Theory of General Relativity (Napoli: Bibliopolis). Cassirer, E. (1921). Zur Einsteinschen Relativitiitstheorie: Erkenntnistheoretische Betrachtungen. Berlin: Bruno Cassirer. Reprinted in Ernst Cassirer (1957), Zur moderne Physik. Darmstadt: Wissenschaftliche Buchgesellschaft, 1-125. English translation by W. and M. Swaby (1953) in Substance and Function) and Einstein)s Theory ofRelativity. Reprinted. New York: Dover, 347-460. Clark, P. and B. Hale (eds.) (1994). Reading Putnam. Oxford: Blackwell. Corry, L. (2004). David Hilbert and the Axiomatization of Physics (1898-1918): From Grundlagen der Geometrie to Grundlagen der Physik. Dordrecht-BostonLondon: I
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Friedman, M. and R. Creath (eds.) (2007). The Camb7'idge Companion to Ca7'nap. Cambridge: Cambridge University Press. Friedrichs, K. (1927). "Eine invariante Formulierung des Newtonschen Gravitationsgesetzes und des Grenzuberganges vom Einsteinschen zum Newtonschen Gesetz." Mathematische Annalen 98: 566-75. Hilbert, D. (1899). Gl,ttndlagen del' Geomet7'ie. As reprinted in Hilbert (1930). ___. (1915). "Die Grundlagen del' Physik: Erste Mitteilung." Nach7'ichten. Konigliche Gesellschaft del' Wissenschaften zu Gottingen, MathematischePhysikalische Klasse, 55-76.
- - - . (1923) "Die Weltgleichungen." Lectures delivered in Hamburg. To appear in a volume in the new Hilbert edition published by Springer Verlag, U. Majer and T Sauer (eds.). - - - . (1930). G7'undlagen del' Geomet7'ie. Siebente Auflage. Leipzig und Berlin: B.G. Teubner. - - - . (1991). N atu7' und mathematische E7'kennen. Gottingen lectures from 1919-1920 as transcribed by P. Bernays and edited by D. Rowe. Basel-BerlinBoston: Birkhauser. Husserl, E. (1900). Logische Untersuchungen. Erste7' Band: P7'0logomena ZU7' 7'einen Logik. Halle: Max Niemeyer. English translation by J.B. Findlay as Logical Investigations (1970), vol. 1: 41-248. London: Routledge and Kegan Paul. - - - . (1913). Ideen zu eine7' 7'einen Phiinomenlogie und phiinomenologischen Philosophie. Erstes Buch. Allgemeine Einfuhrung in die reine Phiinomenologie. Jahrbuch fur Philosophie und phiinomenologische Forschung. Bd. 1.
- - - . (1929). "Formale und transzendentale Logik. Versuch einer Kritik del' logischen Vernunft," Jahrbuch fUr Philosophie und phiinomenologische Forschung 10: 1-298. English translation by David Carr as Formal and Transcendental Logic (1969). The Hague: Martinus Nijhoff. - - - . (1931). "Author's Preface to the English Edition," Ideas: General Introduction to Pure Phenomenology. Translation of Husserl (1913) by W.R. Boyce Gibson, 11-30. London: George Allen and Unwin Ltd.. Text of German original in Husserl (1952),138-62. - - - . (1952). Ideen zu einer reinen Phiinomenlogie und phiinomenologischen Philosophie. Drittes Buch. Die Phiinomenologie und die Fundamente del' Wissenschaften. M. Biemel (ed.). The Hague: Martinus Nijhoff. Malament, D.B. (ed.) (2002). Reading Natural Philosophy: Essays in the History and Philosophy ofScience and Mathematics. Chicago: Open Court. Nozick, R. (2001). Invariances: The Structu7'e of the Objective W07'ld. Cambridge,
MA: Harvard University Press. Reichenbach, H. (1920). Relativitiitstheorie und Erkenntnis A Priori. Berlin: Julius Springer. English translation by M. Reichenbach as The The07'Y of Relativity and A Priori Knowledge (1965). Berkeley and Los Angeles: University of California Press. Richardson, A. and TE. Uebel (eds.) (2007). Cambridge Companion to Logical Empiricism. New York: Cambridge University Press. Ricketts, T (1994). "Carnap's Principle of Tolerance, Empiricism, and Conventionalism." In Clark and Hale (eds.) (1994), 176-200.
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Ryckman, T. (1991). ((Conditio Sine Qua Non: Zuordnung in the Early Epistemologies of Cassirer and Schlick." Synthese 88: 57-95. - - . (2005). The Reign of Relativity: Philosophy in Physics 1915-1925. New York: Oxford University Press. - - - . (2007a). "Logical Empiricism and the Philosophy of Physics." In A. Richardson and T.E. Uebel (eds.) (2007), 193-227. - - - . (2007b). "Carnap and Husser!." In M. Friedman and R. Creath, (eds.) (2007), 81-105. Schlick, M. (1918). Allgemeine Erkenntnislehre. Berlin: Julius Springer. English translation of 2nd Gerll1an ed. (1925) by A.Blull1berg as The General Theory of ](nowledge (1985). LaSalle, IL: Open Court. Sieg, W. (1999). "Hilbert's Programs: 1917-1922." Bulletin of Symbolic Logic 5: 1-44. Trautman, A. (1980). "Generalities on Geoll1etric Theories of Gravitation." In Bergmann and DeSabbata (eds.) (1980), 1-4. Wald, R. (1984). General Relativity. Chicago: University of Chicago Press. Weyl, H. (1918). Raum) Zeit) Materie. Berlin: Julius Springer. English translation of 4th German edition (1921) by H.L. Brose as Space-Time-Matter. London: Meuthen Press. Reprint edition (1953) New York: Dover. - - - . (1921). "Zur Infinitesimalgeometrie: Einordnung der projektiven und der konformen Auffassung." N achrichten. ](ijnigliche Gesellschaft der Wissenschaften zu Gottingen) Mathematische-Physikalische ](lasse) 99-112. As reprinted in Weyl (1968), 195-207. - - . (1968). Gesammelte Abhandlungen. Vol. 2.
PART V Post-IZuhnian Philosophy of Science
[21J The Role of the Foundations of Mathematics in the Development of Carnap's Theory of Theories 1 WILLIAM DEMOPOULOS 2
Abstract I-lilbert's approach to the nature and significance of the axiomatization of mathematical theories had a profound influence on the program of rational reconstruction that we associate with Carnap and the logical empiricists; their views can be seen as a natural extension of Hilbert's ideas to the case of empirical theories. Mter briefly reviewing its historical context and philosophical motivation, Carnap's mature proposals for the rational reconstruction of the language of physics are expounded, together with an in1portant emendation ofWinnie's proposal. The paper concludes with the formulation of a basic limitation of the approach.
Introduction By 'Carnap's theory of theories'-also known as his reconstruction of the language of physics-I mean the theory foreshadowed in "Testability and meaning" (Carnap 1936/1937), and later elaborated in the book based on his UCLA seminar, his Replies in the Schilpp volume (Schilpp, ed. 1963), and elsewhere. From the point of view of the foundations of mathematics, the principal influences on Carnap's approach to reconstruction are twofold. There is first the influence of modern logic and its deployment by Frege and Russell in their articulation of logicism, the philosophy of arithmetic that is based on the definition of a cardinal number Card ( a) of a concept a as the class of all concepts in one to one correspondence with a. The definition implies what, in §62 of Grundlagen, Frege called the "criterion of identity" for number, namely,
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For any a and [3) Card(a) = Card([3) iff a is in one-one correspondence with [3.
The cardinal nurnbers of finite concepts (the inductive cardinals in Russell's terminology) can be shown to satisfY the Peano-Dedekind axioms for arithmetic; indeed, they constitute the domain of an isomorphic representation of any model of the axioms. This definition of cardinal number is explicative in the sense that it captures those components of our concept of cardinal number that are indispensable to its scientific employment, a requirement that is compatible with the definition's reconstructive character, exhibited by its divergence in various respects from common usage. The definition of cardinal number is the simplest illustration of logicism's contribution to the n1ethod of explication and rational reconstruction. The centrality of logical forn1 to Frege's and Russell's conception of analysis is perhaps the central distinguishing feature that separates it from the classical tradition of philosophical analysis. Indeed, Russell himself cites the work of the nineteenth century mathematical analysts as the proper historical antecedent and inspiration for the innovative conception of analysis the view advances (Russell 1946, 783-84). The role of logical form is particularly important for understanding the criterion of identity. On Frege's articulation of logicism, the criterion of identity employs a cardinality operator Card(.) that relates concepts to their cardinal number by a correspondence that is type-lowering, since in Frege's hierarchy of concepts and objects, numbers are of lower type than concepts. By contrast, in Russell's simple type hierarchy, numbers are of higher type than the objects to which they are assigned, so that for him, the correspondence is type-raising. This difference in logical form-a type raising vs. a type-lowering cardinality operator-is important: it facilitates Frege's derivation of the PeanoDedekind axioms of arithmetic without an axiom of infinity of the sort Russell was driven to postulate, and it yields a successful analysis of our knowledge of pure arithmetic in terms of a principle which underlies its applicability. In this respect, logicism endeavored to advance a theory that is more explanatory of our knowledge of arithmetic than a mere axiomatization of it. However, Frege's failure to address the paradox that beset his introduction of classes left the account of the connection between arithmetic and logic obscure. In Russell's development oflogicism, arithmetic is reconstructed within a consistent logical theory of classes, namely, simple type theory; however, it is only on the assumption of a special axiom of infinity that the PeanoDedekind axioms are recoverable within Russell's theory. Thus, in Russell's case, there is a consistent development of a logical theory of classes, but one with the unfortunate consequence that the arithmetical theory reconstructed in it must be judged a posteriori. Although simple tyl'~!E~~Y
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cannot secure the existence of sufficiently many numbers without a transparently elnpirical assumption, given the existence of a non-inductive class of individuals, Russell can prove, without any additional assumptions, the fact that the numbers can be mapped one-one into a proper subset of themselves. The attempt to account for our knowledge of arithlnetic by representing it as an extension of logic foundered, in Frege's case, on the inconsistency of his system, and in the case of Russell, on the need to assume a posteriori principles. Logicism's legacy was therefore mixed: its central development-polyadic logic with multiple generality-proved an indispensable tool for the logical analysis of the language of science. But logicism left unsolved the fundamental problem of responding to the lZantian claim that mathematics represents a counter-example to any theory of knowledge that seeks to minimize the importance of the synthetic a priori. This brings us to the second influence on Carnap's theory of theories, namely Hilbert's "first program," by which I mean his program of axiomatization and the methodology it introduced for understanding the nature of mathematical theories, especially the role this approach assigned to axioms for our understanding of the primitives of a mathematical theory. The view was stated with great cogency in Hilbert's response to Frege in their well-known correspondence over geometry: ... you say that my concepts, e.g., 'point,' 'between,' are not unequivocally fixed. 'Between' is understood differently on p. 20, and a point is there a pair of points. But it is surely obvious that every theory is only a scaffolding (schema) of concepts together with their necessary connections, and that the basic elements can be thought of in any way one likes. E.g., instead of points, think of a system of love, law, chimneysweep ... which satisfies all the axioms; then Pythagoras' theorem also applies to these things. Any theory can always be applied to infinitely many systems of basic elements. 3 Hilbert's Geometry ([1899] 1971) suggested an account of our knowledge of mathematics that is both independent of its relation to logic and free of the rationalist excesses of the synthetic a priori. The idea that emerged from Hilbert's work was that mathematical axioms define the primitives that occur in them, and the truth of the axioms consists in their satisfiability. The axioms of a mathematical theory are known a priori because they are free stipulations; hence, the correct explanation of our knowledge of them is one that renounces any claim to their being synthetic truths about the world. My focus in this paper is Carnap's development of this model of mathematical theories into an account of empirical theories, theories which, if
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known, are known a posteriori. It is important to bear in mind that Carnap's target was much broader than a theory of empirical theories. This point was insightfully recorded by Hempel when he wrote that Carnap sought to illuminate-indeed, set in relief-the classical empiricist project of showing that all our knowledge of the world derives from what is immediately given to us in the data of our direct experience. Stated in these general terms, the idea could be construed as a psycho-genetic claim concerning the development of man's conception of the world; but Carnap characteristically presented empiricism as a systematic-logical claim to the effect that all concepts suited to describe the worldand thus, all the concepts that could ever be required by empirical science, from physics to sociology and historiography-can be reduced, in a clearly specifiable sense, to concepts serving to describe the data of immediate experience or observation .... (Hempel 1975, 1-2) The fundamental concept of this project is that offactual content. Its explication depends on Carnap's proposals for reconstructing the language of science. There are two phases to such a reconstruction.
1. Phase One: Theoretical and Observational Vocabulary Factual content is expressed in terms of the observational vocabulary of a theory, where, it is in1portant to emphasize, the notions of observational vocabulary item and observational sentence are artifacts of the rational reconstruction. Putnam (1962) persuasively argued that the unreconstructed vocabulary items and the sentences formulated in terms of them are not easily classified as observational or theoretical in the required sense: they do not refer to just observable or unobservable events, a point that has been widely conceded within the logical empiricist and neo-logical empiricist traditions. 4 But as we will see, Putnam's observation, though correct, is largely irrelevant to the successful execution of a reconstructive program like Carnap's. As just noted, the distinction that is required concerns the vocabulary of the language ofthe reconstruction. Having the distinction available at the level of the vocabulary of the reconstructed language of physics proves to be vital to the articulation, n1otivation and philosophical significance of the logical empiricist view. To draw such a distinction in vocabulary, it suffices that it should be possible to distinguish between observable and unobservable events, their properties and relations. So long as this distinction is admitted, it doesn't matter that the unreconstructed vocabulary in actual
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use is not a suitable instrument for drawing the intended dichotomy. 5 Given a division of the domain of a possible model for the language of a theory into its observable and unobservable parts, it is always possible to introduce relation symbols whose interpretation is restricted to the observable part of the domain. The intended interpretation of the relation symbols of the theoretical vocabulary is restricted to the unobservable part of the domain. We call the observation vocabulary of the reconstruction the O-vocabulary, its theoretical vocabulary the T-vocabulary. An O-sentence is formed using only a-vocabulary items and a T-sentence is formed using only T-vocabulary items. Primitive relation symbols are either wholly observational or wholly theoretical-i.e., there are no "mixed" relations, a point to which we will return. & we have noted, there are however mixed sentences, and these are of great importance, as we will soon see. On Carnap's account of theories, theoretical terms make no contribution to the factual content of the theory to which they belong. Rather, the theoretical terms acquire a content from the a-vocabulary of the language of the theory by the occurrence of mixed sentences, called correspondence rules, which contain both T- and a-vocabulary items. A statement involving theoretical terms is properly regarded as a statement of an empirical theory only if there are correspondence rules that connect its constituent theoretical terms with observation terms. Correspondence rules (C-rules) establish a correspondence between the theoretical relations in the domain of unobservable events and the relations among observable events. They differ from the T-sentences by containing both 0- and T-terms. Indeed, in the absence of C-rules, the T-postulates are true provided they are consistent. Without the provision of the additional element C-rules provide, there is no non-arbitrary answer to the question of whether a theoretical claim is true, a contention that would be entirely commonplace (at least in a first order context) if, instead of 'C-rules,' we wrote 'semantic interpretation,' thereby taking the relativization to a specific semantic interpretation to be the additional element that distinguishes truth from mere consistency. But when formulated in terms of C-rules, the claim is a very different one, since it relativizes the truth of a T-sentence to an "epistemic interpretation" of the theoretical vocabulary in terms of the vocabulary belonging to the evidentiary basis of the theory. The logical empiricists rejected the idea that giving an intended semantic interpretation can address the problem of interpreting a theory as a theory about the material or actual world. From their perspective, the knowledge that the provision of such a semantic interpretation requires is precisely what is expressed by the C-rules. The C-rules are therefore principles of epistemic interpretation; without them, we would have no reason to suppose that we could even understand a semantic interpretation of the T-vocabulary in the domain of actual events. This of course is entirely in keeping with the
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semantically privileged role that Carnap and the logical empiricists assigned to the O-vocabulary because of its connection with observable events. Here, too, the influence of Hilbert's Geometry was decisive: Hilbert had proposed that the axioms of a mathematical theory define the theory's primitives. But by contrast with a mathematical theory, a physical theory requires principles of epistemic interpretation. It is possible to specify the "meanings" of the prin1itives of a mathematical theory by addressing only their logical category: this exhausts what provision of a semantic interpretation of a mathematical theory achieves. In contradistinction to the principles of semantic interpretation, C-rules bridge the theoretical and observational vocabularies and secure the evidential basis for the theoretical claims of physics in empirical observations; in doing so they elevate the theoretical statements of physics to the status of genuinely synthetic claims about the world. As we noted earlier, the division between T- and O-predicates is exhaustive: there are no mixed primitive predicates-predicates that apply to both observable and unobservable events. Having the observation-theory distinction at the level of the vocabulary of the reconstruction is important to the theory of lexical meaning that underlies the logical empiricist view of theories. According to this essentially concept-empiricist theory, our understanding of the meaning of a primitive vocabulary item rests on our acquaintance with its referent. Notice, however, that a theory of this kind allows for the forn1ation of expressions that are about items that transcend our observation. In particular, a sentence built up out of an exclusively O-vocabulary may well be about unobservable events. This is not only entirely compatible with the view we are expounding but has long been regarded by its proponents as one of its principal strengths. Only the sharp division in primitive 0- and T-vocabulary is based on the observability or otherwise of the referents of the 0- and T-terms. This point is often missed in discussions of the logical empiricist conception of theories, but it is characteristic of both it and its classical antecedents. 6 What is vital to the episten10logical point of the account is the sustainability of the dichotomy in prin1itive vocabulary on the basis of its reference. By contrast, sentences are o or T merely on the basis of the primitive vocabulary they contain. If the restriction on mixed primitive predicates were relaxed, a correlative question would naturally arise at a later stage of our discussion, namely, "Is our understanding of mixed predicates unproblematic in the way in which our understanding of O-predicates is held to be unproblematic, or do mixed predicates, like T-predicates, pose a special difficulty?" The reconstruction of theoretical knowledge we are exploring requires that we exclude mixed primitive predicates from the language of the reconstruction or, if we allow them, that we classify them with T-predicates as requiring special consideration in the account of their meaning. We wi1l2i~~~~
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on the simpler assumption that there are no n1ixed primitive predicates. One may of course choose to weaken this assumption. But to abandon the T- and O-vocabulary dichotomy altogether would be to engage in a completely different reconstructive project. For the present, our goal is to see where a sharp adherence to the dichotomy n1ight lead.
2. Phase Two: RalTISey and Carnap Sentences There are, as I have said, at least two phases to Carnap's reconstruction that we will need to consider. The first phase introduces a distinction between T- and O-vocabulary which extends to a distinction in theoretical and observational sentences in the intended way. It must be stressed that the formulation of the language of physics this assumes is already heavily reconstruction-dependent. The conjunction of the correspondence rules and theoretical postulates comprising a theory is then given by
where 01) ...) Om and T l ) ...) Tn are the 0- and T-predicates introduced at the first phase of the reconstruction. 7 The second and distinctively Carnapian phase of the reconstruction consists, in the first instance, in replacing a theory TCwith its Ramsey sentence R(TC), i.e., in replacing the theoretical predicates T p ... ) Tn of TC with variables X p ...) X n of the appropriate type and arity, and existentially generalizing over the new variables. Such sentences were tlrst discussed by Ramsey in his posthumously published "Theories" (1929), but the general idea of expressing what we would today call "satisfiability in a n10del" by a higher-order existentially quantified sentence was a con1mon practice in the logical tradition of the 1930s. From a model-theoretic perspective, the innovation of the Ramsey sentence consists in this: it uses a higher-order sentence to express satisfiability in a model relative to a fixed interpretation of a part of the language, namely, the O-vocabulary. The Ramsey sentence of a theory is important for Carnap because it and the theory imply the same O-sentences. (That every O-sentence which is a consequence of TC is a consequence of R(TC) follows from the fact that if there is an assignment to the variables X p ...) X n that satisfies TC(Op ...) 0mJ Xl) ...) X n)) then this assignment can form the basis of an interpretation of the non-logical constants T l ) .'.J Tn under which TC is true.) It is this fact that motivates the proposal that the Ramsey sentence represents the factual content of the theory TC. Carnap's account of the conventional or analytic con1ponent of TC re uires the notion of the Carnap sentence C ( TC) of a theory, namely, the
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conditional whose consequent is TC and whose antecedent is TC's Ramsey sentence. 8 The conjunction R( TC) /\ C( TC) is obviously logically equivalent to TC. In the presence of its Carnap sentence, a theory is equivalent to its Ramsey sentence. Thus, if it could be shown that the Carnap sentence is plausibly regarded as analytic, we could conclude that a theory is its Ramsey sentence, since their equivalence would depend only on an analytic truth. Carnap argues that the Carnap sentence is analytic on the ground that all of its O-consequences-all the sentences in the O-vocabulary it implies-are logically true (L-true); the Carnap sentence is in this sense observationally uninformative, a fact that can be readily verified. 9 Thus, there is an obvious sense in which the Carnap sentence can be said to have no factual content and can, therefore, properly be regarded as analytic. Modulo an analytic truth, TC is R(TC). The reconstruction that emerges from these considerations thus divides TC into two components, R(TC) and C(TC), the first expressive of TC's factual content, the second merely a stipulation controlling the use of its theoretical vocabulary T1) ...) Tn and expressive of the theory's analytic component. More generally, a sentence is analytic-or more precisely, analytic in TC-if it is a consequence of just C(TC). Carnap's enthusiasm for this proposal had many sources. But what is perhaps the chief virtue of the proposal is the simplicity with which it promises to draw a sharp and non-arbitrary division between the factual and analytic components of a theory. There are three desiderata Carnap imposes on a reconstruction that seeks to incorporate such a division into analytic and factual components: (i) The conjunction of the factual and analytic components of TC is logically equivalent to TC. (ii) The factual component is O-equivalent to TC.
(iii) The analytic component is observationally uninformative. We have just seen how naturally these desiderata are fulfilled byCarnap's proposed rational reconstruction in terms of the Ramsey and Carnap sentences of a phase one reconstructed theory-of a theory whose formulation respects the division into T- and O-sentences.
3. Winnie's Emendation of Carnap In an important paper, Winnie (1970) noted that Carnap's desiderata are not con1plete, and that this undermines the contention that the suggested division into analytic and factual components is non-arbitrary. To appreci-
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ate Winnie's contribution, suppose we take one of the T-postulates, Ti say, and propose the new and obviously equivalent reconstruction, R(TC) /\ [C(TC) /\ T i ], which takes [C(TC) /\ T i ] as its analytic component. To satisfY Carnap's third desideratum, [C(TC) /\ T i ] must be shown to be observationally uninformative. This is the content of Winnie's observation:
Suppose TC is satisfiable and that T i is a T-sentence logically implied by TG Then C(TC) /\ T i is O-uninformative.lo Notice that this observation regarding T i and C(TC) can be iterated through all the T-postulates of TG There is, therefore, nothing to exclude the acceptability of a reconstruction that, like R(TC) /\ [C(TC) /\ T] /\ ... /\ Tk], takes the conjunction of all the T-postulates T] /\ ... /\ T k of TC to be part of the analytic component of its reconstruction. It is an early objection of Quine (1951) that Carnap fails to characterize a non-arbitrary analytic-synthetic or fact-convention dichotomy. An initially plausible response (cf. Maxwell 1963) holds that the arbitrariness is harmless since it attaches only to the unreconstructed sentences of a science. But we have just seen that the objection applies equally to the sentences of the second phase of Carnap's proposed reconstruction, and this appears to be a complete vindication of Quine. To address this difficulty, Winnie proposes revising Carnap's reconstruction by adding a fourth desideratum:
(iv) The analytic component of TC is observationally noncreative in TC, where X is observationally noncreative in TC just in case TC logically implies X, and for any Y such that TC logically implies Y, every O-consequence of X /\ Y is an O-consequence of¥. (Winnie 1970,150) It can be shown that desideratum (iv) is satisfied by Carnap's original pro-
posal but not by any of the problematic extensions [C(TC) /\ Ii] of the analytic component: every such extension is observationally creative in TC whenever T i is a T-postulate not implied by C(TC). Indeed, the consequence class of the Carnap sentence characterizes exactly the sentences that are O-noncreative in TC: any sentence not implied by the Carnap sentence will, when added to C(TC), be O-creative in TG This is arguably the principal interest of the Carnap sentence for the project of rational reconstruction. The consequence class of the Carnap sentence is the largest subclass of the class of O-uninformative sentences of TC that are O-noncreative in TG The condition of O-noncreativity in TC rules out adding to the analytic component any sentence not implied by C(TC); a fortiori, it rules out adding any T-sentence not implied by the Carnap sentence.
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There are several positive features of Carnap's reconstruction when it is revised in the manner just reviewed. We may note the following: First, the holism of the approach means that the division of factual and analytic components is independent of the formalization. Since the consequence class of the Carnap sentences of two distinct, but logically equivalent, formulations of a theory will be the same, so also will their characterizations of analytic and factual sentences. As Winnie has ren1arked, the relativity that attaches to the notion postulate of T does not attach to the notion analytic in T (Winnie 1970, 149). Moreover, as Hempel has emphasized, the ambiguous status of C-rules-they appear to be both law-like statements (hence, factual) and interpretive (hence, analytic )-makes the application of the analytic-synthetic distinction to the sentences of a phase one reconstruction highly problematic (Hempel 1963, 703). Indeed, it calls into question the very possibility of a non-arbitrary dichotomy of sentences of this phase into analytic and factual statements. These difficulties are completely avoided when the distinction is applied at the second reconstructive phase-when the analytic sentences are represented as the consequence class of C(TC). Second, the approach locates precisely the factual component of a theory and it does so without employing predicates that refer to unobservables. Hence, the account of factual content is shown not to require a solution to the vexed problem of the meaning of the theoretical vocabulary, the problem of how terms which purport to refer to unobservable entities can be understood. Carnap's proposal addresses this problem by replacing all the theoretical predicates with variables: aside from the matter of their logical category, their elimination means that there is no need to appeal to the meanings of terms belonging to the theoretical lexicon in order to express the factual content of a theory. This is perhaps the point at which Hilbert's influence is most evident. Third, the account subsumes Carnap's 1936 analysis of theoretical terms in "Testability and Meaning." That analysis, which isolated the analytic and synthetic assumptions underlying the application of disposition predicates, can be seen as an early anticipation of the present reconstruction in terms of Ramsey and Carnap sentences. (This is observed by Carnap in his Replies; see also the Demopoulos (2007) mentioned in footnote 1.)
4. Factual Content and the "External" Problem of Underdetermination Borrowing a terminology Carnap employs in another context (Carnap 1950), let us distinguish two problems of underdetermination, one "exter-
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nal," the other "internal." We understand the internal problem to be the price of empirical inquiry under the constraint of finite resources. The data accessible to us can always be improved, and the parameters employed in their description can always be replaced with new parameters. Underdetermination at one stage in this process of refinement might be resolved at a later stage by the expansion of the available observational information, but some underdetermination is always present because of the incompleteness of our data. By contrast, the external problem of underdetermination concerns the ideal case where all observational data are in. On Carnap's explication of factual content, the notion that there could be an external problem of underdetermination rests on a mistaken supposition. For Carnap, given complete observational knowledge, two theories can be incompatible only if there is a sentence expressible in their conlmon observational vocabulary which is implied by one while its negation is implied by the other. In the case of first order theories, with disjoint T-vocabularies but coincident O-vocabularies, a Carnapian solution to the external problem of underdetermination is a consequence of standard metalogical results. English (1973) observed that if two theories T l and T2-two possibly infinite sets of sentences in a first order language-with disjoint T-vocabularies but coincident O-vocabularies are inconsistent with one another, then there is a sentence X in their common O-vocabulary which is such that T l implies X while T2 implies -,X. Hence, T l and T2 cannot be compatible with the same "data," if by 'data' we mean what is expressible in the common O-vocabulary of these two first order theories. The argument for English's observation is based on the Craig Interpolation Theorem which tells us that if Xi (i = 1) 2) are first order sentences in L i such that L = L l n L 2 and Xl /\ X2 has no nlodel, then there is a sentence X of L such that Xl implies X and X2 implies -,X. By the Compactness Theorem, if T l U T2 is inconsistent, there are finite subsets Li ~ Ti (i = 1, 2) such that 1:1 U L 2 is inconsistent. Let Xi be the conjunction of the sentences in L i . Then Xl implies -,X2 . Hence, by Craig's Theorem there is a sentence X such that Xl implies X and X implies -,X2 ' and X is an O-sentence formulated in L, the common observational vocabulary of T I and T2 . Hence, Tl and T2 cannot both be conlpatible with all observations: if X holds, then T2 is false, and ifit does not hold, TI is false. But now Carnap's Ran1sey sentence reconstruction allows this fact about first order theories to be exploited for any two first order theories which are such that their formulation respects the distinction between T- and O-vocabulary and their O-vocabularies are coincident. To see this, notice that the essential idea behind the use of the Ramsey sentence in reconstructing a theory is that only the logical category of the theoretical terms is relevant to their theoretical role. It follows that so long as the
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logical category of the original vocabulary items is preserved, someone who grants that the Ramsey sentence of a theory captures its factual content can have no objection to a uniform replacen1ent of the T-vocabularies of T1 and T2 with new non-logical constants. Such a replacement makes the T-vocabularies of the theories disjoint and thereby insures that T1 and T2 satisfy the hypothesis of English's observation. The identification of a theory \vith its Ramsey sentence therefore implies that the notion that two theories might be compatible with the same data and yet conflict with one another must be given up, so that a conflict in some T-postulate is always reflected in an O-sentence. Given Carnap's conception of factual content, complete observational knowledge excludes underdetermination. Of course, we cannot suppose that we will ever be in the position of knowing all possible data. But we should not dismiss the external problem and its solution just because it does not arise in practice. This would n1iss its point for our understanding of Carnap's theory of theories: for Carnap there is no external problem of underdetermination because empirical adequacy with respect to all possible evidence is the same as truth.
5. The Near Analyticity of the Theoretical Component We saw that Carnap's conception ofwhat is factual can be defended against Quine's objection that the line between the factual and the analytic is necessarily arbitrary. But although Quine's objection can be met, there is a more basic difficulty: Without calling into question the viability of a factconvention dichotomy, the conception of the factuality of the theoretical postulates that emerges from Carnap's reconstruction is insufficiently robust. The difficulty is that theoretical sentences, though factual, are almost logical truths, and hence, are almost analytic. This is a consequence of the reconstruction's conception of the theoretical vocabulary, since it is this that justifies replacing theoretical terms by variables and identifying a theory with its Ramsey sentence. It is often suggested that the effect of Ramseyfication is to call into question the status of realism about the referents of the theoretical vocabulary. But as we will soon see, a more basic difficulty is whether, having identified a theory with its Ramsey sentence, one can claim to have captured pre-analytic intuitions about the nature of our knowledge of the truth of theoretical statements. This is arguably the fundamental methodological issue that a reconstruction of our theoretical knowledge must successfully address. We can begin to see how the "quasi-analyticity" of theoretical claims arises by reviewing the proof of a corollary to Craig's Interpolation Theorem, a corollary that figures importantly in the a~~y~i~ ?~ ~?~~~P~S__
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notion of observational uninformativeness, since it implies that on a reconstruction like Carnap's, a purely theoretical statement-one with no 0vocabulary items-has only L-true observational consequences. Its proof shows how disjointness of the T- and O-vocabularies permits a type of model construction that is relevant to our discussion. Let X and Y be sentences of a first order language without equality such that (i) X and Y share no non-logical vocabulary, (ii) X logically implies ~ and (iii) X is satisfiable. Then Y is logically true. Proofll Arguing toward a contradiction, suppose that there is a model M in which Y fails so that -,Y holds in M. Since X is satisfiable, it too has a model N, which we may assume is of the same cardinality as M. Letfbe a one-one onto map from the domain N of N to the domain M of M, and for each n-ary relation symbol R occurring in X define an n-ary relation ..In RM·ff . lTIOn, IS 1 ·IS In R M on Mb y t he cond·' RN. Since X holds in N, this expands M to a model M* for the vocabulary of X in which X holds when its relation symbols are interpreted by the relations RM. Since X and Y have disjoint vocabularies, -,Y is true in M* iff -,Y is true in M. Thus X /\ -,Y holds in M*, contrary to the hypothesis that X implies Y. Notice that the language L for which the corollary holds is restricted: it is a language without equality. If L contained equality, -,Y might hold only in a model of finite cardinality n; but if, for example, X holds only in infinite models, the argument will break down. Restricting the language prevents this, since satisfiability then implies satisfiability in a countably infinite model. This restricts the generality of the corollary, but it does not restrict its philosophical interest. The only effect of the restriction on L that we require is
(*) The reconstruction applies to sentences which are true in countably infinite models if they are true at all. Taking L to be without equality is a simple way of insuring (*), but it is not strictly necessary: we can simply impose the requirement (*) directly and proceed to avail ourselves of equality and the expressive resources it brings. Now suppose there is a model M in which the O-sentences hold. Then provided only that the cardinality of M is not unduly restricted by the O-sentences, we can, just as in the proof of the corollary, expand M to a model M* in which the T-sentences are also true. The sense of "almost analytic"-and even, "almost L-true"-that applies to the T-sentences is this:
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Modulo a logical assumption of consistency and an empirical assumption about cardinality) it follows that if the O-sentences are true) the T-sentences are also true.
The proof of the technical part of this claim follows exactly our argument for the corollary to Craig's Theorem. The philosophical interest of the claim derives from the fact that the relations which, when assigned to the T-predicates occurring in aT-sentence T, make Ttrue-in-M* also make T true. This follows fron1 Carnap's reconstruction because the content of T is reduced to the purely existential assertion that there are relations corresponding to the relational expressions of Twhich make it true; but so long as T is satisfiable, this will hold as a matter of logic in any model of the 0sentences, provided only that the model is large enough. To see why this situation is unsatisfactory, consider the following elementary example. (The example is deliberately over-simplified, but misses no point of principle; indeed, it is essentially the special case that arises in connection with Carnap's analysis of dispositional properties in "Testability and Meaning.") Let T be a theory consisting of the single sentence,
T For every x) if Fx) then Rx and if Rx) then lex, where F and IC are O-terms and R is a T-term. Its Ran1sey sentence is R(T) There is an X such that for every x) Xx) then lex.
if Fx,
then Xx and if
For Carnap, the factual content of T is that every F is a IC. T goes beyond this by telling us that the theoretical property R weakly separates F and I( in the sense that Fis contained in Rand R is contained in IC. This is a synthetic truth that we can only know a posteriori; it tells us something about the connection between the three properties F, IC and R that may of course hold or not, but in any case is not a logical consequence of the fact that F is contained in IC. This is not true of R(T) since we know a priori that for any F and IC if F is contained in IC, there is an X which wealdy separates them: If IC-F is empty, let F = X; if it is not empty and there is an x in IC but not in F, put X = Fu {x}. Notice, by the way, that contrary to an often-cited response to this line of objection,12 the difficulty it raises has nothing to do with the "reality" of the properties and relations which the Ramsey sentence quantifies over: Even if our theory Twere concerned with an "artificial" property R-in almost any sense of 'artificial'-it would still make perfect sense for us to distinguish the fact that R wealdy separates F and IC, both from the claim that some property or other separates them, and from the claim that a "real"
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property separates them. Our theory lacks the triviality of the former claim and the obscurity of the latter; neither claim is a faithful representation of what our theoretical knowledge aspires to tell us.l 3
6. Conclusion In his Geometry, Hilbert articulated a new way of thinking about mathematical theories, one that shaped the way mathematics came to be written in the twentieth century, and one that put the last nail in the coffin of the Kantian formulation of the idea that intuition has a role to play in the epistemological basis of pure mathematics. Hilbert's ideas had an influence beyond pure mathematics: Einstein famously remarked that without them he would never have arrived at the General Theory.l4 We have seen in some detail how Hilbert's conception of pure mathematics was elaborated by the logical empiricists into a theory of empirical theories, and more broadly, into a general scheme for reformulating the claims of traditional empiricist epistemology: The entire account revolves around the use it makes of the division of vocabulary into 0- and T-predicates. The meaning of a-sentences is taken as unproblematic; the account of the meaning of T-sentences is an adaptation of Hilbert's view of the axioms of geometry, while the presence of C-rules makes the theory under reconstruction one of physics rather than pure mathematics. The near analyticity of the theoretical component that emerges from Carnap's reconstruction is reminiscent of the conventionality Poincare (1905) attributed to the axioms of physical geometry. For Poincare the vocabulary of a geometrical theory has the status that the T-vocabulary has for Carnap, and like the T-vocabulary, the language of geometry stands in need of epistemic interpretation. For Carnap, it is only by their association with observable events and their properties and relations that the T-sentences acquire a factual content. For Poincare, the primitive geometrical relations are defined over a domain of geometrical objects-points, lines, planes-which are autonomous with respect to the physical processes that are indicative of them. Just as the T- and a-vocabulary are disjoint, so also for Poincare, the primitive vocabulary of the theory of the geometry of space is disjoint from that of physics; but it is only by its association with physical processes that geometry acquires the character of an applied science. Given this framework of assumptions, the argument by which we established the near analyticity of a consistent set of T-sentences can be adapted to show the conventionality of the theory of the geometry of space. The comparison with Poincare helps illuminate what is an otherwise puzzling feature of Carnap's conception of the domain of the higher order
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variables of the Ramsey sentence. In his response to Hempel, Carnap says of the Ramsey sentence that while it does indeed refer to theoretical entities by the use of abstract variables[, ... ] it should be noted that these entities are ... purely logico-mathematical entities, e.g. natural numbers, classes of such, classes of classes, etc. Nevertheless, R( TC) is obviously a factual sentence. It says that the observable events in the world are such that there are numbers, classes of such, etc. which are correlated with the events in a prescribed way and which have among themselves certain relations; and this assertion is clearly a factual statement about the world. (Carnap 1963 b, 963) In his Introduction to the Philosophy ofScience (1966), Carnap offers a clarification of this contention: [ ... physicists may, if they so choose,] evade the question about [the existence of electrons] by stating that there are certain observable events, in bubble chambers and so on, that can be described by certain mathematical functions, within the framework of a certain theoretical system.... [T]o the extent that [the theoretical system] has been confirmed by tests, it is justifiable to say that there are instances of certain kinds of events that, in the theory, are called "electrons." (Carnap 1966,255) Thus, on Carnap's view, the theoretical existence clain1s peculiar to R(TC) concern only logico-mathematical entities built upon a suitable numbertheoretic domain, just as for Poincare the claims of applied geometry retain a merely forn1al significance. Although the identification of theoretical properties and relations with such entities is at variance with the idea that the higher order variables range over a domain built from unobservable physical events, it comports well with the picture of the theoretical as merely instrumental to the derivation of observational consequences. In my view, the abstractness of Carnap's account, which is often cited as a defect, is in fact one of its most attractive features, since it facilitates the precise isolation of both the strengths and weaknesses of the approach. Regarding strengths, perhaps the chief among them is the sharpness and clarity with which it separates the factual and analytic components of an empirical theory. But if our critical analysis is correct, this is inseparable from the account's central defect, namely, a conception of the theoretical sentences of the language that is inimical to a satisfactory account of the knowledge they purport to express. A successful empiricism must yield the conclusion that our theoretical claims, when true, are at least sometimes significant extensions of the true observation sentences. This requirement
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is not satisfied by Carnap's reconstruction: given a model of the O-sentences of the right cardinality, it is possible by a purely metalogical argument to transform it into a model of any set ofT-sentences, provided only that the T-sentences are consistent.
NOTES
1. This paper complements and partially overlaps a longer study, "Carnap on the reconsu'uction of scientific theories," which appears in Friedman and Creath, ed~. (2007). Both papers are developments of Demopoulos and Friedman (1985). 2. Work on this project was begun during my tenure as a Visiting Fellow of All Souls College. My thanks to the College for providing me with such an ideal work environment. I would also Wee to thank Mary Domski and Michael Dickson for their editorial and organizational work and for their invitation to contribute to this salutary project. Support of my research by the Social Sciences and Humanities Research Council of Canada is gratefully acknowledged. 3. Draft or excerpt of Hilbert's letter to Frege of 29.xii.99; in McGuinness (1980), p. 42, translation by Hans Kaal. Hilbert's page reference corresponds to p. 28f, Section 9 of Hilbert (1971). 4. "There was never a stage of language at which it was impossible to talk about unobservables. Even a three-year-old child can understand a story about 'people too little to see' and not a single 'theoretical term' occurs in this phrase" (Putnam 1962,242). 5. This observation and its elaboration into a division between the theoretical and observational vocabulary of the reconstructed theory can be found in Friedman's (1982) review of van Fraassen (1980). The relation of Carnap's account of theories to the semantic view of theories and constructive empiricism is also addressed in Demopoulos (2003), 389-93. 6. The historical background is discussed in Demopoulos and Friedman (1985) and in Demopoulos (2003). 7. We are supposing the correspondence rules and theoretical postulates to be finite in number; this is an assumption that can always be met, though doing so may incur the cost of increasing the strength of the underlying logic of the theory. For an overview of relevant results, see van Bentham (1978). 8. See Psillos (2000) for an early formulation. 9. IfC(TC) implies X, then ---,R(TC) implies X and TCimplies X. But since X is an O-sentence, R( TC) also implies X. Hence, R( TC) Y ---,R( TC) implies X, and X is L-true. Notice also that since for any T, R( T) is O-equivalent to T, the O-uninformativeness ofC(TC) is equivalent to the claim that the Ramsey sentence of the Carnap sentence of TC is L-true. 10. See section V of the appendix to Winnie (1970), and the discussion on pp. 149-50 of its reprinting in Hintikka (1975). It should be noted that I have departed from Winnie's terminology. Winnie's argument is also expounded in the appendix to my Demopoulos (2007), mentioned in footnote 1, as are the arguments for several other of tlle technical claims made in the text.
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11. My proof-sketch is based on Robinson (1974), 5.1.8. 12. See for example, Lewis (1984), 227. Lewis's view has been highly influential; see Hochberg (1994) for its elaboration in connection with a difficulty closely related to the one discussed in the text and expounded in Demopoulos and Friedman (1985). 13. This simple example of a theory and its Ramsey sentence has been used by Worrall and Zahar to argue that the proper explication of the "observational content" of T should not employ an unrestricted quantifier (Worrall and Zahar 2001, 240-41). So understood, the a-sentence
For every x) Fx only if I6c is strictly stronger than the observational content of T It follows that on Worrall and Zahar's conception of observational content, R( T) is also strictly stronger than the observational content of T. Worrall and Zahar take this observation to undermine criticisms ofRamseyfication like the one raised in the text. But the central critical point does not depend on the exact explication of observational content, but on the difference, to which we have called attention, between T and R(T). 14. For the historical context and reception of Hilbert's Geometry, see Freudenthal (1962). For an overview of its content in the context of the philosophy of mathematics at the turn of the twentieth century see Hallett (1994). And for a discussion of its influence on Schlick and on the development of the General Theory see Friedman (2002).
REFERENCES Braithwaite, R.B. (ed.) (1960). The Foundations ofMathematics and Other Logical Essays. Paterson, NJ: Littlefield and Adams. Carnap, R. (1936/1937). "Testability and Meaning." Philosophy of Science 3: 419-71, and Philosophy of Science 4: 1-40. - - - . (1950). "En1piricism, Semantics and Ontology." Revue internationale de philosophie 11: 20-40. - - . (1963a). Autobiography. In Schilpp (ed.), 3-84. - - . (1963b). Replies. InSchilpp (ed.), 859-1013. - - . (1966). An Introduction to the Philosophy ofScience. Edited by M. Gardner. New York: Basic Books. Cohen, R. and M. Wartofsky (eds.) (1970). Boston Studies in the Philosophy of Science Vol. 8. Dordrecht and Boston: Reidel. Demopoulos, W. (2003). "On the Rational Reconstruction of Our Theoretical IZnowledge." British Journal for the Philosophy ofScience 33: 371-403. - - - . (2007). "Carnap on the Reconstruction of Scientific Theories." In M. Friedman and R. Creath (eds.) (2007), 248-72. Demopoulos, W. and M. Friedman (1985). "Russell's Analysis of Matter: Its Historical Context and Contemporary Interest." Philosophy of Science 52: 621-39.
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English, J. (1973). "Underdetermination: Craig and Ramsey." Journal of Philosophy 70: 453-61. Feigl, H. and G. Maxwell (eds.) (1963). Minnesota Studies in the Philosophy of Science VOl. 3. Minneapolis: University of Minnesota Press. Freudenthal, H. (1962). "The Main Trends in the Foundations of Geometry in the 19th Century." In E. Nagel et al. (eds.), 613-21. Frege, G. (1884). Die Grundlagen der Arithmetik) eine logisch rnathematische Untersuchung uber den Begriff der Zahl. Breslau: W. I(oeber; English translation by J.L. Austin: The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number (1980), second revised edition. Evanston, IL: Northwestern U niversity Press. Friedman, M. (1982). "Review of Bas van Fraassen's The Scientific Image." Journal ofPhilosophy 79: 274-83. - - - . (2002). "Geometry as a Branch of Physics: Background and Context for Einstein's 'Geometry and Experience.'" In D.B. Malament (ed.) (2002), 193-229. Friedman M. and R. Creath (eds.) (2007). The Cambridge Companion to Carnap. Cambridge: Cambridge University Press. George,A. (ed.) (1994). Mathematics and Mind. Oxford: Oxford University Press. Hallett, M. (1994). "Hilbert's Axiomatic Method and the Laws of Thought." In A. George (ed.), 158-200. Hempel, C.G. (1963). "Implications of Carnap's Work for the Philosophy of Science." In Schilpp (ed.), 685-709. - - . (1975). "RudolfCarnap, Logical Empiricist." In J. Hintikka (ed.) (1975), 1-13. Hilbert, D. ([1899J 1971). Grundlagen der Geometrie; revised and enlarged by P. Bernays and reprinted with an English translation by L. Unger as Foundations of' Geometry (1971). La Salle: Open Court. Hintild(a, J. (ed.) (1975). Rudolf Carnap) Logical Empiricist. Dordrecht and Boston: Reidel. Hochberg, H. (1994). "Causal Connections, Universals, and Russell's Hypothetico-Scientific Realism." Monist 77: 71-92. Lewis, D. (1984). "Putnam's Paradox." Australasian Journal of Philosophy 62: 221-36. Malament, D.B. (ed.) (2002). Reading Natural Philosophy: Essays in the History and Philosophy ofScience and Mathematics. Chicago: Open Court. Maxwell, G. (1963). "The Necessary and the Contingent." In Feigl and Maxwell (eds.) (1963), 398-404. McGuinness, B. (ed.) (1980). Philosophical and Mathematical Correspondence of Gottlob Frege. Translated by H. IZaal. Oxford: Blackwell. Nagel, E., P. Suppes and A. Tarski (eds.) (1962). Logic) Methodology) and Philosophy of Science. Stanford: Stanford University Press. Poincare, Henri ([1905] 1952). Science and Hypothesis. First English translation originally published by Walter Scott Publishing Co. Ltd., New York: Dover. Psillos, S. (2000). "Rudolf Carnap's 'Theoretical Concepts in Science. '" Studies in IIistory and Philosophy of Science 31: 151-72.
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Putnam, H. (1962). "What Theories are Not." In E. Nagel et al. (eds.) (1962), 215-27. Quine, W.V.O. (1951). "Two Dogmas of Empiricism." Philosophical review 60: 525-34. Ramsey, F.P. (1929). "Theories." In R.B. Braithwaite (ed.) (1960),212-36. Robinson, A. (1974). Introduction to Model Theory and the Metamathematics of Algebra. Second revised printing of the 1963 edition. Amsterdam: ElsevierNorth-Holland. Russell, B. (1946). History of Western Philosophy. London: George Allen and Unwin. Schilpp, P.A. (ed.) (1963). The Philosophy RudolfCarnap. LaSalle, IL: Open Court. van Bentham, J. (1978). "Ramsey Eliminability." Studia logica 37: 321-36. van Fraassen, B. (1980). The Scientific Image. Oxford: Oxford University Press. Winnie, J. (1970). "Theoretical Analyticity." In R. Cohen and M. Wartofsky (eds.) (1970) 289-305. Worral, J. and E. Zahar (2001). "Ramseyfication and Structural Realisn1." Appendix IV in E. Zahar, Poincare)s Philosophy: From Conventionalism to Phenomenology, Chicago and LaSalle, IL: Open Court, 236-51.
[22J The Construction of Reason:
IZant, Carnap, Kuhn, and Beyond RICHARD CREATH
In this essay my theme is human reason-I'm for it. But to defend reason we must have some understanding of it: of its shape and structure and of its limitations, both real and alleged. Specifically, I want to discuss an idea about the structure of our reason that has its roots in the eighteenth century and reached maturity in the twentieth. Especially in its mature form it seems one of the most provocative and promising developments that we have in philosophy. Naturally, an idea so bold is bound to have its detractors, so I shall address as well one common and in1portant objection. And I shall discuss a well-known alternative. The idea, the objection, and the alternative are each major themes of a recent book by Michael Friedman (2001), and I shall be addressing his book throughout. Indeed, he is the 'Beyond' in my title. In the opening section of my remarks we shall consider some very special parts of our intellectual commitments, such as geometry and arithmetic, and consider as well IZant's response to them in the eighteenth century. Second, I shall examine the mature form of what I(ant started. This is the idea that in some sense human reason is an ongoing construction of our own, that we have some choice in the n1atter. This idea might be called IZantianism with moveable categories. Third, I discuss the charge of relativism. Is this allegation as damaging as some would have us believe? And can Friedman's book show us a way to defend the core idea from the claim that it mal(es human reason, and science as a prime example, utterly incoherent? Fourth and finally, I explore a popular alternative, namely Quine's naturalism.
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I. The Idea: IZant and the Constitutive A Priori Our beliefs about arithmetic and geometry have always seemed a little weird. When we n1ix one deciliter of water with a similar volun1e of sugar to get one and one half deciliters of mixture, no one, regardless of philosophic or scientific persuasion, thinks that we thereby disconfirm the arithmetic tables. Logic seems weird too, for when we try to argue for a system of logic, we seem forced to presuppose it and thereby beg any interesting question. For the time being, however, I shall set the peculiarities of logic aside. The idea, one thread of whose history I want to trace, has its beginnings in the domains of geometry and mathematics. In order to account for the weirdness of these two, this apparent disconnect between these special beliefs and experience, rationalists like Plato and Descartes postulated a mysterious mode of access to a mysterious world of objects beyond the ordinary empirical objects of science. In order to avoid such mysteries I(ant suggested that the weird beliefs did not represent an otherworldly domain, but rather gave form and structure to ordinary empirical representations. This structure, he thought, is imposed by the human mind and so will hold for all of our empirical representations forever. Stripped to its essentials, the idea is that our beliefs fall into two tiers. One, call it the A-level, is not empirically tested in any straightforward way (for Kant, not at all) and so is in some sense a priori. The other level, B, presupposes the first for its (B's) intelligibility, identity, and testability. In this sense A constitutes B, and since the A-level beliefs are often called a priori, the whole approach can be called "the constitutive a priori" (De Pierris 1992). The constitutive a priori is an arresting and suggestive idea; it promises to provide a way of coping with the peculiarity of some of our beliefs without resorting to something even more mysterious. Note that I have drawn the distinction between the A and B levels as a distinction among judgments, and thus, in I(antian terms, entirely within the d0111ain of the understanding. Kant can and does recognize the distinction as I have drawn it, but he is also at pains to set it in a larger and more complicated framework. As Friedman himself argued in !(ant and the Exact Sciences (1992, 56ff.), geometry and arithmetic cannot be derived from the Aristotelian logic that is available to IZant, and the only other source of validity that IZant recognizes is intuition, that is, the domain of sensibility rather than the understanding. But IZant is also convinced that ordinary empirical intuition is insufficient to account for the necessity and independence of experience, what I earlier called the weirdness, of geometry and n1athen1atics. Since IZant correctly sees that diagrams are essential components of the geometrical proofs of his day, it is not hard to appreciate why he postulates a special, "pure" form of intuition corresponding to such diagrams as underlying the special validity of geometry (and mathe-
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matics). It is precisely in the realm of intuition, both pure and empirical, that the human mind imposes restrictions on our representations. And this has the effect of guiding what can be proved in a Euclidean demonstration, thus ensuring the A-level judgment of the correctness of Euclidean geometry, and also the effect of guaranteeing that all future empirical intuitions, and all future science, for that matter, will likewise conform to it. In the two centuries after the Critique developments in geometry, mathematics, and physics, especially in the form of general relativity, made IZant's assumptions difficult to sustain; they had to be revised. At the same time, the modern logic of relations developed to the point that at least the formal structures of geometry and mathematics might be derived fron1 this logic, either in its higher-order form or together with set theory. Some have thought that the rise of modern logic undermined the need for the IZantian strategy and that the application of non-Euclidian geometries undermined its tenability. The development of logic, however, emphasized rather than erased the weirdness of logic itself, and it did nothing to improve the prospects for the Platonist epistemology that I
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Einstein. Ironically, this would n1ake Quine's naturalism self-defeating. For reasons of time we will not be able to treat this part of Friedman's argument fully, but it is important nonetheless. It is not an easy book, and one might quibble with this or that. Indeed, I shall. But I want to emphasize from the outset my fundamental agreement with it, my agreement that exploring the constitutive a priori not only illuminates large stretches of the recent past in philosophy of science but also throws a strong and pron1ising light toward its future. We may proceed as Friedman does, namely, by exploring certain themes in the history of recent philosophy of science. I said earlier that developments in the nineteenth and early twentieth centuries in geometry, logic, physics, and so on undermined the traditional IZantian story. But even for a I
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role? This problem arises only if the various logics are genuinely different, that is, not mere notational variants of one another. By the 1930s, when Carnap was writing The Logical Syntax of Language (1934, 1937), this seemed to be the case. Fully assin1ilating the idea that there really are alternative logics is what brings the two-tier approach to its more mature phase.
II. The Idea Updated: IZantianism With Moveable Categories In looking at Logical Syntax, one n1ight well think that the main thesis of the book is that all philosophy is logic or, in the terminology of the book, syntax. But this is, by itself, little more than the idea that insofar as philosophy is viewed traditionally as a priori, it is here viewed as part of the constitutive a priori. As the "Foreword," "Principle of Tolerance," and Carnap's careful examination of alternative logics makes plain, however, what is really innovative about the book is the principle that there is no uniquely correct logic. Logics are in effect languages, for which the notion of correctness does not apply. The choice among these languages is conventional, but it does not follow that the choice is a matter of indifference. Alternative conventions will differ in their pragmatic utility. Our choice, if wise, will be useful to us. Even the limits of these linguistic conventions are conventional. Since all philosophy is logic, and there is no uniquely correct logic, there is no uniquely correct philosophy either. Philosophic "systems" are best viewed as proposals for structuring our language, our scheme of representation, in a specific way. What happens to the constitutive a priori in all this? The B-Ievel is still constituted by the A-level, that is, by the language. Now, however, we have a choice of A-levels, a choice that is conventional, though it can be guided pragmatically. This is what I(uhn would later call "IZantianisn1 with moveable categories."l When I say that the A-level constitutes the B-level, I mean simply that B-level claims presuppose those of the A-level. Presupposition is asymmetric here. Moreover, a given B-level claim is the claim it is only by being so-structured by the A-level. Of course, every claim is in some language, and we can make that more explicit by saying that each claim is relativized to a language . Now since a language is not something that we even can be right or wrong about, this relativisn1loses its sting. This is the payoff for the linguistic turn. There is nothing subjective here, and the relativization is not to a culture but to an abstract structure. A judicious choice for an A-level, Carnap thinks, would include an empiricist structure. Thus, empiricism is a convention, albeit a wise one. An empiricist structure is one in which all claims not settled by the A-level
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alone would be appropriately tied or tieable to observation reports. Carnap calls the ties correspondence rules. Others have called them by other names, including operational rules (P.W. Bridgman), a dictionary (Norman Campbell), and coordinative definitions (Reichenbach); the differences here are immaterial for present purposes. What ties would be appropriate and what the form and structure of observation reports should be requires further engineering decisions. Presumably, the ties should include measurement and testing procedures for quantitative and theoretical notions. Presun1ably as well, a wise choice of A-level would involve a sufficiently rich conceptual scheme to express general and comparatively simple theories of the world. By 1936 Carnap publicly recognized that major scientific changes such as the advent of the general theory of relativity often involve meaning change such that the new theory is not translatable into the vocabulary of the old (1936). Carnap was to develop this view over the following decades, but in essentials it remained stable. The chief challenge to it came, as everyone knows, from W.V.O. Quine (1951), a challenge addressed specifically to Carnap's distinction between analytic and synthetic claims. Quine doubted that behavioral or other empirical criteria could be provided for analyticity, so what Quine is demanding is that analyticity satisfy an empiricist criterion of significance. If Quine is right, one could not go to an actual community, such as the community of physicists in 1916, and discover that some sentences in that community were analytic while others were synthetic. I think that to the extent that it is legitimate Quine's challenge can be met. But this is neither the time nor the place to do that. Friedman, who respects both Quine and Carnap, as indeed I do as well, expresses some doubt as to whether Carnap can satisfy Quine's demands (Friedman 2001, 41). There is enough difference, however, between what Friedman says Carnap cannot do and what I say he can that it is unclear whether there is a genuine issue between us. There is at least a little reason to think that we might agree. If I am right in characterizing Quine's demands, and Quine is right in thinking that they cannot be met, then a traveling anthropologist could not discover for our chosen community which claims within it are among the constitutive a priori either. So if we have historical evidence that allows us to say reasonably that certain claims were a priori, Quine's challenge must in principle have been met. Friedman thinks that he has such evidence, and on this aspect of the case, I am inclined to agree. In any case, Friedman was among the first to recognize and appreciate that the central message in Carnap's philosophy involves the idea of a constitutive a priori with a moveable A-level. Friedman has also been able to articulate, perhaps more forcefully than anyone else in our time, the advan-
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tages that this approach can have as we try to frame an adequate understanding of science and why the various statements of our current science have some claim on our belief. This does not con1n1it him to agree with everything that Carnap says, only that here is a central idea that is worth developing. Another figure of central importance for Friedman is Thomas IZuhn. IZuhn started as a physicist and then as a historian of science, eventually backing his way into some very provocative philosophic work. His historical research (1957) into the Copernican revolution suggested two ilnportant conclusions, which he expressed in Structure. First, at certain decisive periods in history different parties to scientific debates often talked past one another. They did not find each other intelligible, and they did not agree on what counted as relevant evidence. It was as if they used different languages, the same words perhaps, but with different n1eanings, even in the description of the evidence itself. Second, IZuhn's research also suggested something that on Quine's view IZuhn could not possibly discover or even have evidence for. This is that many of the commitments of actual scientists do not meet experience in the same way that their more mundane beliefs do. These special commitments are revisable but not in the short run. For the time being they are not testable; they are effectively a priori-for now. These commitments IZuhn calls a paradigm. There is, moreover, a connection between his two results. When the a priori commitments change (and this can happen in some sense under the pressure of the evidence) so do the meanings. That is to say that when some change these basic beliefs, those who do not change find those who do change difficult, if not impossible, to understand. There is an obvious structural analogy here to Carnap's work. The IZuhnian paradigm plays much the same role as Carnap's languages. We get meaning change in scientific revolutions because paradigms/languages change. Both are versions of a constitutive a priori with an A-level that is revisable on broadly pragmatic considerations. These important similarities were not always obvious. The similarities are consistent, however, with there being important differences. Transparently, IZuhn is Inore sociological and psychological, and he develops particular cases historically. Carnap is much more abstract, even meta-mathematical. Given the differences, the similarities are even more surprising. That philosophers as influential as Carnap and IZuhn (as well as several other important twentieth-century philosophers of science) gave us versions of the constitutive a priori suggests that we ought to take that approach very seriously: to explore both the opportunities it affords and the potential problems it poses.
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III. The Objection: Relativism One such problem is relativism. Early in IZuhn's career, and here I am thinking specifically of Structure, it seemed to many of his readers that, because paradigms contained standards for appraising lower level theories, there neither would nor could be standards of rational appraisal for the paradigms themselves. Some took this as an invitation to declare science no better than superstition. The Ouija board and the Hubble telescope are equally good ways of finding out about the world. They are just instruments from different paradigms with their different standards of appraisal. Let us call this, so as not to beg any questions, "irrational relativism"; there may be other kinds of relativism, but we will set that question aside for the n1oment. Kuhn was appalled (1970, especially 259-64) that anyone would use his work to reach such a conclusion and so was at pains to counteract it. There were, he said, universal standards by which paradigms could and should be measured. These include predictive accuracy at the observation level, fruitfulness, and simplicity. This raises difficult questions about the epistemic status and origin of these universal standards. Are they results of deep but mysterious metaphysical insights into the nature of reality a la Descartes and Russell? Are they permanent features of the human mind a la IZant? Friedman suggests in Dynamics ofReason that there are in fact no such universal standards (Friedman 2001, 51-52). Quantitative accuracy ofprediction, he says, appeared as a norm in physics only with the scientific revolution. And simplicity as a norm in physics, he suggests, appeared only with the advent of the general theory of relativity. So the version of IZuhnianism that Friedman wants to address is the earlier one in which the availability of universal standards for appraising the "correctness" of paradigms is not being assumed. Friedman is not about to accept irrational relativism, so we need to see whether it can be avoided. The only way this can be done, he thinks, would be if, in the context of a revolutionary period, one and only one of the paradigms that arise therein becomes the rational choice. Not any old unique outcome will do. If it results from mere herd instinct, that is, from the workings of fashion, science would be irrational. The same if it results from pure power politics. Of course any human interaction has a political dimension. There are those who read IZuhn's acknowledgement of this fact as saying that all major scientific change is a matter of the new theorists gaining control of the journals, granting agencies' and other levers of professional power and using them to squeeze the older theorists out of the profession to the point that they die without reproducing their kind. I suppose some of that does go on. But the real issue is not whether there are political aspects to a scientific life; of course
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there are. The real and pressing issue is whether there is also a rational dilnension that makes the choice among paradigms a reasoned one. But what kind of rationality is available for the purpose? Universal standards have already been ruled out. Here Friedman turns to Habermas (53 ff.). A distinction is drawn between instrumental rationality, which is practical reasoning concerning means and ends, and communicative rationality, which is some sort of non-coercive reason born of serious discussion (communication) in a comn1unity of free and autonomous agents. The non-coercive aspect is to rule out unalloyed power politics, and that this aspect is to be rational rules out the vagaries of fashion. Friedman is doubtful about the ability of instrumental rationality to provide the required agreement. Our interests are just too varied and idiosyncratic. But he is hopeful about communicative rationality. Is there any reason to think that there is any such thing or that it could yield agreement? At this point Friedman appeals to IZuhn who claims that, as a matter of historical fact, periods of crisis and revolution are rather brief and have as their outcome complete agreement on a paradigm and cessation of methodological disputes. If the argument were left at this point, I do not think it should be very convincing. Of course, Friedman does not leave it here, but to see why what he adds is important, we need to probe the weaknesses of what we have seen. To begin with, the sort of relativism that the constitutive a priori approach faces is not as problematic as one might suppose. Remember, the A-level constitutes the B-Ievel, so (to put it in semantic terms) there is not a proposition that the one framework affirms that the other framework denies. For both Carnap and IZuhn, there has been a change of meaning of the sentences involved. Nor are we forced to say that using the Ouija board is just as good as using the Hubble telescope. If the Ouija board is the only methodological principle, not only could we never learn what it said (ordinary observation having been denied us), but the words of its "results" would be so far from their ordinary meanings that they would neither offer characterizations nor be about anything at all. The idea that it could provide answers on any topic that we might care about is parasitic on the assumption that it is being used in conjunction with more prosaic methods that serve to fix the referents and give some content to the descriptions. But on this assumption the Ouija board will be in such massive conflict with those other methods that it will be even n10re worthless than an outright guess. At this point, though our interests do differ, instrumental rationality is more than enough to rule out the sillier forms of superstition in comparison with the methods of modern physics. And it is enough to rule out many of the conclusions of the more extreme forms of social constructivism. In a sense, what I am arguing is that Friedn1an's own basic position is less open to charges of irrational relativism than Friedman himself fears.
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I have just noted that instrumental rationality can be a powerful tool. Friedman minimizes such rationality by saying that our interests differ. This is true, but not I think decisive. Your tastes in food may differ from mine, but I suspect that nearly all of us would prefer a dinner at a Michelin three star restaurant to a sharp stick in the eye. Interests differ but not always radically. Where the differences in interests are extreme, as between sybarites and masochists, it is difficult to form a con1munity in order for communicative rationality to get off the ground. Where the variations are more modest, the same or similar methodological principles may be useful to everyone alike. On the whole, it seems likely that accuracy of prediction and ease of forn1ulation and calculation will in general be handy to all but a few. Now I cannot guarantee that uniformity of methodology will be the result, but I cannot guarantee either that everyone will be rational, whether that rationality is instrumental or of some other sort. We now have detailed accounts of instrumental rationality, though no doubt we are well shy of a final theory. By contrast, the account of communicative rationality that Friedman gives us frOll1 Habermas is rather thin. 'Communication' and 'comn1unity' are freighted words, seriously in need of explication. And we need help as well in deciding whether that which results from discussion is this special form of rationality rather than an instrumental accommodation or, less respectably, the workings of fashion and coercive politics. The mere fact that revolutions end does not say how they end. Nor should we take the claim that they do end in unanimity as a given. Remen1ber that Einstein never won the Nobel Prize for his work on relativity; there were too many holdouts against it. Kuhn says that revolutions are brief and quickly followed by normal science during which there is no debate on methodological issues. IZuhn's critics have emphasized that if that is what normal science is, then there is no normal science. No period has been free from debate on fundamental methodological issues. Nor should we be too confident of the speed of such revolutions as do oc:cur. It took eighty or more years for the Copernican heliocentric model to be widely accepted even by the vanguard of scientific opinion. And Newton's theory of universal gravitation was widely resisted on the Continent until about 1750. I have no idea how long it will take the Darwinian theory of natural selection to penetrate all the way to Crawford, Texas. The net result of these reflections, to put it bluntly, is that if Friedman's argument were left at this point, the appeal to communicative rationality would be an ineffective solution to a problem that may not arise. Of course, Friedman does not stop at this point. What he gives us in addition is some very detailed historical analysis of revolutionary episodes, most notably, episodes surrounding the introduction of the general theory of relativity. What the history shows is that the constitutive a priori approach
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can apply in detail to and illun1inate the reasoning involved in the revolutionary changes. Because the changes of theory were so dramatic and the constituting frameworks so fundamentally different, here if anywhere there should be a failure of understanding, a failure of communication, and a breakdown of the rational enterprise. When we examine the cases in detail, I shall argue, understanding does not fail. It is difficult, but it does not fail. It is not achieved by translating one framework into the other or both into a supposedly neutral third fralnework. Translation is just the wrong.model of understanding. Quine is surely n1isguided in thinking that it is the only model available. In saying this, I an1 coming down far short of what IZuhn is often taken to say, namely, that scientists from different paradigms cannot understand one another. I think that what IZuhn is more likely to have meant is that there is no exact translation between the theoretical terms of the respective theories and that moreover the changes in meaning also infect the most basic observational judgments. There is thus no theory-neutral observational vocabulary in which the empirical content of the two theories can be appropriately expressed. If this is all that IZuhn meant, I see no reason to deny it in order to argue that we can sufficiently understand one another, and that partisans of different paradigms can sufficiently understand one another, to allow the rational appraisal of the alternative theories to proceed. 2 What one can do, and humans are fairly adept at this, is to see what one's interlocutor takes as a reason within his or her own framework. Analogously, a chess player can come to understand the rules of checkers without translating them (whatever that would be) into the game of chess. This is facilitated on one side by the fact that the practitioners of the new framework are, so to speak, bilingual; they were for the most part also practitioners in the prior framework. In turn, the new framework can be presented to the traditionalists by taking models that they already accept and extending or truncating them in various ways to yield a new model. Indeed, the new framework itself n1ay have been developed by taking resources developed in the old fran1ework and extending or otherwise modifying them. In just this way Einstein used and reshaped ideas of Poincare. This is greatly aided if all or part of a prior theory can be mapped into the new, as when the Riemannian physical space of non-constant curvature from general relativity is said to be Euclidean in the small, that is, as the extent of the region described gets arbitrarily close to zero. Another example is that relativistic laws of motion converge to the classical laws when the velocities are small in comparison to the velocity oflight. Perhaps Friedman is optimistic in thinking that the old framework will in general provide descriptions that find their place as special cases in the new framework and theory. I doubt that this always or even generally happens. But still it does sometimes happen, and when it does, it aids mutual under-
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standing, so Friedman is right to point it out. IZuhn, Feyerabend, and Sellars were always keen to insist that even in these special cases we do not have exact agreement because of meaning change. But here I am not talking of exact translation or even truth. What is at stake here is a structural analogy between the inference rules applicable within a limited domain. The grasp of this analogy is often the opening wedge for a lin1ited but sufficient form of n1utual understanding. Only where this is possible to a substantial degree is one forced to choose among the frameworks. Given a measure of n1utual understanding, this choice is possible in a rational way. I can come to see what my interlocutor takes as evidence and what the relevance of that evidence should have according to his or her own standards. If necessary I can then find evidence that the interlocutor takes to be relevant. I can come to see the overall structure of the alternative theory as well as inconsistencies or tensions between the framework's ambitions and the theories to date, again by the alternative's own standards. Of course, even in revolutions as massive as that surrounding relativity there is often a considerable overlap in the standards of appraisal being used. They do not have to be universal and timeless; all we need are sufficient standards that apply at least locally to move the argument forward. Friedman's historical narrative is, in effect, an explication of the role of reason within the particular historical context in which these revolutions take place. There are, of course, many other ways of developing and using a history of science. One might, for example, be concerned with the outcomes of scientific research as when philosophers have tried to show that those outcomes are generally contradicted by later ones or contrariwise to show that they are generally incorporated as special cases into later ones. Friedman is concerned, not so much with the outcomes, but with the force and structure of the reasoning within the particular contexts he examines. No wonder he calls his book Dynamics ofReason. What Friedman's detailed historical narrative does then is to show that even major revolutions can be, so to speak, rational in the small. That is to say, if we look closely enough at the understanding that the various parties had at any given moment of one another and the arguments they used to incline one side to a change of position and then of framework, that process is shown to be rational. So the historical narrative does real work for Friedman's account. The history is not an illustration of his argument that we are not forced into the relativists' and irrationalists' dead end; it is the argument itself-a powerful one. I am under no illusion that even a successful defense of the constitutive a priori against any dangers that relativism or irrationalism may pose shows that it is right or even the best available strategy. One would at least have to consider the most prominent one-tier alternative, namely, Quine's.
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IV. The Alternative: Qlline's Naturalism We come then to another major theme of Friedman's book, to an evaluation ofW.V.O. Quine's one-tier approach to our epistemic commitments. In "Two Dogmas of Empiricism" (1951) Quine developed a view according to which all our beliefs meet experience together. This body of belief includes logic and mathematics, methodology, physics and biology, and all the most casual matters of history and personal biography. There are no distinctions of level. When experience contrary to expectation happens, as often it does, we must revise. But all our beliefs are in it together, and each is open to revision. We can make our revisions in various ways; indeed, Quine says that we can save any of our beliefs that we choose, ifwe are prepared to make drastic enough changes elsewhere. Other than empirical adequacy, the only two principles guiding our choice are conservatism (we want to preserve as much of our prior belief as possible) and simplicity (we want to have the most economical system possible). Simplicity and conservatism often pull in opposite directions, and where they do, simplicity generally triumphs. That is Quine's one-tier view, virtually in its entirety. It proves both disarmingly subtle and remarkably powerful. Simplicity, though it is not much articulated, is used to defend both physical realism and theoretical realism. And conservatism is used to show why logic and mathematics seem to be immune to empirical disconfirmation. Logic and mathematics are central in the sense that changing one of our logical or mathematical beliefs requires changes in many other beliefs as well. Since we can save any of our beliefs and since, by conservatism, we want to change as few as possible, we will in general not even consider revising our logic or mathematics. They thus have the appearance of not meeting empirical test, and they become so deeply entrenched as to seem certain. Of course, according to Quine, they do n1eet empirical test like any other claim; they are just more entrenched and more resistant to change. This is the view that Friedman calls Quinean naturalisn1. Though 'naturalism' did not enter into Quine's own vocabulary in any systematic way until the late sixties, thereafter he was happy to use it as a general term for all of his views, that of "Two Dogmas" included. Besides a handy general term that Quine acknowledged, there is something to be gained by calling the view naturalism. Quine introduces the term in his famous paper "Epistemology Naturalized," where he tells us to reject Carnap's rational reconstruction as "make believe" and to "settle for psychology" instead (Quine 1969, esp. 75). Presumably he means to reassure us that his own epistemological writings are meant as faithful descriptions of the actual psychological processes whereby we initiate or alter our beliefs. The original formulations in "Two Dogmas" can be and have been read either as
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descriptions of our actual habits or as prescriptions for habits that we ought to acquire. These are quite different interpretations, and different arguments would be relevant to them. In calling his view naturalism, Quine seems to come down clearly on the side of the descriptivist reading. And in calling the subject of his discussion "Quinean naturalism," Friedman is underscoring the authoritativeness of this interpretation of "Two Dogmas" and signaling his intention to evaluate the view with arguments relevant to Quine's view so understood. If Quine's one-tier approach is taken as a description of actual processes, then it is utterly legitimate to turn to history to see whether major scientific advances are actually in accord with the proffered description. Friedman again turns to the history surrounding Newton and Einstein and concludes that Quine's model is not a satisfactory description of the actual events. The specific target of Friedman's attack is the idea, clearly implicit in Quine's writing, that our body of belief is a conjunction, each part of which could be altered or removed, leaving the others intact. The vast bulk of Friedman's argument is designed to show that one part of our belief in fact presupposes another in a serious way. In Newton's case, without the mathematics of the calculus, the mechanics would be unintelligible and even unstatable, and without the mechanics, the theory of gravitation would have no empirical meaning. For Einstein, the general theory of relativity presupposes the principle of equivalence, and that in turn presupposes Riemann's mathematical theory of manifolds for intelligibility and empirical meaning, that is, for testability. Friedman's argument is a powerful one, with an impressively meticulous grasp of the scientific details. Unfortunately for his overall project, Quine would have available to him a reply that is also powerful. He would, I think, agree that the theory of universal gravitation and the general of relativity do indeed presuppose the corresponding mathematical theories for their intelligibility and testability (though he might not distinguish the intelligibility from the testability). Having conceded what seemed.to be Friedman's central point, Quine would then insist that the mathematics is neither to be really understood nor tested apart from the non-mathematical theories. Thus, the presupposition goes both ways, and symmetry is restored. Friedman does not explicitly consider such a reply, but he does give some relevant hints. The first is quite explicit: It is a profound n1istake of Quinean holism, however, to view this procedure [of combining the mathematics, physical laws, and correlations between them and deriving observational consequences] as empirically testing the other two [non-mathematical] parts in the same way. For, in the first place, it is clear that the mathematical part of our theories, considered independently of the empirical. application in question is in
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no way empirically tested by such a procedure; .... Thus, for example, the theory of Riemannian manifolds is used. in formulating both general relativity and versions of classical mechanics (in Hamiltonian formulation). But the theory of Riemannian manifolds itself is empirically tested by neither of these applications (each of which employs its own characteristic coordination); it remains a purely abstract description of certain mathematical structures, ,,,,hose distinctive theorems and principles are justified purely mathematically. (2001, 80) . This last sentence is not otherwise supported, but presumably it is not meant as a mere philosophic assertion balancing Quine's equally unsupported counter-assertion. Since the overall passage is historical, one tal(es it that Friedman is reporting that the participants of the then current debates took the mathematics to be justified independently of any empirical results. In short, they treated the mathematics and the physical laws asymn1etrically. I think the idea that we should look to community standards of justification in order to determine whether the epistemic system that is actually in place has one tier or two is exactly right. Unfortunately, it is not a suggestion that we can develop further in these pages. In addition to this Friedn1an ren1inds us that Quine's account of the apparent independence of the n1athematics from any empirical results simply does not apply in either the Newtonian or Einsteinian context. The mathematics is too new and controversial to be entrenched. And because the physical theories that would be disturbed by disturbing the mathematics are also new, conservatism is not a relevant reason. Plainly, what Quine is offering is not a straightforward description of actual science, but rather a somewhat imaginative reconstruction, rational or otherwise. Note that Friedman hin1self need not claim that Newton thought about his enterprise in IZant's way nor that Einstein thought about his enterprise in Carnap's or Friedman's way. Given what Newton, Einstein, and their contemporaries did say about their respective theories and the justification thereof, Friedman needs only to point out that his own reconstruction is closer to the phenomena than is Quine's. He could argue in addition that his own two-tier account is more nuanced, more detailed in its application to actual science, and clearer than anything that Quine had offered for simplicity and the like. Quine would, of course, have been free to ignore the actual practice of science and frankly recommend that it be altered. To do this, however, he would have to abandon the banner of naturalism and admit that his one-tier approach, construed descriptively, was a failure. For Quine seriously to have followed through on a project of rational reconstruction, he would first have had to articulate his central notions, such as sin1plicity, more fully than he ever did and also to give an argument that the alteration thus recommended is an improvement over
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actual practices and over alternative reconstructions. In fact what Quine seems to have done instead in his later decades is to embrace a weak form of analyticity (1973, 78-80), demote simplicity to the point that it can no longer help select an10ng alternatives before us (1960, 19), and to move cautiously toward a two-tier account of confirmation (1990, 14).
v.
Conclusion
In this essay we have been discussing a picture of human reason. We have seen the structure it was said to have by I
NOTES 1. IZuhn used the phrase in the presentation of his Presidential Address to the 1990 Biennial Meeting of the Philosophy of Science Association. When the lecture was published, the phrase was omitted, but the idea was plainly there, including the explicit reference to Kant (Kuhn 1990, 12). 2. Thus far, I am in agreement with Scott Tanona's very interesting paper in this volume. There are still some differences of strategy. For him, the appraisal of the new theory progresses by correlating the claims of the new theory with obser-
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vations laden by the old theory. I, in contrast, see the process of understanding and appraisal as involving a kind of semantic ascent and pragmatic evaluation. In saying that scientific rationality can proceed in this way need not imply that it cannot do so in the way Tanona suggests. I simply do not know without looking at cases whether a failure of a new theory to correlate with the "old" observations ought always to be taken as a decisive objection against it.
REFEREI CES Carnap, R. (1934). Logische Syntax der Sprache. Vienna: Springer. - - - . (1936). "Wahrheit und Bewahung." Actes du Congres internationalede philosophie scientifique, Sorbonne, Paris 1935. 4:18-23. - - - . (1937) The Logical Syntax of Language. Translation by A. Smeaton with additions of Carnap (1934). London: Kegan Paul Trench, Trubner & Co .. Creath, R. (ed.). (1990). Dear Carnap, Dear Van: The Quine-Carnap Correspondence and Related Work. Los Angeles: University of California Press. De Pierris, G. (1992). "The Constitutive A Priori." Canadian Journal of Philosophy, Supp. Vol. 18, 179-214. Fine, A., et al. (eds.) (1990). PSA 1990: Proceedings of the Biennial Meeting of the Philosophy of Science Association. East Lansing: Philosophy of Science Association. Friedman, M. (2001). Dynamics ofReason. Stanford, CA: CSLI Publications. - - - . (1992). Kant and the Exact Sciences. Cambridge, MA: Harvard University Press. Kuhn, T. (1957). The Copernican Revolution. Cambridge, MA: Harvard University Press. - - - . (1962). The Structure of Scientific Revolutions. Chicago: University of Chicago Press. - - - . (1970). "Reflections on My Critics." In 1. Lakatos and A. Musgrave (eds.), Criticism and the Growth of Knowledge. Cambridge: Cambridge University Press, 231-78. - - . (1990). "The Road Since Structure." In Fine, et al. (eds.) (1990), Vol. 2, 3-13. Quine, W.v. (1934). "Lectures on Canup". In Creath (1990), 45-103. - - - . (1951). "Two Dogmas of Empiricism". Philosophical Review 60: 20-43. - - . (1960). Word and Object. Cambridge, MA: MIT Press - - . (1969). "Epistemology Naturalized." In Quine (1977), 69-90. - - . (1973). Roots ofReference. La Salle, IL: Open Court. - - - . (1977). Ontological Relativity and Other Essays. New York: Columbia University Press. - - - . (1990). Pursuit of Truth. Cambridge, MA: Harvard University Press.
[23J How Should We Describe Scientific Change? Or: A Neo-Popperian Reads Friedman NORETTA IZOERTGE
1. Introduction Michael Friedman's project is to "IZantianize" IZuhn. In this essay I will compare that approach with the efforts of Imre Lakatos and others to "Popperize" IZuhn. By drawing parallels we may be able to improve both accounts. But it may turn out that we decide that Friedman's dynan1ics of reason best describes only the special case of mechanics. Ernan McMullin has remarked that mechanics is not the only-and perhaps not the bestexample to look at when we study the structure of science'! IZuhn's Structure of Scientific Revolutions (1962) was enorn10usly successful in introducing philosophy of science to academic culture. (When I arrived at Indiana University in 1970 it was being used as a textbook in seven different courses, including one in the School of Nursing.) The idea of pre-paradigm science was attractive to social scientists who worked in areas where there was little agreement on basics. The idea of normal science in which scientists solved problems by using exemplars looked like a helpful idea for science educators. For those who viewed science as a boring, monolithic, juggernaut, it was exciting to learn that the whole system could end up in a state of crisis that could result in revolution. And IZuhn's explicit denial that science was progressing towards anything like an ultimate truth meant that science was not as epistemically privileged as some had claimed. IZuhn also posed serious challenges to the existing philosophical attempts to account for the growth of scientific knowledge. Let us briefly consider a few philosophical ideal types and encapsulate their reactions to IZuhn's monograph:
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To a naive inductivist, who believes that science progresses by accumulating and systematizing facts, any suggestion that revolutions are a recurring feature of science is shocking. Any substantial revisions are indications of methodological incompetence. A traditional hypothetico-deductive (H-D) type, who believes that science grows through the testing of hypotheses, is shocked at how thoroughgoing IZuhnian revolutions turn out to be. It is not surprising when some of our speculations turn out to be false, but at most, only the top layer of what Feyerabend characterized as the layer-cake model of science should be impermanent. The bottom layer of observations and the middle layer of laws should be accumulative. To a traditional synthetic a priorist, who, like IZant, believe in a unique, basic conceptual framework, only the Einsteinian revolution is shocking (the other upheavals IZuhn describes are viewed as occurring in the empirical realm). To a Quinean, who views knowledge as a web of beliefwith logic in the center and observations at the periphery, it is mildly surprising that such big chunks of the web are ripped out at once. The normal expectation is that revisions will conserve most of the network connections. On the other hand, to a radical Popperian, who calls for bold conjectures and rapid refutations, what is shocking about Kuhn is his documentation of the importance of normal science. (The motto of Popper's Conjectures and Refutations was "revolutions in permanence.") To everyone except some conventionalists and constructionists, who were already prepared to explain scientific change in terms of convenience or social negotiations, IZuhn's account of the arational nature of theory choice was shocking.
2. Comparing Accollnts of Structure/Stratification Friedman's response (if I may be permitted to continue to paint with a very broad brush) was generally sympathetic to IZuhn's account of the development of science as a series of paradigm shifts. However, he would revise IZuhn's account in two crucial ways: First, he offered an alternative to IZuhn's quasi-psychological/sociological notion by postulating that the paradigms governing normal science contain at their core "relativized yet still constitutive a priori" principles (2001, 71). These principles make normal science possible but occasionally hav~_~~_Q~_g.Lsm~l1tl~d_and _
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revised, a revolutionary process. Secondly, Friedman, unlike IZuhn, emphasized that the radical conceptual revolutions that sometimes take place in science exemplify rational discourse of the most sophisticated variety (2001, xii). Although they come from quite different philosophical traditions, there are some striking parallels beniVeen Friedman's IZantian-inspired approach and Lakatos's Methodology of Scientific Research Programmes account (MSRP). It will be instructive to compare them. First, some background on Lakatos's MSRP, which seems to have grown out of his participation in a 1965 Conference held at Bedford College in London. 2 Although Popperians could mount telling arguments against IZuhn's psychologistic approach and his wishy-vvashy attitude towards the truth content of scientific claims, Lakatos felt that Popperians did have to come up with an alternative account of the phenomenon of normal science . .Although he first presented his MSRP as a friendly amendment to Popper's methodology, it turned out to be quite different. John Watkins (1958) had already emphasized the role in science of what he called "irrefutable, but influential metaphysics," and Popper had often described how metaphysical views posed S0111e of the problems which scientists set out to solve. 3 Popper had also rather reluctantly acknowledged that, given the Duhemian problem of deciding where to pin the blame in the event of an empirical refutation, it might be necessary to have some ten1porary dogmatism enter into scientific inquiry (1972, 30). And, notoriously, he described the so-called Rationality Principle that served as the basis for his account of social science as so indispensable that we should never challenge it even though it is probably false! Thus we render the principle that people act appropriately to their problem-situations as they perceive them immune to revision, even in the cases of madness or perseverative behavior. Popper felt that the heuristic value of the assumption of minimal rationality was so important that we should always try to explain away apparent deviations from it. (See Popper 1967 and the discussion in IZoertge 1979.) So Lakatos set out to describe the way metaphysics operated in science and to explain the rationality of certain kinds of dogmatism. He postulated that science progressed through Research Progran1mes, which were characterized by what he called a Metaphysical Hard Core that was accompanied by a Positive Heuristic. The semi-permanence of the Hard Core was ensured by a Negative Heuristic that directed the arrow of modus tollens away fron1 the Hard Core and toward the Protective Belt of auxiliary hypotheses. (See Lakatos 1970 and the analysis in IZoertge 1995.) Clearly there are some striking functional similarities between Friedman's relativized, constitutive a priori principles (RCAP) and
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Lakatos's Hard Core. Both tell us about the fundamental entities of the domain we're studying and both are logically prior to the empirical research on which we are embarking. But their philosophical status is not the same. Lakatos's running example is Newtonian mechanics. However, unlike Friedman, he includes the law of universal gravitation as part of the Hard Core. Was this just a historical mistake? After all, commentators have pointed out that scientists did fiddle with that law and so it wasn't so "hard" after all. I think it was not a mistake. What this example shows is that Lakatos conceives of Hard Core conjectures merely as plausible ways to describe the entities and processes we are studying, not as axioms or principles that define or constitute the don1ain. The empirical law of gravitation gets into the Hard Core because it animates the models of bodies in motions that we want to test. Later it serves as an analog for the development of other force laws. Thus it plays a variety of crucial roles in the Newtonian Research Programme. This interpretation is supported by the way Lakatos and his students talk about competing Research Programmes. 4 The wave and particle theories postulate con1peting ontologies; proponents argued about which accounts are better at explaining what we already know about light and better at suggesting new experiments (cf. Worrall 1976). But I doubt ifit would illuminate the debate if we were to characterize either conception as definitive of what we mean by light or as constituting an a priori space of possibilities. But maybe I'm wrong. I guess I might simply pose the question to Michael Friedn1an: Which Lakatosian Hard Cores could be, or should be, re-interpreted as RCAPs? Or, to leave Lakatos aside, which debates about competing lines of research (wave vs. particle, Stahlian principles vs. Lavoisierian elements, circular vs. non-circular motion, nature vs. nurture, steady-state vs. expanding) should be analyzed in terms of incompatible RCAPs and which should simply be construed as disagreements within a framework about how the world works? If Friedn1an's notion of ReAP only applies to notions like space and time, then he will not have the analytical resources to discuss other kinds of paradigm shift.
3. Retrospective accounts The fact that Friedman describes the most basic parts of a scientific theory as constitutive and a priori (relative to the study at hand) might lead us to anticipate that incommensurability issues would loom large when we try to con1pare theories. His favorable gloss on Habermas's conception of communicative rationality (although he faults Habermas for not insisting on its applicability to science) and his requirement of "prior agreement on a constitutive framework" (Friedman 2001, 95) heightens these concerns.
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1 will not attempt a detailed summary of Friedman's retrospective account here but simply remark on the fact that it hinges on the mathematical component of theoretical systems. He describes space-time-motion theories as consisting of "a mathematical part, a mechanical part, and a (properly) physical or empirical part" (2001, 79). The mathematical part plays a crucial role in describing the relation of a current theory to its predecessor through devices such as specifYing limiting cases. But what resources can we use when talking about non-quantitative theories? An obvious candidate is the work done on the idea of a General Correspondence Principle (see Post 1971 and Koertge 1968), which was also utilized by John Worrall (1996) in his writings on Structural Realism. One thereby compares theories by looking at their explanatory structures. A5 an example, let's look briefly at the phlogiston account of certain chemical reactions and see what is retained by the oxygen account. 5 The ontologies in the two theories are very different. The phlogistonists talked about active principles that did not necessarily have weight. (I should note, however, that despite Lavoiser's emphasis on weighing products and reactants, his list of elements included caloric and light.) The explanatory success of the phlogiston theory came from the relationships it drew between chemical reactions. When a metal is "analyzed" using the instrument of fire, it loses its metallic properties as phlogiston escapes and turns into a powdery calx. To restore or "synthesize" the metal, we must add phlogiston to the calx. The process of the combustion or "analysis" of charcoal indicates that it is rich in phlogiston. So according to the phlogistonists, it comes as no surprise that metallurgists use charcoal when they extract metals from ores. Stahl used this schema of analysis-synthesis pairs for other reactions. His student Johann Juncker expressed the basic idea as an "indubitable" axiom: "[AJ body is composed of the materials from which one forms it and into which it is resolved" (quoted in Koertge 1968, 368). They emphasized that both analysis and synthesis were necessary. Analysis alone is not reliable-the starting materials may be impure, leading us to infer too many constituents; on the other hand, some of the principles may be lost in the process of analysis, e.g., if they combine with a solvent, thereby leading to errors of omission. Thus their insistence on using the joint methods of analysis and synthesis. Stahl thought that when sulfur was "analyzed" by fire, forming vitriolic acid, phlogiston was released. To "complete the proof' of his account of the composition of sulfur, Stahl needed to re-synthesize it by adding phlogiston to the vitriolic acid. He finally succeeded using a series of reactions that involved charcoal (Koertge 1968, 375-85). A5 is evident in Figure 1, Lavoisier's account reversed Stahl's assignment of reactions as analyses and syntheses. Calcination and combustion
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became syntheses, not analyses. But parts of the explanatory structure of the phlogiston theory were retained in the new theory. The relationship is not as crisp as the ones that figure in the history of mechanics, but it does illustrate how we can give a retrospective account of a transition between qualitative theories. And I find this account much more illuminating than one which would focus on the ReAPs, such as Juncker's axiom about the constituents of mixt bodies or Lavoisier's axiom that the quantity of matter is conserved. The most telling differences between the oxygen vs. phlogiston accounts are not captured by an emphasis on the differences between the above axioms. Lavoisier would not deny the truth ofJuncker's claim about constituents-his famous mercury/oxygen experiments were an instantiation ofJuncker's principle. And phlogistonists agreed with Lavoisier about the principle of quantitative conservation. Stahl chided metallurgists for not realizing that the more ore that was to be reduced, the more phlogiston-containing charcoal would be needed to complete the reaction. To understand the crucial differences between these approaches, what is most revealing is a comparison of their paradigms of successful explanation and their heuristic methods.
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4. Prospective Accounts Retrospective accounts use hindsight to argue that there are objective reasons to prefer the successor theory over the earlier attempt. In a typical revolution the later theory explains n10re phenomena, makes more accurate predictions, and has a stronger heuristic with increased problem-solving potential. From the standpoint of the new theory, we can understand why the old system had limited success but also why we no longer consider its core to provide an adequate picture of the world; its RCAPs are no longer taken to be constitutive. Once we arrive, the superiority of the new systen1 may be obvious. But is there a rational process that leads from one paradigm, or set of relativized, constitutive a priori principles, or research programme, to another? That is the challenge to those who would give prospective accounts of scientific change. According to IZuhn (1962), the prime motive for revolutionary change is the build-up of anomalies, that is, empirical results that are at odds with the paradigm. Lakatos (1970) introduces the notion of a degenerating research programme; following the dictates of the positive heuristic no longer leads to more accurate predictions and there may be phenomena in need of explanation for which the positive heuristic offers no helpful hints. Larry Laudan (1977) emphasizes the role of conceptual problems, cases of internal inconsistency or tensions between "adjacent" theories. These are all good reasons for abandoning one theoretical system-or for at least beginning to shop around for a replacement. But can we say more about the actual transition? Heinz Post, in an important 1971 paper called "Correspondence, Invariance and Heuristics," pointed out that the General Correspondence Principle could be run forwards as well as backwards; by demanding that the new theory relate in a certain way to the old theory, we put heuristic constraints on the search space. Conservation laws and invariance principles are good candidates to be retained, at least as a limiting case. He also talks of ways in which the better theory can leave a "footprint" in its predecessor. To avoid the illusion of time-reversal, I think of these as cases where there are gratuitous simplicities or other unexplained features in the old theory. Friedman (2001) mentions one such case: the equality of inertial and gravitational mass in Newtonian theory. Ptolemaic astronomy offers another example: Why should the positions of the sun, earth and the center of the epicycle for the inferior planets be co-linear?6 It is the job of historians of science to describe the actual meanderings, false starts and blind alleys that the scientific con1ffiunity follows in times of revolution. Lakatos proposes that these episodes can be "rationally reconstructed" so as to show the reasoning underlying these moves. Friedman's prospective account of the history of mechanics reminds me a
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bit of Neurath's challenge to rebuild a boat while at sea. Friedman reminds us of the various "spare parts" lying around-alternative geometries and mathematical work on what we now call the Lorentz group, old philosophical discussions about the principle of equivalence-and then describes what he calls a "series of natural transformations" (Friedman 2001, 63). Ifwe follow Friedman's account closely, we are supposed to understand how it happened that one moves step-by-step from one set of constitutive a priori principles that supposedly lin1it what we can conceptualize to another, radically different set of framing principles. Saul is no longer struck blind on the way to Damascus and instantly transformed into Paul. Instead he is subjected to a series of gentle shocks while attending a consciousnessraising group. But if scientists can survive without constitutive a priori principles during the transition period, why do we need to assume that they are locked in the grip of constitutive a priori principles at other times? During periods of normal science, researchers do share a great deal and do think in very similar ways. However, I don't think we should give a special epistemic status to the core of such a consensus. Two examples: David Hull's historical study of the variation in views of people identified as Darwinians reminds us that not all scientists agree on everything, although I agree that the degree of heterogeneity 'waxes and wanes. (See Hull 1988, 204.) More importantly (and I guess this is a Quinean point), even when scientists agree on core claims, they may not agree on which are definitions, or constitutive, and which are contingent. Consider the chemical concept of element. (See Figure 2.) Chemists use it in two quite different contexts-the first, as in Lavoisier's list of elements, to refer to the last products of analysis, and the second to designate the first principles of matter (to adopt Stahl's terminology), the substances that persist throughout chen1ical reactions (as when we say oxygen and hydrogen are still present when they form water) and which are somehow supposed to help explain the properties of the compounds. Certain German writers have introduced different terms to resolve the ambiguity: einfacher Stoffvs. Grundstoff (see Paneth 1962). Dalton's atomic theory formed a bridge between these concepts. Atoms were really the fundamental principles of matter. Each of Lavoisier's elements consisted of one kind of atom, which was characterized by a distinctive atomic weight. Atoms were the building blocks that persisted in compounds and simple substances were just big collections of atoms of one kind. But when isotopes were discovered, thus challenging the idea that elements were homogeneous on the atomic level, chemists painlessly shifted the links that one might have thought were analytic. They blithely reverted to the Lavoisier idea of last products of analysis (but limited themselves to chemical analysis!), kept Mendeleev's Periodic Table of Elements though shifting emphasis from atomic weight to atomic number, rejected Dalton's
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idea that all the atoms of elements were the same (isotopes differ in atomic weight), but replaced it with the claim that they all had the same atomic number, dithered about how to describe the Grundstoffen that persist in water (is it just the nucleus, the inner shells of electrons, or what?) and left it to the fundamental particle physicists to find out the true building blocks of the universe. I would say that during this entire period the concept of element was tightly connected to a variety of other notions. When it came time for a change, the network of concepts shifted and was modified-and our prospective account should describe these meanderings, but it seems to me that it would be artificial and that we would gain no understanding of this episode by trying to declare certain links constitutive or a priori. I dare say a similar pattern would obtain if we were to describe the history of the gene concept-or energy, for that matter.
5. Conclusion Contra to Habermas's communicative conception of rationality, which requires a common framework, Friedman's beautiful, detailed account of the transition from Newton to Einstein makes it seem to be a period of the most sophisticated type of rational discussione As Popper noted in his 1965 essay "The Myth of the Framework," as long as people share the goal of communal problem solving, they needn't come at those problems from within a common framework. 8
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There are occasions in which it can be extremely useful to formalize a theory, to regiment its claims. Often there are competing ways to do this and we can learn a lot by comparing them. I am concerned that the process of rationally reconstructing the progress of science by looking for Relativized Constitutive A Prioris, or even for the looser notion of Hard Cores, for that matter, can destroy our appreciation of the con1plexity and fluidity of the ideas that were actually operating at the time. Population geneticists remind us that evolution requires variation; chelnical reactions occur because not all of the molecules have the same energy. Perhaps it is because scientists do not share precisely identical frameworks that they are able to break out of old n10lds and revolutionize science. Using the case of revolutions in mechanics as a point of entry, Friedman has proposed a comprehensive account of the "dynamics of reason." Other essays in this volume will evaluate the success of his analysis of the paradigm shifts in our conceptions of space and time. My purpose in this article was two-fold. First, as an exercise in the history of philosophy of science, I examined some interesting parallels between Friedman's reaction to IZuhn and those oflmre Lakatos and Heinz Post, two philosophers coming out of the Popperian tradition who have commented on multiple examples of paradigm shifts. Secondly, I looked in some detail at two episodes of scientific change from the history of chemistry and asked whether Friedman's approach, which requires us to identify relativized constitutive a priori principles that are said to form indispensable frameworks for theoretical inquiry, could help us understand the reasoning that went on as our concept of chemical element changed. My conclusion was that the principles and concepts underlying these paradigms (to revert to IZuhn's terminology) do not function as if they were either constitutive or a priori, even in a temporary or relativized sense. These examples raise questions about the domain of applicability of Friedman's analysis of the structure and evolution of scientific theories.
NOTES
1. McMullin made this point at some length in the 2001 Coffa Memorial lecture, which he gave at Indiana University, titled "Forn1alism and Ontology in Early Astronomy." It is also implicit in his paper on Structural Explanation. When I requested a citation, Professor McMullin replied as follows: "You set me quite a challenge, as it turned out!! I have been malcing the point for years in my teaching and also in various lectures that mechanics is a very bad choice as a model for philosophers of science who want to tall( about the nature of the sciences generally. But you asked me where I had said that in print. I could have sworn that I had
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,indeed said so more than once but when I checked, I could not find any explicit text laying out the point. (The late Jim Cushing used to kid me that I never gave a talk on philosophy of science without laboring the issue of ho\v poor a model mechanics was for the sciences generally!)" (e-mail correspondence, 14 March 2005 ). 2. Lakatos's landmark paper (1970) was published in volume 4 of the Proceedings of the International Colloquium in the Philosophy of Science, London, 1965. As a graduate student in attendance I remember Lakatos as having two concerns as a host. One was to make prominent guests as comfortable as possible-at one point graduate students were drafted to replace a short dormitory bed vvith a longer one for Carnap. The second concern was to defend the Popperian account against both logical positivist criticisms and the new challenge presented by IZuhn's historical approach. 3. See especially his discussions of Greek atomism and the mind-body problem in Conjectures and Refutations. Popper's own metaphysical program for social science is described in IZoertge (1975). 4. See Spiro Latsis (ed.) (1976), Method and Appraisal in Economics, and Colin Howson (ed.) (1976), Method and Appraisal in the Physical Sciences. Each volume includes proceedings from a conference held in Napflion, Greece. 5. For a detailed discussion of the historical situation see IZoertge (1968), 131-91. Resemblances between the explanatory structures of the successive theories are spelled out in IZoertge (1968), 192-234 and 361-74 6. For a fuller discussion of Post's account and additional detailed examples see the Post Festschrift edited by French and IZamminga (1993). 7. Friedman gives a sympathetic reading of Habermas's conception and uses it to show why having shared constitutive a priori principles facilitates communication (Friedman, 53-54 and 93-94). It is ironic then that Friedman's prospective accounts of scientific change describe interchanges that violate Habermas's criteria for rationality. 8. This essay is reprinted in Popper (1994).
REFERENCES Classen, E.M. (ed.) (1967). Les Fondements Philosophiques des Systemes Economiques. Paris: Payot. French, S. and H. IZamminga (eds.) (1993). Correspondence) Invariance and Heuristics: Essays in Honour of Heinz Post. Dordrecht: IZluwer Academic Publishers. Friedman, M. (200 1). Dynamics ofReason. Stanford: CSLI Publications. Howson, C. (ed.) (1976). Method and Appraisal in the Physical Sciences. Cambridge: Cambridge University Press. Hull, D.L. (1988). Science as a Process. Chicago: University of Chicago Press. IZoertge, N. (1968). A Study of Relations Between Scientific Theory: A Test of the General Correspondence Principle. Doctoral Thesis, London University.
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- - - . (1975). "Popper's Metaphysical Research Program for the Human Sciences." Inquiry 18: 437-62. - - - . (1979). "The Methodological Status of Popper's Rationality Principle." Theory and Decision 10: 83-95. - - - . (1995). "Lakatos and the Methodology of Scientific Research Programmes." In H. Stachowiak (ed.), Pragmatik: Handbuch Pragmatischen Denkens 5: 103-22. Hamburg: Felix Meiner Verlag. IZuhn, T.S. (1962). The Structure of Scientific Revolutions. Chicago: University of Chicago Press. Lakatos, 1. (1970). "Falsification and the Methodology of Scientific Research Progran1n1es." In Lakatos and Musgrave (eds.) (1970),91-196. Lakatos,1. and A. Musgrave (eds.) (1970). Criticism and the Growth ofI(n 0 wledge. Can1bridge: Cambridge University Press. Latsis, S. J. (ed.) (1976). Method and Appraisal in Economics. Cambridge: Cambridge University Press. Laudan, L. (1977). Progress and Its Problems. Berkeley: University of California Press. McMullin, E. (1978). "Structural Explanation." American Philosophical Quarterly 15: 139-47. Paneth, F.A. (1962). "The Epistemological Status of the Chemical Concept of Element." British Journal for the Philosophy of Science, 13: 1-14,144-60. Papineau, D.(ed.) (1996). The Philosophy of Science. Oxford: Oxford University Press. Popper, IZ.R. (1963). Conjectures and Refutations. London: Routledge and IZegan Paul. - - - . (1967). "La rationalite et Ie statut du principe de rationalite." In Classen (ed.) (1967), 142-50. - - - . (1972). Objective I(nowledge. Oxford: Clarendon Press. - - . (1994). The Myth of the Framework. London: Routledge. Post, H.R. (1971). "Correspondence, Invariance and Heuristics." Studies in History and Philosophy of Science 2: 213-55. Watkins, J. (1958). "Influential and Confirmable Metaphysics." Mind, N.S. 67: 344-65. Worrall, J. (1976). "Thomas Young and the 'Refutation' of Newtonian Optics." In Colin Howson (ed.), 107-79. - - - . (1996). "Structural Realism: the Best of Both Worlds?" In Papineau (ed.) (1996), 139-65.
[24] Synthesis, the Synthetic A Priori, and the Origins of Modern Space-Time Theory ROBERT DISALLE
1. Introduction: Conceptual Change and the A Priori Logical empiricism was often faulted for an excessive emphasis on physics. Such a criticism supposes that the positivists took physics to be uniquely representative of science, and argues that in doing so they developed a conception of method that does not apply well to other scientific disciplines-if' indeed, it applies well even to physics. But the supposition, I think, is a misleading way of looking at the positivists' aims. Their fundail1ental concern was not philosophy of science, as it is now understoodan analysis of what is methodologically rational about science-but philosophy; physics had a unique place, not as the archetypal science, but as the science in which certain philosophical questions most urgently arise. And these were not particularly about the nature of rationality, but about meaning: they concerned how the theoretical concepts of science acquire their empirical content. When Einstein introduced radically new conceptions of space and time, it was not by following ordinary scientific methods of empirical research, but by a searching epistemological analysis of the old conceptions and their tenuous connections with experience. The chief lesson of this scientific revolution was that science begins by assigning empirical meanings to its fundamental concepts. All subsequent empirical inquiry presupposes the theoretical framework that these concepts constitute. If other sciences appeared to have less philosophical interest than physics, in short, it was not because they seemed to be epistemically inferior, or less "scientific." Rather, it was because they seemed to be able to take for granted just those philosophical issues that, in physics, had forced a conceptual revolution. This general account of science appeared, at first, to be a compelling synthesis of a IZantian view of the a priori, including the role of the a
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priorI In defining a framework for scientific inquiry, with the postEinsteinian awareness of the contingency and variability of such a framework. For IZant, the constitutive principles of scientific inquiry were the permanent "metaphysical foundations of natural science" (cf. IZant 1786), emerging from the in1mutable categories of the understanding. Evidently no principle could clain1 to be metaphysical without being synthetic a pri0ri: it must n1ake some synthetic claim about the world whose truth must yet be independent of mere empirical facts. The synthetic a priori indeed, was the characteristic domain of metaphysics (Bxvii): if not a priori, then metaphysics would be hypothetical and uncertain; if not synthetic, then trivial. 1 But Kant saw that traditional "dogmatic" metaphysics-the endeavor to discern the nature of things as they are in themselves, independently of how they appear to us-had failed in both senses. On the one hand, the metaphysicians had sought a priori certainty only in principles of pure reason that could not (in spite of temptations such as the ontological argument) yield any claims about the nature of things; on the other hand, such explanations as they could give for the nature of things depended, finally, on essentially arbitrary hypotheses such as the Leibnizian monadology. Kant inferred from this failure that metaphysics could address its proper subject matter-the synthetic a priori-only by abandoning its traditional dogmatic aim. This means replacing the requirement that "our knowledge must conform to objects," with the requiren1ent that "objects must conform to our knowledge" (Bxvi). In other words, metaphysics grasps the synthetic a priori, not in the nature of things in themselves, but in the nature of our own cognitive faculty. Synthetic a priori principles express the forms imposed on sensible intuition by the concepts of our understanding. Physics, in particular, by imposing laws that determine matter, force, and interaction, invests the metaphysical concepts of "substance," "cause," and "community" with concrete empirical meaning. Thus physics is able to approach nature, "not as a pupil ready to accept whatever the teacher should recite, but as a judge compelling a witness to answer the questions that he sets" (Bxiii). Given the radical changes in the conceptual foundations of science that came after Kant, and especially in the early twentieth century, the link between the a priori principles of science and the nature of our cognition could hardly be maintained. The logical positivists' account of the a priori embraced those radical changes. According to Reichenbach, the essential lesson was to distinguish and to separate two senses of a priori: the recognition of a priori principles as "constitutive of the object of knowledge" had to be separated from the now-untenable view that they must be "valid for all time" (Reichenbach 1920). The constitutive a priori, moreover, came to be understood as essentially constitutive of meaning for the fundamental concepts of scientific theories-that is, as essentially analytic j
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rather than synthetic a priori. The most important constitutive principles are the "coordinative definitions" that assign empirical content to elements of the formal structure of a theory. Einstein himself had put this view into practice, defining simultaneity by light-signaling. For the positivists, then, a proper understanding of the a priori must abandon metaphysics altogether, Kantian or dogmatic; the constitutive principles must be introduced by arbitrary decision, in order that empirical questions can be framed and answered by the facts. Precisely this "relativized a priori," however, was the casualty of the most important reactions against logical positivism. From one side, Quine's holism questioned the very distinction between constitutive and empirical principles: if all of science is demonstrably prone to revision under the pressure of new empirical evidence, the distinction between the a priori and the a posteriori appears to be a matter of degree. From the other side, Kuhnian historiography of science embraced the idea of constitutive frameworks ("paradigms") for the sciences, but inferred the radical incommensurability of one framework with another, in a seemingly effortless slide from the relativized a priori to radical epistemological relativism. The views of Quine and Kuhn, between them, have been more or less definitive of post-positivist philosophy of science. This malces it especially noteworthy that the leading ideas of Kant and the logical positivists, especially the idea of the relativized a priori, have found a new life in the work of Michael Friedman. Against Quine, he has rehabilitated the distinction between the empirical hypotheses of science and the constitutive principles that make such hypotheses possible-the linguistic, mathematical, and conceptual resources without which empirical principles could not even be meaningfi.llly formulated. When such principles are revised, therefore, there results a fundamental change in meaning that differs, in its method, motivation, and consequences, from the acceptance or rejection of any empirical claim. Against Kuhn, however, Friedman has shown how competing constitutive frameworks-clashing paradigms-can be, and historically have been, rationally discussed within a wider and more inclusive conceptual setting, a metascientific framework in which radically opposed scientific frameworks are discussed in a shared conceptual language. Friedman's view emphasizes that, in such times of conceptual transformation, a peculiar and crucial role is played by philosophy. A new theory that redefines the fundamental concepts of a discipline may eventually transform, not only scientific principles, but also the aims and methods of the discipline, and perhaps even the discipline's conception of what constitutes a scientific fact. It may not seem an exaggeration to call the end result a new worldview that is incommensurable with the old. Nor does it seem surprising that Kuhn would have seen just such a transformation as characteristic of twentieth-century physics, in the aftermath of relativity
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theory and quantum n1echanics. But IZuhn tended to portray such changes as relatively sudden, leaving even those scientists who experienced them with only the vaguest sense of how it all unfolded, and how they came to be "living in a different world." The most important historical cases, Friedman argues, suggest otherwise. For example, the context in which the Newtonian physicist encountered relativity was not only defined by the principles of the Newtonian framework itself, and the empirical difficulties that they confronted at the time. It was also shaped by decades of philosophical reflection on the foundations of physics and geometry-reflections that were not connected with any special concern about "anomalies" facing the framework, or with any special sense of crisis, but that arose simply from the idea that the connections between physical and mathen1atical concepts ought to be better understood. Quantum mechanics arose in nearly the same setting, further informed by the widespread view of relativity as a model of philosophically motivated physics that was well worth emulating. Of course, to recognize such philosophical influences on science is not a novel idea of Friedman's. It was among the defining ideas of logical positivism that relativity embodied in itself a kind of philosophical revolution, one that emerged from the application of strict principles of epistemological analysis to the Newtonian conceptions of time and space. The analysis finally replaced those questionable n1etaphysical conceptions with precise empirical concepts, defined by their "coordination" with specific procedures of empirical measurement. But the positivists' account of the philosophical motivations of relativity, especially of general relativity, came to appear naive and simplistic by the lights of later philosophy of science. The weaknesses of their account undoubtedly enhanced the appeal of IZuhn's picture of the role of philosophy in science: an important historical influence on the creation and reception of theories, but certainly not a source of rational motivation for conceptual change. Rather, IZuhn placed philosophical ideas among the subjective and psychological factors that lead individual scientists to prefer one sort of theory to another. Scientists might, for example, have a philosophical predilection in favor of "relativism" that would predispose them to toward Einstein over Newton, or a philosophical inclination toward "uncertainty" that would make them especially receptive to quantum mechanics. But such philosophical motivations can hardly be rationally persuasive to someone who doesn't share them in the first place. They can only figure in the "necessarily circular" arguments that those who presuppose a given paradigm can give for the paradigm itself (cf. IZuhn 1970a, 94). On this picture, in short, the influence of philosophical ideas is part of the problen1 of incommensurability, rather than a possible key to its solution. What distinguishes Friedman's approach, then, is a re-examination of the role ofphilosophy in scientific change, restoring philosop~r£o_i~_syl~~
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as a field for the rational discussion of theoretical alternatives. But this is far from a n1ere rehabilitation of the positivists' claim that physics is transformed by modern epistemological insight. Einstein's philosophical case for relativity, for example, did not merely invoke the need to replace metaphysical notions of space and time with en1pirically sound coordinative definitions. It drew upon a complex of philosophical traditions, stretching back over decades and even centuries-represented by, among others, Poincare, Mach, and Hume-that emphasized the critical analysis of scientific concepts, and of the connections betvveen mathematical concepts and empirical knowledge. These traditions provided physicists with something other, and much more important, than a subjective predisposition in favor of revising the Newtonian framework: the occasion for rational and self-conscious reflection on the fundamental concepts of the framework, and the conceptual resources to think of otherwise bizarre Einsteinian notions, such as relative simultaneity and space-time curvature, as intelligible theoretical possibilities (cf. Friedman 2002). These are pursuits for which there is no proper place either in "normal" or in "revolutionary" science as seen by IZuhn: normal research by definition accepts the basic concepts uncritically, while the "debates about fundamentals" that may happen in revolutionary periods-the only times when scientists "behave like philosophers"-involve, again, the clash of circular arguments among adherents of opposing conceptual frameworks (cf. IZuhn 1970b, 6). As Friedman points out, however, these philosophical discussions have the effect of enlarging the space of conceptual possibilities. They allow for the rational contemplation and communication of ideas that fall outside the lin1its of accepted theory, in an enlarged conceptual language . In this setting, the IZuhnian problem of incoll1mensurability problem poses no serious threat. Friedman's account of conceptual change places the idea of a relativized a priori in a new light. He n1akes essential use of the positivists' idea of coordinative definition, as defining the fundamental concepts of a theory and their empirical interpretation, thereby defining a set of empirical questions and empirical procedures for answering them. Unlike the positivists, he combines this idea with an understanding of the philosophical context that ll1akes such seemingly arbitrary conventions intelligible; such an understanding exhibits radical conceptual transformations, like those introduced by Einstein, as matters of rational deliberation. Yet this general approach may be pursued even further. The role of philosophy in conceptual transformation, it seems to me, goes beyond its contributions to enlarging the COlnmon conceptual space, and providing for intelligible discourse across theoretical divides. In the most in1portant historical cases-those cases in which, as in that of relativity, the most fundamental concepts are most radically redefined-we find that philosophical argument is essential, not only
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to the rational evaluation of a new theory, but to the very construction of the theory; the conceptual transformation is precisely the product of a conceptual analysis. In such a case, in other words, philosophical analysis is more than a means of articulating and defending the new theory against its predecessor; the philosophical analysis is the new theory itself. Deeper philosophical understanding is then not only the accessory to revolutionary scientific understanding; the new theory itself represents a revolution in philosophical understanding. To realize what this might mean, we have to reconsider the constitutive role of the coordinative definition, and the emergence of new coordinative definitions in times of revolutionary change. In hindsight, it is easy to see such definitions as arbitrary, given the possibility in principle of coordinating a concept in arbitrarily many ways. The definitions of simultaneity by light-signals, for example, and of the spatial straight line by the path of a light ray, are most commonly represented as arbitrary choices from among infinite possibilities. They can be justified in the end only by the simplicity that they introduce into the general system of physics. But in their historical context, we find that such definitions are not presented as free choices, but arrived at through a process of analysis. To call this process a conceptual analysis, indeed, is to understate the force of the argument: it is typically a dialectical argument from the prevailing definition to a new one, revealing the hidden presuppositions of the old conception, and exhibiting the internal difficulties that must be resolved by the new. Indeed, such an argument deserves to be called a transcendental argument. If a certain way of defining a concept is shown to be a condition of the possibility of en1ploying that concept at all, and thereby a condition of the possibility of the scientific reasoning on which that concept depends, then it can hardly be seen as an arbitrary coordination. Nor, therefore, can the argument for it be seen as an a posteriori appeal to the convenience or simplicity of the framework that the definition constitutes. The argument, rather, reveals the new conception in its transcendental role, as uniquely making possible the synthesis of experience under formal scientific principles. This account of conceptual change in physics is not as far-fetched as it may seem at first. It is possible, in fact, to see such dialectical arguments at work throughout the history of modern physics. 2 When Galileo urged the replacement of Aristotle's conception of motion with a novel one, involving the persistence of bodies in uniform motion, he did not claim the freedom to adopt a new definition for the sake of simplicity; he argued, in dialectical fashion-not only in literary form, but in actual method-that his novel definition was in fact already in use, as a presupposition of the everyday treatment of relative motions. When Newton introduced his concept of absolute rotation, he was directly opposing the prevailing Cartesian view that motion is essentially relative. But he did not need to arg~~
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against the Cartesians that a simpler overall system would result; he pointed out, instead, that absolute rotation in his sense was already an implicit, and sometimes explicit, presupposition of Cartesian reasoning about the causes of planetary motion. The distinction between uniform and non-uniform motion, in fact, constituted a condition of the possibility of mechanistic explanation in general. In the present paper, I will show that Einstein's account of simultaneity has this same dialectical and transcendental form. To acknowledge all of this, however, is not to repeat the error of Kant, mistalcing empirical principles of physics for universal and necessary conditions of the possibility of experience in general. The transcendental necessity revealed in these arguments is of an inescapably relative kind: the conditions of possibility for theoretical understanding evolve with our theoretical knowledge, and so make different demands at different moments in the history of physics. And this is why the a priori principles of physics are revisable. It is not because we are free to adopt whichever ones we wish, but, arguably, precisely because we are not free; a certain a priori framework cannot be maintained in the face of arbitrary new evidence, and so at certain points in the history of physics the transcendental principles must be revisited, and a new transcendental argument is required. This historical process is no doubt what Reichenbach originally hoped to understand through the relativized a priori (1920), before moving toward the conventionalist view that became associated with logical positivism in general. By examining this process more carefully, we begin to see that the relativized a priori and the synthetic a priori, excessively narrow conceptions taken individually, are complementary parts of a clearer view of conceptual change.
2. The Philosophical Argument for Special Relativity Absolute space and time were among the central points of philosophical controversy for Newtonian physics. By the time Einstein began his work on electrodynamics, however, the problem of absolute space had been essentially solved. From Newton's time, the dynamical distinction between inertial and accelerated motion had been alleged to support a general distinction between motion and rest in absolute space. But the general distinction was widely understood to be incompatible with the Galilean principle of relativity. In the 1880s, the solution to this difficulty was clearly expressed: Newton's mechanics requires not a single, privileged absolute space, but an equivalence-class of distinguished inertial frames. Thomson (1884) introduced the notions of "reference-frame" and "reference-dial-traveler," i.e. a spatial frame and a temporal standard relative to
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which motion may be measured, so that the laws of motion may be stated thus: For any system of particles moving anyhow, there exists a frame and a time-scale with respect to which every acceleration is proportional to and in the direction of an applied force, and every such force belongs to an action-reaction pair. Moreover, any frame in uniform rectilinear motion relative to such a frame is also an inertial frame. Independently, Lange (1885) offered an essentially equivalent conception, the "inertial system" and "inertial tinle-scale," and Lange's version (and terminology) was more prominently discussed in the German-language literature that Einstein might have read, above all in Mach's Die Mechanik (1889). Einstein never explicitly referred to this literature, and so how much or how directly it influenced him is unclear. It is clear, however, that by 1905 it seemed to him completely uncontroversial that mechanics has no need of absolute space, but needs only "a coordinate system in which the equations of mechanics are valid" (1905, 892); the relativity principle was naturally interpreted as the claim that any system that is in uniform motion relative to such a system is physically equivalent to it. The only open question concerned the relation of electrodynamics to this relativity principle. Accustomed as we are to Einstein's theory, we may suppose that light presented a prima facie violation of the principle, by treating electromagnetic processes as waves propagating with a definite velocity in a stationary medium, the ether. But this was not at all obvious at the time. Central supporters of the ether theory, such as Maxwell, asserted that the relativity principle was in no danger of being contradicted: the velocity of light relative to the ether is, after all, still a relative velocity (Maxwell 1877, 35). Therefore there was no inherent tension between the laws of mechanics and the laws of electrodynamics. One could certainly maintain the equivalence of inertial frames, while acknowledging that one subset of them happens to represent the rest-frame of a certain physical object whose states determine electromagnetic phenomena. In short, one could treat the velocity of light as having no particular bearing on the invariance-group of space-time. At the san1e time, Lorentz and his contemporaries confronted the peculiar fact that n10tion relative to the ether is impossible to detect: the effects of such motion on the relative velocity of light had been calculated, and the Michelson-Morley experiment was sensitive enough to produce the effects of the motion of the earth through the ether, but no effects were detected (then or since). Lorentz explained this null result by the contraction of all objects, including our measuring apparatus, in proportion to their velocity relative to the ether. For Einstein, rest in the ether was tantamount to absolute rest, and the extension of the relativity principIe to electrodynamics was therefore an open problem. From a Lorentzian point of view, however, it was not necessary to infer that a profound conceptual issue had arisen.
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As the logical positivists later reconstructed this situation, general epistemological reasons forced the conceptual issue to prominence; to a physicist who appreciated those reasons, the puzzle of the Michelson-Morley result could only be solved by a re-examination of fundamental concepts. What was required-and what Einstein eventually provided-was an "epistemological analysis of the concept of time": epistemology requires an account of the empirical meaning of the concept, and physics provides a process-light signaling-that enables us to define the concept in empirical terms. So the revolutionary significance of special relativity lay in this: Newtonian physics had an abstract conception of "absolute" simultaneity, but no physical definition of it; Einstein saw the need for a "coordinative definition" by which simultaneous events could be identified. He therefore addressed the need by stipulating that when a light-signal is propagated from a point A and reflected at B, the time of propagation from A to B is the same as the time from B to A. Thus a vague and "metaphysical" conception of simultaneity was replaced by one with a precise empirical meaning. Einstein had given an analysis of the concept that was, at the same time, a rule for verifYing that two events are in fact simultaneous. In his own words, moreover, Einstein sometimes suggested that he viewed the matter from just this verificationist point of view. His 1905 paper asserts that "a mathematical description [of the motion of a material point] has no physical meaning unless we are quite clear as to what we understand by 'time'" (Einstein 1905, 892) and proceeds to consider possible empirical methods of synchronization. He elaborates in his popular account of relativity: "The concept of simultaneity does not exist for the physicist until he has the possibility of discovering whether it is fulfilled in an actual case. We thus require a definition of simultaneity such that this definition supplies us with a method by means of which, in the present case, he can decide by experiment whether the two lightning-strokes occurred simultaneously" (Einstein 1917, 22). Reichenbach drew the verificationist lesson most explicitly: "The physicist who wanted to understand the Michelson experiment had to commit himself to a philosophy for which the meaning of a statement is reducible to its verifiability, that is, he had to adopt the verifiability theory of meaning if he wanted to escape a maze of ambiguous questions and gratuitous complications" (Reichenbach 1949, 290-91). Einstein also said much to encourage the view of coordinative definition, especially in the case of simultaneity, as a matter of conventional choice. In 1905 he asserts that a "common time" for different observers can be defined only if we "establish by definition that the 'time' required by light to travel from A to B equals the 'time' it requires to travel from B to A" (Einstein 1905, 894). But he gives no explicit justification for the use of light-signals in particular. And in later remarks, he speaks as if the isotropy of light-propagation, and its use in time measurement, is fixed by
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an arbitrary stipulation. In his popular exposition of his work (1917), he considers a possible objection to his principle: how can we test the hypothesis that the speed oflight is isotropic, unless we already have a way of measuring time? The answer is that the principle is only a definition. "Only one requirement is to be set for the definition of simultaneity: that in every real case it provide an empirical decision about whether the concept to be defined applies or not"; that light takes the same amount of time to travel in both directions "is neither a supposition nor a hypothesis, but a stipulation that I can make according to my own free discretion, in order to achieve a definition of simultaneity" (1917,15). In his Princeton lectures (1922), he raises the question of why light-propagation should play such a central role in his theory, and gives no better answer than that "It is immaterial what kind of processes one chooses for such a definition of time," except that it is "advantageous ... to choose only those processes concerning which we know something certain" (1922,28-29). From remarks like these, we may infer that there was at least some truth in the positivists' account of special relativity. Mterwards we can justify its acceptance on inductive grounds, or on whatever empirical criteria it seems to satisfy better than its predecessors. In creating the theory, however, Einstein saw himself as investigating the meanings of fundamental concepts, and as linking their meanings to the empirical procedures that determine their application. Moreover, he noted the element of arbitrary convention in the choice of criteria. In short, it is clear that in Einstein's mind, the problems of electrodynamics were connected with the a priori principles of physics, that is, with the concepts of spatial and temporal measuren1ent from which the empirical study of physics must begin. He suggests as much in his 1905 paper: "The theory that is to be developed rests-like all electrodynamics-on the kinematics of the rigid body, since the assertions of any such theory concern the relationships between rigid bodies (systems of coordinates), clocks, and electromagnetic processes. Insufficient consideration of this circumstance lies at the root of the difficulties which the electrodynamics of moving bodies presently has to struggle" (1905, 892). The question is whether, in interpreting Einstein in this way, we are con1pelled to accept the arbitrariness of his starting-point. Einstein deduced the Lorentz transformations from the invariance of the velocity of light, but, of course, the argument can go in the opposite direction: instead of explaining the seeming contraction by the invariance of the velocity of light, we can explain the seeming invariance of the velocity of light by the real contraction of our measuring devices. The logical relation between the two principles cannot by itself determine that one or the other deserves to be regarded as more fundamental. Lorentz's theory, no less than Einstein's, might be regarded as a "natural" (as opposed to an artificial or ad hoc) way to explain the Michelson-M_~_eL!~~_~.!I~§_~j:hQltt_resort: __ --
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ing to arbitrary hypotheses. As Lorentz himself said in describing the aim of his theory, Surely this course of inventing special hypotheses for each new experimental result is somewhat artificial. It would be more satisfactory if it were possible to show by means of certain fundamental assumptions and without neglecting terms of one order of magnitude or another, that many electromagnetic actions are entirely independent of the motion of the system ... The only restriction as regards the velocity will be that it be less than that of light. (Lorentz 1904, 13) Such a remark suggests that Lorentz had some of the same methodological concerns that we typically attribute to Einstein. It casts some doubt, therefore, on the notion that it was Einstein's sensitivity to those concerns, in particular, that moved him to seek a radical theoretical alternative. The essential distinction from Lorentz, and the essential argument against him, lie elsewhere, in the analysis of simultaneity. Einstein's presentation of the problem of electrodynamics, at the opening of his 1905 paper, suggests no reason to suspect that simultaneity is at issue. The starting-point is, rather, the problem of relativity. "Maxwell's electrodynamics ... in its application to moving bodies, leads to asymmetries which do not appear to be attached to the phenomena" (1905, 891). For example, a phenomenon that depends only on the relative motion of a conductor and a magnet-the production of an electric current-is represented by the theory in two completely different ways, depending on whether the conductor or the magnet is talcen to be at rest. Assuming that the conductor is at rest, there is an electric field in the vicinity of the magnet; assuming that the magnet is at rest, there is an electromotive force in the conductor. But the measurable magnitude-the current-is the same in both cases, as long as the relative motions are the same. If the Michelson-Morley experiment had no great influence on Einstein's thinking, it is doubtless because, as Einstein himself suggests, it was only an additional example of the kind of empirical symmetry that had concerned him already on independent grounds. At first glance, this problem may appear to be no more or less serious than the problem of absolute space in Newton's theory, and the analogous asymmetry between absolute space and uniformly-moving frames. By the same token it may seem as if eliminating the asymmetry were no more difficult or serious than asserting the equivalence of all Newtonian inertial frames and thereby eliminating absolute space. For these reasons it might be tempting to assert a straightforward methodological justification for special relativity, as merely eliminating a theoretical distinction that malces no difference. Yet the asymmetries that Einstein names have much more
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profound implications. On the one hand, the theoretical asymmetry is a kind of ontological asymmetry, in which, depending on what is taken to be at rest, a different sort of field is said to exist. One might argue that there is a similar ontological asymmetry in the case of absolute space. But there it was clear even to Newton that, however absolute space n1ay be bound up with his general metaphysical picture, it had no role to play in our account of the physical entities and processes at work in a Newtonian world: the ontology of bodies moving under the influence of accelerative forces does not require any distinction between uniform motion and rest. 3 So the elimination of the distinction, and of absolute space, in no way disturbed the fundamental physical concepts of Newton's theory. On the contrary, it merely brought the theory of space and time into complete harn10ny with those concepts. In the electrodynamical case, the asymmetry is essential to the conception of electromagnetic forces as mediated by waves in the ether. Eliminating that asymn1etry might seem, on general methodological principles, to be a simple acknowledgement of an empirical fact, that there is no such thing as rest in the ether. As Einstein noted, however, this move appears to involve us in a contradiction, namely between the Galilean principle of relativity (the "relativity postulate") and the principle that the velocity of light is independent of the motion of the source (the "light postulate"). In retrospect, we have been convinced by Einstein that this is only an apparent contradiction. But it is entirely genuine if we presuppose a framework of concepts about space, time, and electrodynamics that seemed perfectly reasonable to presuppose at the time. So Einstein faced a twofold task of philosophical analysis: first, to bring to the surface those assumptions that the contradiction rested upon; and only then, to discern the basis on which an alternative could be constructed. The alternative, too, had a twofold burden: not only to construct a framework in which the contradiction would not arise, but also to show that the fundamental concepts of this framework were well defined in a way that the previous concepts were not. For, without an argument for the second point, Einstein could only make the subjective argument, namely that his hypothesis could explain the same phenomena as Lorentz's in a more "natural" way. To his own mind, at least, he was doing something more than this. Within Lorentz's framework, it is natural to interpret the MichelsonMorley result as posing a problem of physical interpretation. Any velocity is essentially relative, including the constant c; furthermore the peculiar characteristics of motion relative to the ether must have some effect on the observed velocity of light. Therefore the invariance of the velocity of light can only be a misleading appearance with a definite underlying cause. It is an implication of the Newtonian space-time background, and a requirement of the Newtonian methodology, that any such unexpected appear-
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ance informs us of the presence, and enables us to measure the efficacy, of some yet-unknown factor. Hence Lorentz offers the quite plausible comment on the contraction hypothesis, that we shall have to admit that it is by no means far-fetched, as soon as we assume that molecular forces are also transmitted through the ether, like the electric and magnetic forces of which we are able at the present time to make this assertion definitely .... Now, since the form and dimension of a solid body are ultimately conditioned by the intensity of molecular actions, there cannot fail to be a change of dimensions as well. (1895,6) Einstein eventually argues that this entire project of explanation rests on questionable grounds, namely, on assumptions about space and time that have not been sufficiently examined. But this is not, as the positivists afterwards claimed, because the Newtonian framework altogether lacks a coordinative definition of simultaneity. Rather, it is that the coordinative definitions that do exist have a very problematic status. His argument starts by taking for granted an inertial system, or "a system of coordinates in which the equations of mechanics hold good (i.e. to first approximation)" (1905, 892). But this starting-point is quickly revealed to be ironic. The equations of mechanics concern the motions of a material point relative to such a coordinate system. But before we can describe the motions of a material point, we need to define what we mean by time. And if we can't take for granted the kinematical description of motion yet, then we can't take for granted the inertial coordinate system either; both stand in need of a definition of time. This is something quite different from the problem of having to accept the equations of motion "to first approximation," with the expectation of small revisions to the equations as the analysis proceeds. The problem is, rather, that the starting-point simply can't be taken at face value, because we haven't defined its basic terms. Hence Einstein's attack on the notion of simultaneity is essentially a dialectical one. This account of the problem may not seem plausible at first. In principle-and in spite of the criticisms of the positivists-there is a perfectly good definition of a Newtonian inertial frame, and a perfectly good definition of simultaneity. Newton's conception of simultaneity is instantiated by the instantaneous propagation of gravitational force; in principle, it ought to be possible to know something immediately about spatially distant states of affairs. That there is no practical application of this criterion does not, by itself, justifY the claim that absolute simultaneity had no empirical meaning, and that Einstein had provided a coordinative definition of simultaneity where none had existed before. It would be reasonable
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to suspect, indeed, that light-signals could provide a kind of stand-in for an infinitely fast signal, as long as we can take into account the travel-time of the light signals, or of any signal whose velocity is known and reasonably constant-though electromagnetic waves apparently stand alone in this regard. This situation, and its significance for the classical view ofspace and time-or, more precisely, its seeming lack of significance-were articulated very clearly by James Thomson. He noted that the objective measurement of distance presupposes that we can objectively determine simultaneous events. This in turn raises the difficulty as to imperfection of our means of ascertaining or specifying' or clearly idealizing, simultaneity at distant places. For this we do con1monly use signals by sound, by light, by electricity, by connecting wires or bars, and by various other means. The time required in the transmission of the signal involves an imperfection in human powers of ascertaining simultaneity of occurrences at distant places. It seems, however, probably not to involve any difficulty of idealizing or imagining the existence of simultaneity. Probably it may not be felt to involve any difficulty comparable to that of attempting to form a distinct notion of identity of place at successive tin1es in unmarked space. (Thomson 1884, 380) This passage states remarkably clearly what is at stake with the notion of simultaneity: the very idea of a kinematical frame of reference. Without an objective standard of simultaneity, we could not determine a set of spatial coordinates, nor express the motions of bodies as functions of time; Thomson's definition of a "reference-frame" would be empty. The passage also makes clear how natural it was, at the time, to assume that the dependence of simultaneity on signal propagation casts no doubt on the view that simultaneity is an objective relation, and suggests no need for the revision of any fundamental principle. As late as 1910, Simon Newcomb saw the matter in purely practical terms: "Were it possible by any system of signals to compare with absolute precision the times at two different stations, the speed [of light] could be determined by finding how long was required for light to pass from one station to another at the greatest visible distance. But this is in1practicable, because no natural agent is under our control by which a signal could be communicated with a velocity greater than that of light" (Newcomb 1910, 623). In other words, this problem of dependence was noted long before there was any reason to recognize a corresponding problem of relativity. It is quite unfair, then, to suppose that Newtonian physicists did not appreciate the role of the concept of simultaneity within the Newtonian system, nor the need to attach some physical meaning to it. Nonetheless,
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in order to understand the force of Einstein's challenge, we must understand on what precarious ground the Newtonian concept of simultaneity really stood-and why the entire conceptual system was thereby at stake. Again, the best possible instantiation of absolute simultaneity was universal gravitation. From the law of velocity-addition, it is true, one could deduce the possibility of particles accelerated to arbitrarily high velocities. But the only known case of instantaneous propagation, even in theory, is gravitation. Yet gravitational signaling is, obviously, not a constitutive principle of the Newtonian framework; gravity itself was discovered by "reasoning from phenomena," according to Newton, a kind of reasoning that necessarily takes the Newtonian spatio-temporal framework for granted, in order to infer properties of forces from properties of motions. More generally' interpreting the solar system as a dynamical system already presupposes that we can measure the relative distances of the planets, their satellites, and the sun, as well as their angular positions relative to the fixed stars. All of this presupposes the theoretical possibility, at least, of determining simultaneous events. Our ability to determine all of these circumstances, however, depends on a physical process-light-signaling-that is not only finitely propagating, but also essentially extraneous to the Newtonian theory. The latter is not necessarily a problem in itself; on the contrary, it might be seen as an advantage to have an observational technique that does not rely upon the theory being tested. But it does reveal rather starkly how the Newtonian approach to celestial mechanics begins with no criterion of simultaneity other than the "intuitive" criterion of visual perception, namely light-signaling. It is a peculiar feature of this situation that, once a physical picture of the solar system is given-as a set of relative motions constructed from relative positions of the planets and the fixed stars-the role of light in establishing the background framework may be forgotten. Indeed, the framework now permits the en1pirical study of light propagation as a process unfolding within this framework. A familiar example is the estimate of the speed of light from the delays in the eclipses of Jupiter's moons; by observing how the timing of the delays depends on the relative positions of the earth and Jupiter, Ole Roemer was able to calculate the time of light-propagation from Jupiter and to reach a remarkably accurate result. Such a calculation begins, in effect, from the assumption that certain events are objectively simultaneous even though they appear to us successively. It assumes, moreover, that the relative velocity of Jupiter and the earth is small enough (relative to the speed of light) to ignore; the important factor is the immense distance. In other words, the visual criterion of simultaneity is revealed, in this development, to be only a kind of stand-in for an instantaneously-propagating signal, the kind of signal that gravity represents but does not provide for any practical purpose. The criterion
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can be assumed to be perfectly adequate for a system of bodies that are relatively at rest (when we correct for different distances), and approximately accurate for systems where the relative velocities are negligible. And one could reasonably expect that, if the relative velocities became too great to be neglected, the criterion could still be applied by applying the Newtonian rule of velocity-addition. Absolute sin1ultaneity, then, is an abstraction from the empirical practice of detern1ining simultaneity by signals. The abstraction involves either neglecting the tin1e involved in signaling, when it is sufficiently small, or extrapolating that any measurable time-delay may be accounted for by a simple calculation. This process of extrapolation illustrates one aspect of absolute time as a measurable quantity. The other aspect is the notion of "equable flow" of time, or equal time-intervals, and it is warranted by an analogous abstraction: if SOll1e cyclic process approximates the condition of moving inertially (such as a rotating sphere or moving particle not subject to any forces), the intervals measured by it are approximately equal. This could be seen as a mere definition, but it yields the practical result that, the more closely clocks approach that ideal condition, the more nearly they will agree with each other, and the more nearly proportional will be the intervals that they measure (cf. DiSalle 2002). So the theory of absolute time is en1pirically quite sound, insofar as absolute simultaneity and equal timeintervals are concepts to which we can expect to arrive at increasingly good approximations. It is precisely this expectation that turns out to be so precarious. If the velocity of light turns out not to obey the classical addition law, then this entire line of approximative reasoning is fatally undermined. If that is true, then the other supposed instantiation of absolute simultaneity-universal gravitation-becomes essentially hypothetical. This does not mean that gravity no longer instantiates absolute simultaneity, or that the latter is no longer a meaningful concept. It does mean, however, that the concept is no longer integrated systematically with our account of its physical measurement; the gravitational definition now stands on its own, no longer connected with the en1pirical definition by a series of approximative steps. This is not a conclusion to which we are forced by the MichelsonMorley experiment. Indeed, to someone intent on interpreting that experiment, such as Lorentz, simultaneity might not appear to be in question at all. We noted that the unexpected null result of the experiment, from Lorentz's point of view, must be seen as providing information about the influence of some unknown factors. Lorentz himself suggested that the results reveal an effect of motion through the ether on inter-molecular forces, and that this effect produces the contraction of moving objects. In fact this is probably the only reasonable approach to take if the underlying conceptions of simultaneity, length, and time are not questioned. Einstein,
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however, was not primarily concerned with interpreting the MichelsonMorley experiment. Rather, he was concerned with understanding the fact that no electrodynamical processes seem to distinguish between uniform motion and rest, even in cases where the theoretical treatment of them explicitly appeals to absolute velocities. Understanding, in this context, does not mean finding an alternative theoretical explanation to Lorentz's theory; it means examining the framework of assumptions within which the fact appears to be contradictory, and articulating a theoretical context in which the contradiction is resolved. Here is where the need arises to examine common assumptions about spatial and temporal measurement and the ways in which the relevant concepts are defined. It is contradictory for the laws of electrodynamics to respect the relativity principle, if the relativity group of physics is the Galilean group, whose invariants include mass, acceleration, length, and time-and therefore simultaneity; no velocity can be an invariant of this group. But if the underlying spatio-temporal concepts are ill-defined, then the relativity principle itself has to be defined in completely different terms, within a completely different kinematical framework. This is why the "Kinematical Part" of Einstein's 1905 paper begins just as it does, by arguing that the concept of a Galilean coordinate system, taken for granted in the Galilean principle of relativity, cannot be accepted without an adequate definition of time. It is in the context of this argument that the definition of simultaneity by light-signals emerges, not merely as convenient, but as uniquely satisfying the requirements imposed by the nature of the problem. Einstein's series of proposed definitionsincluding seemingly naive references to "the hands of my watch" and the like-have an apparently "operationalist" aspect that has often been remarked upon. But Einstein's discussion does not really invoke a truly operationalist view. The requirements of the problem are theoretical as well as practical, and the assumptions that eventually go into the definition involve both empirical facts and the theoretical structure of Maxwell's electrodynamics. In fact Einstein proceeds through a brief series of possible operations for determining simultaneity, each time showing that theoretical requirements make the proposed definition inadequate, until he arrives at a satisfactory one from which his own theory can be developed. The aim of Einstein's analysis, indeed, is to arrive at a definition of "absolute" simultaneity, in the general sense in which Newton used the term "absolute." Obviously he does not impose in advance the Newtonian relation of simultaneity, which does not depend on the state of motion of the observer. But he seeks a criterion of simultaneity that is not merely relative: one that is independent of position and motion, that is based in physical laws that are independent of any observer, and that makes simultaneity a symmetric and transitive relation. It is a kind of Socratic irony that the cri-
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terion of simultaneity that eventually satisfies these requirements, and so deserves to be called "absolute" in this special sense, finally mal(es the relation of simultaneity a relative one. Einstein begins by proposing two practical procedures, each of which supplies an "operational" criterion of simultaneity: to define "time" by "the position of the small hand of my watch," and to coordinate the time of every event with a watch at a fixed location, by the time at which a lightsignal from each event reaches the watch. The first obviously fails to meet the requiren1ent of defining simultaneity for distant events; the second meets that requirement, but "has the disadvantage that it is not independent of the standpoint of the observer" (1905, 893). Thus each of these proposed criteria fails the theoretical requirements in some way. But then Einstein introduces his final criterion: a common time for points A and B can be defined when we "establish by definition that the 'time' required by light to travel from A to B equals the 'time' it requires to travel from B to A" (1905, 894). Placing "tin1e" in quotation marks emphasizes the fact that we are not to associate some pre-theoretical or intuitive n1eaning with the concept; again, if there were a principled way to do that, Einstein's statement could be an en1pirical claim rather than a definition. But this evidently is a definition, and one that evidently does satisfy Einstein's requirements. It exploits the invariance of Maxwell's equations as the basis on which an objective theoretical concept can be constructed. Up to this point in the argument, ,nothing about the true significance of the empirical facts, including the null results of the Michelson-Morley experiment, need be assumed. It need not be ruled out, in particular, that such facts are the effects of physical contraction as bodies move through the ether. Whatever its true significance, however, the n1erely apparent invariance of the speed of light guarantees that light-signaling will satisfy Einstein's requirements for an adequate definition of sin1ultaneity. Whether it uniquely satisfies then1 is another question altogether, and one that could never be answered conclusively; no argun1ent of Einstein's could prove that there is nothing in the world that travels faster than light, or no more appropriate signal for determining simultaneity. But it is clear, at least, that the general den1and that Einstein articulated later-"that in every real case it provide an empirical decision about whether the concept to be defined applies or not"-is satisfied, and, moreover, that its satisfaction involves much more than specifying an operation. In fact it involves specifying the place of the definition within our systematic knowledge of physical laws. The Newtonian definition, as we saw, is deeply embedded in the entire framework of Newtonian mechanics and gravitation theory. But its connection with empirical criteria was always somewhat tenuous, and, again, dependent on assumptions extraneous to the Newtonian program. Once that connection was called into question, by unexpected facts about
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light propagation, Einstein's definition stands as an implicit challenge to the Newtonian conception of time: what physical principles can define a common relation of simultaneity for observers in relative motion? Taken simply as a problem of coordinative definition, this challenge might seem to be a simple appeal to the positivists' verificationist account of meaning. Having considered the dialectical argument that raises the challenge, and its theoretical dimension, we can see that it is considerably more complicated and interesting than the positivists took it to be. It cannot be characterized as "an epistemological analysis of time" in some general sense, as a demand for an empirical account of "what we mean by simultaneity." In fact Einstein's analysis does not question what we ordinarily mean by time, since the most common way of identifying simultaneous events is by observing them at the same time; once the velocity of light is acknowledged to be finite, it is a minor matter of error-correction to adjust this criterion for the different distances at which events talce place. What Einstein does question is the relation of this common-sense notion to our systematic knowledge of physics, and the peculiar status it acquires in the contemporary state of physics. On the one hand, because of the apparent failure of velocity-addition, Newtonian physics no longer provides a theoretical context that connects the commonsense criterion with the "absolute" conception. On the other hand, and for the same reason, the invariance properties of electrodynamics place this criterion in a peculiar "absolute" position. For now it stands on its own, rather than serving as a local or slow-moving approximation to some more fundamental criterion. It provides a definition of simultaneity, not in virtue of being a reliable velocity somewhere in between zero and infinity, but in virtue of the special symmetry that characterizes its behavior. In other words, we could say that Einstein's analysis respected the intuitive criterion of simultaneity, but recognized that its newly discovered properties had disconnected it from the intuitive theory of simultaneity. On the intuitive theory, it was supposed to be an objective fact whether any two events happened at the same moment, or successively (and in which order). But it turned out to be only a shaky theoretical edifice, after all, one that had connected ordinary judgments of simultaneity with the absolute distinction of past, present, and future.
3. The A Priori, Beyond Space and Time Einstein's argument for special relativity, then, does much more than propose to elevate light-signaling to a constitutive role in the theory of space and time. It actually exhibits light-signaling in its transcendental role, its place within the very conditions of the possibility of a spatio-temporal
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framework. The argument is similar in form, therefore, to Newton's arguments for absolute motion, as a presupposition of mechanical reasoning about physical forces. Einstein's works at a more fundamental level, however, since it radically examines the kinematical framework that Newton's argument could only take for granted. Moreover, the same form can be seen again in Einstein's argument for general relativity. The coordination between the geodesics of the space-time manifold and the motions of falling bodies is not an arbitrary choice, but the product of a dialectical argument: that the equivalence principle exhibits a fatal ambiguity in the definition of an inertial frame (cf. DiSalle 2006, chapter 4). The common element in the arguments for successive theories of space and time, then, is not the role of conventional stipulations that connect geometry with physics. It is, rather, the role of philosophical analysis-a transcendental argu'ment-that makes explicit the presuppositions on which geometrical concepts are used, and forces new interpretations of those concepts from recalcitrant empirical facts. What the present account adds to Friedman's, therefore, is a proper appreciation of philosophy as something that, in a conceptual transformation of this kind, is something more than an external influence on physics. In fact the philosophical analysis is integrated with physical reasoning: the en1erging new theory expresses, in itself, a deeper philosophical understanding of the nature of space and time and how their structure is exhibited by physical phenomena. In Newton's theory, for example, the question of "the frame of the system of the world"-that is, the question between Ptolemy and Copernicus, or Tycho and IZepler-was radically redefined. One might say simply that Newton found a way for physics to settle the question. But it is much more accurate to say that Newton recognized, and argued both in the Scholium on space and time and in the argument of Book III of the Principia, that physical knowledge had forced a new understanding of the question. From a question about the simplest hypothesis to explain the phenomena of planetary motion, it was transformed into one about the position of the center of mass and the relative masses of the bodies involved, one that could be answered by an empirical investigation. By the development of new mechanical laws, the central concepts underlying the question-the concepts that crucially determine the sense of the question-had been explicated in completely novel ways. So Newton's argument to his contemporaries, controversial as it may have seemed to them, was simply to make explicit the consequences of the laws that they all accepted. But all the argun1ents of Newton, and everyone who followed in the path of his physics for the next two centuries, presupposed a background structure of space and time incorporating absolute sin1ultaneity, the succession of instantaneous states, and Euclidean geometry of space at each instant. And the relation of this structure to physical processes had never been ade-
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quately examined. It remained for Einstein to show how the spatia-temporal background itself depended on the physical assumptions underlying the definition of simultaneity. The revision of those assumptions was made possible by nineteenth century insights into the foundations of physical geometry, and forced by what the nineteenth century had learned about electrodynamics. Given the historical mutability of fundamental principles, one might ask what sort of Kantian view, if any, might still be reasonably maintained. One response might be to seek a weaker version of the a priori, weak enough to be compatible with a succession of new physical theories. But this seems to miss an important historical lesson. The fact is that such fundamental principles-understood as a priori principles of synthesis-have always been extremely weak and general, so much so as to be practically empty. This would not affect Kant, however, because the concepts of substance, cause, and community came to him already schematized; that is, they were already interpreted with the help of principles borrowed from spatial and temporal intuition. 4 So Kant's idea that the Newtonian form of these principles was immutable-reflecting permanent conditions of the possibility of comprehending phenomena under the concepts of substance, cause, and community-ret1ects his conviction that the underlying spatia-temporal framework is simply given to us in the forms of outer and inner intuition. It therefore stands independently of all physical principles, as their transcendental condition; Kant could plausibly regard the Newtonian principles as uniquely characterizing the physical concept of causality-determining the schema of the concept-within that kinematical framework. But this view of Kant's is difficult to maintain once we recognize that the space-time framework requires, as it were, a schematism of its own, an interpretation that depends on particular physical principles. Kant held that what we call our idea of, say, a triangle is not any image of it, or a mere general term. Instead it is the schema, or the rule for the construction of the concept in spatial intuition (A141/B180); similarly, our idea of causality is embodied in the rules (ultimately based in Newton's laws) by which a causal relation is defined in space and time (A142/B181). In the nineteenth century, however, through the work of Helmholtz and Poincare especially, it became known that spatial and temporal relations, too, are not immediately given to us. Rather, they are constructed in accord with some rules borrowed from physics: distance is given by the displacement of rigid bodies, and simultaneity by some process of signaling such as light-propagation. Kant himself pointed out that placing events in an objective temporal order depends on comprehending them in an objective causal order, which in turn depends on causal laws (A192/B237). But the idea that the very framework presupposed by such a causal analysis requires a further analysis-an uncovering
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of the empirical principles through which its basic relations are knownwas beyond Kant's reckoning. "This schematism of our understanding," Kant wrote, "is an art concealed in the depths of the human soul, whose real modes of activity nature is hardly likely ever to allow us to discover, and to have open to our gaze" (A140jB181). The remark seems to suggest that the schemata have a purely psychological origin, inaccessible to any philosophical investigation. Helmholtz and Poincare revealed, instead, through an inquiry at once psychological and philosophical, that implicit physical concepts underlie our seemingly immediate acquaintance with space and time. What IZant did not anticipate, in short, was that the fundamental principles of synthesis are not fully determined by the nature of our cognition. They require a physical interpretation, which, as such, necessarily hangs on contingent features of the physical world. The problem faced by postNewtonian physics, and physical geometry, was to discover how the a priori concepts of substance, cause, and community must be interpreted, or re-interpreted, in light of new and surprising empirical knowledge. In other words, the principles that constitute such concepts are a priori and synthetic only in the sense that they represent the demands of the understanding for a synthesis of appearances under rules of necessity and universality; their synthetic character in the deeper sense-as principles that express something about the nature of the world-depends on their empirical interpretation. Thus the form that these concepts take, at a particular moment in the history of physics, reflects the particular physical processes that are then exploited (e.g., rigid displacement, light-signaling) in imposing these general principles of order. There may be a kind of transcendental argument for the interpretation, as Einstein's case for special relativity suggests. But it does not rest on general features of the understanding. Instead, it identifies a particular interpretation as the condition-in the circumstances-of our ability to impose any such order at all. There is an element of profound truth in IZant's comparison of the physicist to "a judge compelling a witness to answer the questions that he sets." What makes it seem shortsighted is our recognition, since I
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such as light and motion. Once we understand this, we can understand that transcendental arguments about space and time do not concern the nature of the cognitive faculty in itself; they concern, instead, the relation between our cognitive faculty and the contingent phenomena that it exploits in forming a representation of the world. It would seem to follow that the subject matter of metaphysics is not, after all, only the nature of the cognitive faculty, but the relation between the cognitive faculty and the contingent world that is the object of its cognition. Once that is understood, the idea that the synthetic a priori must be "relativized" to a particular state of empirical knowledge, at a particular stage in the history of science, does not seem so paradoxical after all. This is not, in other words, the replacement of the synthetic a priori by purely analytic principles determined by convention. It is just the recognition that the principles of synthesis, the means that we may exploit for the synthetic grasp of the phenomena of nature, are contingent on the nature of the world and the always-imperfect, evolving state of our knowledge of it. This last remark may shed some light on a question raised by Friedman (2001), and mentioned at the beginning of this essay: is the problem of a priori knowledge a general problem in the philosophy of science, or something uniquely raised by theories of space and time? Or, what comes to the same thing, do these reflections have any interesting implications for other theories than those of space and time? As I suggested before, there is good reason to suppose that some other fields of science have the privilege of ignoring the kind of question about meaning that arises in theories of space and time. But I don't think it is quite right to suggest that what particularly distinguishes space-time theories, in this regard, is the need for coordinative definitions; surely every scientific inquiry starts from definitions of basic theoretical terms and somehow coordinates them with experience. It would be more precise to say, rather, that the question of a priori knowledge arises just in those fields where interpretive principles have to be introduced as a precondition for further inquiry, and those principles must be established by a kind of transcendental argument. Space-time theories do have a special place among such fields, both because of the transcendental arguments on which they depend, and because of the transcendental role that spatio-temporal principles have played in our conceptions of "substance, cause, and community." As Friedman has emphasized, other sciences frame their principles within the constraints imposed by these concepts. This historical fact does not imply, however, that spacetime theories necessarily play that role. It should be evident, from the foregoing, that they can represent only a particular interpretation, or kind of interpretation, of the fundamental concepts; to take space-time in the broadest sense as a universal and necessary framework, therefore, would be to repeat Kant's mistake. Indeed, history suggests that the boundary
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between interpretive principles and empirical principles must shift as circumstances change. Both Newtonian gravitation theory and classical electrodynamics were, more or less, empirically derived within the interpretive framework of Newtonian space-time by inferences from phenomena to the forces that cause them. And yet, through Einstein's work, both gravitational and electrodynamical motions-free-fall and light-propagationbecame the foundations for radically novel theories of space and time. The foregoing suggests that this general approach to the role of a pri0ri principles in science, if it is not completely misguided, certainly ought to have something to say about the foundations of quantum mechanics. Like gravity and electrodynamics, the theory of the constitution of matter ought to have emerged as an empirical theory framed against the background of Newtonian space and time; like those theories, it posed problems that ultimately could not be solved within the existing framework. Moreover, those who constructed quantum mechanics had before them the example of Einstein, and claimed to be applying to the properties of particles the very same kind of epistemological analysis that Einstein had applied to spatio-temporal relations. This fact by itself, however, may not seem to be a reason for optimism. From the point of view of post-positivist philosophy of science, the philosophical lessons that quantum physics claimed to derive from relativity were precisely the most naive aspects of logical positivism: the view of physical theory as nothing but a calculus for generating predictions, and the general prejudice against unobservable entities. Against Einstein's objections to quantum mechanics, Heisenberg defended "the idea that a good theory is no more than a condensation of observations," as a principle that had inspired special relativity (1958, 64). Views of this sort do little to challenge Friedman's assessment, that quantum mechanics has so far lacked a "timely interaction with an ongoing tradition of philosophical reflection" (Friedman 2001, 121). Yet there is something more to be said, I think, for the philosophical perspective of the founders of quantum mechanics, and for the philosophical work that the theory's foundation involved. At least Heisenberg and Pauli were convinced, not only that they were engaged essentially with a philosophical tradition, but also that the philosophical tradition they were engaged with was precisely the one that had guided the revolutionary development of the theory of space and time-a tradition of dialectical analysis of the presuppositions underlying our application of certain concepts' and of the empirical circumstances under which those concepts can be well defined. As Pauli put it, the exemplary philosophical achievement underlying special relativity was "a conceptual analysis of the correspondence between observational data and the mathematical quantities in a theoretical formalism" (Pauli 1958, vi). That this philosophical motivation was never convincingly articulated, in the case of quantum mechanics, ---------------------
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should not be taken as a reason to dismiss the possible relevance of a priori principles. Rather, it should be taken as a way of sharply defining the problem of interpreting quantum mechanics. If the relativized a priori does indeed matter to quantum mechanics, it can only be because there is a relativized transcendental argument for its basic principles, analogous to the sort of argument given by Einstein for his conception of simultaneity. And, as in the case of relativity, the principles singled out by such an argument would amount to something more than just one possible interpretation of quantum theory; they would have a reasonable claim to represent the essential content of the theory, the essential departure from classical mechanics that the theory represents. That such an interpretation exists even in the case of space-time theory may be doubted, given the persistence of serious interpretive debates. But these seem to me to have a fundamentally different character from those surrounding quantum theory. They concern the metaphysical implications of a theory whose basic structure and content are more or less settlednot, as in the latter case, profound questions about how to frame the theory's fundamental principles. Disagreement on such questions, indeed, lends the interpretation of quantum mechanics a certain air of arbitrariness: the multiplicity of interpretations seems to reflect a freedom of choice regarding which aspects of quantum mechanics ought to be regarded as fundamental. For special relativity, there was little controversy on this point, since Einstein and Minkowski had convincingly identified the theory's most essential departures from the Newtonian and Lorentzian theory-what Minkowski referred to as "a novel understanding of the time-concept that is forced upon us by the appearances" (1908, 56). As Minkowski's work (1908 and 1909) suggested, from this perspective on Einstein's central physical insight, we can immediately see the theory's most natural formal representation. One might see reasons to cling to Lorentz's theory, but this would be a theoretical alternative rather than a competing interpretation of special relativity. For quantum mechanics, however, there are divergent views on what the theory "is about," and corresponding disagreement on what aspect of the classical framework is most essentially challenged. It is easy to sympathize with Friedman's view; that this problem of interpretation will be solved only within some philosophical framework that has yet to be fully articulated. And in the absence of that framework, even the right formal representation might seem open to doubt. In order to clarifY the interpretive problem of quantum mechanics, it may be helpful to compare it with special relativity in more detail. Another way of posing the question of interpretation is by the familiar question, "why did the gods need the quantum?" Or, more precisely, why is it that any theory of fundamental particles, even one whose underlying ontology
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is deterministic and classical, must approximate the indeterminism of quantum mechanics in every possible empirical circumstance? In the case of special relativity, the analogous question would be, why must electrodynamical phenomena, even if their underlying ontology is that of Lorentz's theory, behave as if the velocity of light is invariant? And to understand the origins of special relativity, it is crucial to recall that Einstein did not try to answer this question. Instead, he proposed to raise the apparent invariance of the velocity of light to the status of a postulate. As we already noted above, however, this is a somewhat simplistic description of Einstein's reasoning. The more essential point was the transcendental analysis, showing that the question itself rests on problematic grounds-on concepts that are poorly defined, especially the concept of simultaneity. Therefore "raising the light-principle to the status of a postulate" means more than just making a stipulation that will define the concept of simultaneity. It means exhibiting the role of the light principle in our ability to apply the concept-or, rather, in our ability to synthesize appearances under the concept of simultaneity, in a manner that is not merely subjective or local. What would be analogous to this argument in the case of quantum n1echanics? It would be an argument to the effect that central principles of quantum mechanics playa transcendental role in our ability to define certain fundamental properties of physical systen1s. This is an idea toward vvhich, at least, Heisenberg was groping in his analysis of the uncertainty principle (e.g., Heisenberg 1927), but his attempt is somewhat blurred by his appeal to the disturbances inherent in measurement. The latter notion purports to answer just the kind of question that Einstein chose to set aside, and to give an intuitive explanation for a counter-intuitive feature of the theory; it therefore obscures the possible role of this feature as (analogously to the light principle in special relativity) constitutive of the theory's explanatory framework in the first place. In other words, the purportedly causal account of uncertainty obscures its role in imposing an entirely new understanding of causality on the physical world. Similarly, one can think of quantum entanglement as imposing an entirely new understanding of "community." In the framework of Newton and lZant, the only conceivable form of coexistence for two bodies is mutual determination of motion in accord with the Newtonian principle of causality, as in the mutual gravitational attraction of two masses. Special and general relativity may represent profound revisions of this idea, but they are continuous with it insofar as objects still determine one another's space-time trajectories. Only with quantum entanglement do we find the properties of physical systems standing in a radically new sort of relation, and the possibility that such strange relations are constitutive-in this context-of any grasp of those properties that we can achieve.
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A compelling argument for an interpretation of quantum mechanics, then, would do more than show that certain principles are a plausible foundation from which the theory may be derived. It would also show that those principles, whatever they might be, play an indispensable constitutive role in our understanding of fundamental concepts. I don't pretend to have such an argument or such an interpretation; I have only hoped to sketch the kind of conceptual analysis from which such an interpretationif it is possible at all-might emerge. I also hope to have shown that the work begun by Friedman, re-examining the problem of a priori knowledge, is profoundly suggestive in ways that go far beyond the history and philosophy of space-time theories.
NOTES 1. All references to I
REFERENCES DiSalle, R. (2002). "Reconsidering Ernst Mach on Space, Time, and Motion." In D. B. Malament (ed.), Reading Natural Philosophy: Essays in the History and Philosophy of Science and Mathematics to Honor Howard Stein on his 70th Birthday, Chicago: Open Court, 167-191. - - . (2006). Understanding Space-time: The Philosophical Development of Physics from Newton to Einstein. Cambridge: Cambridge University Press. Einstein, A. (1905). "Zur elektrodynanuk bewegter IZorper." Annalen der Physik 17: 891-921. - - - . (1917). Uber die spezielle und die allgemeine Relativitiitstheorie (Gemeinverstiindlich). Second edition. Braunschweig: Vieweg und Sohn.
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- - - . (1922). The Meaning ofRelativity. Princeton: Princeton University Press. Einstein, A., H.A. Lorentz, H. Minkowski, and H. Weyl (1952). The Principle of Relativity. W. Perrett and G.B. Jeffery (trs.). New York: Dover Books. Friedman, M. (1992). ](ant and the Exact Sciences. Cambridge, MA: Harvard University Press. - - . (2001). The Dynamics of Reason: the 1999 ](ant Lectures at Stanford \ [Jniversity. Stanford, CA: CSLI Publications. - - - . (2002). "Geometry as a Branch of Physics: Background and Context for Einstein's 'Geometry and Experience'." In Reading Natural Philosophy: Essays in the History and Philosophy ofScience and Mathematics to Honor Howard Stein on his 70th Birthday. D. Malament (ed.). Chicago: Open Court, 193-229. Heisenberg, W. (1927). "Ober den anschaulichen Inhalt der quantentheoretischen I
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Nevvton, 1. ([1729J 1962). The System of the World. In Sir Isaac Newton)s Mathematical Principles of Natural Philosophy and his System of the World, ed. F. Cajori, trans. A. Motte, 2 vols. Berkeley: University of California Press. Pauli, W. (1958). Theory of Relativity. Oxford: Pergamon Press. Reprint, Nevv York: Dover Publications, 1981. Reichenbach, H. (1920). Relativitatstheorie und Erkenntnis Apriori, Berlin. Translated as The Theory of Relativity and A Priori I(nowledge (1965) by M. Reichenbach. Berkeley and Los Angeles: lJniversity of California Press. - - - . (1927). Philosophie der Raum-Zeit-Lehre) Berlin. Translated as The Philosophy of Space and Time (1957) by M. Reichenbach. New York: Dover Publications. - - - . (1949). "The Philosophical Significance of Relativity." In Albert Einstein) Philosopher-Scientist, ed. P.A. Schilpp. 289-311. Thomson, J. (1884). "On the Law of Inertia; the Principle of Chronometry; and the Principle ofAbsolute Clinural Rest, and of Absolute Rotation." Proceedings ofthe Royal Society ofEdinbur;gh 12: 568-78.
[25J Baclz to "Baclz to IZant" MAR!( WILSON
While the building of nature is growing spontaneously from within, the model of it we seek to construct in our descriptive science can only be constructed by means of scaffolding fron1 without, a scaffolding of hypotheses. While in the real building all is continuous, in our model there are detached parts, which must be connected with the rest by temporary ladders and passages, or which must be supported till we can see how to fill in the understructure. To give the hypotheses equal validity with the facts is to confuse the temporary scaffolding with the building itself. -J.H. Poynting 1
Philosophers, by wont, frequently complain of intellectual barrenness amongst their contemporaries, even in periods that strike us later as abundant in novelty (thus Wittgenstein saw nothing but "poverty" in the philosophically fertile 1930s). Perhaps induction assures us that future generations will discover a comparable bounty within our present day efforts. However, when I grade our productions according to the usual internal signs of scholasticism (lack of contact with other fields of endeavor; absence of salient examples; sweeping "philosophical analyses" of uncheckable content), I wonder if our prospects for persisting glory are so auspicious. However, a key exception immediately emerges from this depressing tally, for the remarkable work of Michael Friedman participates in none of the dismal tendencies I have noted, particularly in the manner in which a concern with broad philosophical sweep is invariably tempered by a careful attention to scientific details. In this regard, he imitates his chief avatar, Immanuel IZant, and in this little essay I will argue for the in1portance of the particular blend of sweep and detail we find in both authors. For this sermon I take as my text the following passage from Friedman's Dynamics ofReason (2001):
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For what is needed is precisely the creation and stimulation of new fran1eworks or paradigms, together with what we might call metaframeworks or meta-paradigms-new conceptions of what a coherent rational understanding of nature might amount to-capable of motivating the revolutionary transition to a ne\v first-level or scientific paradigm. (Friedman 2001, 23) So stated, the recommendation is quite abstract, perhaps to an uncomfortable measure. However, we can quicldy arrange solid flesh on its rarified bones if we follow Friedman further into his historical work, especially that which pertains directly to Iunt himself. At first glance, the IZonigsberg titan seems an unlikely fount of detail and definiteness and many of us, asked to select a great philosopher who errs on the side of "sweeping analyses of possibly uncheckable content," might very well choose Iunt as our exemplar. Indeed, his devoted students usually approach his corpus only as a mighty edifice designed according to the most rarified forms of architectural principle, where even the decorative figurines upon the mantlepieces have been deduced from the Table of Categories. It is the singular merit of Friedman's historical scholarship to have recovered the hard and sweaty toil with respect to classical physics that partially underpins the completed domicile and which originally inspired Iunt to frame its timbers as he has. This vital scientific context is often hard to discern, not because it is disguised or hidden, but because Iunt writes in an extremely compressed manner and makes great presumptions of his readers, creating interpretative barriers that the passage of time has only increased. Friedman's perspicacity in this difficult salvage project is truly astonishing. Insofar as I can determine, most contemporary philosophers who strive to "bring Kant alive in our times" (in the field of ethics, say) are content to wander through his mighty mansion and appropriate the little pieces of thematic bric-a-brac that suit their fancies (I imagine them stashing homiletic samplers underneath their coats). But, insofar as the interests of philosophy of science and modern descriptive metaphysics go, we are better advised to imitate the Master in his keen appreciation of delicate methodological oddity rather than simply appropriating his more imperial mottos. Indeed, our best prospects for executing Friedman's recommended program of "meta-paradigmatic" investigation lie in replicating, as best we can, the basic patterns of critical scrutiny that characterize Kant's own endeavors. Many of the vital philosophers who parade through the pages of Dynamics ofReason (Helmholtz, Reichenbach, Carnap, Cassirer, et al.) originally marshaled their agencies under a banner of "Back to IZant!" believing that the extravaganzas of later nineteenth century idealism had strayed too far from the sterner discipline of their progenitor. And now seen1S like a good time to renew such efforts once again. As myself a
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scientific realist at heart, I hope that a IZantian answer will not represent the final outcon1e of our reflections, but I firmly believe that eavesdropping within the tents of a modern "Back to IZant" revival provides an excellent recipe for obtaining guiding questions that might eventually extract us fron1 the doldrums that hold us captive. Although I am not aware of any place where Friedman himself articulates a comparable claim, I believe that, even today, it will be I(ant himself who provides the best exemplar for how Friedman's meta-paradigmatic tasks ought to be approached, rather than any of the worthy succession of later semiIZantians mentioned above. Later in the essay I'll explain why I think this is so. Let me offer a few words of apology before I proceed. I am scarcely an expert in IZant: almost everything I state here derives from a seminar in which we read a work-in-progress draft of Friedman's massive commentary on The Metaphysical Foundations of Natural Science. 2 However, I have studied the history of continuuD1 D1echanics (especially in nineteenth and twentieth century contexts) in some detail and I was surprised to learn from Friedman's work that IZant clearly enjoyed a substantial grasp upon the rather subtle puzzles that affect n1athematical coherence within this arena (in this context, "continuum physics" covers the physics of fluids and flexible bodies whose matter is continuously distributed over a finite region of space). It happens that the puzzles indigenous to this field have historically driven many observers independently to roughly IZantian conelusions on their own recognizance, whether or not they were familiar with his works or not (not surprisingly, Leibniz had a good grasp on these same problems and IZant understood the presence of these themes in his predecessor better than modern commentaries often do). I will be able to articulate my central observations about a IZantian revival most aptly if I sketch a bit of this background in general terms. It may also aid future students of Friedman's commentary UPOD the Metaphysical Foundations to see the basic puzzles of continuum mechanics initially laid out in stark and pungent terms, because Friedman's own discussion must perforce pay greater attention to the rococo details of IZant's fuller architecture, potentially leading to an outcome where some readers may not see how forest and trees fit together, overcome by Critical kudzu. Not burdened with such exegetical responsibilities myself, I can concentrate upon the basic pattern ofsweep combined with details that provides an excellent paradigm for our o\vn researches. In the roughest terms, IZant's reflections upon physics revolve around a division between descriptive characterizations that pertain to nature along three scales of size: (1) the global or infinitely large (i.e., as a con1pleted "absolute space"), (2) the infinitesimal or infinitely small (the "smallest particles" into which Nature is potentially decomposable), and
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(3) the mesoscopic or approximately human-sized scale at which physicists are able to categorize (or "intuit" in IZant's jargon) concrete physical objects. Kant n1aintains that it is only the type (3) descriptive "constructions" that we properly understand, although these mesoscopic procedures internally force an intellectual movement towards the infinitesimal and global levels of description, although we humans cannot actually comprehend these levels of inquiry in a directly meaningful way. As Friedman has helped us appreciate in very concrete terms, 3 IZant's response in the global case is to reconfigure Newton's "absolute space" as a regulative ideal that governs how physicists working at the mesoscopic scale must proceed, rather than allowing "absolute space" to stand as an ungraspable form of realistic object. As I0nt himself explains the doctrine: The idea of reason can therefore do no more than prescribe a rule for the regressive synthesis in the series of conditions; and in accordance with this rule the synthesis must proceed from the conditioned, through all subordinated conditions, up to the unconditioned, although the absolutely unconditioned is not to be met with in experience. (Kant 1781/1787; A510/B538)4 Now it is plain that some analogous accommodation must arise for IZant between the infinitesimal and mesoscopic scales, although the discussion of these issues in the Metaphysical Foundations resists ready interpretation (much of what one encounters in the earlier literature is patently wrong). I shall eventually argue that it is within the infinitesimal aspects of IZant's concerns that the considerations of greatest direct relevance to our current philosophical concerns lie. At the outset the reader should recognize that the "classical world view" articulated in the average modern physics primer is not the dominion we seek, for it does not articulate a domain of flexible continua with clearly defined shapes as 1Zant wanted. Instead, its base ontology consists of a swarm of localized-to-a-point singularities that act upon each solely through action-at-a-distance forces. From this point of view, "atoms" do not possess true primitive shapes, but merely define rough regions where the potential repulsion exerted against an intruder becomes high. As such, this point of view was clearly articulated by Boscovich in the 1740's and was then adopted with some variation by the French atomists. But sound experimental considerations forced abandonment of a literal point-mass picture in favor of continua after 1840 (which is one reason the Victorians were especially alive to the conceptual difficulties of continua). Pedagogically, the point-particle approach became again prominent in the 1920's, largely because it serves as a convenient handmaiden to quantum mechanics. s IZant's early (1756) "Physical ~~~~~l~~~':_a~~~~c~~ ~_vi~~ __
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point close to Boscovich's and its interpretation of matter as singularities of inertia captures what Kant has in mind when he contrasts his Critical period opinions with those of "physical monadology" (Leibniz's original monads, on the other hand, are not of this point-particle type, but instead serve as shifting centers of control over extended regions of continuous matter). Mathematically, the working principles of this point-mass mechanics can be formulated in terms of ordinary differential equations quite easily and this fact often creates an illusion that "classical mechanics" is conceptually straightforward. But if, following the trend of most atomists (including Newton), we deviate from Boscovich and puff up the volume of our singular centers so that they become rigid bodies of finite dimension, many anomalies tracing to their possibilities of contact action immediately emerge (e.g., Newton's third law runs the danger of incoherence in such settings-an observation that Kant exploits deftly in his arguments against complete rigidity). The most rational approach to handling these problems of finite size is to deny absolute fixity to our newly extended objects and require that their bounding surfaces must distort when stretched or compressed through exterior agencies (see Figure 1). As this happens, body 9J must exert contact (or, synonymously, traction) forces directly upon across their mutual interface. Because and 9J are in true contact, such forces can't be aetion-at-a distance in character, since no distance separates the surfaces. A little further thought persuades us that similar traction forces must also arise across any imaginary interior surface that we might mentally draw inside itself (such cutouts are often called "free body diagrams"). In fact, the vast behavioral differences between a plank of wood, a blob of clay and a glass of water must turn on the details of how these internal contact forces are created and transmitted through a body's interior, because most materials respond to action -at-a-distance forces like gravitation in more or less the same manner. The recognition that the laws of mechanics must apply to these internal subregions as well as the integral body to which they belong is usually credited to Euler and sometimes dubbed "Euler's cut principle" for this reason. In the Metaphysical Foundations Kant often argues in cut principle-like fashion, as when he argues that localized points can't assume finite mass values lest the free body regions that surround them be forced to carry infinite masses. Such "measure theory" arguments linking infinitesimal and mesoscopic behaviors are vital to the integrity of any viable form of continuum mechanics, although their particulars are often deceptively subtle in character. Indeed, the key ingredient of modern continuum mechanics that Kant (along with everyone else in his era) missed in this vein lies in the fact that traction forces can sometimes pull obliquely across a bounding surface rather than merely acting in the perpendicular fashion of an inviscid fluid (a
a
a
a
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FIGURE 1.
proper utilization of cut principle argumentation then establishes the local existence of a stress state, but this wasn't recognized until Cauchy's work of the 1820's). This oversight left Kant without a plausible mechanism for solidity,? for his normal traction assumptions can only underwrite (or "construct," to employ the typical Kantian phrase) a local notion of internal pressure, rather than the more complex stress. 8 Earlier generations of physicists had commonly assumed that a rubber ball can manifest elastic behavior only if it represents a matrix of spatially separated atoms whose intervening spaces close up when compressed and then re-expand after the load is diminished (the mechanism of reconstitution was credited either to the re-entrant pressure of some surrounding fluid (Descartes) or, more plausibly, to action-at-a-distance repulsions between atomic centers (Newton)). But in his embrace of true continua, Kant stoudy rejects this point of view and argues that elastic behavior can be scientifically comprehended as an intrinsic capacity of extended bodies, without any need to posit far-from-observation hidden mechanisms of an atomist stripe: Everything that relieves us of the need to resort to empty spaces is a real gain for natural science, for they give the imagination far too much freedom to malce up by fabrication for the lack of any inner knowledge of nature. (Metaphysical Foundations, 4: 532) To this day, continuum-based descriptions usually provide a more accurate and trustworthy treatment of bulk materials, free from the vicissitudes of inaccuracy to which atomic models are generally heir. In a nineteenth-century context, the often-mocked arguments of the anti-atomists (e.g., Ernst Mach and Pierre Duhem) were often directed against faulty arguments that were a priori in a manner reminiscent of "elastic bodies must contain unseen spaces for their compression" (I'll cite Boltzmann in this context
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shortly). Like Mach and Duhem, Kant is eager to establish the coherence of a chemisu-y that allows commingling substances to inhabit exactly the same spatial regions, exactly as modern continuum mechanics handles mixing and phase change to this day.9 Indeed, the affinities between Kant's physical concerns and those evident a century later are quite su-iking and I wish I had space to pursue these parallels more fully. It is a pity that the virtues of pro-continuum reasoning generally pass unremarked in most modern commentaries. However, the acceptance of flexible continua presents us with a problem in coordination of different types of force that is wholly absent from the simple Boscovichean picture, for we have now allowed bodies to interact in two quite distinct manners: (i) through traction forces arising from contact along their adjoining surfaces and (ii) through body forces such as gravitation. But these two kinds of force act in different locations and are dimensionally incomparable as well. Suppose an elastic blob a is pressed by various loads around its perimeter and its innards are gravitationally amacted to distant matter (see Figure 2). Consider a point p. within this sphere and carve out a little "free body" surface S around p.. If we hope to develop a conventional law of motion for the material in the ball (that is, formulate an appropriate set of partial differential equations), we must explain how p. will respond to its surrounding environment of forces. Any action-at-a-distance force 10 g like gravity will work directly upon a in an unproblematic manner and partially determines how a accelerates, but not completely, for there is a schedule of traction forces around the surface of S with which we must also contend. But S is not located at p.: how do the infinity of contact forces on S manage to affect p. itself? Well, we can mathematically u-avel over the surface of S and employ the parallelogram rule to add up forces as we go, until a final resultant force t* is obtained. Perhaps we should simply carry this summed t* inward to point
tJf. (sum of the t)
How should p be combined with tJf.? FIGURE 2.
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Il and combine it with the gravitational g? If this simple rule were allowed , our traction + body force resultant g + t* can coherently inform point Il how it must accelerate, following Newton's F = mao But is this simple scheme for force combination viable? Well, such a rule would be appropriate if S represents a completely rigid haUl! but we've already rejected Such fixity as problematic. In any case, no single summation procedure can possibly provide the right answer because we have observed that the chief difference between an ivory billiard ball and a ball of putty lies in how they transmit contact forces through their interiors. But our "treat S as rigid" procedure pays no attention to how this transmission occurs and would therefore force wood and putty to act exactly alike. But lacking a rationale for bringing t* into coordination with g, we are stuck with two types of force that can't be put into sensible accommodation.
The problem of combining forces doesn't simplify at smaller scales FIGURE 3.
Here's another way of articulating the conundrum. It is often remarked that "physics is simpler in the small," indicating that laws of material behavior can be formulated elegantly only at the infinitesimal (that is, differential-equation) level. Let's try to apply this motto to our blob. We begin with a large volume (the sphere a) surrounded by surface tractions and attracted by gravity in its interior (Figure 3). Let us now concentrate upon the "small," viz., how the local point Il behaves. As we saw, Il'S response to gravitational influences is quite straightforward, but to handle the internal stresses, we must cite Euler's cut principle to carve out a nested sequence of "free body" volumes S around Il. But no matter how far we press these cuts, we wind up with a problem of exactly the same type as that with which we began: how does a volume of matter react when it is surrounded by surface tractions and pulled upon by body forces in its interior? Moving towards the small within a homogeneous continuum carries absolutely no simplification in its wake: we merely reproduce our original problem at a smaller size scale.
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Plainly we must break this chain of unprofitable descent if we hope to get anywhere in our continuum physics. How might we do this? Historically, a number of different resolutions have been advanced to this end, virtually all of those before 1950 evoking some policy of essential idealization along the way. By "essential idealization" I mean the thesis that, at an appropriately small size scale, flexible matter can be intentionally misdescribed in a manner that permits the available tools of applied mathematics a handhold upon their behavior. Many of the early advances in classical mechanics depended upon techniques where some complex system is replaced by a simpler equivalent to a reasonable degree of accuracy. In particular, Kant would have been familiar with the fact that a rigid, spherical earth will act like a simple point center from far away with respect to its gravitational field and that a setup involving a single bob will prove equivalent to a more dispersed compound pendulum. Such examples suggest that, at some tiny scale of size, suitable point centers can substitute for small volumes S inside our blob and S's ambient tractions can be replaced by a suitable schedule of action-at-a-distance forces. Such replacement policies are commonly found in textbook presentation even to the present day and were ubiquitous in Kant's time. For example, in his early attempts to write a differential equation for a bent wooden beam, Leibniz decomposes the timber into small parallelepipeds, concentrating upon their center-of-mass locations. He then represents the interior traction forces by little springs acting in a gap between the block-centers (I have marked these traction surrogates by an arrow that I have superposed upon what is otherwise Leibniz's original diagram; see Figure 4). This replacement policy allows the beam's traction forces to be replaced by what is, mathematically, an action-at-a-distance force, codified in the fictitious form of externalized spring forces acting as what Kant calls a "solicitation within an infinitely
D
~rna] st;esses ~ repres~nted by
spnngs
FIGURE 4.
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small space." But, plainly, this altered description of our beam and the ersatz externalization of its traction forces runs contrary to our original account of what the bean1 is like, viz., that it comprises an uninterrupted elastic whole. Accepting this technique requires that we have subscribed to a policy of essential idealization, for some methodological rationale must explain why we are allowed to n1ake this descriptive substitution when we approach the beam on an infinitesimal level. Only by malling a descriptive substitution can the infinite descent of non-simplifying physics observed above be arrested. Or so presumed virtually all physicists before the midtwentieth century. It is n1Y belief-although I do not pretend to con1prehend the fine details of how it all unfolds-that I(ant seeks to rationalize these odd descriptive replacements by treating internal points like 12 in the sarne manner as he does Newton's absolute space: not as a real component within the billiard ball, but merely as a regulative ideal that directs our improving mesoscopic (that is, S-scale) descriptions of its behavior. In lieu of a detailed IZantian interpretation (which we must leave to Friedman's forthIroming masterwork), let me quote a particularly full articulation of the "replace by point mass" policy provided by IZarl Pearson in 1892: Ifwe take a piece of any substance, say a bit of chalk, and divide it into small fragments, these still possess the properties of chalk. Divide any fragment again and again, and so long as a divided fragment is perceptible by aid of the n1icroscope it still appears chalk. Now the physicist is in the habit of defining the smallest portion of a substance which, he conceives, could possess the physical properties of the original substance as a particle. The particle is thus a purely conceptual notion, for we cannot say when we should reach the exact limit of subdivision at which the physical properties of the substance would cease to be. But the particle is of great value in our conceptual model of the universe, for we represent its motion by the motion of a purely geometrical point. In other words, we suppose it to have solely a motion of translation ...; we neglect its motions of rotation and strain....Vvhat right has the physicist to invent this ideal particle? He has never perceived the limited quantity, the minimum esse of a substance, and therefore cannot assert that it would not produce in him sense-impressions that could only be described by the concepts spin and strain. The logical right of the physicist is, however, exactly that on which all scientific conceptions are based. We have to ask whether postulating an ideal of this sort enables us to construct out of the motion of groups of particles those more complex motions by aid ofwhich we describe the physical universe. Is the particle a symbol by aid of which we can describe our past and predict our future sequences of sense-impressions with a
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great and uniform degree of accuracy? If it be, then its use is justified as a scientific method of simplifYing our ideas and economizing thought. (Pearson [1892] 1991,335-36)12 On this basis, the necessity of essential idealization became central to Pearson's broader philosophical conclusions (which greatly influenced later generations of British philosophy): We interpret, describe, and resume the sequences of this real world of sense-impressions by describing the relative positions, velocities, accelerations, rotations, spins, and strains of an ideal geometrical world which stands for us as a conceptual representation of the perceptual worl,zi. (ibid., 285) To be sure, Pearson's proposals are far cruder than Kant's, but there is plainly a family resemblance: it is important that we appreciate that both proposals were driven by similar initial reflections upon physical technique. Indeed, programs of "broad philosophical sweep" often gather their robust substance from the puzzling nitty-gritty "details" of a circumstances like that of our wooden beam. It is worth noting that there were schools in the late nineteenth century (e.g., that of Ludwig Boltzmann) who regard the "essential idealization" appearance of point-mass replacements in the treatment of continuous matter as evidence that discretely separated atoms are actually present in the real life beam. Quasi-Kantian doctrines to the effect that "science must engage in policies of essential idealization" were widely defended in the late Victorian period precisely as a means of opposing such literalist thinking (indeed, this is Poynting's point when he complains of mistalcing "temporary ladders and passages" for realities in the epigram to this paper). Instead of positing Boltzmanian atoms, we should recognize that, as human agents of limited conceptual capacity, we are forced to impose certain mathematical constructions upon reality "by aid of which we can describe our past and predict our future sequences of sense-impressions with a great and uniform degree of accuracy." But that descriptive necessity is entirely our own foible-there is nothing in nature that truly answers to such constructions. There is a complementary line of thought that is also worth noting. One of the basic lessons of Zeno's paradoxes is that extensive quantities which attach to finite regions of a homogeneous continuum (mass, applied force, etc.) cannot be built up from points bearing the same quantities (because any finite mass value assigned to the points will total to an infinity over any extended region, no matter how small). In modern terms, a point can serve as a carrier of density, but in normal measure theoretic
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practice such density distributions will be induced upon the points in a top down manner, viz., we first assign mass to a schedule of enclosing finite regions and then allow this assignment to precipitate densities upon the local points. This epistemological order is not easily reversed: once Humpty Dumpty has been atomized into points, we'll not be able to put him back together rightly, for most of his vital features will have been unrecoverably erased by the decomposition into dust. As such, these problems of measure date to the ancients, but, in the special case of the traction forces of mechanics, these reassembly-from-points problems become even more pressing. For if a pressure wave happens to pass through point !2, we may not see equal traction vectors upon any surface S around !2, no matter how small. But then the localized density-like surrogate for traction force must prove rather queer in character, for we require a "point" whose perpendicular "sides" are pulled upon in different manners. But what might an infinitesimal "point with perpendicular sides" be like? In truth, many of us have grown accustomed to this strange idea-it lies encapsulated in what we now call a stress tensor-often without realizing the oddities we have gamely swallowed. Although Kant wrote before the notion of stress had been clearly articulated (he dealt instead with "pressures"), he possessed a vivid awareness that physical descriptions can become quite weird when pressed to the infinitesimal level and would have argued, in complete symmetry with his strictures upon "absolute space," that the stress notion cannot be interpreted realistically, but instead qualifies as a regulative ideal shaping the ever improving series of essential idealizations we provid~ for the smallish regions of a continuous material. Indeed, he expresses this conviction in more or less the same terms as Leibniz did before him: physics' talk: of infinitesimal states should not be interpreted as reflecting matter's true "metaphysical composition" but merely provides a decomposition useful for human thinking: For if matter is divisible to infinity then (concludes the dogmatic metaphysician) it consists of an infinite aggregate ofparts; for a whole must already contain in advance all of the parts in their entirety ... And this last proposition is undoubtedly certain for every whole as thing in itself. But one cannot admit that matter, or even space, consists of infinitely many parts . .. One would therefore have to conclude ..., to the annoyance of the metaphysician, that space is not a property of a thing in itself, ... but merely an appearance of our outer senses in general. (Metaphysical Foundations, 4: 506) But since a decomposition into points is inherent in the very notion of objective space, Leibniz and Kant conclude that "things in themselves" therefore cannot be truly located within a spatial arena (and similar argu-
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ments can be run with respect to time). For essentially the same reasons, Victorian physicists, confronted with the same problems of contact force and measure, also deny the "reality" of their point-based descriptive vocabulary, even if they don't always go so far as to abandon space and time as well. A modern expert in continuum physics will argue that such "essential idealization" rationalizations for standard procedures in continuum mechanics are mistaken, although the mathematical tools for avoiding the artificial replacement policies we surveyed were not articulated until well into the twentieth century.1 3 Nonetheless, there remain many common techniques elsewhere in current physical practice that, at first blush, seem as if they rely upon some allied policy of describing portions of a target object in terms of mathematical structures foreign to its anticipated nature. For example, the boundary conditions typically assigned to an atomic system strike me as possibly of this nature: they seem too rigidly specified to represent a realistic condition upon the system. However, in the same "essential idealization" manner as before, we can't apply our differential equation modeling coherently to such a system until such side conditions are set. Working physicists are commonly cavalier about such assumptions, just as typical textbook instruction in continuum mechanics often brushes over the conceptual tensions to which Kant paid eagle-eyed attention two hundred years before (only experts know of the more sophisticated resolutions I have mentioned). We moderns would do well to imitate Kant in his careful methodological scrutiny, for isn't it the job of philosophers to pay attention to conceptual issues that might escape others in a greater hurry to reach practical destinations? Indeed, I understand Friedman's call for meta-paradigmatic investigation to invite philosophical work of precisely this nature: he expects that informative sweep will originate in picayune details of exactly the sort to which Kant carefully attended. But such an investigative model would generate proposals of a considerably different character than those that dominate philosophical activity today, where "sweep" considerably outruns motivational "detail." If so, successful meta-framework meditation requires that we pay closer attention than we commonly do to the tools employed in modern mathematics: if a novel "mathematical object" (such as the stress tensor) is defined through limits and some complex form of equivalence class construction, have we produced an assembly that nature can coherently exemplifY or have we merely confused a regulative ideal with genuine description? Plainly, a range of interesting answers are possible here and if philosophers can articulate such proposals with genuine ingenuity, they will have contributed to a worthy revival of Kantian concerns, in exactly the mode Friedman encourages. And these seem projects that might especially benefit from philosophical training, whereas I think it is unrealistic for us to
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expect that we will able to resolve the problems of quantum gravity on Our own recognizance. As such, it is to be expected that our labors will often take the form of scrutinizing established doctrine with a close eye for methodological oddity, in exactly Kant's manner. Much of the nineteenthcentury philosophical thinking that Friedman extols as paradigms of metaparadigmatic investigation were motivated by concerns within continuum mechanics such as we have surveyed (Mach's historical/philosophical writings representing a case in point). Such musings eventually achieved their greatest effects within an entirely different quarter (Einstein's work), but this is exactly the sort of unexpected consequence that Friedman expects from a fruitful meta-framework investigation. However, our survey has indicated a motive for shifting Friedman's own emphases slightly. We observed that Kant himself contrasted the human-sized level of physical description with both infinitesimal and global scales, each level supplying a distinct intellectual engine that drives us through regulated sequences of improving mesoscopic representations. In his large scale thinking, Kant is plainly captivated by the siren voice of a globally specifiable geometry such as Euclid provides: we grasp its global features at the very same time as we learn the constructions deemed possible within its framing. Certainly, as a matter of philosophical aesthetics, Kant's picture of human thought as hedged on two sides by the very large and the very small is quite charming. However, it is precisely on the global side of things that Kantian expectations with respect to physics have most plainly come to grief. Pace Bertrand Russell and other commentators, it was not the discovery of the early non- Euclidean geometries that utterly dashed orthodox Kantian hopes, for, as Friedman has nicely documented, Helmholtzian assumptions of a free mobility type can nicely temper Kantian insight with a richer dose of empiricism. 14 The deeper obstacle to Kant's contentions lies in the fact that a workable physics needn't lay down a global geometrical structure ahead of time: that is the harsh lesson that differential geometry and General Relativity teaches. Instead, the geometry of space-time can naturally grow by the same differential equation processes that cause a puddle of fluid to alter its shape over time and the global features of such a universe might prove inaccessible to any local observer. It strikes me (although such a vague claim would be hard to defend) that most of the great neo-Kantian philosophers have taken their inspirations primarily from the global portions of Kant's thinking; this is obvious in Helmholtz and Poincare's cases, but the same broad scale consideration seems active, derivatively, in the framework principles advocated by Reichenbach and Carnap. However, if the drift of my musings is correct, this large scale emphasis may represent a mistake: a modern reviver of Kant should scrutinize the infinitely small in his spirit and abandon his global
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aspirations. Friedman borrows the Reichenbachian designation "the relativized a priori" as a label for what we should seek in our meta-paradigmatic researches, but perhaps this choice of terminology is not ideal. "A priori," after all, suggests structures that we know in advance must be fit to the world and this line of thought attaches mainly to the global aspects of the Kantian project. In contrast, the infinitesimal branch doesn't insist upon any fixed impositions at all, even of a relativized nature: its concerns revolve mainly around the possibility that human understanding is parochially limited to an orbit of mathematical tools that display their descriptive limitations on their sleeves. Careful scrutiny of physical technique might therefore persuade us of the inevitability of Kantian essential idealization without requiring any relativized a priori principles in the "prime-the-pump" vein of the logical empiricists at all. As a scientific realist at heart, I hope that these investigations will not break the idealist's way, but it would be foolish to wax dogmatic on these matters, as the subtle and tortured history of classical continua should warn us. And thus Friedman's program of critical "meta-paradigmatic" appraisal, placed in conjunction with a concrete appreciation of how the historical Kant went about his business, truly offers a novel route that may allow philosophy to escape the listless latitudes in which it currently languishes. Following this recipe will not be easy, for it requires the rest of us to imitate, as best we can, Friedman's remarkable capacity to combine modern mathematical and physical understanding with a rich appreciation of philosophy's traditions and the vicissitudes of scientific development.
NOTES 1. Cited in Freund (1960), 227. 2. I am indebted to the seminar's members for many helpful suggestions, as well as to Friedman's manuscript and the comments he has provided. 3. See Friedman (1992). 4. All citations from the Critique of Pure Reason refer to the Norman Kemp Smith (1965) translation. 5. See Truesdell's "Goldstein's Classical Mechanics" in Truesdell (1984). 6. The full title of the essay is "The Employment in Natural Philosophy of Metaphysics Combined with Geometry, of which Sample I Contains the Physical Monadology." The English translation of this work is included in Kant (1992), 47-66. 7. As Kant himself comments in the Metaphysical Foundations at 4: 528. 8. Friedman's work-in-progress contains an excellent discussion of the role of pressure in Kant's thinking. In this essay I pursue the easier task of highlighting the basic difficulties that one encounters in setting up both pressure and stress as viable physical constructs within a continuum.
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9. See K. Hutter and K. Johnk (2004). 10. Actually, it must be a force density, for the measure theoretic reasons we shall soon consider. 11. Historically, another popular route for reaching continua travels through rigid bodies, rather than point mass replacements as highlighted here. See Casey (1992). 12. Although Pearson is best remembered today for his work in bio-statistics he began his career as an elastician. ' 13. See Antman (1980). 14. See Friedman (1996).
REFERENCES Antman, S. (1980). "The Equations for the Large Vibrations of Strings." American Mathematical Monthly 87: 359-70. Casey, J. (1992). "The Principle of Rigidification." Archive for the History of the Exact Sciences 43: 329-83. Friedman, M. (1992). Kant and the Exact Sciences. Cambridge: Harvard University Press. - - - . (1996). "Poincare's Conventionalism and the Logical Positivists." In Greffe, Heinzman and Lorentz (eds.) (1996), 333--44. Reprinted in Friedman (1999),71-86. - - - . (1999). Reconsidering Logical Positivism. Cambridge: Cambridge University Press. - - - . (2001). Dynamics ofReason. Stanford: CSLI Publications. Freund,1. (1960). The Study ofChemical Composition. New York: Dover. Greffe, J.L., G. Heinzman and K. Lorentz (eds.) (1996). Henri Poincare: Science et Philosophie, Nancy 1994. Berlin: Academie Verlag. Hutter, K. and K. Johnk (2004). Continuum Methods ofPhysical Modeling. Berlin: Springer. Kant, 1. ([1781/1787] 1965). Critique of Pure Reason. Translated by Norman Kemp Smith. New York: St. Martin's Press. - - . (1992). Theoretical Philosophy 1755-1770. Edited by in David Walford. Cambridge: Cambridge University Press. - - . ([1786] 2004). Metaphysical Foundations ofNatural Science. Edited by M. Friedman. Cambridge: Cambridge University Press. Pearson, K. ([1892] 1991). The Grammar ofScience. Bristol: Thoemmes. Truesdell, C. (1984). An Idiot)s Fugitive Essays on Science: Methods, Criticism, Training, Circumstances. New York: Springer-Verlag.
PART VI Concluding Essay
[26J Synthetic History Reconsidered * MICHAEL FRIEDMAN
The Newtonian Era
574
I(ant
585
Philosophy of Mathematics
585
Philosophy of Natural Science
599
From I
621
IZant and N aturphilosophie
622
Helmholtz
628
M~h
637
Poincare
642
Einstein
653
Carnap and Logical Empiricism
664
Transcendental Philosophy and Modern Physics: Neo-l
678
Synthetic History and the Dynamics of Reason
696
Post-IZuhnian Philosophy of Science
712
Notes
729
References
802
571
would like to begin by expressing my heartfelt gratitude to the editors of this volume and to the contributors. Very few things could possibly be as gratifying as seeing such a large number of friends, colleagues, and former students participate so eagerly, and so insightfully, in a common intellectual endeavor we have shared over the years. The editors have dubbed this endeavor "synthetic history," which means, in the broadest sense, an approach to the history of science giving special significance to its relationship with the history of philosophy and an approach to the history of philosophy giving special significance to its relationship with the .history of science. This endeavor is motivated, more generally, by the conviction that neither an adequate philosophical understanding nor an adequate historical understanding of science can be achieved without paying particular attention to the manifold vvays in which philosophy and the sciences have interacted throughout our intellectual history. According to a ll10ttO attributed to Norwood Russell Hanson, the founder of the Department of History and Philosophy of Science at Indiana University, philosophy of science without history of science is empty, history of science without philosophy of science is blind. The motto of synthetic history, by contrast, in the spirit of the recently founded discipline of the History of the Philosophy of Science (HOPOS), is that neither history of philosophy without history of science nor philosophy of science without such integrated history· is possible. My own work over the last thirty years has ranged over a number of topics and episodes involving the historical interaction between philosophical and scientific themes, many of which are represented in this volume. There have been two principal centers of gravity in my work: IZant and his relationship with Newtonian science, on the one side, the logical empiricists and their relationship with Einsteinian relativity, on the other. Moreover, since my approach to the logical empiricists has emphasized their IZantian and neo- IZantian roots, it is not too much to say that IZant's place within the history of both modern philosophy and modern science has been my central focal point. Finally, my own approach to extracting morals for contemporary post- IZuhnian philosophy of science from this intellectual history is distinctively framed by IZantian and neoIZantian preoccupations (which I see as still present in logical empiricism as well)-involving both a special place for the modern mathematicalphysical theories of space, time, and n10tion (from Newton to Einstein) and a special role for a priori constitutive principles (in both geometry and mechanics) within these theories. This particular approach, which I call the dynamics of reason, is by no means shared by all (or even most) contributors to the present volume, and, as the editors rightly emphasize, it is by no means definitive of the enterprise of synthetic history more generally.
I
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Accordingly, and very properly, many of the contributions to the present volume do not directly engage with my work, even when they treat topics, such as I(ant or modern physics, which are central to my approach. For the point of the volume, as explained by the editors, is not so much to comment on what I personally have done as to illustrate the scope and value of the enterprise of synthetic history in all (or at least many) of its varied forms. What I shall offer here, therefore, is not an essay-by-essay "reply" to the various contributions, but rather a reconsideration of my own approach to synthetic history as seen from n1Y current perspective. Nevertheless, since my current understanding has been influenced over the years by all of the contributors, I shall talce every opportunity that offers itself to locate my remarks in relation to their contributions to synthetic history, both here and in earlier work.
The Newtonian Era Like all of the historical periods and episodes on which I focus, I view the Newtonian era-the seventeenth century through the Leibniz-Clarke correspondence-in relation to Kant and IZantian themes. In particular, in my historical work on IZant, as noted above, I have focused largely on his relation to Newtonian physics. In my most recent work on both IZant and the seventeenth century, however, I have also emphasized the importance of theological questions-about God, his creation of the material world in space, and the consequences different views of such creation have for the metaphysical foundations of physics. For a defining event within both the philosophy and the science of this period was the condemnation of Galileo for defending the Copernican system; and the task of then showing that the new mathematical science descending from Galileo's work was consistent, after all, with both the spirit and the letter of the Christian religion became central for virtually all natural philosophers beginning with Descartes-including, of course, both Newton and Leibniz. Mary Domski's contribution very helpfully and instructively considers some of the fundamental differences between Descartes and Newton on the foundations of geometry. Whereas Descartes proposes a revolutionary new classification of "geometrical" curves, based on the extended sense of geometrical construction underlying his new analytic geometry, Newton, following ancient geometrical practice in such authors as Pappus and Archimedes, rejects both Descartes's distinction between "geometrical" and "mechanical" curves and the intrusion of algebraic methods into the subject-matter of geometry n10re generally. The result, for Newton, is a more tolerant conception of the class of curves admitted as the objects of geometry and, accordingly, a more intimate relation between the subject-
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matters of geometry and mechanics. Domski suggestively links this fundamental difference to a disagreement about philosophical methodology and its relation to history. Whereas Descartes, in the Discourse on Method, proposes to break with all previous historical authorities and follow his own method of clear and distinct ideas, Newton has no room for systematic philosophy pursued independently of the "wisdom of the ancients." For Newton, the constructive appropriation of this earlier wisdom in the context of recent discoveries in both mathematics and mathematical physics rather constitutes the essence of his philosophical reflections on the foundations of geometry. Without engaging directly with Domski's methodological thesis, I would like to suggest an important sense, in a somewhat different context, in which both Descartes and Newton appropriate ancient wisdom on behalf of their own distinctive projects in mathematics and mathematical physics. What I have in mind, more specifically, is the sense in which both Descartes and Newton put neo-Platonic themes to work on behalf of the theological credentials of their projects. Thus Descartes, as Stephen Menn has recently argued in detail, appropriates ideas about our knowledge of God and the soul developed by Plotinus and Augustine in articulating what Descartes himself calls a "metaphysical foundation for physics." In particular, Descartes follows the "method of ascent," through which we come to a knowledge of God from a knowledge of ourselves: we first grasp that our essential nature is intellectual (and thus non-sensible and incorporeal); we then appreciate the fact that our intellect is finite and imperfect; and we finally arrive at the existence of a fully perfect, infinite intellect-God. This method, in Descartes's hands, then privileges purely intellectual knowledge, as paradigmatically exemplified in pure geometry, over sensible and empirical knowledge, and it thereby privileges modern mathematical physics (especially as developed by Descartes, of course) over Aristotelian physics. It is modern-Cartesian-cosmology and physics (which, it so happens, are essentially Copernican) rather than AristotelianPtolemaic cosmology and physics which best accord with the Faith. l As Domski points out, the main target of Newton's rejection of "relationalism" in favor of an "absolutist" metaphysics of space is Descartes, not Leibniz, and the locus classicus for Newton's own metaphysics of space is his unpublished De Gravitatione. 2 What was most important for Newton was decisively to reject Descartes's identification of matter with extension and to defend, accordingly, the concept of absolute (empty) space existing independently of matter. Yet Newton, like Descartes before him, also appropriated nea-Platonic ideas (from a somewhat different tradition) in the course of this very rejection. For Newton, the most salient source of such ideas was the Cambridge Platonism represented especially by Henry More, and Newton employs them in his doctrine that absolute
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space is neither a substance nor an accident but what he calls "an emanative effect of God and an affection of every kind of being" (p. 21). 3 In particular, absolute space or pure extension is even an affection of God himself, since God is omnipresent or everywhere. God can thereby create matter or body (as something quite distinct from pure extension) by endowing certain determined regions of space with the conditions of mobility, impenetrability, and obedience to the laws of motion. God can do this anywhere in space, in virtue of his omnipresence, by his imn1ediate thought and will, just as our souls can move our bodies by our in1n1ediate thought and will. It is essentially this doctrine which surfaces in Newton's well-known published statements, in the General Scholium to the Principia and Query 31 of the Optics, that space is the "sensorium" of God. There is no doubt that the sharp differences between Descartes's and Newton's metaphysics of space-their different conceptions of the relationships among space, God, and matter-are of fundamental importance. For Descartes, since space is simply identical with matter, God creates matter by creating space itself, and it is precisely this act of creation of space at successive moments of time that is responsible for the laws of motion. In particular, the conservation of what Descartes called the total "quantity of motion" results from the unity and simplicity of God, whereby God continually recreates the entire universe (the whole of pure extension, whose various parts may have different instantaneous tendencies to motion at any given time) at each instant while constantly expressing the very same divine essence. For Newton, by contrast, matter and space have radically different statuses vis-a-vis God's creation. Space is "an emanative effect of God and an affection of every kind of being," including God, while matter is the result of God's creative activity in space wherein certain determined regions are then endowed with the conditions ofn10bility, impenetrability, and obedience to the laws of motion. By instituting the laws of motion, in particular, God thereby endows certain regions of space with Newtonian mass or quantity of matter (vis inertiae), and the presence of this quantity, specifically, clearly distinguishes matter from empty space. This not only leads, following earlier work of Wren, Wallis, and I-Iuygens, to a much n10re adequate formulation of the laws of impact (whereas Descartes's inadequate formulation had no room for the quantity of mass, and thus no room for momentum or quantity of motion in the Newtonian sense), it eventually leads to the theory of universal gravitation of Book III of the Principia. In particular, the famous Scholium on absolute space, time, and motion at the beginning of the Principia refers to the procedure for distinguishing true from apparent motions developed later in Book III, where the theory of universal gravitation then puts the notions of absolute space, time, and motion to real physical work in determining the center of mass of the solar system as the true "center of the world."
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Nevertheless, despite these fundamental differences, both Descartes and Newton are using neo-Platonic ideas to support an essentially mathematical approach to physics over the older qualitative approach of Aristotelian physics. For Descartes, the world described by physics is, in its essence, the object of pure geometry. God, in creating this world, not only brings about (what Descartes talces to be) the (mathematical) laws of . motion of the new physics, but he also guarantees, in creating us as mindbody composites located within this world, that we can use our purely intellectual mathematical knowledge in successively correcting and refining our knowledge of the material world-as we apply pure mathematics, that is, to the initially misleading deliverances of our senses. 4 For Newton, although the world described by physics is not, in its essence, the object of pure geometry, space (which is the object of pure geometry) nonetheless constitutes the "frame of the world"-an emanative effect of the divine existence wherein God then creates matter by an immediate act of his will. The bare existence of space suffices for the existence of all the shapes and figures studied in pure geometry (De Grav., 22): "there are everywhere all kinds of figures, everywhere spheres, cubes, triangles, straight lines, everywhere circular, elliptical, parabolic, and all other kinds of figures, and those of all shapes of sizes, even though they are not disclosed to sight." And thus pure geometry is ipso facto applicable to all material bodies as well (pp. 22-23): "the delineation of any material figure is not a new production of that figure with respect to space, but only a corporeal representation of it, so that what was formerly insensible in space now appears before the senses." Therefore, in virtue of their (differently) neo- Platonic conceptions of a metaphysics of space, neither Descartes nor Newton has any room for a necessary gap (as there was in Plato's original "Platonism") between pure mathematics, on the one side, and the sensible and material world, on the other. 5 The significance of this point becomes clearer if we contrast the conceptions of both Descartes and Newton with the quite distinct approach of Leibniz, who was explicitly opposed to both Descartes and Newton in correspondingly different ways. Leibniz began, in fact, by reacting to Descartes's failure adequately to formulate the basic laws of impact which were supposed to govern, according to the then dominant paradigm of the mechanical natural philosophy, all phenomena in the material or corporeal world. Leibniz responded to this problem, in his "Brief Demonstration of a Notable Error of Descartes and Others Concerning a Natural Law" (1686), by emphasizing the importance of a new, essentially dynamical quantity, which he called vis viva or living force (in modern terms, mass times the square of the velocity), where the basic law of motion is now formulated as the conservation of the total quantity of vis viva. Beginning with his Discourse on Metaphysics (written in the same year), Leibniz also
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strongly emphasized that living force is not purely geometrical or mechanical, so that, in particular, this quantity (unlike Descartes's purely mechanical "quantity of motion") reintroduces an element of Aristotelian teleology into the mechanical philosophy. For vis viva, on Leibniz's view, is the counterpart of the Aristotelian notion of entelechy: namely, that internal (non-spatial) principle by which an ultimate simple substance or monad determines (by a kind of "appetition") the entire future development of its own internal state. Moreover, in accordance with this same renewed emphasis on Aristotelian teleology, Leibniz then articulated a doctrine of divine creation in terms of God's choice of the best among all merely logically possible worlds. The distinction between what is logically possible and what is actual-between all merely thinkable worlds available to the divine intellect and the best and most perfect of these worlds as determined by the divine will-thereby corresponds to the distinction between principles of pure mathematics (including geometry) and principles of natural science or physics (the laws of motion). In particular, the laws of motion, unlike the principles of pure mathematics, precisely express the divine wisdom in actualizing or creating the best and most perfect of all possible worlds. Leibniz thereby breaks decisively with Descartes's metaphysics of space, for the actual world of material substances results from a special act of the divine will which introduces additional non-spatial and essentially teleological elements into the mechanical laws of motion. Indeed, Leibniz's break with Descartes on this issue is deeper still, for, on Leibniz's view, the entire mechanical physical world (including the space in which bodies move) is a secondary appearance or phenomenon (a "well-founded phenomenon" like the rainbow) of an underlying metaphysical reality of simple substances or monads-substances which, at this level, are not spatial at all but rather have only purely internal properties and no external relations. This point, in turn, is closely connected with a fundamental disagreement with Descartes about the nature of the intellect: whereas Descartes entirely rejects traditional Aristotelian logic and instead takes purely intellectual knowledge to be exemplified by the procedure of his new analytic geometry, Leibniz self-consciously returns to the idea that purely intellectual knowledge is essentially logical. And, although Leibniz appears to have envisioned some sort of extension of Aristotelian logic capable of embracing the new algebraic methods of his calculus, there is no doubt that the traditional subject-predicate structure of this logic pervades his monadic metaphysics: it is precisely because ultimate metaphysical reality is essentially intellectual in the logical sense that the entire mechanical world, including space, is a merely secondary reality or phenomenon. Thus, although Leibniz, like everyone else in the period, holds that there are exact mathematical laws governing the sensible and material world, he
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reintroduces a new kind of necessary gap between reality as known by the intellect and this sensible world. For Newton, by contrast, space-the very space in which bodies exist and move-is metaphysically fundamental, for, as we have seen, it is "an affection of every kind of being," including God himself. Indeed, Newton puts the point even more strongly several pages later (De Grav., 25): "Space is an affection of a being just as a being. No being exists or can exist which is not related to space in some way." In particular, God, through his omnipresence, creates matter in space by endowing certain determined regions with mass (vis inertiae), and God thereby institutes the (Newtonian version of) the laws of motion by singling out momentum (mass times velocity) as the fundamental dynamical quantity governing all changes of motion of matter. For ewton, moreover, the basic dynamical quantity causally responsible for such changes is impressed force (vis impressa), where this refers to any action of a second body on the body in question by which a change of momentum of the first body is produced. Force, in the Newtonian sense, is thus an external action of one body on another, not an internal principle of change like Leibnizean vis viva, and, what is more, it is an action not intrinsically limited to the condition of contact. On the contrary, the principal instantiation of this concept, in the Principia, is precisely the force of universal gravitation, whereby one body exchanges momentum with another immediately and at a distance; and it is the theory of universal gravitation, as we have seen, which then puts the notions of absolute space, time, and motion to real physical work in determining the true "center of the world.,,6 It is by no means surprising, therefore, that Newton also entirely rejects the u'aditional Aristotelian notion of substance, and replaces it, in effect, with space itself-or, more precisely, with space plus God (De Grav., 29): "For the existence of these beings [bodies] it is not necessary that we suppose some unintelligible substance to exist in which as subject there may be an inherent substantial form; extension and an act of the divine will are enough. Extension takes the place of the substantial subject in which the form of the body is conserved by the divine will; and that product of the divine will is the form or formal reason of the body denoting every dimension of space in which the body is to be produced." For Leibniz, by contrast, space, as we have seen, is a mere "well-founded phenomenon," and pure intellectual knowledge is explicitly modeled on Aristotelian subjectpredicate logic: (a modified version of) the Aristotelian concept of substance must be metaphysically fundamental? The early eighteenth century witnessed a great stage-setting intellectual debate, the correspondence between Leibniz and Clarke, which sharply focused attention on the opposition between the Leibnizean and Newtonian philosophies. This debate paid equal attention to both
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technical problems in physics and natural science (such as the laws of impact and the nature of matter) and more general issues within metaphysics and theology (such as the principle of sufficient reason and God's choice to create our world). Leibniz objected to the Newtonian doctrine of direct divine intervention in the phenomena of the n1aterial universe (such as specially adjusting the orbits in the solar system, for example, so as to ensure that they all lie in approximately the same plane) and defended his own version of the principle of sufficient reason, whereby God's creative activity is exercised only in his initial choice of the best of all possible worlds. Clarke (representing Newton) replied that this would entail an unacceptable limitation on God's freedom of action. In particular, he defended Newtonian absolute space against Leibniz's use of the principle of sufficient reason to argue that such a space is impossible because God would then have no reason to place the material universe in one position rather than another within absolute space. In mid-eighteenth-century Germany this great debate between Leibnizeans and Newtonians dominated the intellectual agenda within both natural science and metaphysics, and lunt was no exception. Indeed, Iunt's earlier "pre-critical" writings were overwhelmingly concerned with problems of natural philosophy in general and the project of reconciling Leibniz and Newton in particular; and the main thrust of my own approach to lunt is that this same project continues into the "critical" period-although in a radically transformed way. Here is not the place to engage in an extended discussion of the transition between IZant's pre-critical and critical periods,8 and I shall present a more detailed discussion of controversial issues of Iunt-interpretation in the next section. But I shall outline here, in concluding this section, how IZant's continuing efforts at integrating the Leibnizean and Newtonian philosophies finally results in a fundamental transformation of the very meaning of metaphysics as practiced by Descartes, Newton, and Leibnizof their characteristic mode of investigation, that is, into the relationships among space, God, and matter. IZant, in the pre-critical period, attempts to fashion a direct unification of Leibnizean and Newtonian ideas by starting with a Leibnizean metaphysics of monads and then building a Newtonian metaphysics of space, as it were, on top of this n10nadic metaphysics. The primary reality remains a non-spatial realm of ultimate simple substances, but these substances, for IZant, now have both purely internal, intrinsic properties and external or extrinsic relations. In particular, such relations among the monads are not necessary for them to be the simple substances that they are, but they are necessary if all of these n10nads are to exist-or, more precisely, to coexist-together in a comn10n world. In this way, God's creative activity has two distinguishable aspects: one act by which the simple substances themselves are created in the first place, and a second act by which a number of
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such simple substances are joined together into a single world. The second occurs in conformity with what Kant calls a "schema of the divine intellect," and it is in virtue of just such a schema, in the end, that what we know as the laws of nature arise. More precisely, what we know as the fundamental forces of matter (attraction and repulsion)-together with the laws that govern them-are a direct expression of the divinely instituted external relations (of co-existence) between monads; and what we know as space is then the phenomenal expression of this same system of divinely instituted relations. Space is thus a secondary reality, derivative from the monads and their external relations, but, since the external relations between monads, for Kant, are just as real as their internal properties, it is a reality nonetheless-and not, as in Leibniz, a merely ideal "well-founded phenomenon." Indeed, since the fundamental force of attraction, for Kant, is explicitly modeled on Newtonian universal gravitation (as an immediate action at a distance through empty space), Kant explicitly links his pre-critical conception of space with the Newtonian conception of divine omnipresence. 9 It is in the Inaugural Dissertation of 1770 that Kant first makes a fundamental break with the Leibnizean philosophy-and, in a somewhat different fashion, with the Newtonian philosophy as well. In particular, Kant here first articulates his characteristic distinction between two independent rational faculties of the human mind-the pure understanding or pure intellect, on the one side, and pure sensibility or pure intuition, on the other. The former embodies the traditional categories and concepts of rational (Leibnizean) metaphysics, but it is the latter, for Kant, which now embodies the concepts and principles of pure mathematics. In particular, Kant now holds that mathematical knowledge is in no way purely intellectual, but is rather essentially intuitive or sensible, requiring the forms of pure sensibility, space and time. The world as we know it therefore bifurcates into two: the intellectual world described by traditional metaphysics (i.e., by the Leibnizean metaphysics of ultimate simple substances as modified by the earlier Kant), and the sensible world as described by mathematics and mathematical physics in space and time. Although something like Newtonian space therefore remains as the foundation of this sensible world, space can no longer be conceived, as in Newton, as the sensorium of God-it is rather, as it were, the form of our sensorium, the form of our pure sensibility. Yet it is an unresolvable problem, in the Inaugural Dissertation, how these two worlds are now supposed to be related, and, in particular, how the world described by mathematics and mathematical physics (the world as it appears to us) is related to the ultimate metaphysical reality of the intellectual world. It is precisely this problem, then, which, after more than ten long years of Kant's "silent decade," finally gives birth to the critical philosophy in
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1781.1 0 I(ant now declares that purely intellectual, metaphysical knoWledge-whether of immaterial things like God and the soul or of the ultimate simple substances which (according to both Leibniz and the pre-critical I(ant) underlie the material world-is completely impossible, at least from a theoretical point of view. The pure intellect, considered entirely on its ovvn and independently of any possible relation to sensibility, can issue only in the empty logical forms of (a slightly extended and modified version of) Aristotelian syllogistic. The pure intellect, as lZant now conceives it, produces only what he calls the "logical forms of judgment." And, while it is true that these forms then yield, in what IZant calls the "metaphysical deduction," the pure concepts or categories of the understanding (substance, causality, community, possibility, actuality, necessity, and so on), such pure concepts of the understanding are themselves entirely empty and without any "relation to an object" (again trom a purely theoretical point of view) considered independently of our particular (human) forms of sensibility-space and time. Such concepts can be used for genuine (theoretical) knowledge only when applied to spatio-temporal "appearances," and thus only when "schematized" in terms of space and time (substance in terms of permanence, causality in terms of succession, and so on). In short, it is only in virtue of a sensible "schema" produced by the pure intellect that rational knowledge of the phenomenal world is possible, and the task of showing how the pure intellect thereby injects itself into pure sensibility (space and time) so as to apply the pure categories of the understanding ·to sensible experience then becomes the problem of the "transcendental deduction." Such an injection of our pure intellect into our pure forms of sensibility-in accordance with what Kant calls the transcendental synthesis of the imagination-now takes the place, as it were, of the pre-critical doctrine that a schema of the divine intellect, by an analogue of Newtonian divine omnipresence, is ultimately responsible for the order we perceive in the physical world. II The transcendental deduction in both editions of the Critique ofPure Reason (1781 and 1787, respectively) emphasizes that, in thus injecting itself into the forms of pure sensibility so as to apply the pure categories of the understanding to sensible experience, the understanding is also the ultimate source of the laws of nature. And, in the most important treatments of this matter published between the two editions of the Critiquethe Prolegomena to Any Future Metaphysics of 1783 and the Metaphysical Foundations of Natural Science of 1786-IZant strives to illustrate the crucial relationship between the understanding and sensibility more concretely. In the Metaphysical Foundations, IZant derives more specific laws of what he calls "pure natural science" from the more general transcendental principles of the understanding of the first Critique, and he tries to show that we thereby obtain a "metaphysical" foundation for what is essentially
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the physics of Newton.!2 In §38 of the Prolegomena, Kant illustrates the "seemingly bold proposition" that the understanding "prescribes" laws to nature by the Newtonian law of universal gravitation and, as I have argued in detail elsewhere, by the Newtonian derivation of this law from Kepler's laws of planetary motion in the Principia. 13 We can then understand more precisely the sense in which the pure understanding is the source of the law-governedness of nature in this case. In the Metaphysical Foundations, the most important laws of pure natural science Kant derives from the corresponding transcendental principles of the understanding are the conservation of the total quantity of matter, the law of inertia, and the equality of action and reaction-corresponding, respectively, to the transcendental principles of substance, causality, and community (these same three laws are also prominently adduced in §Vof the Introduction to the second edition of the Critique as clear and uncontroversial examples of synthetic a priori pure natural science). Moreover, the "deduction" of the law of universal gravitation in Book III of the Principia from the "phenomena" recorded in Kepler's laws proceeds, as Kant understands it, by presupposing these three "laws of mechanics," along with the (Euclidean) geometry of space, and then arriving at the law of universal gravitation by a kind of constructive procedure on the basis of both the Keplerian "phenomena" and these a priori presuppositions. It is only in this way, in particular, that we can understand the law of universal gravitation as what Kant characterizes as a strictly universal and necessary law, as opposed to a merely empirical regularity possessing merely inductive or "comparative" universality.!4 Corresponding to this procedure for arriving at the law of universal gravitation, for Kant, is a parallel constructive procedure for arriving at the true motions of bodies from their apparent motions. We begin with our parochial perspective here on earth, from which we can record both the observable phenomena governed by Galileo's law of fall and the observable relative motions of a variety of satellites in the solar system with respect to their primary bodies (the moon relative to the earth, the planets relative to the sun, the moons of Jupiter and Saturn relative to their planets). The latter are just the phenomena expressed in Kepler's laws, and what we now find-again presupposing the laws of motion and the geometry of space-is that we can first determine the true state of rotation of the earth (using small deviations from the law of fall manifesting what we now call Coriolis forces); and we can then determine the masses of all the primary bodies in the solar system (at least those actually having satellites), with the result (as Newton shows) that the center of mass of the solar system is always very close to the center of the sun. In this way, in particular, we can empirically determine, from the observable phenomena themselves, the true center of motion of the solar system, and this thereby counts as
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an approximation, for Kant, of Newtonian absolute space. However, since it is also true, for Kant, that the solar system itself rotates around the center of the Milky Way galaxy, this galaxy rotates around the center of a larger system of such galaxies in turn, and so on ad infinitum, absolute space (the true center of motion of the entire universe) is in the end what he calls an "idea of reason"-a forever unreachable regulative ideal we can only successively approximate in experience but never fully attain. IS Since Newtonian absolute space is thus viewed as a regulative idea of reason, there is then a final reconfiguration, for the critical lZant, of the relationships among space, the laws of nature, and the idea of God. For the idea of God, too, is a regulative idea of reason. Indeed, there is an important sense in which it is the ultimate regulative idea, since all human activity, together with the whole of nature, is ultimately subject to the idea of the Highest Good-the idea of a perfect community of all rational beings in a n10ral realm of ends, for which our only ground even to hope this could actually be achieved in nature (or, more precisely, actually successively approximated) is the idea of God (or, more precisely, divine providence). Moreover, lZant saw a deep analogy between the comn1unity of all rational beings in a moral realm of ends and the thoroughgoing community effected among all material bodies in the universe by the law of universal gravitation, and this is the basis, in fact, for his final re-interpretation of the Newtonian doctrine of divine omnipresence in a footnote appended to the General Remark to the Third Part of Religion Within the Limits of Reason Alone (1793): When Newton represents [the universal gravitation of all matter in the world] as, so to speak, divine universal presence in the appearance (omnipraesentia phenomenon), this is not an attempt to explain it (for the existence of God in space contains a contradiction), but rather a sublime analogy, in which it is viewed n1erely as the unification of corporeal beings into a world-whole, in so far as we base this upon an incorporeal cause. The same would happen in the attempt to comprehend the self-sufficient principle of the unification of the rational beings in the world into an ethical state and to explain the latter from the former. We know only the duty that draws us towards this; the possibility of the intended effect, even when we obey this [duty], lies entirely beyond the limits of all our insight. (6, 138-39) For the critical lunt, in other words, the only possible meaning the idea of divine omnipresence (and divine providence) can now have is a purely practical meaning, in so far as we unconditionally obey the comn1and of morality to strive to realize the realm of ends here on earth, and, accordingly, we take the whole of that material nature of which we are a part to
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be capable of such a realization in principle (or, more precisely, its successive approximation). Kant thereby brings the characteristic mode of metaphysical investigation into the relationships among space, God, and matter practiced by his predecessors to a close-and with it the Newtonian era.l 6
Kant The previous section has already begun to treat Kant in a substantial way, primarily in relation to three central figures of the Newtonian eraDescartes, Leibniz, and Newton himself. We have seen, in particular, how Kant follows Newton in taking the space of geometry (now understood as a pure form of sensibility) as an autonomous foundation for both purely mathematical investigations and their application to the empirical world in mathematical physics. Moreover, in thus privileging space and geometry in the foundations of mathematics and physics, Kant is also, in a way, following Descartes's devaluation of traditional logical reasoning on behalf of the inferences of pure mathematics-and he is certainly rejecting the opposing Leibnizean view according to which purely logical reasoning suffices. Nevertheless, Kant agrees with Leibniz (and disagrees with Newton) that a prior "metaphysical" consideration of purely intellectual concepts such as substance, causality, community, and so on (concepts which, for both Leibniz and Kant, are intimately related to traditional Aristotelian logic) is still necessary for properly understanding the metaphysical foundations of physicsP In the present section I focus on controversial matters of Kant-interpretation from the critical period. I shall concentrate on issues in both the philosophy of mathematics and the philosophy of natural science that have been further discussed by the contributors to the present volume (among others), and concerning which, in particular, I have reconsidered my own views partially in response to this ongoing discussion. I hope also to show how some of these issues are related to the themes from the Newtonian era treated in the previous section.
Philosophy of mathematics I begin with an issue concerning the interpretation of Kant's philosophy of mathematics raised at the beginning of Charles Parsons's contributionthe dichotomy between an emphasis on the logical role of Kantian pure intuition in facilitating certain mathematical inferences and the more traditional view that the role of intuition is largely confined to verifYing the fundamental axioms of mathematics from which such inferences proceed.
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In Friedman (1985)-reprinted, slightly revised, in Friedman (1992a)-I self-consciously aligned myself with the former approach, as previously developed by Evert Beth and Jaakko Hintild
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struct a concept is to present the intuition corresponding to it a priori" (A713/B741). Accordingly, Kant concludes, "[philosophy] confines itself merely to universal concepts, [mathematics] can effect nothing by mere concepts, but hastens immediately to intuition, in which it considers the concept in concret£r-not, however, empirically, but merely in an [intuition] that it presents a priori, that is, which it has constructed, and in which that which follows from the universal conditions of construction must also hold universally of the object of the constructed concept" (A715-16/B743---44 ). Exactly what, however, is a pure or non-empirical intuition corresponding to a general concept-a singular instance of this concept that is nonetheless presented purely a priori? Moreover, how can any singular instance of a general concept (no matter how it is supposed to be produced) possibly be an additional source, over and above purely conceptual knowledge, of universal and necessary knowledge? Immediately after the just quoted sentence defining the construction of a concept as the a priori presentation of the corresponding intuition, Kant says (ibid.): "For the construction of a concept we therefore require a non-empirical intuition, which consequently, as intuition, is a singular object, but which nonetheless, as the construction of a concept (a universal representation), must express universal validity, in the representation, for all possible intuitions that belong under this concept." But how, once again, can an essentially singular representation (no matter how it is supposed to be produced) possibly express such truly universal validity? Problems of precisely this kind, of course, underlie the contrary conviction, common to all traditional forms of rationalism, that mathematical knowledge must be conceptual or intellectual as opposed to sensible. Kant illustrates his meaning, in the continuation of our well-known passage, by an example of a Euclidean proof, Proposition 1.32 of the Elements, where it is shown that the sum of the interior angles of a triangle is equal to two right angles (180°). Given a triangle ABC one extends the side BC (in a straight line) to D and draws the line CE parallel to AB. One then notes (by Proposition 1.29) that the alternate angles BAC and ACE are equal, and also that the external angle ECD is equal to the internal and opposite angle ABC. But the remaining internal angle ACB added to ACE and ECD yields two right angles (the straight line BCD = 180°), and these last two angles have just been shown to be equal to the first two internal angles. Therefore, the three internal angles together also add up to two right angles. This construction and proof obviously has universal validity for all triangles, because the required inferences and auxiliary constructions (extending the line BC to D and drawing the parallel CE to AB) can always be carried out within Euclidean geometry, no matter what triangle ABC we start with. I8
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It appears, in fact, that the proof-procedure of Euclid's Elements is paradigmatic of "construction in pure intuition" throughout I
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In fact, schemata rather than images of objects are what lie at the basis of our pure sensible concepts. No image at all would ever be adequate to the concept of a triangle in general. For it would never attain the universality of the concept, which makes it hold for all triangles, whether right-angled, acute-angled, and so on, but would always be limited to only a part of this sphere. The schema of the triangle can never exist anywhere but in thought, and it signifies a rule of synthesis of the imagination with respect to pure figures in space. (A140--41/B180) This "rule of synthesis," therefore, appears to be nothing more nor less than the Euclidean construction of an arbitrary triangle considered in the Axioms of Intuition as a "mere [universal] function of the productive imagination. "21 More generally, then, we can take the Euclidean constructions corresponding to the fundamental geometrical concepts (line, circle, triangle, and so on) as what Kant means by the schemata of such concepts. We can understand the schema of the concept of triangle as a function or constructive operation which takes three arbitrary lines (such that two together are greater than the third) as input and yields the triangle constructed out of these three lines as output (in accordance with Proposition 1.22); we can understand the schema of the concept of circle as a function which takes an arbitrary point and line segment as input and yields the circle with the given point as center and the given line segment as radius as output (in accordance with Postulate 3; compare A234/B287); and so on. 22 Such constructive operations have all the generality or universality of the corresponding concepts: they yield, with appropriate inputs, any and all instances of these concepts. Unlike general concepts themselves, however, the outputs of a schema are indeed singular or individual representations-particular instances, or what Kant calls images, which fall under the concept in question. The outputs of a schema are thus neither propositions nor truth-values, and so Kantian schemata cannot be identified with either Russellian propositional functions or Fregean concepts. This last point represents a first step in understanding why Kant tal<es pure mathematics essentially to involve non-discursive or non-conceptual cognitive resources which, nonetheless, possess all the universality and necessity of purely conceptual thought. Characteristic of conceptual thinking, for Kant, is the logical procedure of subsumption, whether of an individual under a general concept or of a less general concept (species) under a more general concept (genus). The basic principle of purely conceptual reasoning is therefore the dictum de omni et nullo, according to which whatever holds of the more general representation must also hold of the less general (and, similarly, whatever contradicts the more general repre-
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sentation must also contradict the less general). 23 Characteristic of n1athematical reasoning, by contrast, is the procedure of substitution-by which, as we would now put it, an object is inserted into the argument place of a function, yielding another object that can be inserted into the argun1ent places of further functions, and so on. Reasoning by substitution is therefore essentially iterative, and it is precisely such iterative thinking, for Kant, that underlies both pure geon1etry (in the guise of Euclidean proof) and the more general calculative manipulation of magnitudes in algebra and arithmetic. Unlike in geometry, however, where we begin our (Euclidean) proof-procedure by postulating certain initial constructive functions (given by Postulates 1-3) and then iterating them (as in the construction of a triangle in Proposition 1.22), in arithmetic we postulate no particular initial functions but presuppose only our general ability successively to iterate indefinitely any initial functions of an appropriate kind (such that new objects always result: compare note 24 below). It is in this sense, for I
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would use in formulating a Hilbert-style axiomatization in modern quantificationallogic, where all such Skolem functions are explicitly consu·ucted by finite iterations of the three initial constructive operations laid down in the first three postulates. We can take a third step, finally, by noting that the discursive structure of the understanding or intellect, for Kant, is delimited by the logical forms of traditional syllogistic logic. Moreover, and in explicit opposition to Leibniz, Kant takes these logical forms to be su·ictly limited to essentially finitary representations: there are, for Kant, no Leibnizean "complete concepts" comprising within themselves (that is, within their defining sets of characteristics [Merkmale] or partial concepts [Teilbegriffe J) an infinite manifold of further conceptual representations. But mathematical representations (such as the representation of space) can and do contain an infinite manifold of further (mathematical) representations within themselves (as in the representation of infinite divisibility). So such representations, for Kant, are not and cannot be conceptual. 27 Of course, we now have an entirely different conception of logic from Kant's, one that is much more powerful than anything either he, or even Leibniz, ever envisioned. Nevertheless, we can still understand Kant's fundamental insight, from our own point of view, if we observe that no infinite mathematical structure (such as either the space of Euclidean geometry or the number series) can possibly be represented within monadic quantificationallogic. 28 Such infinite structures, in modern logic, are represented by the use of nested sequences of universal and existential quantifiers (where existential quantifiers depend on universal) using polyadic logic. These same representations, from Kant's point of view, are instead made possible by the iterative application of constructive functions in the "productive imagination," where, as we have seen, Skolem functions for the existential quantifiers we would use in our formulations are rather explicitly constructed. 29 We now see, from Kant's point of view, why mathematical thinking essentially involves what he calls "the pure productive imagination" and why, accordingly, this type of thinking essentially exceeds the bounds of purely conceptual, purely intellectual thought. However, we have not yet malce an explicit connection between this feature of mathematical representation and the Kantian perceptual faculty of sensibility, and, in particular, we have not explained how pure mathematics is grounded in the a priori structure of our (specifically human) sensibility via the pure forms of sensible intuition, space and time. Yet it is centrally important to Kant's theory of mathematics that all possible objects of human sense-perception, all objects of what Kant calls empirical intuition, must necessarily conform to the a priori principles of mathematics established in pure intuition (A165-66/B206): "The synthesis of spaces and times, as the essential form of all intuition, is that which, at the same time, makes possible the
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apprehension of appearance, and thus every outer experience, [and] therefore all cognition of the objects thereof; and what mathematics in its pure employment demonstrates of the former necessarily holds also of the latter." Unless we can explicitly connect the functions of the pure productive in1agination with the Kantian forms ofpure intuition (space and time), this crucial link to our perceptual faculty of sensibility remains entirely unexplained-and this is the most fundamental lacuna, I now believe, in the purely logical approach to IZant's theory of mathematical intuition. I attempt to articulate such a connection in Friedman (2000a), building on earlier work by Parsons (1992) and Emily Carson (1997). Parsons and Carson suggest, in particular, that there are basic perceptual or phenomenological features of space underlying the axioms (and I presume inferences) of Euclidean geometry. These features of space are delineated in the third and fourth arguments of the Metaphysical Exposition of Space (in the second edition of the Critique), where the former emphasizes the singularity or unitary nature of our representation of space (that space is one single all-encompassing unitary whole, which, as such, precedes any smaller spatial parts) and the latter (as we have seen in note 27 above) emphasizes the infinity of space. These two features, for Parsons and Carson, are then intimately connected, in so far as the infinity of space rests on the phenomenological fact that any perceived spatial region belongs within a larger "horizon" as part of a single, uniquely given perceptual space-so that, as Parsons (1992, 70) puts it, "places, and thereby objects in space, are given in a [single] space, therefore with a 'horizon' of surrounding space." Moreover, this unitary and infinite perceptual space then supplies the framework within which geometrical constructions are undertaken and which guarantees, in particular, that the constructions postulated by Euclid can indeed be carried out (and iterated indefinitely). As Parsons (1992, 77-78) puts it: "[Euclidean constructions] are constructions in intuition; space is, one might say, the field in which the constructions are carried out; it is by virtue of the nature of space that they can be carried out." It is in this sense that the phenomenological features of perceptual space (of our form of outer intuition) underlie the axioms (and I presume inferences) of Euclidean geometry_ In Friedman (2000a) I basically agree with this view. In particular, I appeal to a passage from lZant's controversy with Eberhard in 1790, where IZant explicitly contrasts the (successively constructed) space of the geon1eter with the "subjectively given" space of our pure form of outer sensible intuition. IZant begins by asserting that "to say that a line can be continued to infinity n1eans that the space in which I describe the line is greater than any line that I may describe in it," so that "the geometer grounds the possibility of his problem-to increase a space (of which there are many) to infinity-on the original representation of a single, infinite, subjectively
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given space." "[G]eometrical ~nd obje~tive ~pace," Kant, c~ntinues, "is always finite," for "the latter IS only gIven In so far as It IS generated [gemacht]." And this geometrical space is then explicitly contrasted with what Kant calls "metaphysical space":
To say, however, that the metaphysical, i.e., original but merely subjectively given space-which (because there are not many) cannot be brought under any concept capable of construction but which still contains the ground of all possible constructions-is infinite means that it consists of the pure form of the mode of sensible representation of the subject as a priori intuition. Therefore, the possibility of all spaces, which proceeds to infinity, is given in this space as a singular representation. 30 Thus, "metaphysical space" is the space considered in the Metaphysical Exposition in the Critique-space as the pure form within which objects (including geometrical objects) may be given (in this case by construction)-whereas geometrical space consists of the indefinitely extendible (but always finite) manifold of geometrical objects which may (at any finite stage) be actually constructed. Yet this important passage contains something not found in the Metaphysical Exposition itself, namely, an explicit connection between space as the pure form of outer intuition and geometrical construction. The question naturally arises, therefore, of exactly how the former is supposed to ground the latter. Is the given infinity of space as a pure form of sensible intuition supposed to be directly seen, as it were, in a simple act of perceptual or quasi-perceptual acquaintance? Are we supposed to have direct perceptual or quasi-perceptual access to such infinity entirely independently of geometry itself-which access we can then use to justifY or verifY the possibility of Euclidean constructions? This last idea appears to be very doubtful, for we are certainly not perceptually acquainted with an actually infinite space as a single given whole. Fortunately, however, Kant himself discusses the question of justifYing or explaining the possibility of geometrical construction several pages earlier: [I]t is very correctly said that "Euclid assumes the possibility of drawing a straight line and describing a circle without proving it"-which means without proving this possibility through inferences. For description, which takes place a priori through the imagination in accordance with a rule and is called construction, is itself the proof of the possibility of the object.... However, that the possibility of a straight line and a circle can be proved, not mediately through inferences, but only
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immediately through the construction of these concepts (which is in no way empirical), is due to the circumstance that among all constructions (presentations determined in accordance with a rule in a priori intuition) some must still be the first-namely, the drawing or describing (in thought) of a straight line and the rotating of such a line around a fixed point-where the latter cannot be derived fron1 the former, nor can it be derived from any other construction of the concept of a magnitude. (20, 410-11 )31 What is immediate, therefore, is the activity of our a priori imagination by which we draw or describe a straight line in thought and then rotate such a line around a fixed point (in a given plane); and we have imn1ediate a pri0ri knowledge of the truth of the corresponding Euclidean postulate simply because it is only by means of this activity itself that the corresponding propositions can even be thought. 32 It is not as ifwe had immediate, quasiperceptual acquaintance with some independent spatial fact, as it were, such as the (supposed) fact that perceptual space is infinite (either actually or potentially), and that we could then justify the possibility of geometrical construction, in particular, by deriving it from such an independently ascertainable fact. Space as a pure form of intuition does not, by itself, provide us with any objects of acquaintance at all, and, a fortiori, it does not, by itself, provide us with any facts with which we could be perceptually acquainted. Only geometrical space can provide us with both objects and facts concerning them, and we have immediate a priori knowledge of these precisely because they can be constructed (and thereby demonstrated)within the a priori, "subjectively given" form of outer sense. 33 It is in1portant to see, however, that the circumstance that geometrical constructions are necessarily carried out within the given structure of space as a form of outer intuition-a form with its own antecedently given whole-part structure, as described in the third argument of the Metaphysical Exposition-has significant implications, in turn, for the inferential structure of Euclidean proof. Indeed, we can clearly see the interplay between geometrical construction and this antecedently given whole-part structure in the example I(ant presents there (which, as we pointed out above, involves Proposition 1.20): Given a triangle ABC, we extend side BA to D such that AD is equal to AC; we then draw CD and note (by Proposition 1.5) that the two angles ACD and ADC are equal, so that BCD (~) is greater than BDC (a); since (by Proposition 1.19) the greater angle is subtended by the greater side, it follows that BD is greater than BC; but BD is equal to the sum of BA and AD (= AC). Now, we know that the angle ADC = BDC (a) = ACD (a') by Proposition 1.5, but how do we know that BCD (~) > ACD (a')? We know this because a' is a (proper) part of~, and, by Common Notion 5,
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D
B
the whole is greater than the part. That ex' is a part of~, however, cannot itself be derived from any of the Postulates and Common Notions; it is rather a necessary feature of the whole-part structure of the limited spatial region (figure) we have now constructed. Yet it would not be correct to say that this fact is simply read off the particular figure (image) we have constructed, because, if the proof is to be generally valid, it must by true of any and all such figures constructed in accordance with the given sequence of constructions (schemata).34 And, in fact, it is evident that any and all figures constructed in this way will exhibit the same relevant wholepart relationships: because BD, by construction, is an extension of BA, both the region ABC and the region DAC are necessarily proper parts of the region DBC.35 Let us pursue the relationship between the space of pure geometry and space as the pure form of outer intuition a bit further, by now turning our attention to the first and second arguments of the Metaphysical Exposition. The first argument attempts to show that space is an a priori rather than empirical representation by arguing that all perception of outer (empirical) objects in space presupposes the representation of space: Space is no empirical concept that has been derived from outer experiences. For, in order that certain sensations are related to something outside me (that is, to something in another place in space than the one in which I find myself), and, similarly, in order that I be able to represent them as outside of and next to one another-and thus not merely as different but as in different places-the representation of space must already lie at the basis. Therefore, the representation of space cannot be obtained from the relations of outer appearance through experience; rather, this outer experience is itself only possible in the first place by means of the representation in question. (A23jB38) This argument emphasizes that space as the form of outer sense enables us to represent objects as outer precisely by representing them as spatially external to the perceiving subject, so that the space in question contains
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the point of view fron1 which the objects of outer sense are perceived and around which the objects of outer sense are arranged. Empirical spatial intuition or perception occurs when an object spatially external to the point of view of the subject affects this subject-along a spatial line of sight, as it were-so as to produce a corresponding sensation; and it is in this sense, therefore, that the pure form of (spatial) sensible intuition expresses the manner in which we are affected by (outer) objects. 36 The second argument goes on to clain1 that space is a necessary a pri0ri representation, which functions as a condition of the possibility of all outer experience: Space is a necessary a priori representation, which lies at the basis of all outer intuition. One can never n1ake a representation [of the supposed fact] that there is no space, although one can very well think that no objects are to be found therein. It must therefore be viewed as the condition of the possibility of appearances, not as a determination depending on them, and is an a priori representation, which necessarily lies at the basis of outer appearances. (A24/B38-39) The crux of this argument is that one cannot represent outer objects without space, whereas one can think this very same space as entirely empty of such objects. And, since the first conjunct may appear to be tautological, the burden of the argument seems to fallon the second conjunct. What exactly does it mean, therefore, to represent space as empty of outer objects, and in what precise context, moreover, do we succeed in doing this? A very natural suggestion is that we think of space as empty of outer (en1pirical) objects just when we are doing pure geometry.37 This would accord very well, in particular, with the concluding claim that space thereby functions as a necessary a priori condition of the possibility of outer appearances, for they would then be necessarily subject to the a pri0ri necessary science of pure geometry. 38 But what is the precise relationship between the a priori structure attributed to space in the first argument (the structure given by the notion of a spatial point ofview or perspective from which, and in accordance with which, outer objects are perceived) and that attributed to space in the second (the structure of pure geometry)? It is natural, in the first place, to view the former structure as itself a priori and geometrical, since it does not depend at all on the particular (empirical) outer objects actually perceived from any particular point of view. On the contrary, this perspectival structure is invariant under all changes in both the 0 bjects perceived and the point of view fron1 which they are perceived, and, in this sense, it thereby expresses the form rather than the matter or content of outer intuition. Moreover, and in the second place, these possible changes in perspective
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themselves constitute what we now take to be a geometrical object: namely, a group of (Euclidean) motions or transformations, comprising all possible translations of our initial point of view through space and all possible rotations of the perspective associated with this point of view around the given point. In particular, any perceptible spatial object, located anywhere in space, can in principle be made accessible by an appropriate sequence of such translations and rotations starting from any initial point of view and associated perspective. Finally, and in the third place, there is a clear connection between this (modern) geometrical structure and geometry in Kant's sense, for, as Kant himself emphasizes, the two fundamental Euclidean constructions of drawing a line and constructing a circle are generated precisely by translations and a rotations-as we generate a line segment by the motion (translation) of a point and then rotate this segment (in a given plane) around one of its endpoints. 39 On the present interpretation, therefore, the first and second arguments of the Metaphysical Exposition forge the crucial connection between Kant's theory of space as a form of outer intuition or perception (what we might call perspectival space) and his theory of pure mathematical geometry as the successive execution of Euclidean constructions in the pure imagination (resulting in what Kant calls geometrical space). The third argument then illustrates how the whole-part structure of space as a form of intuition is presupposed in the proof-procedure of Euclidean geometry (as a procedure of successive construction), and the fourth argument appeals to this same whole-part structure in clearly differentiating such geometrical reasoning from purely logical or conceptual reasoning (as in the proof of infinite divisibility, for example). Thus, I now accept, as suggested above, central features of the phenomenological approach to Kant's theory of spatial intuition-especially as developed in Parsons (1992) and Carson (1997). I do not agree, however, that we thereby obtain an independent, quasi-perceptual justification of Euclidean geometry available prior to the science of geometry itself (see again the paragraph to which note 33 above is appended, together with the preceding paragraph, as well as the paragraph to which note 35 is appended). Rather, geometrical construction and the phenomenological structure of perceptual space are, for Kant, mutually and reciprocally dependent on one another; and, since an emphasis on geometrical construction is characteristic of (my version of) the logical approach to Kant's theory of intuition, this approach and the phenomenological approach are mutually dependent on one another as well. We can deepen our appreciation of the peculiar intertwining of logic and phenomenology in Kant's theory of space and geometry by returning to Kant's attempts to combine the insights of both Leibniz and Newton considered in the last section. For, in the first place, Kant's emphasis on
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the perceptual and intuitive aspects of geometry (and mathematics n10re generally) corresponds to Newton's approach, in contrast to the n10re algebraic approach ofLeibniz. 4o And, in the second place, I
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the way for the more formal and axiomatic conceptions of both logic and pure mathematics that were to come. 43
Philosophy ofnatural science I now turn to Kant's critical philosophy of natural science, especially as presented in the Metaphysical Foundations of Natural Science of 1786. Beginning with the Prolegomena (1783), and continuing with the Introduction to the second edition of the Critique ofPure Reason (1787), Kant divides the "general problem of pure reason"-"how are synthetic judgments a priori possible?"-into two more specific sub-problems: "how is pure mathematics possible?" and "how is pure natural science possible?" Kant's critical philosophy of mathematics is supposed to be the answer to the first question, and it is in precisely the Metaphysical Foundations that Kant most fully presents his answer to the second. As Kant explains in §§10-11 of the Prolegomena, the answer to the first question is found in the pure intuitions of space and time (4, 283): "for mathematics must first present all of its concepts in intuition, and pure mathematics [must do this] in pure intuition, that is, construct them-in the absence of which it is impossible for it to take a single step (since it cannot proceed analytically, through the analysis of concepts, but rather [proceeds] synthetically), so long, namely, as it lacks pure intuition, in which alone the material for synthetic judgments a priori can be given." As we have seen, however, since what is constructed in pure intuition are mathematical concepts, the pure understanding, via its schematization in terms of the pure intuitions of space and time, is also essentially implicated in Kant's answer to the first question-as Kant himself emphasizes in §20 of the Prolegomena. 44 In her very illuminating contribution to the present volume, Alison Laywine explains how Kant, like his friend and correspondent J.H. Lambert, took geometrical construction in accordance with the Euclidean postulates as a model for the reform of metaphysics much more generally. In §3 of her contribution Laywine suggests, in particular, that Kant took the model of geometrical construction to hold the key to the difficulty raised in the famous letter to Marcus Herz of 21 Fe bruary 1772 concerning how the intellect can represent its objects a priori: it might be able to do this, that is, by somehow constructing these objects, just as, in pure mathematics, we can construct a priori lines, circles, and so on. Of course, as Laywine points out, there is an obvious difficulty with this suggestion, because the objects of nature-the objects of what Kant shall later call pure natural science-are not given to us a priori, but only empirically, in experience. They are objects of empirical rather than pure intuition, and
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so, if we are somehow able to represent them a priori nonetheless, it must be by a method at most analogous to geometrical construction. This, in fact, is precisely Laywine's suggestion,45 and she concludes by urging us to read the transcendental deduction with this suggestion in mind. As Laywine also points out, I
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This constructive procedure is analogous to the procedure of Euclidean geometry in so far as we begin with a given initial input and reach a desired output by successive application of a finite number of constructive operations corresponding to a finite number of well-defined general rules. In the case of Euclidean geometry, as we know, the given initial input is any pair of arbitrary points (or any single given line segment), and the desired output is the entire Euclidean plane (all the points generated by straight edge and compass) arising from successive application of Euclid's first three postulates. In the case of Kant's successive approximations to Newtonian absolute space, however, the given initial input consists of the observable relative motions in the solar system seen from our parochial perspective here on earth, and the desired output is the privileged center of motion of the solar system arising from successive application of Kant's version of the three Newtonian laws of motion (his three "laws of mechanics" derived in the preceding chapter or Mechanics). The all-important disanalogy, of course, is that the given initial input, in this latter case, is empirically given, and it can in no way be constructed a pri01'i. As a result, there is no a priori guarantee that the latter construction can in fact be successfully carried out. On the contrary, we can successfully carry out our constructive determination of the true motions only if the initial a posteriori given phenomena-the observed relative motions as seen from our parochial perspective here on earth-are themselves sufficiently well-behaved. 48 Kant's answer to the question "how is pure mathematics possible?" rests, as we have seen, on our faculty of pure intuition. By contrast, Kant's answer to the question "how is pure natural science possible" rests, as explained in §36 of the Prolegomena, on our faculty of pure understanding (4,318): "it is only possible by means of the constitution of our understanding, in accordance with which all [the] representations of sensibility are necessarily referred to a consciousness, and through which the peculiar manner of our thinking is first possible-and by means of the latter experience [is first possible], which is entirely distinct from insight into the objects in themselves." However, as we have also seen (see note 44 above, together with the paragraph to which it is appended), both pure intuition and pure understanding are implicated in the answer to the first question, for the categories of quantity are unavoidably involved in all the judgments of pure mathematics. The crucial difference, therefore, is that all the categories-those of quantity, quality, relation, and modality-are necessarily involved in the answer to the second question and thus in the construction of nature. Further, the application of the modal categories in the three Postulates of Empirical Thought involves the successive application of the three earlier classes of principles-the Axioms of Intuition, the Anticipations of
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Perception, and the Analogies of Experience. For, in the course of explaining that the Postulates, like the Analogies, are only regulative with respect to the appearances (compare note 48 above), Kant speaks of the Postulates as those principles "which together concern the synthesis of mere intuition (the form of appearance), of perception (the matter of appearance), and of experience (of the relation of these perceptions)" (A180/B223). Similarly, the four chapters of the Metaphysical Foundations correspond to the categories (and principles) of quantity, quality, relation, and modality, respectively, and the three propositions of the Phenomenology themselves correspond, accordingly, to the three earlier chapters: the first "determines the modality of motion with respect to phoronomy" (556); the second "determines the modality of motion with respect to dynamics" (557); the third "determines the modality of motion with respect to mechanics" (ibid.). Thus, the first chapter or Phoronomy is concerned with the conceptualization of motion (which, according to the first explication, is the primary predicate of matter) as a quantity or magnitude [Grofle]. Indeed, "[p ]horonomy is thus the pure theory of magnitude [GrOflenlehre] (Mathesis) of motions" (489), and is thereby concerned with "the construction of motions in general as magnitudes" (487). This construction, in accordance with the concept of quantity or magnitude in general articulated in the first Critique, involves the construction of a composite motion [zusammengesetzten Bewegung] out of two (or more) given motions (486): "To construct the concept of a composite motion is to present a motion a priori in intuition, in so far as it arises from two or more given [motions] united in one movable. "49 More generally, the aim of the Phoronomy is "to determine these motions a priori solely as magnitudes, with respect to both their speed and direction, and, indeed, with respect to their composition" (487). Our object, then, is to show how motion can be mathematically conceptualized as what we would call the directed (vector) quantity of velocity, and we do this, in accordance with the traditional notion of (continuous) magnitude descended from ancient Greek mathematics, by exhibiting in pure intuition how any two given motions can be combined or added together into a single composite motion. 50 This exhibition or construction is the subject of the single proposition of the Phoronomy, and the result is that motion is now mathematically conceptualized and measurable as a (continuous) magnitude. 51 All other quantities introduced in subsequent chapters-such as force, elasticity, and mass-are then to be measurable in terms of the effects they have on motion. 52 In the Phoronomy itself, however, no other quantity of matter is considered except motion, and, accordingly, the movable matter or "body" in question can be considered as a mere mathematical point:
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Since in phoronomy nothing is to be at issue except motion, no other property is here ascribed to the subject of motion, namely, matter, aside from movability. It can itself so far, therefore, also be considered as a point, and one abstracts in phoronomy from all inner constitution, and therefore also from the quantity of the movable, and concerns oneself only with motion and what can be considered as quantity in motion (speed and direction).-If the expression "body" should nevertheless sometimes be used here, this is only to anticipate to some extent the application of the principles of phoronomy to the n10re determinate concepts of matter that are still to follow, so that the exposition may be less abstract and more comprehensible. (480)53 It is precisely by deliberately abstracting from all the empirically given properties of Inatter-properties such as force, elasticity, and mass-that vve can, in Phoronomy, consider matter as merely the movable in space and thus represent the mathematical characterization of motion as a magnitude entirely a priori in pure intuition. 54 In the immediately following second chapter or Dynamics, I(ant then explicitly considers the properly sensible and empirical aspects of matter by now considering matter as the movable "in so far as it fills a space" (496). This mal(es sense, in terms of the general correspondence between the Metaphysical Foundations and the first Critique, because we are now considering matter under the categories of quality and thus under the principIe of the Anticipations of Perception. According to this principle (as expressed in the second edition), "in all appearances the real) which is an object of sensation) has an intensive magnitude, that is, a degree" (B207), and so we are here concerned with that which can only be an object of sensation (not mere pure intuition): we are thereby acquainted with something real filling the otherwise empty pure intuitions of space and time. 55 In the present case, in particular, we are concerned with matter as that which fills an otherwise empty space and which, in this very filling, can thereby exert an influence on our senses. How, then, does matter fill a space? The answer is given by runt's crit:ical version of a "dynamical theory of matter." There are two fundamental moving forces of matter, attraction and repulsion, and matter fills the space it occupies by a continuous "balancing" of these two forces exerted by all the continuum of points in the space in question. This theory is "dynamical," therefore, in so far as forces are more fundamental than primitively solid bodies, and properties of bodies like solidity and impenetrability are then not taken as primitive at all, but are rather derived from an interplay of forces. Moreover, the degree to which matter thereby fills a space-its density-is not determined by the ratio between completely full (solid) and completely empty (void) spaces within the region in question (as the
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mechanical natural philosophy would have it), but is rather given as what IZant calls an intensive magnitude pertaining equally to all of the points within this region. 56 Kant's exposition of this theory is long, complex, and controversial, and I have not considered its intricate details in my previ0usly published work. However, my work in progress, !(ant)s Construction of Nature (on which Mark Wilson has been kind enough to comment in his contribution to this volume), attempts to n1alce up for this serious lacuna, and so I shall briefly sketch some of my main conclusions here. The significance of Iunt's dynamical theory of matter and its role within the overall argument of the Metaphysical Foundations can only be properly understood, I believe, if we carefully situate it within two different historical-intellectual contexts. In particular, since this is a work on the "metaphysical foundations of natural science," the relevant background includes both metaphysical issues first considered in Kant's pre-critical period and contemporaneous developments in physical theory relevant to the choice between atomistic and continuum models of matter. Both of these contexts, in turn, are inextricably connected, for IZant, with his ongoing philosophical struggle to integrate the conceptual structure of Newtonian natural science with a conception of the pure intellect or pure understanding ultimately derived from Leibniz. As we have seen (note 9 above, together with the paragraph to which it is appended), IZant's pre-critical philosophy features a modified version of a Leibnizean metaphysics, based on an underlying reality of monads or ultimate simple substances. Unlike in Leibniz, however, the substances in question have both purely internal or intrinsic properties and external or extrinsic relations. The latter, in particular, include causal relations of real "physical influx" between monads, and the most basic of these are phenomenally manifested as what we know as the two fundamental forces of matter, attraction and repulsion. What we know as space, moreover, is the phenomenal expression of precisely those fundamental causal relations between the ultimate simple substances that are manifested as attraction and repulsion. The Physical Monadology of 1756 develops an earlier version of a dynamical theory of matter within this modified Leibnizean framework-which, in sharp contrast to the theory later developed in the Metaphysical Foundations, employs an atomism of discrete force-centers rather than a true continuum view of n1atter. 57 The question dominating the Physical Monadology is a particular metaphysical issue arising in the debate between Leibnizeans and Newtonians: if the ultimate constituents of matter are absolutely simple elementary substances or monads, as the Leibnizeans contend, how can this be reconciled with the geometrical infinite divisibility of space? It would appear that by dividing the space filled or occupied by any given piece of matter, no matter how small, we would also eventually divide the elementary material ---------------
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substances found there as well-contrary to the assumed absolute simplicity of such substances. So how can an elementary constituent of matter or "physical monad" possibly fill the space it occupies without being infinitely divisible in turn? Kant's answer, in 1756, is that physical monads do not fill the space they occupy by being immediately present in all parts of this space; they are not to be conceived, for example, as bodies that are solid through and through. Physical monads are rather to be conceived as pointlike centers of attractive and repulsive force, where the repulsive force, in particular, generates a region of solidity or impenetrability in the form of a tiny "sphere of activity" emanating from a central point. 58 Geometrically dividing this region of impenetrability in no way divides the actual substance of the monad but merely the "sphere of activity" in which the point-like central source manifests its repulsive capacity to exclude other monads from the region or volume in question. So the Leibnizean commitment to ultimate simple substances or monads is perfectly consistent with the infinite divisibility of space after all; but (and here is Kant's characteristic twist) it can only be maintained by explicitly adopting the Newtonian conception of forces acting at a distance (in this case a short range repulsive force acting at a very small distance given by the radius of its "sphere of activity").59 By contrast, Kant's critical version of the dynamical theory of matter is a true continuum view. Kant continues to conceive the basic properties of matter-impenetrability, solidity, hardness, density, and so on-as arising from an interplay of the two fundamental forces of attraction and repulsion. In sharp contrast to the Physical Monadology, however, Kant abandons the idea of smallest elementary parts of matter or physical monads, and he argues instead that all parts of matter or material substances must be infinitely divisible, just like the spaces they occupy. Indeed, in the course of developing this argument, Kant explicitly rejects the very theory of physical monads he himself had earlier defended in 1756. A space filled with matter or material substance, in Kant's new theory, now consists of an infinity or continuum of material points, each of which exerts the two fundamental forces of attraction and repulsion. 60 The "balancing" of the two fundamental forces, which had earlier determined a tiny (but finite) volume representing a "sphere of activity" of impenetrability around a single point-like central source, now determines a definite density of matter at each point in the space in question, effected by the mutual interaction of attraction and repulsion. 61 Thus, in the Metaphysical Foundations, as in the first Critique, material or phenomenal substance is no longer viewed as simple and indivisible but is instead a genuine continuum occupying all the (geometrical) points of the space it fills. Accordingly, the problem posed by the infinite divisibility of space that the Physical Monadology had attempted to solve by invoking
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finite "spheres of activity" is now solved, in the Dynamics of the Metaphysical Foundations, by invoking the transcendental idealism articulated in the Antinomy of Pure Reason of the first Critique-and, more specifically, the argument of the Second Antinomy resolving the apparent incompatibility between the infinite divisibility of space, on the one side , and the presumed absolute simplicity of the material or phenomenal substances found in space, on the other. Matter or material substance is infinitely divisible but never, in experience, ever infinitely divided; hence, since matter is a mere appearance or phenomenon, and is thus given only in the "progress of experience," it consists neither in ultimate simple elements nor in an actual or completed infinity of ever smaller spatial parts. Only by viewing matter as a thing in itself or noumenal substance (which would be necessarily simple) do we obtain a genuine contradiction or antinomy; and, therefore, by an indirect proof or reductio ad absurdum, we obtain a further argument in support of I
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point endeavors to move in all directions with precisely the same force with which it is pressed towards anyone ofthem, a property on which rests the first law of hydrodynamics, although it can never be attributed to an agglomeration [Anhaufung] of smooth and solid [festen] corpuscles, as can be shown by a very easy resolution of its pressure in accordance with the laws of composite motion, thereby proving the original character of the property of fluidity. (528 )64
This argument, although not as sophisticated, reminds us of Euler's rejection of solid corpuscles in 1755. 65 Moreover, Kant's formulation of the defining property of fluidity explicitly emphasizes that every part of the fluid medium is to be subject to the very same conditions, and it thus leads to the representation of the state of fluidity as a true continuum. For Kant, we thereby obtain an additional important reason for rejecting the matter theory of the Physical Monadology. The space inside the sphere of activity of a monad cannot, properly speaking, be characterized as fluid, precisely because it does not possess the property Kant now finds definitive of elastic matter in general-namely, that "in a space filled with matter, every part of it contains repulsive force, so as to counteract all the rest in all directions, and thus to repel them and to be repelled by them, that is, to be moved a distance from them" (503). Thus, in the General Remark, Kant explicitly relates what he has just called the "decisive" property of fluidity to this earlier characterization of elastic matter in general: The above-cited second definition of fluidity, on which the fundamental law of hydrostatics rests-namely, that it is that property of a matter whereby every part of it strives to move in all directions with precisely the same force by which it is pressed in any given directionfollows from the first definition, if one combines it with the principle of general dynamics that all matter is originally elastic. For this matter must then be striving to expand in all directions of the space in which it is compressed, with the same force by which the pressure occurs in any direction, whatever it may be, that is, if the parts of a matter can be displaced along one another by any force, without resistance, as is actually the case with fluids, it must be striving to move in all directions. (529 )66 The moral, in the present context, is that the points within the space occupied by a physical monad are not independently movable (in all directions) under the action of an external pressure. On the contrary, only the central point of a repulsive sphere of activity is movable on its own, and the behavior of all other points of this sphere entirely depends on that of the central point. In particular, all other points of such a sphere behave as if they were
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rigidly connected to the central point, and this is quite incompatible, of course, with the essential nature of fluidity as characterized by Euler. Hence, although the elementary corpuscles of the Physical Monadology are elastic rather than hard, and are therefore, in a sense, "compressible,"67 the "elastic medium" they are supposed to constitute still fails, from Kant's present point of view, to possess the essential characteristics of fluidity. It is not the case that every point within this medium is subject to the same conditions, and it is not the case, in particular, that every such point is independently movable (in all directions). Thus, whereas Kant concludes the Physical Monadology with the claim that the combined elasticities of the physical monads together "constitute a medium which is primitively elastic" (1, 487), it appears that he had here not yet thought this question completely through. 68 Now, against the background of a markedly more sophisticated understanding of the state of fluidity, Kant is finally in a position' in the Dynamics of the Metaphysical Foundations, to replace an atomism of discrete force-centers with a true (dynan1ical) continuum view of matter. 69 For IZant, more generally, the most basic state of matter is that of what was called in the eighteenth century a permanently elastic fluid (i.e., a gas). Thus, after first introducing the concept of repulsive force in the first proposition of the Dynamics, IZant goes on to characterize it as an expansive force in the second proposition (499): "Matter fills its space through the repulsive forces of all of its parts, that is, through an expansive force of its own, which has a determinate degree, such that sn1aller or larger degrees can be thought to infinity." Moreover, in the first note to this proposition I(ant explicitly associates the expansive force in question with the fundamental property of matter he calls original elasticity (500): "The expansive force of a matter is also called elasticity[; and,] since it is the ground on which the filling of space, as an essential property of all matter, rests, this elasticity must therefore be called original, because it can be derived from no other property of matter." The original or fundamental state of matter, for lZant, manifested in its continual striving to expand the space it occupies due to the mutual repulsion of all of its parts, is precisely that of a permanently elastic fluid, and what lZant calls the "expansive force of matter" is just what we call (expansive) pressure.70 The "balancing" argument first articulated in the fifth proposition of the Dynamics, which immediately follows the extended discussion of infinite divisibility contained in the fourth proposition and two following remarks (see notes 60 and 62 above, together with the paragraphs to which they are appended), should now be considered in precisely this context. The point of the argument, in particular, is that the matter which (according to the first explication) fills a space is only possible by an interplay or interaction between the two fundan1ental forces of repulsion and attrac-
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rion. Thus, for example, IZant begins by claiming that repulsive force alone \vould issue in an indefinite expansion of matter towards a state of zero density (508): "matter, by its repulsive force (containing the ground of impenetrability [i.e., resistance to compression]), would [by itself] alone and if no other moving force counteracted it, be confined within no limit of extension, that is, it would disperse itself to infinity, and no specified quantity of matter would be found in any specified space." In order, then, that a specified quantity of n1atter may be found in a specified space (so that this space is actually filled rather than empty), it is necessary that there be a fundamental force of attraction serving as precisely the counterbalancing force in question. 71 This argument may easily appear mysterious or even fallacious. When placed in the context of IZant's conception of the original or fundamental state of matter, however, it makes perfect sense. In the simplest case of hydrostatic equilibrium, for example, atmospheric air is maintained in an equilibriun1 state by a balancing of its internal expansive pressure by the gravitational attraction of the earth-the "weight" of the air towards the (center of) the earth-where this weight or compressive pressure depends on the height above the earth's surface: the higher the region of the atmosphere under consideration, the smaller is the weight of the air. It is necessary for a state of equilibrium, then, that the air form concentric layers above the earth's surface where, at equal distances from the surface, all points of a given layer have the same pressure and density, such that, as the distance increases, the pressure and density decrease accordingly.72 Where the compressive pressure or weight would become less than the air's expansive pressure, therefore, the atmosphere can no longer exist-any remaining air would drift away from the earth's gravitational field and disperse itself in cosmic space. Moreover, if the earth's gravitational field were somehow to be turned off, the entire atmosphere would disperse. The result would be an "atmosphere" of zero density, just as is claimed in IZant's argument. 73 IZant's critical version of the balancing argument, in any case, plays an absolutely central role in the systematic structure of the Metaphysical Foundations as a whole. It is crucial, in particular, for understanding lZant's critical conception of the fundamental concept of quantity of matterwhich, as IZant emphasizes, now represents the quantity of substance (540-41). This link between quantity of matter and quantity of substance is only established in the Mechanics, and I shall return to it below. Nevertheless, the concept of quantity of matter, as we have seen, is first introduced in the context of the balancing argument of the Dynamics, where lZant claims that, without a counterbalancing force of attraction, matter "would disperse itself to infinity, and no specified quantity of matuld be found in any specified space. "74 Here, therefore, the concept
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of quantity of matter is linked to the concept of the degree of filling a given space or density (525), where density and quantity of matter are reciprocally related through the concept of volume (534): density is just "quantity of matter at the same volume.,,75 Finally, this relationship between quantity of matter, density, and volume is also connected to the idea that density, on a true continuum view, can only be (immediately) compared among matters of specifically the same kind, where "one can be generated from the other by mere compression" (526): the same quantity of matter, in the dynamical sense, can be compressed into a smaller volume, where it then manifests an increased expansive resistance to compression and thus fills this (smaller) space to a greater degree. Conversely, as Iunt argues in the balancing argument, the expansion of the same quantity of matter into a greater volume results in a (larger) space of lesser density, such that, were it to expand into infinite space, it would then manifest infinitely small density, "and no specified quantity of matter would be found in any specified space.,,76 This relationship between quantity of matter, density, and volume links quantity of matter, in the dynamical sense, to the fundamental force of repulsion which, as expansive resistance to compression or pressure (compare the paragraph to which note 70 above is appended), can fill a space to varying degrees. But quantity of matter, in the Dynamics, is equally (and perhaps even more essentially) linked to the fundamental force of attraction. In particular, the quantity of matter in a given space is directly proportional to the attractive force exerted by that n1atter (at a given distance) on other matter-a result, according to Iunt, of the circumstance that the original attraction, unlike the original repulsion, is a "penetrating" rather than "surface" force (516).77 Moreover, this special property of attraction, as Iunt also en1phasizes in the Dynamics, plays a central role in the Newtonian theory of universal gravitation and, in particular, in Newton's determination of the quantities of n1atter of the prin1ary bodies in the solar system by the (gravitational) accelerations effected on their satellites (515).78 What, exactly, is the relationship between the idea that the fundamental force of attraction is a penetrating rather than surface force and the idea that is directly proportional to quantity of matter? In the General Remark to Dynamics (in the course of the same passage from which I quoted in note 77), Iunt explains the relevant difference between attractive and repulsive force as follows (524): "attraction rests on the aggregate of matter in a given space, whereas its expansive force, by contrast, rests on the degree of filling of this space, which can be very different specifically (as the same quantity of air, say, in the same volume, manifests more or less elasticity in accordance with its greater or lesser heating)." Here, it seems, Kant is equating "aggregate of matter" with "quantity of matter." And this makes sense, because, when Iunt finally provides an official definition of
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"quantity of matter" in the first explication of the Mechanics, he characterizes it precisely as "the aggregate of the movable in a determinate space" (537). But what, exactly, does Kant mean by this characterization? He cannot, as in other contexts (see note 56 above, for example), simply mean volume by "aggregate," for Kant is perfectly clear, as we have seen (note 75 above, together with the paragraph to which it is appended), that quantity of matter depends on both volume and density-and, in any case, the notion of volume already appears in Kant's characterization in the guise of "a determinate space.,,79 Moreover, Kant is similarly clear, as we have also seen, that densities, in turn, can only be (immediately) compared among matters of specifically the same kind, so that "one can be generated from the other by mere compression" (526). Indeed, Kant emphasizes a parallel limitation on the relationship benveen quantity of matter and volume in the immediately following proof of the first proposition, according to which, although "quantity of matter is proportional to the size of the volmile" for matters "of the same kind," this by no means results in a universally valid standard of comparison that would also work for matters of "specifically different" kinds (537-38).80 The first proposition itself states that the desired universally valid standard of comparison can only arise from the concept of quantity of motion which, in turn, has just been linked to the concept of quantity of matter at the end of the first explication: The quantity of matter is the aggregate of the movable in a determinate space. This same [quantity of matter], in so far as all its parts in their motion are considered as acting (moving) together, is called mass, and one says that a matter acts in mass, when all its parts, moved in the same direction, together exert their moving force externally. A mass of determinate figure is called a body (in the mechanical meaning). The quantity of motion (estimated mechanically) is that which is estimated by the quantity of the moved matter and its velocity together; phoronomically it consists merely in the degree of velocity. (537) Of course quantity of motion is just what we now call momentum: mass times velocity. But Kant is malting a much more substantive point. He is claiming that, in order to apply the concept of quantity of matter to bodies in our experience, we must consider the "aggregate" of points constituting the body as acting, through t11eir combined motion, on another body (in such as way that "all its parts, moved in the same direction, together exert their moving force externally"). Kant is claiming, in other words, that, in order empirically to "estimate" the quantity of matter contained in any given body, we must explicitly consider exchanges of momentum between this body and other bodies. 8l
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The parallel passage in the Anticipations of Perception at A172-74/B214-16, quoted in note 56 above, suggests that there are two basic methods for determining quantity of matter by means of exchanges or momentum. lZant first says that quantity of matter is determined "partly through the moment of gravity or weight, partly through the resistance to other moved 1l1atters." He then speaks of "impenetrability or weight" and finally illustrates how "each reality could still have its own qualitative degree (of resistance or of weight)." It seems clear, therefore, that the two basic methods I(ant considers for determining the quantity of matter in experience involve, on the one hand, the weighing of bodies (in a balance), and, on the other, cases of momentum exchange between two bodies interacting by impact in collisions. The second depends on the conservation of momentum in all cases of impact (elastic as well as inelastic), enabling us to determine the ratio of the masses of the two colliding bodies from the ratio of their changes in velocity: m 1/m2 = !1V2 / !1V1 , where !1V1 and !1V2 represent the changes in velocities. The first method also depends on the conservation of momentum-but now applied to a static situation where the downward (infinitesimal) change of momentum (mass times acceleration) of one heavy body counterbalances the downward (infinitesimal) change of momentum of another. 82 The two methods are equally good in theory, and both depend on an "exertion" of (mechanical) "moving force"-i.e., an exchange of momentum (see note 81 above)-in just the way I(ant suggests. In practice, however, IZant gives decided preference to the first method and thus to the determination of quantity of matter in terms of weight. 83 Thus, in IZant's remark to the first proposition, he says that "quantity of matter (the aggregate of the movable) manifests itself in experience only by the quantity of motion at equal speed (for example, by equilibrium [Gleichgewicht])" (540; compare again note 81). The parenthetical illustration is a reference to static equilibrium, and thus to the determination of quantity of matter by means of the balance. 84 Indeed, it is for precisely this reason that Kant insists that the velocities involved in the comparison of quantities of motion here are equal to one another. For, according to Galileo's law of fall, all bodies fall with the same acceleration and thus acquire equal velocities in equal times. As we would now put it, the weights (gravitational forces) exerted by two masses m 1 and m 2 on the two (equal) arms of the balance are given by ill l al and ill2 a2 ; since, by Galileo's law, a = a2 , it folg lows that these two weights are as the corresponding masses. 5 If the two bodies are in equilibrium, then (by the equality of action and reaction) the two weights-and therefore the two masses-must be equal. 86 One important reason I(ant gives preference to determining quantity of n1atter by weight is that doing so mirrors the procedure by which Newton then extends the terrestrial concept of weight to a concept of quantity of
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matter or mass applicable in both terrestrial and celestial contexts in his theory of universal gravitation. In this way, in particular, the traditional statical concept of weight is conceived as a manifestation of the gravitational amaction of the bodies in question (at equal distances) by the earth; and, when we thus incorporate terrestrial gravity within the context of universal gravitation, we see that there are actually three relevant exchanges of momentum here: each of the attracted bodies (in the two equal arms of the balance) exchanges momentum with the earth, and they thereby exchange momentum with one another (because of the assumed rigidity of the arms of the balance). Nevertheless, in this situation of static equilibrium we can ignore the exchange of momentum with the earth itself (which, in any case, is quite negligible) and attend only to that involving the two bodies in equilibrium. 87 Now, as we know, when Newton extends his concept of quantity matter to the heavenly bodies in the solar system (including the earth), he does this by means of the gravitational accelerations exerted by such bodies on their satellites. Kant refers to this procedure, as we also know (see note 78 above, together with the paragraph to which it is appended), in the Dynamicswhere, in particular, he considers the determination of the masses ofTupiter and Saturn via the gravitational accelerations produced on their moons. But, in this procedure, we are also ignoring the motions of Jupiter and Saturn themselves, and so it is by no means clear how this determination or estimation is "mechanical" in the sense of the first proposition of the Mechanics. In Kant's remark to this proposition, therefore, he returns to the case of dynamical estimation of quantity of matter via universal gravitation, and he argues that it, too, is, in an appropriate sense, "mechanical": Nevertheless, original attraction, as the cause of universal gravitation, can still yield a measure of the quantity of matter, and of its substance (as actually happens in the comparison of matters by weighing), even though a dynamical measure-namely, attractive force-seems here to be the basis, rather than the attracting matter's own inherent motion. But since, in the case of this force, the action of a matter with all its parts is exerted immediately on all parts of another, and hence (at equal distances) is obviously proportional to the aggregate of the parts, the attracting body also thereby imparts to itself a velocity of its own inherent motion (by the resistance of the attracted body), which, in like external circumstances, is exactly proportional to the aggregate of its parts; so the estimation here is still in fact mechanical, although only indirectly so. (541) Thus, if we determine the quantity of matter ofJupiter from the acceleration produced by Jupiter's force of attraction on one of its moons, Jupiter
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and its moon interact with one another by universal gravitation. In particular, Jupiter's force of attraction, because of the resistance of the moon to this force and the equality of action and reaction, produces a corresponding acceleration-and therefore change of n10mentum-in Jupiter itself; and this change of momentum, in the given circun1stances, is also proportional to Jupiter's mass. Just as the moon falls towards Jupiter, Jupiter falls towards its moon-and Jupiter's "weight" towards this moon (like all gravitational forces) is, at a given distance, directly proportional to Jupiter's mass. 88 Determining the quantity of matter of an attracting body by the attractive force it exerts on another body is therefore mechanical in IZant's sense, because it rests, like all mechanical comparisons, on an exchange of momentum between the two interacting bodies; it is "only indirectly so," however, because the change of momentum of the attracting body itself, despite the fact that it is indeed proportional to this same body's quantity of matter, is not what is actually measured. 89 The general point, in any case, is that all comparisons of quantity of matter of interest to IZant-employing either the "moment of resistance" in impact or the procedure of weighing or the action of universal gravitation-essentially depend on the conservation of the total quantity of momentum (or, equivalently, on the equality of action and reaction applied to the bodies in question). And this point, for IZant, then links the concept of quantity of matter to what he calls the quantity of substance. Substance, according to the First Analogy of Experience, is just what is conserved or "persists" in all "change in the appearances" (B224). Indeed, the second edition formulation of this principle states it as a quantitative conservation law (ibid.): "In all change in the appearances substance persists, and the quantum of substance is neither increased nor decreased in nature." And the "first law of mechanics" of the Metaphysical Foundations IZant derives from the First Analogy closely mirrors the second edition formulation (541): "In all changes of corporeal nature the total quantity of matter remains the same, neither increased nor diminished." Finally, sincequantity of matter, according to the first proposition of the Mechanics, can only be estimated, in turn, via exchanges of quantity of motion (momentum), it follows that conservation of the total quantity of matter-conservation of the total quantity of material substance-is equivalent to the conservation of momentum. 90 This, in the end, is why lZant defines quantity of matter, in the first explication of the Mechanics, as "the aggregate of the movable in a determinate space"-as a continuously distributed totality of points in a given spatial region which, in particular, can "together exert their moving force externally." And it is for precisely this reason, accordingly, that matter cannot be conceived as consisting of unextended point-centers or physical monads (539-40): "That the quantity of matter can only be thought as the
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aggregate of movables (external to one another), as the definition expresses it, is a remarkable and fundamental proposition of general mechanics. For it is thereby indicated that matter has no other magnitude than that consisting in the aggregate of manifold [elements] external to one another, and hence has no degree of moving force at a given speed that would be independent of this aggregate, and could be considered merely as intensive magnitude-which would be the case, however, if matter consisted of monads, whose reality in every relation must have a degree that can be larger or smaller, without depending on an aggregate of parts external to one another." For, as Kant explains in his remark to the second proposition (his "first law of mechanics"), it is only if we assume this characterization of the quantity of matter that we can then link it to the concept of substance and thereby derive a quantitative conservation law (542): "What is essential in this proof to the characterization of the substance that is possible only in space, and in accordance with its condition, and thus possible only as object of the outer senses, is that its quantity cannot be increased or diminished without substance arising or perishing. For, since all quantity of an object possible merely in space must consist of parts external to one another, these, if they are real (something movable), must therefore necessarily be substances. "91 But why exactly does a quantitative conservation law for substance depend on an aggregate of parts external to one another? Why can an unextended point-center or physical monad not have a conserved quantity of substance? Kant's critical understanding of the categories of relation corresponding to the Analogies of Experience essentially involves a relationship between these concepts and the categories of reality corresponding to the Anticipations of Perception. In particular, the schema of the category of substance is "the permanence of the real in time, that is, the representation of [this real] as a substratum of empirical time determination in general, which therefore remains while everything else changes" (A144/B183). An instance of this schematized concept is therefore a "reality" serving as "the substratum of the empirical representation of time itself, in which alone all time determination is possible" (A183/B226): only a "reality" that is permanent in this sense can be a (phenomenal) substance. However, any instance of the schematized category of reality is an intensive magnitude having a degree (see the paragraph to which note 55 above is appended), and it must therefore have a degree or "quantum of reality" such that continuous transitions from this (finite) degree to zero are possible (A143/B182-83). On the one hand, then, any "reality" is an intensive magnitude possessing a degree that can be continuously diminished to zero, and, on the other, it is just such a "reality" which, as permanent, must realize the schematized category of substance.
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How can these two apparently conflicting demands be met? Kant's critical version of the dynamical theory of matter as a spatially extended and continuous "aggregate of movables" supplies us with a concrete example. Any matter that fills a space does so as a spatially extended continuum characterized by a balancing of repulsive and attractive forces at every point in the region in question whereby matter has a determinate degree of filling of space or density at each such point. But this degree of density, in I
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Indeed, Kant's remark to his "first law of mechanics" stating the conservation of the total quantity of matter takes pains to distinguish material substance, in this respect, from the presumptive substantial soul constituting the object of inner rather than outer sense: By contrast, that which is considered as object of inner sense can have a quantity, as substance, which does not consist ofparts external to one another; and its parts are thus not substances; and thus their arising or perishing need not be the arising or perishing of a substance; and their augmentation or diminution is therefore possible without violating the principle of the permanence of substance. So consciousness, namely, and thus the clarity of representations in my soul-and therefore the faculty of consciousness, apperception, and even, along with this, the very substance of the soul-has a degree, which can be greater or smaller, without any substance at all needing to arise or perish for this. But since, from its gradual diminution, the complete disappearance of the faculty of apperception would finally have to result, the very substance of the soul would still be subject to a gradual perishing, even if it were of a simple nature; for this disappearance of its fundamental force [Grundkraft] could result, not by division (separation of substance from a composite), but rather, as it were, by expiration [Erlo'schen]and this, too, not in a moment, but by a gradual waning of its degree, whatever the cause of this might be. (542) Precisely because the soul, unlike matter, is not spatially extended, the only candidate for quantity of substance here would be a merely intensive quantity (like degree of consciousness )-which, for precisely this reason, could be continuously diminished "without violating the principle of the permanence of substance." We therefore cannot infer from the principle of the permanence of substance (the First Analogy) to the permanence of the soul as the object of inner sense, and, for just this reason, it cannot be a substance in the critical sense. 96 Yet the critical concept of substance (and its quantity) is necessarily more general than the concept of material substance (and its quantity) deVeloped in the Metaphysical Foundations. In a passage from the Anticipations of Perception we have looked at several times before (see note 56 above and the paragraph to which note 82 is appended), Kant criticizes the purely extensive conception of density and quantity of matter characteristic of the mechanical natural philosophy for making an unwarranted metaphysical assumption (Al73/B215): "namely, that the real in space (I may not here call it impenetrability or weight, because these are empirical concepts) is everywhere of the same kind, and can only differ according to extensive magnitude." In particular, because impenetrability
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and weight are both empirical concepts (corresponding to the fundamental forces of repulsion and attraction, respectively), these concepts, at the "metaphysical" or "transcendental" level of the first Critique, cannot be constitutive of the critical concept of substance. Nevertheless, since (as Kant came to realize with increasing clarity between the first and second editions of the Critique) substance, in the critical sense, is necessarily spatial, the concept of the real in space does directly instantiate the critical concept of substance. 97 More precisely, the "reality" instantiating the schematized concept of substance (see again the paragraph following the one to which note 91 above is appended) also instantiates the concept of the real in space. We know that, in the Metaphysical Foundations, there is a reciprocal relationship of dependence between the concepts of quantity of matter (mass) and quantity of motion (momentum): the latter provides what Kant calls a condition for the "application to experience" of the former whereby "this quantity of matter (the aggregate of the movable) manifests itsdfin experience" (see note 81 above, together with the paragraph to which it is appended). Indeed, as we have also seen, it is for precisely this reason that conservation of quantity of matter, in the Metaphysical Foundations, is equivalent to conservation of momentum. 98 Accordingly, there is a parallel reciprocal relationship in the first Critique between the "transcendental" concept of substance and the concept of action. 99 In particular, the concept of action provides what Kant, in an important passage in the Second Analogy, calls the "empirical criterion" of the concept of substance (A204/B249): "Causality leads to the concept of action, this leads to the concept of force, and thereby to the concept of substance.... I can [therefore] not leave untouched [here] the empirical criterion of substance, in so far as it appears to manifest itself, not through the permanence of the appearance, but better and more easily through action." It appears, therefore, that action provides the "empirical criterion" of the "transcendental" concept of substance in precisely the same way that quantity of motion provides the condition for the "application to experience" of the concept of quantity of matter. As Kant has already indicated at the end of the First Analogy, however, what is required, more specifically, is an "empirical criterion of this necessary permanence [of the substratum of all time determination] and with it substantiality in the appearances" (A189/B232). Since we now know, in the Second Analogy, that this "empirical criterion" is given precisely by the causal action exerted by such a substance,100 it is by no means surprising that Kant goes on to assert that "action, as a sufficient empirical criterion, manifests substantiality, without my having to seek the permanence of the subject first of all by the comparison of perceptions-which could also not proceed, in this way, with the rigor and specificity
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[Ausfiirlichkeit] required for the magnitude and strict universality of the concept" (A20S/B2S0-S1). What Kant appears to be saying, then, is that we do not identifY a candidate for substantiality simply by watching it to see whether, in the appropriate sense, it is in fact unchanging or permanent: neither the "magnitude" (or quantum) of the candidate substance nor the necessity and strict universality of its putative permanence could thereby be rigorously established. Only the causal actions exerted by a substance yield both a quantitative measure of its magnitude and a rigorous proof of the necessary conservation of the resulting total quantum. Kant's discussion of the relationship between quantity of matter and quantity of motion in the Metaphysical Foundations provides us, once again, with a concrete example of what this more general "transcendental" claim actually amounts toJOl It is in precisely this way, therefore, that Kant's critical attempt to adapt the fundamental concepts of the Leibnizean philosophical tradition he inherited to the task of providing a metaphysical foundation for specifically Newtonian physics is deeply implicated, as well, in Kant's more general radical transformation of the very meaning of metaphysics as practiced by his predecessors. Kant undertakes this radical transformation by combining a Leibnizean view of the pure understanding with a Newtonian view of space and time, in so far as Kant now answers the question "how is pure natural science possible?" by appeal to the schematization of the pure intellect in terms of the pure intuitions of space and time (see again the paragraph to which note 11 above is appended, together with the following paragraph). Accordingly, Leibniz's most basic mistake, from Kant's point of view, was entirely failing to appreciate the importance of such schematization (A270/B326): "the famous Leibniz erected an intellectual system ofthe world, or rather believed [to be able] to know the inner constitution of things, in that he compared all objects only with the understanding and the isolated [abgesonderten] formal concepts of its thinking." Leibniz failed, in particular, to make the crucial distinction between the concept of a noumenal substance conceived by the pure understanding alone and that of a phenomenal substance actually found in nature, and he was therefore led to "intellectualize the appearances" (A271/B327) by viewing mindlike ultimate simple substances or monads as the ultimate constituents of matter:
Only that is internal in an object of pure understanding which has no relation at all (with respect to its existence) to anything different from itself. By contrast, the internal determinations of a substantia phaenomenon in space are nothing but relations, and it itself is nothing but a totality of mere relations. We are only acquainted with substance in space through forces that are active in space, either driving others into
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[this space] (attraction) or stopping their penetration into it (repulsion and impenetrability). We are acquainted with no other properties constituting the concept of a substance which appears in space and which we call matter. As object of the pure understanding, on the other hand, every substance must have internal determinations and powers, which pertain to [its] internal reality. However, what can I entertain as internal accidents except those which my inner sense presents to menamely, that which is either itself a thought or is analogous to it? Therefore, Leibniz, after he had taken away everything that may signify an external relation, and therefore also composition, made of all substances, because he represented them as noumena, even the constituents of matter, simple substances with powers of representationin a word, monads. (A265-66/B321-1) IZant's critical conception of the dynamical theory of matter and n1aterial substance then illuminates the precise character of Leibniz's mistake by showing that only the schematized concept ofsubstance can apply to nature as we experience it, and, accordingly, neither absolutely simple substances nor mind-like monads can ever be found there. 102 But this n1eans, for IZant, that neither of the supersensible entities discussed in the metaphysics of his predecessors-neither God nor the soulcan possibly be objects of theoretical cognition. They are instead purely intellectual concepts of noumenal objects, entirely disconnected from the appearances of nature-which concepts, as such, have only a (still absolutely indispensable) practical employment in morally regulating all of our activities (including our inquiries into nature) under the three highest ideas of reason: God, freedom, and irnmortality.lo3 Given the crucial importance, as we have just seen, of the critical conception of the dynan1ical theory of matter and material substance in this radical transformation of the very point and meaning of metaphysics as practiced by his predecessors, it is by no means surprising that, towards the end of his Preface to the Metaphysical Foundations, I(ant clearly alludes to precisely this point. 104 In the course of sharply distinguishing between general metaphysics or transcendental philosophy and the special metaphysics of corporeal nature (that is, the metaphysical foundations of natural science ), I
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tality" (477). Kant thereby clearly anticipates, as well, his famous remark in the Preface to the second edition of the Critique that he had to deny knowledge [Wissen] in order to make room for faith [Glauben]: transcendental idealism must deny theoretical knowledge of the supersensible precisely in order to allow the idea of the supersensible properly to guide us. lOS
From Kant to Carnap-Newton to Einstein As we have seen, Kant's philosophy of human knowledge and experienceincluding his more specific philosophy of(corporeal) nature developed in the Metaphysical Foundations of Natural Science-is based on a number of fundamental distinctions. The most important is the distinction between the passive or receptive faculty of pure intuition or sensibility and the active or intellectual faculty of pure understanding, and it is this distinction which then gives rise to the dichotomy between appearances (spatio-temporal objects given to our sensibility) and things in themselves (purely intellectual objects thought by the understanding alone). Post-Kantian German idealism-as successively articulated by Fichte, Schelling, and Hegel-is characterized by a rejection of these fundamental distinctions, a rejection, that is, of what came derogatorily to be known as "Kantian dualism." In particular, the tradition of post- Kantian German idealism could not accept the sharp dichotomy between practical and theoretical reason (compare again note 103 above, together with the paragraph to which it is appended), and, as a result, the so-called absolute idealism articulated by Schelling and Hegel returned, in some sense, to a more traditionally metaphysical conception of the relationship between spirit and nature closely modeled, in this respect, on the philosophy of Spinoza. This metaphysical turn of post- Kantian idealism has typically struck more scientifically-minded philosophers as scandalous, if not completely unintelligible. Indeed, beginning in the immediate wake of post-Kantian idealism in the mid to late nineteenth century, such reactions led first to a powerful "back to Kant!" movement, typified by the work of such natural scientific thinkers as Hermann von Helmholtz and academic neo-Kantian philosophers as Hermann Cohen,106 and then to the development of the form of "scientific philosophy" we now know as logical positivism or logical empiricism in the early years of the twentieth century. These developments, in turn, are of course extremely complex, and I shall return to them below. 107 But I first want to explain a lesson I myself have learned only in the past several years-partly on the basis of recent work by Frederick Beiser on absolute idealism and what he calls the organic view of nature, partly on the basis of my own research into the connection between
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Schelling's Naturphilosophie and the famous discovery of electromagnetism by Hans Christian Oersted. The point, in particular, is that Naturphilosophie and the organic view of nature did not represent an unaccountable and "unscientific" philosophical regression, but rather a perfectly intelligible and very insightful response to a combination of deep internal problems afflicting I(ant's original system and revolutionary new developn1ents in the sciences of chemistry, electricity, magnetisn1, and biology. Any serious history of scientific philosophy must therefore contain a significant place for these developments as well. I08
I(ant and Naturphilosophie Let us begin with a further important fundamental I
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tef as such into particular species and subspecies-the problem of what he dis "the specific variety of matter" in the General Remark to Dynamicstor the further development of empirical physics and chemistry. Moreover, in the Preface to the Metaphysical Foundations, Kant famously denies scientific status to chemistry and asserts that, at least at present, "chemistry can be nothing more than a systematic art or experimental doctrine, but . " (471) .III never a proper sCIence How, then, is the more empirical study of nature, such as chemistry, to proceed? It is here that Kant invokes the regulative use of reason and, in particular, the idea of the complete systematic unity of all empirical concepts and principles under the a priori constitutive concepts and principles already generated by the understanding. In particular, the idea of systematic unity guides our process of inquiry in the more empirical and inductive disciplines, without constitutively constraining it, as we successively ascend from lower level empirical concepts and principles towards higher level concepts and principles. The goal of this process is an ideal complete empirical science of nature in which all empirical concepts and principles are constitutively grounded in the pure categories and principles of the understanding-an ideal we can only successively approximate but never actually attain. Moreover, the paradigmatic application of the regulative use of reason, in the period of both the Metaphysical Foundations and the first Critique, is precisely to contemporary chemistry. Kant sees this chemistry-primarily Stahlian phlogistic chemistry as supplemented by the new discoveries in pneumatics but not yet including Lavoisier-as a purely empirical or experimental art guided by the regulative use of reason towards an entirely unspecified and indeterminate future state of affairs in which the experimental results in question are finally grounded in the fundamental forces of matter in a way we are not yet (and perhaps never shall be) in a position to anticipate.l l2 Moreover, as is well known, Kant, in the Critique ofJudgment, extends the doctrine of the regulative use of reason to what he now calls the faculty of reflective judgment, and he now applies this faculty, in particular, to the case of biology. The problem here, in a nutshell, is that all matter in general and as such-all matter as the object of our outer senses in spaceis essentially lifeless. This, in fact, is how Kant interprets the law of inertia, which, in turn, is itself constitutively grounded by a further specification of the a priori principle of causality articulated in the first Critique.l l3 Biology, the study of/ife, can therefore never be a science in the strict sense for Kant; it can never be constitutively grounded in the fundamental forces of matter. The best we can do, in this case, is to extend the doctrine of the regulative use of reason via the teleological idea of purposiveness [Zweckmafl~keit]-an idea which already arises for reflective judgment in general as it guides our inductive ascent from particular to universal
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towards the ideal infinitely distant goal of a complete systematic unity of nature. And this idea can now be applied to particular objects of nature or "natural products" (i.e., living organisms) in so far as they are conceived, by reflective judgment, as themselves purposively organized. But such a mode of conception is in no way constitutive of these objects themselves; it is rather a merely regulative device for guiding our empirical inquiry into living organisms as far as it may proceed. From the point of view of post-I
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nate degree results from a balance or equilibrium between the two fundamental forces of attraction and repulsion. From Schelling's point of view, this theory of the most general properties of all matter as such (which embraces, therefore, even the "dead" or inert matter considered in statics and mechanics) has already introduced an essentially dialectical and evolutionary element into nature, in so far as the dynamical constitution of matter in general proceeds fi:om the positive reality of expansive force (repulsion), through the negative reality of contractive force (attraction), to the limitation or balance of the two in a state of equilibrium. Yet this equilibrium, for Schelling, is only a temporary first step, for we now know, as Kant himself did not, that chemistry can be dynamically grounded by a dialectical continuation or extension of this same progression-as we proceed, more specifically, from the magnetic, through the electrical, to the chemical (or galvanic) forms of the basic or original dynamical process grounded in the fundamental forces of attraction and repulsion. us And, once we have gone this far, it is then a very short step (particularly in view of the newly discovered parallel interconnections among electrical, galvanic, and biological phenomena) to view biology, too, as a further dialectical continuation of the same dynamical process. U6 Biology, too, can be a science, for all rational science, as Kant did not see, is grounded in a single dynamical evolutionary dialectical progression. The whole of nature, in this sense, is at once both rational and alive;u7 and this means, in particular, that there actually is life-objectively, not merely regulatively-in even the very simplest forms of organized matter. U8 Schelling's entrancing vision is fueled by a number of dramatic new discoveries in electro-chemistry. In particular, the invention of the Voltaic pile (1800) had led to the discovery of current electricity or galvanism, and current electricity, in turn, had led to the discovery of the electrolytic decomposition of water-whereby oxygen and hydrogen separate out from the water and accumulate, respectively, at the positive and negative poles inducing the electrolytic current. Oxygen and hydrogen were thereby associated with negative and positive electricity, respectively, and this suggested an especially close link between electrical forces and the fundamental chemical forces involved in combustion. Schelling, along with many other researchers at the time, took this as evidence for the electrical nature of chemical affinities quite generally.U9 Finally, the wellknown parallels between electrical and magnetic forces suggested that magnetism, too, is essentially implicated in chemical interactions (including galvanism) and, more specifically, that what Schelling called the basic or original form of the dynamical process is further differentiated, at the level immediately following that of Kant's two fundamental forces of attraction and repulsion in general, into magnetism, electricity, and chemical forces (including galvanism).120
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It is precisely here, for Schelling, that we can unite the concept of matter in general as conceived in Kant's original dynamical theory (the "dead" matter considered in statics and mechanics) with matter as conceived by Naturphilosophie-as an inexhaustible source of rational life. It is in precisely this context that we can view chemistry, in Schelling's words, as a dialectical "middle term" between mechanism, on the one side, and biological (ultimately rational) living purposiveness, on the other. 121 Indeed, even the inert ll1atter considered in statics and mechanics is already at least potentially alive, since I
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vide an a priori grounding for the central theoretical construct of Lavoisier's new chemistry (the caloric fluid or matter of heat) and, at the same time, to serve, in a way that had long been familiar in eighteenth-century matter theory, as the medium or vehicle for light, electricity, and magnetism as well. In this way, the totality of forces or powers of nature-including, above all, the specifically chemical forces-are, at least in principle, systematically unified. The a priori representation in question-a universally distributed caloric fluid or aetherial medium-is, by the standards of Kant's critical philosophy, an extremely peculiar one. As a continuum of forces providing a basis for the further specification of the concept of matter in general, it is a discursive or conceptual representation; as a space-filling continuum, providing what Kant calls a perceptual "realization" of the pure intuition of space, it is a sensible or intuitive representation. Moreover, and by the same token, as an a priori principle for the further specification of the concept of matter in general, it is a constitutive representation; as the ultimate ground for the systematic unity of all of the forces of matter, it is a regulative representation. It is in this way, in fact, that the "top-down" constitutive procedure of the Metaphysical Foundations and the first Critique has a necessary intersection, as it were, with the "bottom-up" regulative procedure of the faculties of reason and reflective judgment; and it is in precisely this way, accordingly, that the skeptical problems arising from the doctrine of the regulative use of reason that so vexed the post- Kantian absolute idealists are finally resolved for Kant himself. 124 It is important to see, however, that Kant's own solution stopped considerably short of Schelling's Naturphilosophie, and, although Kant was right on the verge of the scientific and philosophical situation addressed by Schelling, he did not and could not take the decisive step. Kant had already considered the problem of extending the dynamical theory of matter into chemistry, and, at the same time, he had already subjected the fundamental distinction between constitutive principles and regulative principles to a radical reconceptualization. But the crucial new developments in chemistry, electricity, magnetism, and biology that provided the fertile empirical soil on which Naturphilosophie took root were in fact unknown to him. Kant, to the best of my knowledge, never engaged with even the electrostatic and magneto-static work of Coulomb, to say nothing of the electrochemical researches arising from the Voltaic pile. The central idea of Naturphilosophie in its application to chemistry-that chemical forces are at bottom electrical in nature-never occurred to Kant; and, as a result, the idea that one could extend the dynamical theory of matter by conceiving magnetic, electrical, and galvanic forces as a further dialectical development of the original dynamical process governing the fundamental forces of attraction and repulsion was entirely foreign to Kant's own final attempt
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to solve the problen1 of a "transition from the metaphysical foundations to physics." Moreover, although there may be roon1, with considerable stretching and straining, to find a place in Kant's theoretical philosophy for a representation that combines both constitutive and regulative aspects, there is no room at all for the grand naturphilosophisch vision of nature as a whole as the evolutionary dialectical development of a single (and ultimately divine) rational life. For this idea, of course, entails the total Aujhebung of all of lZant's most fundamental distinctions, along with the critical philosophy itself. That Kant himself, at the end of his career, was right on the verge of the scientific and philosophical situation addressed by Schelling further underscores and illuminates the precise sense in which Naturphilosophie and the organic conception of nature represent a perfectly intelligible and very insightful response to a combination of deep internal problems afflicting Kant's original system, on the one side, and revolutionary new scientific developments at the turn of the century, on the other. Despite the fact, moreover, that his dialectical evolutionary model of magnetic, electrical, and chemical or galvanic phenomena did not manage to survive into the second half of the nineteenth century (let alone into the twentieth), Schelling's speculative physics did in fact fruitfully guide the experin1ental and theoretical work of the principal founders of electro-chemistry and electromagnetism in the first half of this century.125 And it was able to do this, in spite of the shortcoming we might perceive in it today, precisely because the new en1pirical situation at the turn of the century demanded the exploration of non-Newtonian physical ideas, while, at the same time, the lZantian philosophical system, at the very end of the eighteenth century, had pushed such Newtonian ideas to their outermost philosophical lin1its. A new philosophy of nature, like Schelling's, which, as we have said, insightfully responds simultaneously to both the deep internal tensions emerging in lZant's philosophical system and the new empirical results, was therefore, in this specific historical context, precisely what was then needed.
Helmholtz Yet the nineteenth century, as it developed further, fairly exploded with breathtaking new developments involving a succession of deep and lasting transforn1ations in both the sciences and philosophy. As suggested above, the most important initial step beyond Naturphilosophie and post-lZantian idealism was taken by Hermann von Helmholtz, one of the most remarkable minds of his era, who made fundamental contributions to energetics, physiological psychology, the foundations of geometry, electricity and magnetism, and scientific epistemology. Helmholtz came to be identified,
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in particular, as one of the leaders of the emerging "back to lunt!." movement for a scientific philosophy, and his celebrated address, "Uber das Sehen des Menschen," delivered at the dedication of a monument to ICant at IConigsberg in 1855, became one of the primary intellectual models of this movement. Helmholtz begins by asking himself, on behalf of his audience, why a natural scientist like himself (at the time a professor of physi010gy at IConigsberg) is speaking in honor of a philosopher. This question only arises, he says, because of the current deplorable climate of enmity and n1utual distrust between the two fields-a clin1ate which is due, in Heln1holtz's opinion, to the speculative system of Naturphilosophie that Schelling and Hegel have erected wholly independent of, and even in open hostility towards, the actual positive results of the natural sciences. What Helnlholtz is now recommending, by contrast, is a return to the close cooperation between the two fields exemplified in the work of lunt, who himself made significant contributions to science (in his nebular hypothesis of 1755) and, in general, "stood in relation to the natural sciences together with the natural scientists on precisely the same fundamental principles." 126 As we have just seen, the charge that Helmholtz-and, following him, the rest of the "back to lunt!" movement for a scientific philosophy-leveled against the Naturphilosophie of the early nineteenth century is fundamentally unfair. Nevertheless, it is also true that Helmholtz himself had an important new weapon up his sleeve: namely, his own formulation of the principle of the conservation of energy in his great monograph, "On the Conservation of Force," published in 1847.1 27 In the introduction to this work, he describes "the ultimate and proper goal of the physical natural sciences as such" as beginning with an "experimental part," where one seeks to describe "the individual natural processes" by "general rules ... which are obviously nothing but universal generic concepts through which all of the appearances belonging thereto are comprehended," and proceeding to a "theoretical part . . . which seeks, by contrast, to find the unknown causes of the processes from their visible effects; it seeks to conceptualize them in accordance with the law of causality" (1882, 12-13; 1971, 3). This procedure aims eventually to discover the "ultimate unalterable causes" lying at the basis of all the appearances: We are compelled and justified in this task by the principle that every alteration in nature must have a sufficient cause. The proximate causes that underlie the appearances of nature can themselves be either unalterable or alterable; in the latter case the same principle compels us to seek for other causes of this alteration in turn, and so on, until we finally arrive at the ultimate causes that act in accordance with an unalterable law, and which, therefore, bring about at every time, under the
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same external relations, the same effect. The final end of the theoretical natural sciences is thus to discover the ultimate unalterable causes of natural processes. (1882, 13; 1971,4) And the possibility of reducing all of the appearances of nature to this basis, in accordance with the lavv of causality, is then "the condition for the complete conceptualizability of nature" (1882,16; 1971, 6). Thus it is clear that when Helmholtz, in his 1855 lecture on human vision, says that IZant stood "together with the natural scientists on precisely the same fundamental principles," the main principle he has in mind is IZant's formulation of the principle of causality.128 Indeed, the salience of IZant's particular formulation of the principle of causality becomes even clearer when we note that the ultimate unalterable causes mentioned in the introduction to "On the Conservation of Force" turn out to comprise a system of masses interacting with one another solely through time-independent (constant or "unalterable") central forces of attraction and repulsion depending only on the distances between them. And the main burden of the monograph that follows is then to contribute to this program by showing that the phenomenological principle of the conservation of energy-the principle, as Helmholtz phrases it, that a perpetual motion machine (of the first kind) is impossible-is equivalent to the theoretical principle that all actions in nature are in fact reducible to ultimate forces of attraction and repulsion in this way. The principle of the conservation of energy is thereby shown, in particular, to involve a more specific realization of I0nt's general principle of causality corresponding, in this respect, to IZant's own more specific realization of his "transcendental" principles of the understanding in the Metaphysical Foundations Natural Science. For Helmholtz, accordingly, all the manifold phenomena of naturemechanical, thermal, chemical, electromagnetic, and biological-among which we see energy transferred, and in which we see the total quantity of energy conserved, are ultimately explained by an underlying system of invisible and unalterable masses governed by fundamental forces of attraction and repulsion closely modeled on 10nt's original dynamical theory of matter;129 and Helmholtz is then in a position to use 10nt's theory to make a number ofimportant points against Schelling's dialectical extension of it. First, and most importantly, it follows that I0nt's original theory is entirely capable, in principle, of comprehending all the new discoveries appealed to by Schelling. Second, and as a corollary, there is no need at all to incorporate life or any "vital principle" into the fundamental forces of matter. On the contrary, by adopting attractive and repulsive forces as fundamental, we thereby attain a unity of all the forces and powers of nature that is completely different in kind from the organic or teleological unity envisioned by Naturphilosophie-in so far as the underlying constancy of
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the total quantity of "force" is now conceived as a reflection of the priorit)' and foundational status of purely mechanical "force.,,130 Third, and finally, it follows that no essentially non-Newtonian physical principles are required after all, for Newtonian central forces of attraction and repulsion, obeying only the principles of ewtonian mechanics, suffice, in the end, to explain all the phenomena governed by the new principle of the conservatlon 0 f energy. 131 In this context, then, Helmholtz takes the main point of Kant's principle of causality to be that of licensing an inference from the observable phenomena or appearances to the "ultimate unalterable causes" of these "visible effects"-an inference from the observable phenomena to their unobservable causes. 132 However, by the time Helmholtz adds supplementary notes to his monograph in 1881, he has clearly abandoned this view. In the very first note, in particular, Helmholtz corrects the abovecited sentence, where the law of causality is said to lead us to "unknown causes" from their "visible effects," as follows (1882, 68; 1971,49): "The philosophical discussion in the introduction is more strongly influenced by Kant's epistemological views than I would now like to recognize as correct. I only made it clear to myself later that the principle of causality is actually nothing other than the presupposition ofthe lawlikeness of all the appearances of nature.,,133 The principle of causality cannot serve, as Helmholtz understood Kant in 1847 (and 1855: compare note 132 above), as an a priori justification for postulating unknown, invisible objects behind the observable appearances. On the contrary, lawlikeness is now a fundamental principle governing the appearances themselves. 134 This important shift in Helmholtz's thinking was mediated by the further articulation of what he called an "empiricist" theory of space-perception in his monumental Handbook of Physiological Optics, first appearing between 1856 and 1867.1 35 Although he had declared his allegiance to such a theory, and his opposition to "nativism," in 1855, the theory did not acquire a clear articulation until 1865-66. 136 The basic idea is that our ability to see objects around us in space, as localized at particular places, is not an innate capacity of either our consciousness or our nervous apparatus. Rather, this ability is itself gradually learned or acquired-as we learn or acquire our native language, for example-by a process of "unconscious inductive inference" based on regularities or associations among our sensations. For example, my ability to localize a perceived table in threedimensional space is in no way directly given by simple visual or tactual sensations; it rather requires (unconscious) knowledge of a large number of regularities among such simple sensations, generated as I move around the table, reach out and touch it, and so on. In this sense, the ability to see objects in space is primarily an affair of the understanding, and "[t]he fundamental principle of the empiricist view is [that] sensations are signs for o
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our consciousness) where learning to understand their meaning is left to our understanding. "137 By the'same token, therefore, the process of learning to localize objects in space is closely analogous to the conscious procedure of inductive inference characteristic of natural science. Indeed, except for the fact that the former inferences are unconscious, the analogy is perfectly exact (1910, 28; 1925, 30-31): "Now the same great significance that experiment has for the security of our scientific convictions, it has also for the unconscious inferences of our sensory perceptions. Only in so far as we bring our sense organs, in accordance with our own willing, into various relations to the objects, do we learn securely to judge about the causes of our sensations, and such experimenting takes place from the earliest childhood on, without interruption, throughout the whole of life." We thus learn or acquire the complicated system of regularities among initially isolated and fragmentary sensations, which, as a system, first constitutes the perception of an object in space, by the very same procedure, and in accordance with the very same causal or inductive principle, that we self-consciously employ in scientific inference.l 38 Hence, since the primary role for the causal or inductive principle here is precisely to secure our grasp of regularity or lawlikeness on the side of our perceptions, it no longer functions as a bridge to another, "hidden" realm existing behind our perceptions.l 39 Helmholtz's work on the psycho-physiology of space-perception was inextricably connected, as well, with his mathematical contributions to the foundations of geometry, developed mainly in the years 1866-70. These contributions resulted in what we now know as the Helmholtz-Lie theoren}, and they paved the way, in turn, for the mature statement of his episten1010gical position in "The Facts in Perception," first presented as a public lecture in 1878 (compare note 135 above). For the upshot of this theorem, for Helmholtz, is that the same regularities in our sensations, on the basis of which we acquire the ability to localize objects in space, also give rise to the representation of space itself. The voluntary actions of our bodies that allow us to localize objects by moving towards, away, and around them, also make possible a precise mathematical construction of the very three-dimensional space within which this process of localization takes place. In this way, space does not serve as the locus of mind-independent objects existing behind the veil ofperception, but rather as a "subjective form of intuition" in the sense of Kant-so that "space will also appear to us sensibly, clothed with the qualities of our sensations of motion, as that through which we move, through which we can gaze forth"; it is thus the "necessary form of our external intuition ... because we con1prehend precisely that which we observe as spatially determined as the external world.,,140 The world of objects in space is truly "external" only in the sense of being spatial, and space itself is a construction, erected
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entirely on the basis of our inductively acquired ability perceptually to localize objects. Helmholtz's mathematical work was directly inspired by Bernhard Riemann's creation of what we now call the general theory of manifolds in his Habilitationsvortrag of 1854, "On the Hypotheses which Lie at the Basis of Geometry." Helmholtz's goal, in particular, was to derive Riemann's underlying assumption or "hypothesis"-that the line-element or metric is infinitesimally Euclidean-from what Helmholtz took to be the fundamental "facts" generating our perceptual intuition of space. Since our representation of space, as we have seen, arises kinematically, from our experience of moving up to, away from, and around the objects thereby localized, the space in question must satisfY a condition of "free mobility" permitting arbitrary continuous motions of rigid bodies; and from this condition, in turn, we can then derive the Pythagorean form of the lineelement. 141 Finally, since the Riemannian metric thereby constructed has what we now call a group of rigid motions or isometries mapping any point onto any other, the only possible geometries we can construct in this way are the three classical geometries of constant curvature: hyperbolic (constant positive curvature), elliptic (constant negative curvature), and Euclidean (constant zero curvature ).142 But it now follows, for Helmholtz, that specifically Euclidean geometry is not built into what he takes to be the essential or necessary character of space, given by the condition of free mobility. The particular propositions of Euclidean geometry, as Helmholtz now argues explicitly in his 1870 lecture "On the Origin and Significance of the Axioms of Geometry," "are not included in the general concept of a three-dimensional extended magnitude and the free mobility of the bounded structures contained in it[; t]hey are not necessities of thought, which flow from the concept ofsuch a manifold and its measurability, or from the most general concept of a rigid structure contained therein and its freest mobility" (1921,17; 1977, 17). Indeed, it follows from Helmholtz's theory of the origin of our spatial intuition that the particular propositions of Euclidean geometry are also not necessities of intuition. For we can now imagine, for example, the series of sensations we would have if we were to find ourselves moving around in a space of constant negative curvature, and such a series ofsensations, on Helmholtz's theory, would constitute an intuition of space (1921, 22; 1977,23): "We can picture to ourselves the view ofa pseudospherical world going in all directions, just as well as we can develop its concept. We can therefore also not admit that the axioms of our geometry are grounded in the given form of our faculty of intuition, or are in any way connected with such [a form]." Since the axioms of Euclidean geometry are not built into the most general necessary conditions underlying our spatial intuition, Kant's theory of the origin of these axioms in
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our "necessary" and "transcendental" intuition of space is incorrect, and they emerge rather as merely empirical facts about the actual behavior of our measuring instruments. Nevertheless, I
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have in our language a very happy designation for that which influences us [auf uns einwirkt], standing behind the change of appearances, namely 'the actual [das Wirkliche].' Here only the action [das Wirken] is expressed; it lacks the secondary reference to enduring existence as substance that the concept of the real [das Reelen] includes." Since, as we have just seen, what "stands behind" the change of appearances is the enduring existence of lawlike relations among our sensations-and, indeed, "[ t ]he lawlike is thus the essential presupposition for the character of the actual" (ibid.)-it follows that we thereby know the actual but not the real (1921, 132; 1977, 140-41): "I do not need to explain to you that it is a contradiction in terms to represent the real or Kant's 'thing in itself' via positive determinations, but without taking it up into the form or our representing. This is often discussed. But what we can achieve is an acquaintance with the lawlike ordering in the realm of the actual, to be sure only presented in the sign system of our sense impressions." This last clause, as we have seen, does not mean that "the sign system of our sense impression" somehow corresponds to a mind-independent realm existing behind the appearances, for what Helmholtz here calls "the realm of the actual" consists precisely of the lawlike relationships existing within the appearances themselves .146 Helmholtz's mature scientific epistemology therefore represents a rather deep transformation of central doctrines of Kant's critical philosophy. For, in the first place, Helmholtz explicitly borrows the overriding importance of the principle of causality from Kant, and, moreover, he continues to view it as an expression of the fundamental character of our conceptual faculty. To be sure, this principle, for Helmholtz, has more a regulative than a constitutive character: as the "condition for the complete conceptualizability of nature," it sets up what Kant would call a regulative ideal akin to the ideal of complete systematic unity (see the paragraph to which note 112 above is appended). Nevertheless, as we have seen, a blurring of the sharp line between Kantian constitutive and regulative principles becomes increasingly characteristic of the entire post-Kantian period-beginning with the "post-critical" work of Kant himself (see the paragraph to which note 124 above is appended, together with the preceding and following paragraphs). In the second place, however, Helmholtz's explicit appropriation of the conception of space as a "necessary" and "subjective" form of our perception of external objects is perhaps even more striking. Indeed, geometry becomes applicable to both space and external objects, just as it does for Kant, by subjecting our outer perceptions to the a priori demands of our conceptual faculty-for Helmholtz, as we have seen, by successfully subjecting such outer perceptions to the requirement of thoroughgoing lawlikeness.l 47 To be sure, Helmholtz, unlike Kant, is conversant with
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non-Euclidean geometries, and so Helmholtz, in particular, famously argues against the Kantian view that specifically Euclidean geometry expresses the "necessary form of our external intuition." On the contrary, "the most essential features of spatial intuition" include only the much more general geometrical property required by Helmholtz's condition of free mobility: the property of constant curvature common to all three classical cases of hyperbolic, elliptic, and Euclidean geometry (see note 142 above, together with the paragraph to which it is appended).l48 Nevertheless, as we have also seen, Helmholtz's mature epistemology continues to en1phasize what is arguably the most fundamental feature of IZant's conception of space as the form of our perception of external objects: namely, that such objects are only properly "external" in so far as they are thereby located in space, and, accordingly, they are sharply to be distinguished from any truly mind-independent objects that might exist behind the "appearances" entirely independently of our" necessary form of external intuition." The most important divergence between Helmholtz and IZant does not concern particular points of doctrine at all but arises from the radically different historical-intellectual contexts within which their respective doctrines are formulated. Helmholtz, even more so than Schelling, worked in a vastly different intellectual world from lZant's-one in which the empirical sciences, in particular, had taken on a dramatically new and entirely unforeseen systematic reconfiguration. Not only had striking connections begun to emerge between mechanics, chemistry, electricity and magnetism, and biology, but the principle of the conservation of energy had also established a precise lawlike relationship comprehending all of these don1ains, whereby a single measurable quantity of "force" is interchangeable among them and, as a whole, quantitatively conserved in all such transformations of form. Moreover, a new experimental science of psychophysics had been founded by Gustav Fechner and developed, among others, by Helmholtz's teacher Johannes Miiller, and the practitioners of this new science understood themselves, quite understandably, to be standing on the threshold of a profound intellectual breakthrough in which the physical sciences, the life sciences, and the en1erging new science of the mind were all to be comprehended within a single unified scientific framework. Helmholtz himselfwas one of the greatest of these practitioners, and his epistemological doctrines-including both his overriding emphasis on the principle of causality and his conception of space as a form of external intuition-were entirely framed, as we have seen, by his characteristic "empiricist" viewpoint in the psycho-physiology of sense-perception. Helmholtz's scientific episten1010gy and his empirical scientific practice were in this way inextricably intertwined, and his distinctive epistemological doctrines, despite their explicit (and, as we have seen, rather deep)
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indebtedness to Kant, thereby took on a "naturalistic" (and, indeed, "psychologistic") cast quite incompatible with Kant's own conception of "transcend entaI p h 1'l osop h y. "149
Mach This same historical-intellectual context also framed the scientific epistemology of Ernst Mach which, in important respects, was developed in explicit opposition to that of Helmholtz. Yet this particular context was largely forgotten in the twentieth century when Mach became best known, rw"ospectively, as one of the most important ancestors of logical empiricism (the Vienna Circle, for example, explicitly took on the name of the Verein Ernst Mach in the late 1920s). Mach's philosophical motivations were thereby standardly assimilated to the traditional philosophical empiricism of Locke, Berkeley, and Hume, or (a bit later) Bertrand Russell.1 5o Nevertheless, Mach's own primary motivations were not epistemological at all, in this sense, but were rather directed at securing a new kind of unification of the empirical sciences-particularly the physical and the life sciences-as he finds them in his late nineteenth-century context. Thus, in the first sentence of the Preface to the first (1886) edition of The Analysis ofSensations, Mach explains (1886, v; 1959, xxxv-vi): "I have been repeatedly led to this domain by the deep conviction that the whole of science in general, and physics in particular, awaits the next great clarification of its foundations from biology, and, in fact, from the analysis of sensations." And, in an important footnote to §12 of the first chapter (§13 in the third and later editions), where Mach is explaining the intellectual background to the "sensationalist" conception he is presenting there, he describes his motivations as follows (1886, 21; 1959, 30): "I make no claim to the title of philosopher. I wish only to talce up a standpoint in physics that does not need immediately to be changed if one looks over into the domain of another science; for, in the end, all should form a whole. Contemporary molecular physics certainly does not meet this requirement." That the unified science in question is especially concerned with contemporary work in psychology and physiology is made explicit in a parallel passage in the third chapter (added to the enlarged second edition of 1900), where, after emphasizing again that he is "precisely a scientist and not a philosopher," Mach states (1900, 35; 1959,47): "I sought only a secure clear philosophical standpoint, on which no metaphysical clouds were encamped, from which practicable paths were visible into both the domain of psycho-physiology and that of physics." In particular, Mach's distinctive scientific epistemology, just like that of Helmholtz, was inextricably entangled with his own empirical research in
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psycho-physics. Thus, for example, Mach made major contributions to the experin1ental study of visual contrast phenomena (including what are now still known as "Mach bands"), and he worked closely with Ewald Hering on a "nativist" approach to the theory of vision explicitly opposed to Helmholtz's "empiricism" (as discussed below). Accordingly, Mach shared the conviction that the new en1pirical science of psycho-physics constituted the basis for a radically new kind of unification of the sciences in general. Just as the principle of the conservation of energy, as we have seen, had revealed deep lawlike correlations between mechanics, chemistry, electricity and magnetism, and biology, the new science of psycho-physics enabled us to push this process even further, into the hitherto entirely mysterious relationship between the physical and the psychical. Whereas in Fechner, for example, this comn1itment to an ultimate unity of the physical and the psychical had led him to pan-psychism,151 in Mach it led to a new form of scientific epistemology capable of comprehending both the novel developments within the psychological and biological sciences and parallel developments within contemporary physics. To begin with, Mach's scientific epistemology involved a version of what later became known as neutral monism. The fundamental objects of scientific knowledge (in all of the sciences) are neither mental nor physical as traditionally conceived. They are instead neutral entities, best conceived as events or episodes of perceiving (what Mach himself calls "sensations" or "elements"), which can become either mental or physical by being placed in systems of lawlike correlations with other such episodes. If, for example, I collect together all episodes of my perceiving a table from various angles and points of view with all other events of my individual experiential history, I thereby conceive then1 and classify them as mental. But if, on the contrary, I collect together these same episodes with all other events of perceiving this particular table (including all such perceptions of other perceivers), I then conceive them and classify them as physical. The task of the empirical sciences is not to reduce the mental to the physical or vice versa-only metaphysical obscurity and an end to scientific progress can result from such attempts. Our task is rather to investigate the infinitely various lawlike relations or correlations among such neutral "elements" at every level: psychological correlations between elements conceived as mental, physical correlations between these same elements conceived as material, and, finally, psycho-physical correlations between these very same elements conceived as belonging to both realms. 152 An appreciation of Mach's neutral monism, however, is only the first step in a proper understanding of his distinctive scientific viewpoint. For we still need to appreciate the fundamental importance of evolutionary biology for Mach, and how this figures centrally in his conception of the "economical" task of science-as expressed in such passages as the follow-
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ing (added to the end of the first chapter starting in the second edition; 1900, 26; 1959, 37): "The biological task of science is to offer the fully developed human individual as complete an orientation as possible. Another scientific ideal cannot be realized, and also has no sense." And to understand the fundamental importance of evolutionary biology, in turn, we need briefly to consider the famous dispute between Helmholtz, on the one side, and Hering and Mach, on the other, concerning "empiricism" versuS "nativism" in the psycho-physiology of sense perception. As we have seen, Helmholtz originally set the terms for this debate by defending what he called an "empiricist" theory of the origin of our representation of space, and he articulated this theory, as we have also seen, by means of modified and generalized versions of Kant's conceptions of the principle of causality and of space as a "necessary" and "subjective" form of outer intuition. For Hering and Mach, however, a commitment to "nativism" involved a corresponding commitment, in particular, to the theory of evolution, and this resulted, for Mach, in a much less Kantian and more "pragmatic" form of scientific epistemology.I53 In Helmholtz's view, our representation of space is fully learned or acquired by what he conceives of as an individual adaptation, where each organism acquires the representation of space within its own lifetime from the lawlike relations among its own perceptions, and there is absolutely no inheritance, in this connection, from previous generations. For Hering and Mach, by contrast, the representation of space is largely "hard-wired" into individual physiology at birth, so that it is thereby subject to a truly evolutionary adaptation extending across many generations.l 54 The result, for Mach, was a version of scientific epistemology in which all of the sciences (including the mathematical and physical sciences) successively emerge as stages of a fundamentally biological evolutionary process tending towards a better and better adaptation of the human race to its natural environment. I55 Accordingly, Mach (1906) distinguishes three different stages in the emergence of our representation of space. Our biological inheritance yields a physiological representation, which approximately satisfies free mobility and is thus roughly homogeneous. Our early experience with physical measurement (which, for Mach, is a further extension of biological evolution: see note 155 above) then yields a physical representation, in which exact geometrical laws are possible (as idealizations) and which can thereby be taken to be precisely homogeneous. Finally, our theoretical investigation of the resulting structure in reflective scientific inquiry (a still further extension of biological evolution) yields a logical (e.g., axiomatic) representation-where, for example, we discover that geometries other than Euclidean are possible. From beginning to end, however, this entire process is driven by the same fundamentally biological goal: that, in Mach's words, of "offer[ing] the fully developed human individual as com-
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plete an orientation as possible." In particular, there is no room, within this evolutionary process, for I
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are valid. Kant's constructive procedure then yields a sequence of better and better approximations to such a frame, in so far as he, too, appeals to precisely the equality of action and reaction-applied to ever wider systems of (gravitationally) interacting bodies-to define a sequence of more and more adequate centers of motion. In the limit, judged from the point of view of what Kant calls "the center of gravity of all matter" (563), we would attain a true inertial frame of reference embracing all the motions in the universe; and this, for Kant, would be what he calls "absolute space" (as an idea of reason). DiSalle (2002b) very clearly explains (see note 156 above) how Mach, in the first (1883) edition of The Science of Mechanics, substituted a privileged cosmic frame of reference-defined by what he called the fixed stars-for Newtonian absolute space. In particular, as DiSalle (2002b, 172-73) puts it, Mach was thereby attempting to summarize the concrete empirical evidence on which Newton's theory of space, time, and motion was actually based, where, for Mach, "every astronomical application of the law of inertia was, essentially, the resolution of some acceleration relative to the fixed stars into components, each of which is paired with some other acceleration relative to the fixed stars." Thus, Mach, like Kant, appealed to an astronomical application of the equality of action and reaction within a privileged cosmic frame of reference to construct an empirically definable surrogate for Newtonian absolute space.l S7 Moreover, as DiSalle (2002b, 175-78) also explains, Mach became acquainted with the development of the concept of an inertial frame ofreference, culminating in Ludwig Lange's fundamental clarification in 1885, between the first (1883) and second (1888) editions; and Mach explicitly discussed this literature in a lengthy appendix to the second edition which was then largely incorporated into the body of the text by the fourth (1901) edition. ISS Mach thereby came to appreciate, in particular, that the laws of mechanics can thus be formulated more abstractly, without assuming that any real physical system (like the fixed stars) actually defines an inertial system; and, accordingly, Mach also acquired a new appreciation for Newton's own understanding of the situation, in so far as Corollary V to the Laws of Motion already shows that absolute space can be entirely dispensed with in favor of an equivalence class of (inertial) frames of reference (Newtonian "relative spaces") under what we now call Galilean relativity.IS9 It is in this precise sense, therefore, that Mach's discussion of the Newtonian concepts of space, time, and motion in The Science ofMechanics can be viewed, as I have suggested, as a deeply insightful conceptual bridge between Kant's penetrating and original discussion of these same concepts in 1786 and the emergence of the modern concept of an inertial frame of reference approximately one-hundred years later. 160
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Poincare With Mach, then, we stand on the threshold of the even more profound transformation of the Newtonian concepts of space, time, and motion effected by Einstein, and, accordingly, we also stand on the threshold of the parallel transformation of scientific philosophy undertaken by logical empiricism-which, as we know, was very substantially influenced by Einstein's work. It is by no means surprising, therefore, that both Einstein and the logical empiricists explicitly expressed their indebtedness to Mach. Nevertheless, in order fully to appreciate the way in which the work of both Einstein and the logical empiricists grew out of its late nineteenthcentury background, we need first to discuss the intervening contributions of Henri Poincare, who, in several important respects, was much closer to the scientific and philosophical situation addressed by Einstein and the logical empiricists than was Mach. Thus, for example, Poincare himself made very significant contributions to the electrodynamics of moving bodies around the same time as Einstein's publication of the special theory of relativity in 1905-to such an extent, in fact, that there is still an ongoing debate concerning who first discovered this theory. Moreover, Poincare's famous articulation of a "conventionalist" philosophy of geometry deeply influenced the logical empiricists, not only in the philosophy of geometry but also in their conception of scientific epistemology much more generally. Finally, this same "conventionalist" philosophy of geometry, as Einstein himself tells us, formed a central part of the background to his articulation of the general theory of relativity in the years 1912-15. Poincare, unlike Helmholtz and Mach, worked primarily in pure mathematics and mathernatical physics; although he was also acutely aware (like virtually everyone else at the time) of the exciting new discoveries in biology and physiological psychology, they were by no means the main focus of his scientific work. 161 Above all, Poincare was one of the very greatest figures in turn of the century mathematics, and so modern abstract mathematics, in particular, constituted by far the most important background to his characteristic scientific epistemology. Yet Poincare's conception of this mathematics was not abstract in the twentieth-century sense (as exemplified by "formalistic" interpretations of the Hilbertian axiomatic method), and, accordingly, he did not conceive the subject n1atter of mathematics as consisting of purely formal, axiomatically defined abstract structures characterized entirely independently of our intuitive experience. Poincare rather worked squarely within the modern group-theoretic tradition of Felix I
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cept in particular mathematical domains. From this point of view, the virtue of the general concept of a group is precisely that it allows us to establish important and unexpected relationships between these different domains;162 and, entirely abstracted from such particular concrete applications, it is then of comparatively little interest. In this tradition, therefore, one still maintained a balance between the purely abstract use of the modern axiomatic method and the more concrete and intuitive conception of mathematics associated with Kant,163 and, as we shall see, Poincare's scientific epistemology is best seen as an acute and penetrating (if ultimately unsuccessful) attempt to maintain such a balance much more generally. The best known and most influential statement of Poincare's scientific epistemology is Science and Hypothesis (1902), whose subject is mathematics and the mathematical sciences. For it is in precisely these sciences, in particular, that we meet the centrally important examples of statements that "are only hypotheses in appearance and reduce to definitions or conventions in disguise" (1902, 2; 1913b, 28): These last are met with above all in mathematics and the related sciences. It is precisely from them that these sciences acquire their rigor; these conventions are the work of the free activity of our mind, which, in this domain, recognizes no obstacle. Here, our mind can affirm because it decrees; but let us understand one another; these decrees are imposed upon our science; they are not imposed upon nature. Are these decrees then arbitrary? No, otherwise they would be sterile. Experience leaves us our free choice, but it guides us by helping us to discern the most convenient path. Our decrees are therefore like those of an absolute, but wise prince, who consults his council of state. (1902,3; 1913b,28) Poincare concludes his briefIntroduction by a review of what he calls "the series of sciences from arithmetic and geometry to mechanics and experimental physics" (1902,4; 1913b, 28), beginning with the sciences where free conventions are most in evidence-where we are dealing with "frames which we impose upon the world"-and concluding with the more physical and empirical sciences. This series or hierarchy of sciences begins with arithmetic, where, according to Poincare, we find the most characteristic form of mathematical reasoning, namely, reasoning by recurrence or mathematical induction. As is well known, Poincare decisively breaks with the currently fashionable logical approaches to the foundations of arithmetic, and he explicitly argues that arithmetic (and therefore mathematics more generally) is a synthetic a priori science based on our intuitive capacity to represent the (potentially) infinite repetition or iteration of one and the same
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operation-which intuition is then the basis for the characteristic inference by mathematical induction.1 64 It is here, therefore, that Poincare's conception of mathematics is at its most Kantian (compare note 24 above, together with the paragraph to which it is appended), but, as I have suggested, he strives to maintain a balance between forn1al and intuitive elen1ents in all of the mathematical sciences. At the next lower level, for example, where Poincare examines the concept of mathematical magnitude, we find another important frame we impose upon our sense experience. This frame, comprising the full continuum of real numbers, is much too "complex and subtle" to be derived from our sense experience, and it is thus "of our own construction." Nevertheless, it is equally important to Poincare that we become as clear as possible about the intuitive or psychological origin of the relevant concept of (continuous) magnitude, for it is only by doing this, he suggests, that we can then fully understand precisely how it applies to our experience.1 65 At the next lower level of the hierarchy we find the concept of space, a very special example of a three-dimensional mathematical continuum, and one which also arises from our intuitive sensory experience. Following Helmholtz, Poincare tal(es the sensory experience in question to be that involved in our experience of bodily motion-the experience, as Poincare puts it, of bodily "displacements." In order to apply mathematics to this intuitive experience we then need to form a corresponding rigorous concept, and this, for Poincare is the concept of a group-in accordance with which, as an idealization, we assume that our bodily displacements are closed under the operation of con1position, that this operation always has a unique well-defined inverse, and so 011. Moreover, we also assume, as an additional idealization, that we thereby arrive at a continuous group in the sense of Lie, where the displacements under consideration are themselves ordered and arranged in a mathematical continuum (in this case, one of six dimensions). It now follows, by the Helmholtz-Lie theorem, that space, in this sense, has a (metrical) geometry, and so, for Poincare, we now have the beginnings of an explanation for how we are able to apply the exact science of geometry to our sensory experience. 166 In particular, we now IU10w that the space thereby constructed must have one of the three classical geometries of constant curvature: hyperbolic (constant positive curvature), elliptic (constant negative curvature), or Euclidean (constant zero curvature). But how do we then know which of these three different cases actually obtains? Heln1holtz's answer-after he had learned that Euclidean geometry is not, after all, uniquely constructible in this way (see again note 142 above )-was that "the most essential features of spatial intuition" include only free mobility and thus constant curvature, and, therefore, that the specifically Euclidean character of physical space is a merely empirical fact about the actual behavior of our
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measuring instruments as we move them around in accordance with the condition of free mobility (see note 143 above, together with the paragraph to which it is appended and the preceding paragraph). In this sense, Helmholtz's philosophy of geometry was Kantian in so far as space indeed has a "necessary form" expressed in the condition of free mobility, but it was empiricist in so far as which of the three possible geometries of constant curvature obtains is then determined by experience-for example, by measuring the angles of a triangle and determining whether their sum is 180° (at least to a very high degree of approximation). Poincare's revolutionary answer, however, was neither Kantian nor empiricist in this sense. The mathematical form we impose upon our experience of bodily displacements is neither necessary nor empirically determined but a "definition or convention in disguise" resulting from the "free activity of our mind." For, in the first place, the group-theoretic analysis of the condition offree mobility Poincare derives from Lie reveals that it, too, is based on nothing more nor less than a freely chosen mathematical idealization of our rough and approximate sensory experience-whereby we simply stipulate or postulate that our bodily displacements, so idealized, form a continuous group (closed under composition, unique well-defined inverse, and so on). And, in the second place, even with such an idealization already in place, there is still no way, for Poincare, that our actual experience of such displacements can then discriminate sufficiently between the three classical geometries of constant curvature. Rather, we stipulate or postulate by convention, as the simplest and most convenient idealization of our rough and approximate sensory experience, that the resulting geometry is Euclidean; and we do this, moreover, because of its mathematical simplicity.167 The decision to use Euclidean geometry for describing our spatial measurements is therefore precisely analogous, for Poincare, to the decision to use metrical rather than English units or Cartesian rather than polar coordinate systems. Why, however, can our spatial measurements not show, after all, that Euclidean geometry fails-by measuring the angles of a triangle, for exampie, and determining that their sum is not 180°? The answer, according to Poincare, is that such measurements always depend on physical objects and processes-physically rigid bodies experiencing no deformations by physical forces, for example, or light rays propagating along geometrically straightest lines-and the physical theories describing such objects and processes necessarily occur lower down in the series or hierarchy of sciences. In particular, we must already have a spatial geometry in place in order to develop the appropriate theories of physical forces and processes: to show that a putatively rigid body really experiences no physical deformations as we move it around, for example, or to develop a theory of geometrical optics on which (by assumption) light really propagates along
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straightest lines. More generally, for Poincare, each science in the series or hierarchy presupposes that all higher level sciences are already in place: what he calls experimental physics presupposes that the science of mechanics is already in place; this science presupposes that geometry is already in place; the latter presupposes that the theory of (continuous) mathematical magnitude is already in place; and this science, finally, presupposes that arithmetic (and thus what Poincare calls mathematical reasoning) is already in place. There can be no question, therefore, of empirically determining the "true" geometry of space by physical measurements. On the contrary, such measurements are themselves only possible in the first place if we have already stipulated, by convention, an antecedent spatial geometry with which to describe them. 168 Nevertheless, although Poincare is perfectly clear that no physical experiments can determine the geometry of space, he is equally clear that experience, as he puts it, "guides" our choice of conventions "by helping us to discern the most convenient path." In the case of geometry, in particular, "these conventions are not arbitrary, and, transported into another world (which I call the non-Euclidean world and seek to imagine), we would have been forced to adopt others" (1902, 5; 1913b, 29). The world in question is the famous "non-Euclidean world" of Chapter IV, where Poincare imagines a pervasive temperature field within a spherical region that uniformly contracts all bodies as they move from the center towards the periphery in accordance with the laws of hyperbolic geometry.169 And the main point of this particular thought-experiment, for Poincare, is that we thereby have an example of how "beings that would have minds made like ours and the same senses as ours, but without having received any previous education, would receive from a suitably chosen external world impressions such that they would be led to localize the phenomena of that external world in a geometry other than that of Euclid" (1902, 68; 1913b, 66). The point, in other words, is that our experience could have been very different from what it in fact has been, and, in that case, we would have been properly "guided" to a nonEuclidean description of physical space. Does this mean that geometry is empirical after all? No; for, given that we have grown up and been educated in our actual world, "ifwe were suddenly transported into this new world, we would have no difficulty in referring its phenomena to our Euclidean space" (ibid.): for exan1ple, we would now describe this world precisely by a distorting temperature field and not by a non-Euclidean geometry. The picture that emerges, therefore, is that we have grown up and developed in a world where our experiences of bodily displacements are very easily represented by a group of isometries or rigid motions of Euclidean space. We could have grown up and developed in a rather different world, however, in which case we would have arrived
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at what Poincare calls a group of non-Euclidean displacements.l 70 Yet, as we grew up and developed in our actual world, we naturally arrived at Euclidean geometry, and then, in accordance with Poincare's hierarchy, we fashioned systems of mechanics and experimental physics based on the presupposition of Euclidean space. In this sense, it is now completely impossible for us to determine experimentally that a geometry other than Euclidean is "true"-even though our decision to adopt Euclidean geometry in the first place was guided by a certain kind of experience (what Poincare calls "ancestral experience") which might, in fact, have been otherwise.l 71 Finally, and unlike Helmholtz, Poincare is also perfectly clear that the geometries of constant curvature admissible under the conditions of the Helmholtz-Lie theorem represent only a small fraction of the spaces or "manifolds" described by Riemann's general theory (which, in particular, also includes geometries of variable curvature): But this result [Lie's theorem] seems to be contradicted by Riemann, for this savant constructs an infinity of different geometries, and that to which his name is ordinarily given [i.e., elliptic geometry-MF] is only a particular case. Everything depends, he says, on how one defines the length of a curve. Now there are an infinity of ways of defining this length, and each of these may be the point of departure for a new geometry. This is perfectly exact, but the majority of these definitions are incompatible with the motion of an invariable figure, which one supposes to be possible in Lie's theorem. These geometries of Riemann, in many ways so interesting, could therefore never be anything but purely analytic and they would not lend themselves to demonstrations analogous to those of Euclid. (1902,63; 1913b, 63) In particular, Poincare has assumed from the beginning that space must satisfY what he calls the "law of homogeneity," according to which "a motion that is produced once can be repeated a second time, a third time, and so on, without its properties varying" (1902, 82-83; 1913b, 74-75); and it is for precisely this reason, therefore, that space necessarily has constant curvature.l 72 The limitation to spaces of constant curvature is thus a fundamental consequence of Poincare's group-theoretical analysis of what he takes to be the particular intuitive sensory experiences underlying our application of the mathematical science of geometry to the physical world. 173 The next lower science in Poincare's hierarchy, after geometry, is mechanics, which is discussed in the third part of Science and Hypothesis under the heading of "Force." Poincare makes it perfectly clear that he
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means classical or Newtonian mechanics; and this is by no means surprising, of course, for perhaps the greatest of Poincare's own contributions to mathematical physics was his ground-breaking analysis of the stability of the solar system in Newtonian celestial mechanics, based on his equally ground-breaking mathematical work on the qualitative theory of differential equations. Accordingly, the fundamental concepts and principles Poincare considers here are the Newtonian concepts of space, time, motion, force, and mass, governed by the three Newtonian Laws of Motion: inertia, F = rna, and the equality of action and reaction. Moreover, Poincare begins his discussion with the striking assertion (1902, 111; 1913b, 92): "There is no absolute space and we can conceive nothing but relative motions; nevertheless the mechanical facts are usually enunciated as if there were an absolute space to which one will refer them." I shall concentrate on this statement in what follows; for, on the one hand, it provides a striking illustration of the way in which mechanics, for Poincare, depends on the antecedently established science of geometry, and, on the other, it leads directly to the deep problems then afflicting the electrodynamics of moving bodies that were eventually solved, according to our current understanding, by Einstein's special theory of relativity. Poincare's assertion that "we can conceive nothing but relative motions" expresses what he calls "the law of relativity"-which is first introduced, in turn, in Chapter V, "Experience and Geometry," of the preceding second part: The laws of the phenomena which will happen [in a material system of bodies] will depend on the state of these bodies and their mutual distances; but, because of the relativity and passivity of space, they will not depend on the absolute position and orientation of this system. In other words, the state of the bodies and their mutual distances will depend only on the state of the same bodies and their mutual distances at the initial instant, but they will not depend at all on the absolute initial position of the system and its absolute initial orientation. This is what I shall call, for the sake of brevity, the law ofrelativi~. (1902,96; 1913b, 83) Moreover, "in order fully to satisfy the mind," Poincare continues, the phenomena in question should also be entirely independent of "the velocities of translation and rotation of the system, that is to say, the velocities with which its absolute position and orientation vary" (1902,98; 1913b, 85). Thus, because of "the relativity and passivity of space," the absolute position or orientation of a system of bodies in space can have no physical effect whatsoever, and neither can any change (velocity) of such absolute position or orientation.
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But what exactly does Poincare mean by "the relativity and passivity of space"? The preceding Chapter IV is entitled "Space and Geometry," and it is here, as we have seen, that Poincare presents his striking and original analysis of the application of mathematical geometry to physical space, based on group theory and the Helmholtz-Lie theorem. In particular, the centerpiece of this analysis, as we have also seen (compare note 172 above, together with the paragraph to which it is appended), is what Poincare calls the "law of homogeneity"-from which it necessarily follows that space has constant curvature so that, as Poincare puts it, "space is homogeneous and isotropic" (1902, 83; 1913b, 75). And it is this situation, I suggest, which Poincare then has in mind when he speaks of "the relativity and passivity of space" in the following Chapter V. Precisely because space is both homogeneous and isotropic, neither absolute position nor absolute orientation can have any physical effect: all points in space, and all directions through any point, are necessarily equivalent. But it then follows, similarly, that the velocity with which such a supposed absolute position or orientation changes can have no physical effect either, and, accordingly, the laws of the phenomena must be entirely independent of all such absolute velocities (of translation or rotation) as well. What Poincare calls "the law of relativity" is therefore a consequence of what he calls "the law of homogeneity. "174 Yet, as Poincare "unhappily" points out right after formulating his law of relativity, "the law thus enunciated is not in accord with experiments, at least as tl1ey are ordinarily interpreted" (1902,99; 1913b, 85). The problem is that velocities of absolute rotation do appear to have physical effects-Poincare mentions measurements of the oblateness of the earth and Foucault's well-known pendulum experiments-and from this fact, Poincare continues, "Newton inferred the existence of absolute space" (ibid.). Poincare concludes that he himself is "quite unable to adopt this point of view," and he "will explain why in the third part" (1902,99-100; 1913b, 85). When we now return to the third part of Science and Hypothesis, on the science of mechanics, we find that the subject matter of Chapter VII (fol e lowing the introductory Chapter VI on "Classical Mechanics") is "Relative Motion and Absolute Motion." Poincare begins by formulating "the principle of relative motion" (1902, 135; 1913b, 107): "The motion of any system whatsoever must obey the same laws, whether it be referred to fixed axes, or to movable axes transported by a rectilinear and uniform motion. This is the principle of relative motion, which imposes itself upon us for two reasons: first, the most common experience confirms it, and second, the contrary hypothesis is singularly repugnant to the mind." This, of course, is the principle of what we now call Galilean relativity, originally formulated by Newton as Corollary V to the Laws of Motion, and playing
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a central role, as we have seen, in the recent literature on inertial frames of reference (con1pare note 158 above, together with the paragraph to which it is appended). However, as Poincare is well aware, such Galilean relativity holds only for (uniform) rectilinear n10tions and does not extend, therefore, to the case of (uniform) rotational motion that Poincare also wishes to subsume under his "law of relativity."175 Nevertheless, Poincare says, "it seen1S that [the principle of relative motion] ought to impose itself upon us with the same force, if the motion is varied, or at least if it reduces to a uniform rotation" (1902, 136-37; 1913b, 108). Our a priori commitment to the law of relativity, derived fron1 the homogeneity and isotropy of space, stands in prima facie contradiction with the well-known experimental limitations of the principle of relative motion. Poincare therefore devotes considerable attention to the case of uniform rotation in his following discussion. He adn1its that the full law of relativity does not appear to accord with experiments-"at least as they are ordinarily interpreted"-and, moreover, that the simplest description of such an apparently rotating system (e.g., the earth) will attribute to the system a true absolute rotation. Nevertheless, Poincare continues, the simplest description is not the only possible description, and he then explores other (and admittedly more complicated) ways in which what we ordinarily talce to be a uniformly rotating system can be redescribed. Indeed, we now know that it is straightforward to reformulate virtually any laws of mechanics-including Newton's-in a generally covariant way, so that, in particular, the laws in question will then hold equally in all possible coordinate systems, including the coordinate axes defining a uniformly rotating system. 176 Of course the laws will then be somewhat more complicatedthey will include additional terms, which, in uniformly rotating systems, represent centrifugal and Coriolis accelerations.!77 This, however, is precisely Poincare's point: it is not an issue of "truth," but only one of simplicity and convenience, whether or not we say that the system is really rotating. 178 At the lowest level of Poincare hierarchy, finally, "we arrive at the physical sciences properly so-called[; h Jere the scene changes; we encounter another kind of hypotheses [other than the conventional-MF] and we perceive all of their fecundity" (1902,5-6; 1913b, 29).!79 Moreover, the main exan1ples of these "physical sciences properly so-called" Poincare here chooses to investigate come from the development of optics, electricity, and electrodynamics-culminating in Chapter XII, "Optics and Electricity," and Chapter XIII, "Electrodynamics," of the fourth and final part of Science and Hypothesis under the heading of "Nature." And this is by no means surprising, once again, for Poincare's own most important contributions to physical science (as opposed to pure mathematics and mathematical astronomy) concern precisely the sciences of optics, electric-
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ity and magnetism, and electrodynamics-culminating in his very significant contributions to the electrodynamics of moving bodies. The main problem then confronting this latter science, of course, was the continuing failure of experimental attempts to determine the absolute velocity of the earth with respect to the electromagnetic aether by electromagnetic or optical means. James Clerk Maxwell's theory of the electromagnetic field had achieved the brilliant success of showing that electromagnetic waves propagate through this aether with a definite constant velocity c (precisely the well-known velocity of light) and had thereby shown that light, in particular, could be an electromagnetic wave-thus fully unifying the two previously independent sciences of optics and electricity. In Maxwell's theory, unlike in classical mechanics, there is therefore a privileged absolute velocity after all, and the classical principle of Galilean relativity no longer holds. Hence, it should now be possible in principle to determine the absolute velocity of the earth through the aether, yet none of the experiments proposed for this purpose, including the famous Michelson-Morley experiment of 1887, were in fact able to do this-despite their continually increasing experimental accuracy. The great Dutch physicist Hendrik Antoon Lorentz, who had completed Maxwell's theory with his theory of the electron, strenuously attacked this problem in the years 1886-1904. It was well known that one could explain why such experiments were unsuccessful to first order in v/ c (where v is the assumed absolute velocity of the earth) using Augustin Fresnel's theory of the (partial) dragging of the medium through which a body moves (here the electromagnetic aether dragged by the moving earth); and Lorentz began by showing that one could derive Fresnel's results from first principles (in the context of a stationary aether), using Lorentz's own theory of the electron and refractive dispersion. Lorentz initially thought that second order experiments (capable of determining terms of the order of v 2 / c2 ) would still be successful, but this hope was dashed by the Michelson-Morley experiment (which, of course, was precisely of second order). It was at this point that Lorentz appealed to the length-contraction hypothesis (beginning in 1892), and here, too, he hoped that further, higher-order experiments would be successful. Finally, in 1904, he erected a theory in which no experiments of any order would succeed, and he showed that the further hypothesis oflocal time (what we now call time-dilation) accomplishes this-and results, accordingly, in what we now call the Lorentz transformations relating different inertial frames of reference moving relative to one another with any constant velocity v. ISO Poincare's own most important contributions to this problem occurred after the 1902 publication of Science and Hypothesis. In a letter to Lorentz
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written between 1904 and 1905, for exan1ple, Poincare was the first to demonstrate that the Lorentz transformations form a group; and, in his important papers "On the Dynamics of the Electron" of 1905-6, he demonstrated the covariance of the Maxwell-Lorentz equations under the Lorentz transformations and solved central current problems (involving the notions of transverse and longitudinal mass) then afflicting the theory of the electron-which, in accordance with the contraction hypothesis, should now be taken to be deformable. 181 And it is on the basis of precisely these contributions that Poincare still has a serious claim to be considered one of the co-discoverers of special relativity.182 Already in Science and Hypothesis, however, Poincare, as we have seen, had formulated his "law of relativity," and, accordingly, he had already expressed considerable dissatisfaction with Lorentz's ongoing piece-meal attempts to explain the negative results of the aether experiments: I must explain, in fact, why I do not believe, despite Lorentz, that more precise observations can ever make evident anything else but the relative displacen1ents of material bodies [as opposed to displacements relative to the aether-MF]. Experiments have been made which should have disclosed terms of the first order; the results have been negative; could this be by chance? No one has accepted that; a general explanation has been sought, and Lorentz has found it; he has shown that the terms of the first order must destroy one another, but it is not the same for those of the second. Then more precise experiments were made; they were also negative; this, too, could not be the effect of chance; an explanation was needed; it was found, they are always found; hypotheses are never lacking. But that is not enough; who does not feel that this is again to leave too great a role to chance? Would it not also be chance, that singular coincidence, which brought it about that a certain circumstance should come just in time to destroy the terms of the first order, and that another circun1stance, entirely different, but just as opportune, should take it upon itself to destroy those of the second order? No, it is necessary to find the same explanation for the one as for the other, and then everything leads us to expect that this explanation will be equally valid for the terms of higher order, and that the mutual destruction of these tern1S will be rigorous and absolute. (1902, 201-202; 1913b, 147-48) Poincare is here demanding that electrodynamics should rigorously satisfy what Einstein shall soon call the "principle of relativity," and there is a strong case, as we shall see, that Einstein was in fact substantially influenced in this respect by his reading of Science and Hypothesis.
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Einstein Einstein begins his 1905 paper "On the Electrodynamics of Moving Bodies" with the famous example of the relatively moving magnet and conductor, where "[t]he observable phenomenon here [i.e., the current tlowing through the conductor-MF] depends only on the relative motion of the conductor and the magnet, whereas, according to the customary view [i.e., MaAwell- Lorentz electrodynamics-MF], the two cases, in which either the one or the other of these bodies is in motion, are strictly to be separated from one another" (1905, 891; 1923, 37). Einstein then inu'oduces his "principle of relativity" in the immediately following second paragraph: Examples of this sort, together with the unsuccessful attempts to discover any motion of the earth relative to the "light medium," suggest that the phenomena of electrodynamics as well as mechanics possess no properties corresponding to the idea of absolute rest. They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics are valid. We will elevate [erheben] this conjecture (whose content will be called the "principle of relativity" in what follows) to the status of a postulate [Voraussetzung], and also introduce another postulate, which is only apparently irreconcilable with it, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body. These two postulates suffice for attaining a simple and consistent theory of the electrodynamics of moving bodies based on Maxwell's theory for stationary bodies. (1905, 891-92; 1923, 37-38) Note that here, as Einstein explains in a footnote added in 1913,183 he did not yet have knowledge of Lorentz's 1904 paper-nor, it would appear, of either the Michelson-Morley experiment or Lorentz's consequent appeal to the contraction hypothesis beginning in 1892-for Einstein only mentions experimental results up to first order in vic. He is already willing to "conjecture," however, that the same results will be found for all orders, and, as he says, he is willing to "elevate" this conjecture to the status of a fundamental principle or postulate. Finally, he now intends to develop the entire electrodynamics of moving bodies on the basis of two fundamental postulates alone: the principle of relativity and the light postulate. We know that Einstein intensively read and discussed Science and Hypothesis in the years 1902--4, as part of the curriculum of his "Olympia Academy" in Bern. 184 It appears very likely, therefore, that Einstein had
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paid particular attention to Poincare's criticism of Lorentz quoted immediately above, for Einstein, like Poincare, is demanding that we here follow a rigorous extrapolation of the known experimental results, according to which the "principle of relativity" is assumed to be valid for all order of v/ c at once. 18S Moreover, a very general methodological moral of Science and Hypothesis, stated in Poincare's "General Conclusions" to the third part, is that the fundamental principles of mechanics are "conventions or definitions in disguise" which are "drawn from experimental laws; these laws, so to speak, have been elevated [erWees] into principles to which our mind attributes an absolute value" (1902, 165; 1913b, 125). It appears from Einstein's language, then, that he is here following Poincare's methodology quite precisely.186 In particular, the experimental law in question comprises the contemporary results on the failure to determine the motion of the earth with respect to the aether. These, like all such experimental results, can only be valid up to a certain order of approXimation; but, by assuming or presupposing that they are strictly and universally valid, we can thereby "elevate" them to a wholly different status. In this case, the presupposed principle of relativity, together with the light postulate, then issues in what Einstein calls a new "Definition of Simultaneity" in the immediately following §1which arises, as Einstein himself emphasizes, in so far as we "stipulate [festsetzen] by definition" that the "time" light takes in traveling from A to B is the same as the "time" it takes in traveling back from B to A (1905,894; 1923,40). A more perfect implementation of Poincare's general methodology, in the context of Poincare's own conception of the role, in electrodynamics, ofwhat he, too, is now calling the principle of relativity (see note 185 above), could hardly be imagined.l 87 Nevertheless, Einstein proceeds to implement this methodology in a way that Poincare himself was never in fact able to accept. For Einstein uses his new "definition of simultaneity" precisely to show how we can now develop "a simple and consistent theory of the electrodynamics of moving bodies" without appealing to any special hypotheses of the kind currently being investigated by Lorentz. Einstein first shows, as a consequence of his definition, that the relation of simultaneity varies in what we now call different inertial frames of reference (since the constant velocity c, by the principle of relativity, is itself invariant in different inertial frames), and we can then derive, as a direct consequence of this, "the relativity of lengths and times" (§2): from what we now call the relativity of simultaneity it follows that the length of a rigid rod varies in different inertial frames and so does the temporal duration measured by an ideal clock.l 88 Length-contraction and time-dilation are thus direct, purely kinematical consequences of Einstein's new definition of simultaneity, and no special hypotheses involving the assumed electronic con-
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stitution of matter (including the hypothesis of the deformable electron) are needed at all. In sharp contrast to Einstein, however, Poincare's own implementation of the principle of relativity in electrodynamics remained squarely within the Lorentzian tradition, and, accordingly, it still essentially involved all the special Lorentzian mechanisms (local time, the contraction hypothesisincluding the deformable electron, and so on) for explaining why specifically electrodynamic phenomena do not reveal the true absolute velocity of the earth. Thus Poincare did not see, in particular, that adopting a new definition of simultaneity could already implement this principle all by itself, with no appeal at all to additional special mechanisms. 189 Indeed, as late as 1912 (the year of his death), in a lecture delivered at the University of London on "Space and Time," Poincare continued to reject the "entirely new conception of space and time" (including the relativity of simultaneity) which "certain physicists" have suggested as a consequence of the principle of relativity. To be sure, these (unnamed) physicists have a right to propose a "new convention" if they wish. Nevertheless, "those who are not of this opinion can legitimately preserve the ancient one in order not to disturb their old habits[; I] believe, between ourselves, that this is what they shall continue to do for a long time" (1913a, 54; 1963, 24). In this sense, there can be no question of Poincare being counted among the discoverers of special relativity, as we now understand it (compare note 182 above), and the question rather becomes one of why he failed. In particular, since, as we have seen, Einstein implemented Poincare's general scientific methodology so perfectly, why was Poincare himself never able to appreciate Einstein's achievement? The answer, I suggest, is that Poincare, from the beginning, understood what he came to call the principle of relativity in a way that was completely different from-and, indeed, entirely incompatible withEinstein's understanding. This principle, for Poincare, was never a mere empirical law, even one which, in accordance with his general methodology, could later be "elevated" to the status of a convention or definition in disguise. Rather, the idea of the relativity of motion is equally based, for Poincare, on an a priori "law of relativity" which itself rests, as we have seen, on the "law of homogeneity" expressing the "relativity and passivity of space": the "law of relativity," in this sense, is not so much a physical as a purely geometrical principle (see note 174 above, together with the paragraph to which it is appended). In particular, the law of relativity rests squarely on the homogeneity and isotropy of space, which, for Poincare, is a necessary presupposition for our applying geometry to physical space in the first place; and, as we have seen, such a geometry must be already in place before any properly physical or empirical theory can even be formulated (compare note 168 above, together with the paragraph to which it is
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appended). It is for precisely this reason that Poincare is not satisfied with the customarily limited scope of the "principle of relative motion" (the classical Galilean principle of relativity), and he argues that this principle, too, should instead be extended so as to embrace (at least) the case of uniform rotation (see the paragraph to which note 178 above is appended, together with the two preceding paragraphs).l90 Poincare's understanding of the principle of relativity is therefore inextricably connected with his hierarchical conception of how the mathematical and empirical sciences are related. In particular, the "physical sciences properly so-called" presuppose that all higher levels of the hierarchy are already in place, and the most important such (empirical) science, for Poincare, is electrodynamics (see the paragraph to which note 179 above is appended). This science, following Lorentz, treats a particular domain of physical entities (electrons) governed by particular laws of physical interaction (the Maxwell-Lorentz equations describing interelectronic interactions via electromagnetic forces), and, accordingly, it presupposes that a theory of mechanics is already in place-which, for Poincare, is of course Newtonian (see the paragraph following the one to which note 173 above is appended). Similarly, Newtonian mechanics, in turn, rests on an already chosen spatial geometry-which, for Poincare, is of course Euclidean (compare the paragraph to which note 167 above is appended, together with the paragraph to which note 171 is appended). What Poincare comes to call the principle of relativity now gives us powerful reasons-both a priori and empirical (see again note 190 above)-to hold that Maxwell-Lorentz electrodynamics fails to specifY an empirically determinable state of absolute rest, just as the laws of (Newtonian) mechanics similarly fail to do so. The only remaining question concerns precisely how we are to implement the principle in this particular case. Its implementation at the level of geometry is already guaranteed by the homogeneity and isotropy of space; and its implementation at the level of mechanics, similarly, is already guaranteed by the classical Galilean principle of relativity (which Poincare calls the "principle of relative motion"). For Poincare, therefore, there must then be special empirical facts about the basic entities (electrons) and laws of interaction (electromagnetic forces) of Maxwell-Lorentz electrodynamics which implement the principle of relativity at precisely this lowest level: these include the well-known contraction hypothesis (including the deformable electron), and they are jointly expressed mathematically, in the most abstract and general terms, by the covariance of the Maxwell-Lorentz equations under the Lorentz group.191 For Einstein, by contrast, what he calls the principle of relativity is simply a rigorous extrapolation of the then current experimental results on
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determining the absolute velocity of the earth by electromagnetic means. Although, as we have seen, he here follows Poincare in "elevating" such an experimental law to the status of a universal principle or postulate, he definitely does not follow Poincare in attributing any kind of a priori status to this law derived from the relativity and passivity of space. Indeed, for Einstein, there are no a priori (or quasi a priori) considerations already in play here-neither a group-theoretical analysis of space and geometry nor an independent relativity principle for Newtonian mechanics. There is only the experimentally supported "conjecture" that the laws of electrodynamics are rigorously valid in all inertial reference frames-the same inertial reference frames in which the laws of mechanics are presumed to be valid l92 -together with the resulting conceptual problem of showing how to combine this conjecture consistently with Einstein's second empirically motivated presupposition or postulate, that light has a definite constant velocity c independent of the velocity of the source. As we have seen, Einstein implements these two postulates, and resolves the conceptual problem, by articulating a new kinematics of space and time featuring a new "definition of simultaneity" between spatially separated events: we thereby obtain a purely kinematical interpretation of length-contraction and time-dilation (the Lorentz contraction and local time), culminating in a purely kinematical derivation of the Lorentz transformations (see the paragraph to which note 188 above is appended). And it is only after this first "Kinematical Part" of his paper is concluded that Einstein then proceeds to a second "Electrodynamical Part"-where, among other things, he derives the covariance of the Maxwell-Lorentz equations under the Lorentz group. The central point, therefore, is that Einstein, unlike Poincare, does not implement the principle of relativity at the more specific and empiricallevel of electrodynamics. On the contrary, he implements it directly at the level of kinematics (which, for Poincare, would be a part of what he calls mechanics), and Einstein then shows that this new, nonNewtonian kinematics already suffices to solve the current problems of electrodynamics independently of all more specific hypotheses (the Lorentz contraction, and so on) formulated within Lorentz's theory of the electron. Einstein's implementation of the principle of relativity is therefore entirely independent of the theory of the electron, just as it is also entirely independent, as we have already suggested, from Poincare's a priori foundation for the principle of relativity in "the relativity and passivity of space." In this way, Einstein totally ignores the hierarchical conception of the mathematical and physical sciences Poincare had carefully constructed-a conception which, at its very center, rests on Poincare's deep group-theoretical analysis of space and geometry. And it is for this reason, above all, that Poincare himself could never accept Einstein's
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implen1entation of the principle of relativity, despite the fact, as I have argued, that Einstein, in turn, was very definitely inspired by Poincare (see note 186 above, together with the paragraph to which it is appended and the following paragraph).193 Of course, we now follow Einstein in interpreting relativity principlesboth the special principle of relativity and the classical Galilean principleas characterizing the underlying kinematical structure of space and time; and, in particular, we also follow Hermann Minkowski (and later Hermann Weyl and Elie Cartan) in interpreting such principles, from a n1athematical point of view, as expressing the geometrical invariants of certain fourdimensional affine space-times.l 94 From this point of view, therefore, Poincare's fundamental error was to focus primarily on the three-dimensional geon1etry of physical space and, accordingly, to miss entirely the overarching importance of four-dimensional space-time geometry for the relativity of motion. 195 Yet we still may wonder, nonetheless, what the real physical significance of this four-din1ensional perspective is supposed to be, beyond its purely mathematical formal elegance. Indeed, since, as we pointed out above (note 191), Poincare's formulation of the electrodynamics of moving bodies is both empirically and mathematically equivalent to Einstein's, we need to ask what real difference it makes if we allow Poincare his hierarchical conception of the sciences-according to which, in particular, the three-dimensional geometry of space retains a privileged role-and refuse to follow either Einstein or Minkowski in their radical revision of the classical conception of space and time .196 This question is made all the more pressing, moreover, when we note that Einstein himself at first dismissed Minkowski's reformulation of special relativity in terms of four-dimensional space-time geometry as a mere mathematical trick. 197 I suggest that if all that were at issue were the electrodynamics of moving bodies, there would really be nothing to choose, from a physical point of view, between Einstein's (or Minkowski's) formulation and Poincare'sPoincare, in that case, could perfectly well preserve his hierarchy of the science if he liked. But electrodynamics was not, of course, the only branch of what Poincare called experimental physics that was centrally salient at the time: there was also the Newtonian theory of universal gravitation, to which Poincare, as I have said, was himselfn1alcing ground-breaking contributions in his celestial mechanics of the solar system. The question naturally arises, therefore, of how our (Newtonian) theory of gravitational force and the Maxwell-Lorentz theory of electromagnetic force are related to one another. And this problem is especially pressing, in particular, because the former force (according to Newton's theory) propagates instantaneously, with an infinitely fast velocity, whereas the latter propagates with finite speed, at exactly the velocity of light. Indeed, it is precisely because gravitational force, in Newton's theory, propagates instantaneously that the notion
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of absolute simultaneity has genuine physical meaning in a Newtonian context, at least in principle. However, if the notion of absolute simultaneity in fact has genuine physical meaning, then, in the context of Maxwell-Lorentz electrodynamics, a privileged frame of reference at rest in absolute space (i.e., the aether) should also have physical meaning as well, again, at least in principle.l 98 But this is not only incompatible with all the experimental evidence so far, it is also entirely at odds with any plausible version of the principle of relativity-including, of course, Poincare's. Now, as a matter of fact, a number of scientists at the time were quite interested in the question of how Newtonian gravitation was to be related to the new Lorentzian electrodynamics. Most of this interest was due to the circumstance that inertial mass was now known to depend on velocity,199 leading to the question whether the same is true of gravitational (attractive) mass-if so, one might then contemplate an electromagnetic explanation of Newtonian gravitation via the theory of the electron. Indeed, Poincare himself discusses this idea in Chapter III of Book III of Science and Method, published in 1908: he notes, in particular, that, if gravitation is thus explained within Lorentz's theory, the principle of relativity will not be violated, gravitation will not propagate instantaneously but rather with precisely the velocity of light, and, finally, that this would even account for some (but by no means all) of the well-known anomaly in the perihelion of Mercury arising on Newton's theory. But none of these attempts to construct an electrodynamic explanation of gravity were actually successful. By contrast, Einstein devoted himself to the problem of the relationship between gravitation and the new electrodynamics (now understood, of course, in his own preferred way) beginning in 1907, and he finally succeeded in solving this problem in his general theory of relativity, appearing in 1915-16-where, among other things, he unexpectedly succeeded, as well, in accounting for the entire anomaly in the perihelion of Mercury. Einstein's starting point, too, was the relationship between gravitational and inertial mass, but (unlike Poincare and others) he had no interest in explaining gravitation via the theory of the electron. Einstein's crucial physical insight, rather, was that, since gravitational and inertial mass were known to be rigorously equal, one could conjecture that gravitation and inertia are the very same physical phenomenon. It would follow, therefore, that gravitational force is actually of the same kind as the so-called inertial forces (such as centrifugal and Coriolis forces) arising in non-inertial frames of reference, in so far as the accelerations produced by both types of forces are entirely independent of the bodies so affected. 200 Einstein called this new conjecture the principle of equivalence; and it is then fully implemented in his finished general theory by viewing the action of gravitation as a geometrical perturbation in the four-dimensional affine structure of
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Minkowski space-tilne. More specifically, instead of a flat four-dimensional affine space of zero curvature, where the four-dimensional straight lines represent the trajectories of particles not affected by any forces in accordance with the law of inertia, we now have a variably curved fourdimensional affine structure, where the curvature depends on the distribution of mass (and energy) in accordance with Einstein's field equations, and the four-dimensional affine geodesics represent the trajectories of so-called freely falling particles affected by no other forces than gravitation. For Einstein, therefore, gravitation is thus finally united with the new relativistic electrodynamics by being explicitly incorporated into the background kinematics-i.e., geometry-of the four-dimensional space-time within which this electrodynamics, following Minkowski, is now formulated. So it is precisely here, in particular, that Minkowski's geometrical formulation of special relativity acquires genuine physical cash-value after all. Indeed, Einstein himself began using Minkowski's mathematical methods in 1912 (when the basic ideas of the new theory of gravitation had first gelled), and, in 1916, he explicitly acknowledged his debt to Minkowski in facilitating the transition from the special to the general theory of relativity.201 Moreover, it is at precisely this point, as well, that Poincare's hierarchical conception of the sciences, along with his deep group-theoretical analysis of space and geometry, becomes no longer tenable at all: since we are now faced with spaces of variable rather than constant curvaturewhere, in addition, the geometry of space is both affected by and affects the behavior of matter-neither space nor space-tin1e can, in general, satisfy Poincare's law of homogeneity (see again the paragraph to which note 172 above is appended). What Poincare called "the relativity and passivity of space" is now definitely a thing of the past, together with Poincare's own philosophical and mathen1atical motivations for what he called the principle of relativity. It is no wonder, then, that Einstein takes pains to develop an epistemology of geometry explicitly opposed to Poincare's in his most developed account of the specifically philosophical background to the new general theory. This, of course, is the celebrated paper "Geometry and Experience" (1921), which then exerted an enormous influence on the philosophy of geometry throughout the twentieth century.202 On the one hand, Einstein defends a conception of pure or mathematical geometry associated with the modern (Hilbertian) axiomatic method, according to which such geometry is purely formal or "contentless," entirely independent of both intuition and experience: Geometry treats of objects that are designated with the words line, point, etc. No kind of acquaintance or intuition of thes~~~j~~~s_i~J~r~~
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supposed, but only the validity of those axioms which are likewise to be conceived as purely formal, i.e., as separated from every content of intuition and experience .... These axioms are free creations of the human spirit. All other geometrical propositions are logical consequences of the (only nominalistically conceived) axioms. The axioms first define the objects of which geometry treats. Schlick therefore designated the axioms very appropriately as "implicit definitions" in his book on theory of knowledge. The conception represented by modern axiomatics purifies mathematics from all elements not belonging to it, and thus removes the mystical obscurity that previously clung to the foundations of mathematics. But such a purified presentation makes it also evident that mathematics as such may assert nothing about either objects of intuitive representation or objects of reality. In axiomatic geometry we understand by "point", "line", etc. only contentless conceptual schemata. What gives them content does not belong to mathematics.
(1921,405; 1923,30-31) In here defending the modern conception of mathematical geometry as a purely formal, "uninterpreted" axiomatic system, Einstein is thus breaking decisively with the earlier Kleinian tradition in the foundations of geometry within which, as we have seen, Poincare himselfworked. 203 On the other hand, however, Einstein also defends a conception of applied or physical geometry as entirely based on the observed behavior of "practically rigid bodies" and their "situational possibilities [LagerungsmiJ"glichkeiten]." It is in precisely this way, he argues, that geometry becomes a properly empirical science, and, in a striking passage, Einstein says that he had to embrace this conception-and, accordingly, to reject Poincare's opposing "conventionalist" viewpoint-in order to establish the general theory of relativity: I attach particular importance to this conception of geometry, because without it I would have found it impossible to establish the theory of relativity. Without it, namely, the following consideration would have been impossible. In a reference system that is rotating relative to an inertial system the situational laws [Lagerung{qesetze] of rigid bodies, due to the Lorentz contraction, do not correspond to the rules of Euclidean geometry. Therefore, the admission of non-inertial systems as equally justified must lead to the abandonment of Euclidean geometry. The decisive step in the transition to generally covariant equations would certainly not have taken place, if the above interpretation had not been taken as basis. If one rejects the relation between the bodies of axiomatic Euclidean geometry and the practically rigid bodies of real-
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ity, then one easily arrives at the following conception, which the perceptive and deep thinker H. Poincare has, in particular, embraced. Euclidean geometry is picked out from all other thinkable axiomatic geometries by simplicity. And since axiomatic geometry alone contains no assertions about experienceable reality, but only axiomatic geometry in connection with propositions of physics, then it may be possible and rational-no matter how reality may be constituted-to hold fast to Euclidean geometry. For one will prefer to opt for an alteration of the laws of physics rather than alter the laws of geometry in case there are contradictions between theory and experience. If one rejects the relation between the practically rigid body and geometry, one will in fact not easily free oneself from the convention according to which Euclidean geometry is to be held fast as the simplest. (1921, 6-7; 1923, 33-34) Einstein immediately goes on to admit that, "sub specie aeterni," Poincare is correct in thinking that there are no genuinely rigid bodies in nature and, more generally, that the behavior of physically (and only approximately) rigid bodies depends on physical and empirical factors (temperature, the action of various physical forces, and so on) going well beyond geometry proper (see again the paragraph to which note 168 above is appended). Nevertheless, Einstein continues, although "the rigid body and clock do not play the role of irreducible elements in the conceptual framework of physics," "it is my conviction that these concepts must still be called upon as independent elements in the present stage of theoretical physics" (1921, 8; 1923, 35-36). As recent scholarship by John Norton (1985) and John Stachel (1980) has made clear, Einstein is here alluding to a line of thought he had developed in the years 1907-12 in connection with his new principle of equivalence. As Norton has shown, in particular, Einstein did not initially understand this principle as it was later implen1ented in his finished theory: in terms of a perturbation of the flat affine structure of Minkowski spacetime, where freely falling bodies acted on by no forces but gravitation follow geodesics (straightest possible lines) in the variably curved version of four-dimensional Minkowski geometry that results (compare the paragraph to which note 200 above is appended). Indeed, Einstein still understood relativity theory in essentially three-dimensional terms, and, as we have said, he did not accept Minkowski's four-dimensional forn1ulation until 1912 (see note 197 above, together with the paragraph to which it is appended). What Einstein did in the years 1907-12, accordingly, was to use the conjecture that gravitation and inertia are the very san1e physical phenomena to develop relativistic models of the gravitational field by investigating non-inertial frames of reference (accelerating and rotating frames) within the context of special relativity.204 The four-dimensional
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tormulation of general relativity that became standard after its first publication in 1915-16 is therefore a product of Einstein's investigations ofthe principle of equivalence in the years 1907-12, not something that was there all along (compare note 201 above, together with the paragraph to which it is appended). Indeed, Stachel (1980) has shown that Einstein first made the decisive breakthrough to a truly geometrical understanding of the gravitational field in 1912-by means of precisely the example of the uniformly rotating frame of reference to which he alludes in our passage from "Geometry and Experience." In particular, if we consider such a rotating system in the context of special relativity, we find that the Lorentz contraction differentially affects measuring rods laid off along concentric circles around the origin of the plane of rotation (due to the variation in tangential linear velocity at different distances along a radius), whereas no Lorentz contraction is experienced by rods laid off along a radius. Therefore, the threedimensional spatial geometry in such a system will be found to be non-Euclidean (the ratio of the circumference to the diameter ofconcentric circles around the origin in the plane of rotation will differ from 1t and depend on the circular radius).205 It was at precisely this point, as Stachel shows, that Einstein first realized that gravitation could be represented by a non-Euclidean geometry, and, since an analogous effect holds for the temporal coordinate of the non-inertial frame in question (by time-dilation), he quickly realized that anon-Euclidean generalization of the flat (pseudo-Euclidean) metric of Minkowski space-time was exactly what he needed. 206 The idea that the action of gravitation could be represented by a variably curved four-dimensional geometry-by the (metric) affine connection in a perturbation of Minkowski space-time-was finally in place. Why did Einstein, as he says, need explicitly to reject Poincare's philosophy of geometry in order to carry through this (ultimately successful) line of thought? The problem Einstein raises is not the problem we raised above in connection with the finished theory: that general relativity is incompatible with Poincare's hierarchical conception of the sciences and group-theoretical analysis of space and geometry (compare again the paragraph to which note 201 above is appended). Indeed, Einstein entirely ignores these central features of Poincare's philosophy of geometry, and he describes it, instead, as resting on a much simpler argument based on the underdetermination of (pure) geometry by experience independently of the laws of physics. 207 What Einstein explicitly rejects, in particular, is the claim that geometry can therefore not be based on the concept of the "practically rigid body"-since even the existence of approximately rigid bodies, according to Poincare, depends on a variety of physical factors. Einstein's argument, in the passage from "Geometry and Experience" we are considering, is that, although Poincare is of course correct-"sub specie
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aeterni"-that the behavior of "practically rigid bodies" depends on a variety of (largely not yet fully comprehended) physical factors, we must still provisionally take such bodies as primitive ("as independent elements in the present stage of theoretical physics") in order properly to appreciate the radically new conception of the relation between geometry and experience ,vrought by the general theory of relativity. What Einstein is doing here is best understood, I believe, as delicately situating himselfwithin the earlier debate between Poincare and Helmholtz on the foundations of (three -dimensional) spatial geon1etry. In creating the special theory, as we have seen, Ejnstein was very substantially influenced by Poincare's general "conventionalist" methodology, in so far as he "elevated" the principle of relativity, together with the light principle, into a radically new "definition of simultaneity" based on a free "stipulation" (see note 186 above, together with the paragraph to which it is appended, the preceding paragraph, and the following paragraph).208 But now, in creating the general theory, Einstein rather takes the side of Helmholtz, against Poincare, on the direct geo1l1etrical significance of "practically rigid bodies" (see again the paragraph to which note 168 is appended, together with the two preceding paragraphs).209 In particular, Einstein now interprets the Lorentz contraction arising in the case of the uniformly rotating system as having direct geometrical significance independently of any issues about the micro-structure of the rigid rods in question and the physical forces acting upon it, just as he had earlier, in his original interpretation of the Lorentz contraction in special relativity, given it direct kinematical significance independently of all questions about the actual physical constitution of the bodies in question arising within the theory of the electron. Just as Einstein, in contrast with Poincare, had originally implemented the Lorentz contraction at the comparatively high level of kinematics (what Poincare would call mechanics) rather than the lowest level of the "physical sciences properly socalled" (see again note 193 above, together with the paragraph to which it is appended), he now, in explicit opposition to Poincare, implements it at the even higher level of geometry.210 In precisely this way, geometry itself becomes a branch of what Poincare would call "experimental physics," and Poincare's carefully constructed hierarchy of the sciences now definitely becomes, as I have said, a thing of the past. 211
Carnap and logical empiricism The scientific philosophy of the logical en1piricists was very substantially influenced, as I have already suggested, by Einstein's creation of both the special and the general theories of relativity. Indeed, the earliest works of those logical empiricists who were n10st concerned with modern mathe-
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maties and mathematical physics were almost entirely devoted to a searching philosophical engagement with Einstein's new theories: Moritz Schlick's "The Philosophical Significance of the Principle of Relativity" (1915) and Space and Time in Contemporary Physics (1917); Hans Reichenbach's The Theory of Relativity and A Priori Knowledge (1920), Axiomatization of the Theory ofRelativity (1924), and Philosophy of SpaceTime-Theory (1928); Rudolf Carnap's Der Raum: Ein Beitrag zur Wissmschaftslehre (1922). Einstein himself followed these philosophical developments with great interest, and his interactions with Schlick-soon to be the founder and guiding spirit of the Vienna Circle-were especially extensive and significant. Thus, as we have already seen, Einstein prominently refers to Schlick's General Theory of Knowledge (1918) in "Geometry and Experience" (compare note 203 above, together with the paragraph to which it is appended), and we know that Einstein studied both this work and Space and Time in Contemporary Physics rather carefully-in particular, Einstein read, and commented upon, drafts of the latter at several stages of its composition.21 2 This work, in fact, became perhaps the most important explanation of and apology for Einstein's general theory of relativity addressed to a general educated audience in the years 1917-22 (during which it quickly went through four editions), and it was largely on this basis that Schlick was called in 1922 to the Chair for the Philosophy of the Inductive Sciences previously occupied by Mach and Ludwig Boltzmann at the University ofVienna. 213 A philosophical engagement with Einstein's new physical theories (and with Einstein himself) was thus a critical part of the process that gave birth to the Vienna Circle in the first place. Moreover, since it was well known that Einstein himself had attributed a leading role in the genesis of his theories to Mach's criticism of Newton's views on space, time, and motion in The Science ofMechanics (compare the paragraph to which note 156 above is appended),214 it is by no means surprising that the Vienna Circle, too, acknowledged a substantial debt to Mach, and even took on the name of the "Verein Ernst Mach in the late 1920s (compare the paragraph to which note 150 above is appended). Yet the logical empiricists did not find in Mach philosophical tools for comprehending the essential use of modern abstract mathematics-group theory, non-Euclidean geometry, the theory of manifolds-in Einstein's new theory. Their problem, in a nutshell, was how to concede the evident strengths of Kant's theory of pure intuition and the synthetic a priori in the context of Newtonian physics, while also recognizing that Einstein's theories of relativity had decisively undermined Kant's particular commitment to Newtonian mechanics and Euclidean geometry. But they did not respond to this dilemma by adopting the more traditional form of empiricism associated with the classical British empiricists, John Stuart Mill, or Mach himself-which, they (rightly) thought,
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could never do justice to modern mathematics and mathematical physics. On the contrary, they continued to insist on the genuinely a priori character of pure mathematics-as a "free creation of the human mind"-while also clearly recognizing that this same pure mathematics has an ineliminable and necessary role in the formulation and empirical confirmation of modern mathematical physics. Schlick, in his early writings, appeals to both Hilbert's axiomatic method and Poincare's "conventionalism." From Hilbert he derives the central distinction between pure geometry as an uninterpreted formal system and applied geometry as an empirical theory of physical space. This distinction already suffices to explain how pure geometry (and pure mathematics more generally) can be genuinely a priori, while also explaining, at the same time, why I
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lation of Frege and Russell through his later involvement with Kurt G6del and Alfred Tarski in the late 1920s and early 1930s. 217 Yet Carnap, unlike other logicians of this period, was primarily concerned with the application oflogic and mathematics to empirical (and primarily physical) knowledge, and he remained concerned, in particular, with the earlier problems associated with Hilbert, Poincare, and Einstein that had been first treated, within the logical empiricist tradition, by Schlick, Reichenbach, and himself. For Carnap, however, unlike the others, his work in mathematical logic gave him an evolving and eventually coherent Qverarching framework within which to treat these problems, and the result was a radically new conception of the role and character of scientific philosophy in relation to both pure mathematics and (mathematical) empirical science. Carnap's first publication, Der Raum (1922), was based on his 1921 doctoral dissertation at Jena, written under the neo- Kantian philosopher Bruno Bauch,218 and here, characteristically, Carnap attempts to integrate and synthesize ideas from a number of different traditions and approaches: Fregean, Hilbertian, neo-Kantian, phenomenological (in the sense of Edmund Husserl), and "conventionalist." Carnap's argument, in particular, is that the well-known disputes afflicting the philosophy of space and geometry can be dissolved by carefully distinguishing among three different concepts of space: formal space is a pure abstract structure in the sense of modern axiomatics and logic;219 intuitive space is a form of intuition in the sense of Kant, immediately presented to us by Husserlian "essential intuition" (Wesenserschauung); physical space, especially as it functions in the general theory of relativity, is the object of an empirical theory entirely on the par with the rest of physics. Carnap accommodates his view of pure spatial intuition to general relativity by arguing that intuitive space has only the infinitesimally Euclidean structure presupposed in Riemann's theory of n-dimensional manifolds (rather than a global three-dimensional Euclidean structure), and he accommodates his empiricist conception of physical space to Poincare's "conventionalism" (as understood by Schlick: compare again note 216 above) by arguing that only the topological (and differentiable) structure of space is given to us by spatial intuition and experience, whereas the metrical structure, by contrast, is then subject to conventional choice based on considerations of global simplicity. In this way, a commitment to a modified version of the Kantian synthetic a priori is in fact central to Carnap's account of the relationship between mathematical geometry and sense experience. 220 But even this attenuated version of pure intuition and the synthetic a priori was abandoned when Carnap became a leading member of the Vienna Circle in the mid to late 1920s-and it was probably already abandoned while Carnap, in 1924-25, was working on Der logische
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Aujbau der Welt before he joined the Circle. Indeed, Carnap tells us in his "Intellectual Autobiography" (1963a, 11-13) that he intensively studied Whitehead and Russell's Principia Mathematica and Frege's Grundgesetze der Arithmetik in 1919-20; and from Frege, in particular, he "gained the conviction that knowledge in mathematics is analytic in the general sense that it has essentially the same nature as knowledge in logic" (12). For Carnap, however, the significance of this view was not that we can thereby justify or explain mathematical knowledge on the basis of another type of knowledge-logical knowledge-presumed to be antecedently (or better) understood, but rather that logic and mathematics together playa distinctively formal or inferential role in framing our empirical knowledge: It is the task oflogic and mathematics within the total system of know1edge to supply the forms of concepts, statements, and inferences, forms which are then applicable everywhere, hence also to non-logical knowledge. It follows from these considerations that the nature of logic and mathematics can be clearly understood only if close attention is given to their applications in non-logical fields, especially in empirical science. Although the greater part of my work belongs to the fields ofpure logic and the foundations of mathematics, nevertheless great weight is given in my thinking to the application of logic to non-logical knowledge. This point of view is an important factor in the motivation for some of my philosophical positions, for example, for the choice of forms oflanguages, for my emphasis on the fundamental distinction between logical and non-logical knowledge. (1963a, 12-13) The point of the Aujbau, accordingly, is then to depict, in the most general possible terms, the way in which the "forms of concepts" supplied by modern mathematical logic can in fact succeed in structuring our empirical knowledge. Carnap retains the Kantian idea that empirical knowledge is itself only possible in virtue of a priori forms and principles antecedently supplied by thought. But now, in the Aujbau, he defends an empiricist version of this conception, in the sense that such (still indispensable) formal structuring is now seen (in virtue of modern mathematical logic) as analytic rather than synthetic a priori. 221 Another well-known passage in Carnap's "Intellectual Autobiography" describes how a combination of Frege-Russell logicism with Ludwig Wittgenstein's conception oftautology allowed the Vienna Circle to arrive "at the conception that all valid statements of mathematics are analytic in the specific sense that they hold in all possible cases and therefore do not have any factual content" (1963a, 47)-a conception which resulted in a major advance over all earlier forms of empiricism:
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What was important in this conception from our point of view was the fact that it became possible for the first time to combine the basic tenet of empiricism with a satisfactory explanation of the nature of logic and mathematics. Previously, philosophers had only seen two alternative positions: either a non-empiricist conception, according to which knowledge in mathematics is based on pure intuition or pure reason, or the view held, e.g., by John Stuart Mill, that the theorems of logic and of mathematics are just as much of an empirical nature as knowledge about observed events, a view which, although it preserved empiricism, was certainly unsatisfactory. (ibid.) Indeed, this rejection of pure intuition and the synthetic a priori in favor of the view that alllogico-mathematical truth is analytic and has no factual content quickly became definitive of what Carnap and the Vienna Circle meant by their empiricism. 222 In the late 1920s and early 1930s, however, the Circle became involved with the "crisis" in the foundations of mathematics precipitated by L.E.]. Brouwer's development of a Kantian-inspired version of "intuitionism" concerning the objects of arithmetic and analysis and David Hilbert's development of proof-theory in response to Brouwer. In particular, Brouwer gave a famous lecture in Vienna in 1928, and the Circle (as Carnap reports) was appropriately impressed: In the Circle we also made a thorough study of intuitionism. Brouwer came to Vienna and gave a lecture on his conception, and we had private talks with him. We tried hard to understand his published or spoken explanations, which was sometimes not easy. The empiricist view of the Circle was of course incompatible with Brouwer's view, influenced by Kant, that pure intuition is the basis of all mathematics. On this view there was, strangely enough, agreement between intuitionism and the otherwise strongly opposed camp of formalism, especially as represented by Hilbert and Bernays. But the constructivist and finitist tendencies of Brouwer's thinking appealed to us greatly. (1963a, 49) One way to understand the problem with which the Circle was now faced, therefore, is how to acknowledge the evident strengths of Brouwer's viewpoint without becoming entangled with a "non-empiricist" commitment to pure intuition. Carnap's solution is the Logical Syntax of Language, published in 1934. In conformity with the basic meta-mathematical method of Hilbertian proof-theory, we view any formulation of logic and mathematics as a syntactically described formal system, where the notions of well-formed formula, axiom, derivation, theorem, and so on can all be
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syntactically expressed. In light of Godel's recendy published incomplete_ ness theorems, however, we do not pursue the Hilbertian project of Constructing a proof of the consistency of classical mathematics using finitary means acceptable to the intuitionist. Instead, we formulate both a formal system or calculus conforming to the strictures of intuitionism (Carnap's Language I, a version of primitive recursive arithmetic) and a much stronger system adequate for full classical mathematics (Carnap's Language II, a version of higher-order type theory over the natural numbers as individuals). For both systems, moreover, we define a notion of logical truth (analyticity) intended formally or syntactically to express their essential independence from all factual content. Finally, and most importantly, Canup formulates the principle of tolerance: both types of system (classical and intuitionistic) should be syntactically described and investigated, and the choice between them, if there is one, should then be made on practical or pragmatic grounds rather than on the basis of prior, purely philosophical commitments. 223 It might appear, however, that Carnap's attempt thereby to dissolve the dispute between classical mathematics and intuitionism is viciously circular. For Carnap's application of the principle of tolerance to this case involves raising the question, in a syntactic metalanguage, whether to adopt the classical or intuitionist logical rules in a particular object-language-in this case, the language of total science (mathematics plus physics). We weigh the relative safety (from the possibility of contradiction) of the intuitionist rules against the greater fruitfulness and convenience (in physics) of the classical rules and then make our choice. But if the intuitionist, as he claims, cannot even properly understand the rules of the classical framework-and cannot, a fortiori, understand the necessarily even stronger classical metalanguage in which we describe these rulesthen it would appear that our entire procedure simply begs the question against the intuitionist. This argument is certainly tempting, and I must confess that I myself have succumbed to the temptation more than once. 224 I now think, however, that it misses the essence of Carnap's position. In particular, Carnap begins from the presupposition that classical mathematics, as it is standardly practiced, is well understood. Indeed, classical mathematics, for Carnap, is a model or paradigm of clear and exact-scientific-understanding, and there is no room for raising doubts about our understanding of this framework on independent, purely philosophical grounds. To be sure, the foundations crisis sparked by the discovery of the paradoxes, and the failure of Hilbert's proof theory, raise serious technical questions regarding the consistency of the classical framework, and this is precisely why, for Carnap, we should now take intuitionism seriously. To take it seriously, however, means that we entertain the proposal, starting from within
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the classical framework, that we should wealcen its rules to make inconsistency less likely. There is nothing in Carnap's position blocking a classical mathematician from entertaining this option or even deciding then to adopt it. Carnap has therefore not begged the question of the choice between classical and intuitionist mathematics as he understands this question. That an intuitionist mathematician cannot understand the choice as Carnap understands it is irrelevant, for the situation in which we in fact find ourselves has arisen within the paradigmatically well-understood practice of classical mathematics itself. 225 The principle of tolerance, therefore, gives new meaning and significance to Carnap's non-traditional understanding oflogicism. For the point of viewing the terms oflogic and mathematics as factually empty-and the statements of logic and mathematics as therefore analytic-is now seen to lie precisely in our freedom to choose which system of logic and mathematics best serves the formal deductive needs of empirical science. Classical mathematics, for example, is much easier to apply, especially in physics, than intuitionist mathematics, while the latter, being logically weaker, is less likely to result in contradiction. The choice between the two systems is therefore purely practical or pragmatic, and it should thus be sharply separated, in particular, from all traditional philosophical disputes about what mathematical entities "really are" (independent "Platonic" objects or mental constructions, for example) or which such entities "really exist" (only natural numbers, for example, or also real numbers, that is, sets of natural numbers). Carnap aims to use the new tools of meta-mathematics definitively to dissolve all such metaphysical disputes and to replace them, instead, with the much more rigorous and fruitful project of language planning, language engineering-a project which, as Carnap understands it, simply has no involvement whatsoever with any traditional epistemological program. 226 The best way to appreciate this point is briefly to consider W.v. Quine's famous criticism of Carnap in "Two Dogmas of Empiricism," first published in 1951. In particular, the final paragraph of Quine (1951, §6, "Empiricism without the Dogmas") emphasizes that Carnap views the choice between different "language forms" or "scientific frameworks" as entirely pragmatic. But the problem, in Quine's words, is that such "pragmatism leaves off at the imagined boundary between the analytic and the synthetic," so that, Quine continues, "[i]n repudiating such a boundary I espouse a more thorough pragmatism" ([1951] 1953,46). In particular, according to the holistic empiricist epistemology Quine has just presented, all statements of science-statements of logic, mathematics, physics, or biology-equally face the "tribunal of experience" together. When faced with a "recalcitrant experience" in conflict with our total system, we then have a choice of where to make revisions: we normally try make them as
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close as possible to the periphery of our overall "web of belief," but, when the conflict is particularly acute and persistent, for example, we can also revise the most abstract and general parts of science, including even the statements of logic and mathematics, lying at the center this web. In all such cases our criteria of choice are, in the end, purely pragmatic, a matter of continually adjusting our overall web of belief to the flux of sensory experience so as to achieve the simplest total system best adapted to that experience. Thus Quine concludes (ibid.): "Each man is given a scientific heritage plus a continuing barrage of sensory stimulation; and the considerations which guide him in warping his scientific heritage to fit his continuing sensory promptings are, where rational, pragmatic." The difference between Carnap's position and Quine's at this point is rather subtle. For, in a crucial section of Logical Syntax (1934, §82, "The Language of Physics"), Carnap makes two claims which sound rather similar to Quine's. First, Carnap adopts an holistic view of theory testing he associates with the names of Duhem and Poincare: "the testing concerns in principle not a single hypothesis, but rather the whole system ofphysics as a system of hypotheses (Duhem) Poincare)" (1934, 246; 1937, 318). Second, Carnap also claims that, although "some change must be undertaken in the system" when faced with an unsuccessful prediction of an observation sentence or "protocol-sentence" (what Quine would call a "recalcitrant experience") deduced from "certain physical principles," we always have, nonetheless, a choice ofprecisely where to make the needed revisions (245; 317): "one can, for example, change the P-rules [of physics] so that these [physical] principles are no longer valid; or one can suppose that the protocol-sentence is not valid; or one can even change the L-rules [of logic and mathematics] used in the deduction"-and, Carnap adds, "[t]here are no fixed rules for the kind of change that is to be chosen." Indeed, in this regard there is only a difference of degree between the logico-mathematical sentences and the sentences of empirical physics (246; 318): "No rule of the language of physics is definitively secured; all rules are laid down with the reservation that they may be changed depending on the circull1~ stances, as soon as it seems expedient. This holds not only for the P-rules, but also the L-rules, including those of mathematics. In this respect, there are only differences in degree; it is more difficult to decide to give up certain rules than others." Where, then, does Carnap's pragmatism, in Quine's words, "leave off'? Immediately following the last quoted passage, Carnap draws the line this way: If, however, one assumes that a new protocol-sentence appearing within a language is always synthetic, then there is this difference between an L-valid, and therefore analytic, sentence ~l and a P-valid
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sentence 1:2 , namely, that such a new protocol-sentence-independently of whether it is recognized as valid or not-can be, at most, L-incompatible with 1:2 but never with 1:1 , In spite of this, it may come about that, under the inducement of new protocol-sentences, one changes the language so that 1:1 is no longer analytic. (246; 318-19) In other words, although both types of change in our total system induced by what Quine would call a "recalcitrant experience" are possible, and both involve broadly pragmatic considerations about the optimal overall arrangement of this total system, there is, for Carnap, a fundamental difference between the two: one involves changing the analytic sentences of tl1e language, and thus the rules of logic and mathematics, whereas the other involves merely the synthetic sentences of empirical physics. Only the latter, on Carnap's view, have genuine factual content, and only the latter, accordingly, are the exclusive concern of the empirical scientist-here the physicist. But now Carnap's position may easily begin to look arbitrary. If we admit that our ultimate epistemological criteria, for both analytic and synilietic sentences, reduce to broadly pragmatic considerations about the optimal overall arrangement of our scientific system, why in the world should we persist in maintaining a fundamental distinction between them? Are we not simply attaching arbitrary labels to different sentences, with no remaining epistemological significance? Are we not then ineluctably driven to the "more thorough," and apparently more radical, pragmatic empiricism defended by Quine? It is just here, however, that the true philosophical radicalism of Carnap's position emerges. In 1936, at the very beginning of his semantical period, he published "Von der Erkenntnistheorie zur Wissenschaftslogik" (From Epistemology to the Logic of Science), the point of which is to argue that all traditional epistemological projects, including his own earlier project in the Aufbau, must now be renounced as "unclear mixtures[s] of psychological and logical components" (1936, 36). Whereas the broadly pragmatic and holistic epistemology Quine develops under the rubric of "empiricism without the dogmas" is intended as a replacement for, or reinterpretation of, what Quine takes to be the epistemology of logical empiricism (i.e., the Aufbau) , Carnap-despite Quine's persistent attempts to associate him with varieties of epistemological foundationalism 227-is breaking decisively with the entire epistemological tradition. What Carnap calls "the logic of science" is in no way concerned with either explaining or justifYing our scientific knowledge by exhibiting its ultimate basis (whatever this basis might be); it is concerned, instead, with developing a new role for philosophy vis-a-vis the empirical sciences that will maximally contribute to scientific progress
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while, at the same time, avoiding all the traditional metaphysical disputes and obscurities which have constituted (and, according to Carnap, continue to constitute) serious obstacles to progress in both philosophy and the sciences. 228 The first major publication of Carnap's semantical period was Foundations ofLogic and Mathematics, appearing in English in 1939. Here Carnap presents an especially clear and detailed account of the application of logic and mathematics in en1pirical science and, in particular, the central importance of the analytic/synthetic distinction therein. The application of logico-mathematical calculi in en1pirical science principally involves experimental procedures of counting and measurement (§§19, 23), whereby quantitatively formulated empirical laws yield testable statements about particular numerically specified outcomes via intervening logico-mathematical theorems. The scientific theory in question (in physics, for exampIe) can thus be represented as an axiomatic system containing both logical and descriptive terms, where the logico-mathen1atical part of the systen1 (containing only logical terms essentially) is, in its standard interpretation, analytic or L-true (in the semantical sense); and, because of the key role of numerical terms (including terms for real numbers) in the experimental procedure of measurement, this logico-mathematical part is most appropriately formulated as a higher-order systen1 (§14, 18)-as opposed to an elementary or first-order logical system, §13 )-containing a sufficient amount of arithmetic and analysis. 229 Since Carnap is well aware, of course, that such higher-order logicomathematical systems can and do lead to controversy, he immediately inserts a section on "The Controversies over 'Foundations' of Mathen1atics" (§20, compare §15). Carnap's response to these controversies, not surprisingly, is the principle of tolerance, now formulated in a clearly semantical way: Concerning mathematics as a pure calculus there are no sharp controversies. These arise as soon as mathematics is dealt with as a system of "knowledge"; in our terminology, as an interpreted system. Now, if we regard interpreted mathematics as an instrument of deduction within the field of empirical knowledge rather than as a system of information, then many of the controversial problems are recognized as being questions not of truth but of technical expedience. The question is: Which form of the mathematical system is technically most suitable for the purpose mentioned? Which one provides the greatest safety? Ifwe compare, e.g., the systems of classical mathematics and of intuitionistic mathematics, we find that the first is much simpler and technically more efficient, while the second is more safe from surprising occurrences, e.g., contradictions. (1939, 192-93)
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As we have already seen, therefore, Carnap's main reason for regarding interpreted mathematics-arithmetic and analysis in their customary interpretations-as analytic or devoid of factual content is that doing so shifts our attention away from questions of "truth" and towards the purely pragmatic or technical problem of language planning (see note 226 above, together with the paragraph to which it is appended). The case of geometry, however, is essentially different (§21). Here, although it is perfectly possible to give a logical or analytic interpretation of a geometrical calculus (within analysis, for example, in terms of real number coordinates), the standard or customary interpretation is descriptive or synthetic-as a theory of actual space. But the great lesson of Einstein's general theory of relativity (§22) is that the geometry of actual (physical) space is an empirical question and, in particular, that it is therefore necessary sharply to distinguish between mathematical geometry (given some logical interpretation) and physical geometry (under the customary descriptive interpretation). The latter, as Einstein clearly shows, is a posteriori and synthetic, whereas the former is a priori but purely analytic. Moreover, since physical geometry is a quantitative empirical theory like any other, the appropriate logico-mathematical framework within which it is to be axiomatized must also contain sufficient arithmetic and analysis. For Carnap, therefore, it follows from Einstein's work that the key difference between geometry, on the one side, and arithmetic and analysis, on the other, is that the former is synthetic (a posteriori) in its standard or customary interpretation while the latter are analytic. 23o And it is this situation that Carnap has foremost in mind in his repeatedly expressed conviction, characteristic of his semantical period, that the distinction between analytic and synthetic truth "is indispensable for the logical analysis of science," so that "without [it] a satisfactory methodological analysis of science is not possible."231 This case helps us to appreciate, in particular, what exactly Carnap means by a "satisfactory methodological analysis of science." Carnap is not concerned, as is Quine, with developing a very general empiricist conception of justification or evidence simultaneously embracing scientific knowledge, common-sense knowledge, and logico-mathematical knowledge. Carnap is specifically concerned with the modern mathematical physical sciences (especially those of the early twentieth century: compare the paragraph to which note 214 above is appended), which are themselves only possible in the first place if we presuppose a certain amount of sophisticated modern mathematics-arithmetic and analysis-for their precise articulation and empirical testing (compare note 229 above). And the point of Carnapian logic of science, moreover, is not so much to describe the nature of science or scientific method as it has been practiced so far as to open up the possibility for a new kind of ongoing philosophical interaction with the
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sciences, which, in Carnap's eyes, promises to be particularly fruitful for both. Armed with the new logico-mathematical tools of modern logic (especially the new tools of meta-mathematics), the philosopher-that is, the logician of science-can participate, together with the scientists themselves, in the articulation, clarification, and development of formal inferential frameworks for articulating empirical theories and testing them by experin1ental methods. Unlike the empirical scientist, however, the logician of science, as such, is not concerned with then actually testing empirical theories within such inferential frameworks. Moreover, unlike the applied mathen1atician (who also develops formal methods for use in the empirical sciences), the logician of science has a characteristically philosophical interest in developing a systematic method for defusing unresolvable metaphysical controversies (about the "true nature" of numbers, for exan1ple, or the "true nature" of space) which, in Carnap's view, constitute an ever-present obstacle to progress in both the sciences and philosophy. Indeed, it was precisely this enterprise-and not any epistemological project-that Carnap already had in n1ind in §72 of Logical Syntax when he famously proclaimed (1934, 205; 1937, 279): "The logic of science [WissenschaftslogikJ takes the place of the inextricable tangle ofproblems one calls philosophy."
Einstein's new conception of physical geometry then provides a striking illustration, for Carnap, of the fruitfulness of his program for the logic of science. In the first place, Einstein's conception was deeply indebted, as he himself emphasizes in "Geometry and Experience," to the modern axiomatic view of purely n1athematical geometry associated with Hilbert (see again note 203 above, together with the paragraph to which it is appended). In the second place, however, in insisting on the overriding importance of the analytic/synthetic distinction, Carnap is paraphrasing Einstein's own remarks (see the following paragraph) on the distinction between mathematical and physical geometry in that same article: "In my opinion, a sharp analytic/synthetic distinction is of supreme importance for the philosophy of science. The theory of relativity, for example, could not have been developed if Einstein had not realized that the structure of physical space and time cannot be determined without physical tests. He saw clearly the sharp dividing line that must always be kept in mind between pure mathematics, with its many types of logically consistent geometries, and physics, in which only experiment and observation can determine which geometries can be applied most usefully to the physical world. "232 From Carnap's point of view, therefore, Einstein's distinction between mathematical and physical geometry was an early illustration of the fruitfulness of Carnap's program, in so far as Einstein thereby opened up the possibility of a radically new use of mathematics in empirical physics, and, at the same time, definitively dissolved the traditional philosophical dispute about the IZantian synthetic a
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priori (compare note 215 above, together with the paragraph to which it is appended). Carnap also clearly thought (although he never provided a detailed development of this idea) that the logic of science could be fruitfully applied to the problems of quantum theory as well. In particular, the final sections of Foundations of Logic and Mathematics (1939, §§24, 25) suggest that the vexed question of the "interpretation" of the wave-function can be resolved by appreciating that theories of modern mathematical physics operate with "abstract" terms, which acquire their meanings, in the manner of Hilbert, by their roles within a formal axiomatic system (and thus require no "intuitive" or "visualizable" meaning), but which still relate to empirical phenomena (experimental measurements) indirectly. This, in fact, appears to be the origin of Carnap's later conception of the "partial interpretation" of theoretical terms, which eventually resulted in the Ramsey-sentence reconstruction of scientific theories-and of theoretical analyticity-developed in Carnap (1966).233 Indeed, immediately following the passage quoted above about Einstein and general relativity, Carnap continues (1966, 257-58): "This distinction between analytic truth (which includes logical and mathematical truth) and factual truth is equally important today in quantum theory, as physicists explore the nature of elementary particles and search for a field theory that will bind quantum mechanics to relativity." From the mid-1940s on, during the last twenty-five years of his long and fruitful career, Carnap worked principally on a new project in the logico-mathematical analysis of empirical science which he called the logical foundations of probability.234 He here turned his attention away from the more traditional deductive inferential frameworks on which he had so far primarily focused and towards the newer probabilistic and statistical frameworks now being constructed and applied with ever-increasing frequency in the physical, biological, and social sciences. And here, once again, Carnap, as logician of science par excellence, aimed fruitfully to engage with the scientists in question in clarifying existing statistical methods and developing new ones while, at the same time, dissolving residual metaphysical controversies (for example, about the "true nature" of probability-objective or subjective) which still threaten scientific progress. This project, from Carnap's own point of view, appears to have been quite successful. For, on the more philosophical side, Carnap's sharp distinction between two different concepts of probability-logical or epistemic (degree of confirmation) and empirical or physical (long-run relative frequency)-appears to have the same potential for defusing needless metaphysical disputes as the earlier distinction between mathematical and physical geometry. And, on the more technical side, Carnap's own positive COntributions to the theory of statistical inference, although somewhat
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outside of the mainstream, did in fact interact fruitfully with other scientific work in the ongoing development of our formal inductive methods. 235 Nothing more, from the point of view of Carnapian logic of science, could either be hoped for or achieved. Carnap's mature philosophical position therefore provides us with an echo of lZant, in so far as Carnap simply (and rightly) takes it for granted that the kind of empirical knowledge paradigmatically exhibited by modern science is itself only possible in the first place on the basis of a prior formal structuring of our knowledge claims by modern mathematics. And Carnap views such prior structuring, accordingly, as empty of empirical content-just as lZant, for his part, had earlier asserted that the only objects of knowledge are empirical objects ("appearances"), so that mathematics, strictly spealcing, has no actual objects of its own. 236 Finally, Carnap views his conception of logic and mathematics as the other side of the coin, as it were, of a characteristically philosophical enterprise aiming to defuse all unresolvable metaphysical controversies once and for all-just as IZant, for his part, had earlier held that his explanation of how synthetic a priori knowledge is possible is the other side of the coin of the claim that the resulting "critique of pure reason" finally puts philosophy "on the secure path of a science" and sets aside all "mock combats" of the traditional metaphysical schools (B xiv-xv). But precisely here the parallels end: I
Transcendental Philosophy and Modern Physics: Neo-IZantianism, Logical Empiricism, and Phenomenology The project of transcendental philosophy, as originally conceived by lZant, essentially involved an explanation of how purely intellectual concepts can necessarily apply to our (human) sense experience of the natural world. Kant's project also involved an explanation of how pure mathematical concepts can necessarily apply to this same experience, where such an explanation, for IZant, was conceived in a distinctive way. As we have seen, the pure intellect, for lZant, is modeled on the Leibnizean conception of logic,
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\\'hereas the faculty of pure sensibility is modeled on the Newtonian con(eptio n of space and time (compare the paragraph to which note 40 above is appended). Mathematics then necessarily applies to our sense experience in so far as the pure forms of our sensibility-space and time-are themselves the primary sources of pure mathematical knowledge. However, this does not yet show that all natural objects presented to us within these forms of sensibility are also necessarily subject to both pure mathematics "nd the pure concepts of the understanding; and it is precisely this, for Kant, which is shown in the transcendental deduction by demonstrating that there is a necessary harmony or interdependence between the pure intellect, on the one side, and both pure and empirical intuition, on the other. 239 It is only this necessary harmony between two initially independent faculties that fully explains how synthetic a priori knowledge is possible, for Kant, and, in particular, how mathematical natural science (Newtonian physics) is possible. 24o We observed above that a rejection of Kant's sharp separation of the faculties of the mind into understanding and sensibility is characteristic of post- Kantian German idealism, and this is no less true of the Marburg School of neo-Kantianism founded by Hermann Cohen (compare note 107 above, together with the paragraph to which it is appended and the preceding paragraph). Indeed, one of the fundamental ideas of Cohen (1871) is that the separation between two faculties of the mind delineated in the Transcendental Aesthetic of the first Critique is a remnant of the pre-critical doctrine of the Inaugural Dissertation, where the critical theory of the transcendental schematism of the understanding has not yet been articulated (compare note 11 above, together with the paragraph to which it is appended and the preceding paragraph). But this sharp separation must be re-evaluated and revised, according to Cohen, when we arrive at the truly critical doctrine of the Transcendental Analytic, according to which all "synthetic unity," including that of space and time, is due ultimately to the understanding. 241 Cohen then reinterprets Kant's "transcendental method" as one that begins with the "fact of science"-the existence of the mathematical sciences in their modern, post-seventeenth-century form-and then seeks their ultimate preconditions or presuppositions by a regressive argument. Kant himself had performed this task for the fundamentally Newtonian mathematical sciences of the seventeenth and eighteenth centuries, and our new task, as Cohen sees it, is to generalize and extend transcendental philosophy so as to embrace the main developments in the mathematical sciences that have occurred since Kant's time. In particular, in light of our principled rejection of Kant's distinction between two independent cognitive faculties, we are now in a position to avoid the idea that the pure forms of sensibility, space and time, have their own a priori structure-the
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structure of Euclidean geometry and Newtonian space and time-given independently of the synthesizing activities of the understanding: other geometries than Euclid's, and other structures of space and time than Newton's, are therefore perfectly possible products of the a priori synthesizing activities of thought. Moreover, for Cohen and the Marburg School more generally, there is no longer an independent contribution of empirical or a posteriori sensibility either: there is no independent "n1anifold of sensations" that is simply given, entirely independently of the synthesizing activity of thought. What there is, instead, is an essentially dynamical, temporal, and historical process of active "generation" (Erzeugung), as the mind successively characterizes or determines the "real" that is to be the object of mathematical natural science in a continuous serial process. The "real" itself-the true empirical object of mathematical natural science-is in no way independently given as something separate and distinct from this synthesis of thought; rather, it is the necessary endpoint or limit towards which a nevercompleted serial process of successive determination is converging: the "real" empirical object is not "given" (gegeben) , but "set as a task" (aufgegeben ).242 This "genetic (erzeugende) conception of knowledge" is the most characteristic contribution of the Marburg School. And, whereas Cohen himself had modeled the epistemological process in question on the methods of the infinitesimal calculus,243 his most important student, Ernst Cassirer, provided a more abstract and contemporary version modeled on the recent developments in modern mathematics and mathematical logic associated with Hilbert and Richard Dedekind, and also with the early works ofFrege and Russell.244 In particular, in Substance and Function (1910), Cassirer conceives the epistemological process described by the genetic conception of knowledge as the historical evolution of a series or sequence of abstract mathematical formal structures ("systems of order"), which is itself ordered by the abstract formal relation of approximate backwards-directed inclusion. Thus, for example, if we decide that a non-Euclidean geometry is a better representative of physical space than Euclidean geometry, the new mathematical structure contains the old (Euclidean) one as a continuously approximated limiting case as we consider ever smaller spatial regions. But in empirical science, unlike pure mathematics, our further development of such structures has no finite endpoint in principle, and the final object of empirical theorizing can only emerge, as before, as the ideal endpoint or limit structure towards which all earlier members of the series are continuously converging. Although this final limiting structure is only a regulative ideal in the original IZantian sense (compare note 242 above), it nevertheless constitutes the characteristic "general serial form" of our empirical mathematical-physical theorizing and, ~~~~~~~~~Ql~~Q~~~wy
_
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on this theorizing its characteristic form of objectivity: despite all historical variation and contingency, there is still a continuously converging progression of abstract mathematical structures framing, and making possible, all our empirical knowledge. Cassirer thereby arrives at a new interpretation of the synthetic a priori as well. Contrary to the original Kantian conception, even the most fundamental principles of Newtonian mechanics "need not be taken as absolutely unchanging dogmas." Such temporarily "highest" principles of experience-at a given stage of scientific theorizing-may evolve into others, and, in this case, even our most general "functional form" for the laws of nature would undergo a change. Yet such a transition never entails that "the one fundamental form absolutely disappears while another arises absolutely new in its place." On the contrary: The change must leave a determinate stock of principles unaffected; for it is solely for the sake of securing this stock that [this change] is undertaken in the first place, and this shows it its proper goal. Since we never compare the totality of hypotheses in themselves with the naked facts in themselves, but can only oppose one hypothetical system of principIes to another, more comprehensive and radical [system], we require for this progressive comparison an ultimate constant measure in highest principles, which hold for all experience in general. What thought demands is the identity of this logical system of measure throughout all change in that which is measured. In this sense, the critical theory of experience actually aims to construct a universal invariant theory of experience and thereby to fulfill a demand towards which the character of the inductive procedure itself ever more clearly presses. (1910,
355-56; 1923,268) And what this means, for Cassirer, is that we must form the idea of a set of ultimate or limiting scientific principles such that all previous stages of scientific theorizing can be viewed as approximate special cases of precisely these ultimate principles. It is at this point-and only at this point-that we can actually specifY the content of the "universal invariant theory of experience," and so there is no way, in particular, that we can determine the specific content of such ultimate principles in advance: [For example,] the validity of a spatio-temporal dependency between the elements ofwhat happens as such, which expresses itself in the universal causal law, remains untouched by every change in the particular causal laws. The goal of critical analysis would be attained if it succeeded in establishing in this way what is ultimately common to all
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possible forms of scientific experience-in conceptually fixing those elements that are preserved in the progress from theory to theory because they are the conditions of each and every theory. This goal may never be completely attained at any given stage of cognition; nevertheless, it remains as a demand and determines a fixed direction in the continual unfolding and developn1ent of the system of experience itself. The strictly limited objective nleaning of the "a priori" appears clearly from this point of view. We can only call those ultimate logical invariants a priori that lie at the basis of every determination of a lavvlike interconnection of nature in general. A cognition is called a priori, not because it lies in any sense before experience, but rather because, and in so far as, it is contained in every valid judgment about facts as a necessary premise. (1910, 357; 1923,269) Just as the object of natural scientific knowledge is the never fully realized ideal mathematical structure towards which the entire historical development of science is converging, so the a priori form of scientific knowledge, for Cassirer, can only be determined as that stock of "categorial" principles which, viewed from the perspective of the ideally completed developmental process, are seen to hold (retrospectively, as it were) at every stage. For example, we do not know, at any given stage, what the particular content of spatial geometry (Euclidean or non-Euclidean) or of our n1echanical theory of motion (Newtonian or non-Newtonian) must be, but we can now venture the well-supported conjecture that some or another such spatio-ten1poral structure must be present at every stage. The example of spatio-temporal structure turns out to be particularly apt and revealing. Although Substance and Function appeared prior to the formulation of Einstein's general theory of relativity, Cassirer's next important contribution to scientific epistemology, Einstein)s Theory ofRelativity (1921), is devoted to precisely this revolutionary new theory. For Cassirer, in particular, the fact that Einstein now replaces the spatial geometry of Euclid with a n1uch more general spatio-temporal geometry of variable curvature in no way implies the collapse of a properly-understood "critical" theory of the a priori (1921, 101; 1923, 433): "For the 'a priori' of space, which [physics] asserts as the condition of every physical theory, does not include, as has been shown, any assertion about a determinate particular structure of space, but is concerned only with the function of 'spatiality in general,' which is already expressed in the general concept of the line-element ds as such-entirely without regard to its more particular determination." Thus, according to the generalized conception of the lineelement derived from Riemann (see the paragraph to which note 142 above is appended), we postulate only that space is infinitesimally Euclidean (the metric continuously approximates to Euclidean geometry
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as the regions under consideration grow ever smaller); and, more generally, we can view the space-time structures arising in the transition from Newtonian physics to Einstein's theories of relativity as a converging sequence, in Cassirer's sense, whose last member, at the present moment (circa 1921), is the space-time structure of general relativity.24S It is this structure, for Cassirer, which then presents us with our best current candidate for an ultimate spatio-temporal invariant: namely, the infinitesimally Minkowskian space-time metric on the basis of which Einstein erects his general theory (see the paragraph to which note 206 above is appended). A5 we noted above, the logical empiricists, with Einstein's approval, initiated their assimilation of general relativity by decisively rejecting the Kantian synthetic a priori in all of its forms (compare note 215 above, together with the paragraph to which it is appended). In particular, Schlick, in his 1921 review of Cassirer's book-entitled "Critical or Empiricist Interpretation of Modern Physics?"-rejects Cassirer's neoKantian attempt to reinterpret this IZantian notion, together with an earlier such attempt by Reichenbach (see note 216 above).246 Schlick's main argument, against Cassirer, is that the general theory of relativity has not only overthrown the particular geometry of Euclid and the particular mechanics of Newton; rather, it is no longer clear, on the basis of this theory, that any particular geometrical or mechanical statements are fixed and unrevisable. Accordingly, Schlick challenges Cassirer to produce specific examples of such synthetic a priori principles,247 and he concludes that Cassirer has very little of a definite character to say.248 A5 we have seen, however, what Schlick entirely misses here is that Cassirer is not even trying to delineate these kinds of synthetic a priori principles-whose content we can now precisely specifY as fixed and forever unrevisable. Cassirer's version of the a priori is not constitutive in the original Kantian sense, but purely regulative (compare note 242 above): we can only specifY the determinate content of the "universal invariant theory of experience" at the forever unreachable endpoint of scientific theorizing; and Cassirer's talk of "spatiality in general," in this context, is intended only to represent our best current conjecture concerning what the content of this ideal endpoint might actually be. Yet it does not follow that Cassirer's conception is therefore entirely without force. For, using the overarching ideal of continuous convergence towards a limit structure, we can then reasonably require-again purely regulatively-that the direction we pursue next will be continuously connected, by just such a convergent process, with where we are right now. In particular, given that general relativity has now emerged in precisely this way from our preceding space-time structures (compare note 245 above), we can reasonably require that its space-time metric should continue to be preserved, in the same approximative sense, in future theories. Cassirer's purely regulative
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version of the synthetic a priori thereby guides scientific inquiry without constitutively constraining it by fixed unrevisable principles, but it does guide scientific inquiry nonetheless. 249 Lying behind this issue between Schlick and Cassirer about the status of the synthetic a priori is a deeper issue concerning how abstract n1athematical structures relate to concrete empirical reality in the first place. For Schlick, a theory in n1athematical physics is defined, purely formally, by a system of "implicit definitions" modeled on Hilbert's axiomatization of Euclidean geometry; and, since such a system of "implicit definitions" is initially completely "uninterpreted," it must then be somehow related to or coordinated with an independently existing empirical reality. This comes about, for Schlick, when a purely abstract and intrinsically unintuitive mathematical structure acquires a relation of "designation" (Bezeichnung) or "coordination" (Zuordnung) with subjective experiences: directly given objects of intuition or "acquaintance" (!(ennen) rather than genuine objects of "knowledge" (Erkennen). Indeed, as n1ere objects of intuition and acquaintance, the experiences in question are entirely unconceptualized, an10unting merely to fleeting and subjective "images" (Vorstellungen) rather than stable and objective "concepts" (Begrijje). Objective concepts, for Schlick, are given only by formal systems of "implicit definitions," and objective conceptual knowledge of empirical reality therefore only arises through a relation of coordination: a purely formal mapping between an abstract n1athematical structure, on the one side, and sOll1ething wholly subjective, unconceptualized, and ultimately ineffable, on the other. 2so This kind of picture of a direct epistemic confrontation between conceptual thought and unconceptualized empirical reality ("in itself") is precisely what the genetic conception of knowledge characteristic of the Marburg School is most concerned to avoid; and it is for precisely this reason, in particular, that empirical reality, on this view, is "set as a task" (aufgegeben) completed only at the forever unattainable ideal endpoint of empirical inquiry. This is also why Cassirer, in the passage from Substance and Function quoted above (1910, 355-56), argues that, "[s]ince we never compare the totality of hypotheses in themselves with the naked facts in then1selves, but can only oppose one hypothetical system of principles to another, more comprehensive and radical [system], we require for this progressive comparison an ultimate constant measure in highest principles, which hold for all experience in general." For Cassirer, the en1pirical reality to which our conceptual thought corresponds is defined only by a sequence of theoretical systems in which one such system is continually superseded by a "more comprehensive and radical" system, and the only relation of "coordination" (Zuordnung) we can then set up is between a more comprehensive conceptual structure occurring later in the sequence and a less comprehensive structure occurring earlier. 2SI And, by the same
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token, since the only way in which we can define empirical reality in the first place is by means of precisely such a continuously converging sequence of conceptual structures, we must maintain a purely regulative version of the synthetic a priori in accordance with what Cassirer calls the "universal invariant theory of experience." There can be very little doubt, I believe, that the conception of "coordination" between abstract mathematical thought and concrete empirical reality defended by the early logical empiricists-and, in particular, by both Reichenbach and Schlick-leaves us with a completely unacceptable (if not completely unintelligible) picture of the application of mathematics in modern physics. 252 It is noteworthy, then, that Thomas Ryckman's recent book, The Reign of Relativity: Philosophy in Physics 1915-1925 (2005), argues that we should reconsider the traditional claim oflogical empiricism to capture and exemplifY the true philosophical significance of Einstein's theories. In particular, Ryckman (2005, chapters 2 and 3) examines the issues involving Reichenbach, Schlick, and Cassirer we have just discussed in a way that is strikingly favorable to Cassirer,253 and Ryckman then goes on to demonstrate that other forms of "transcendental idealism"-represented by Hermann Weyl and Arthur Eddington-are even more intimately implicated in both the mathematical foundations and the later development of general relativity. Ryckman's discussion of Weyl is especially interesting, from our present point of view, because ofWeyl's explicitly acknowledged debt to Husserl's phenomenological version of transcendental idealism. Ryckman (2005, chapters 5 and 6) documents Weyl's extensive immersion in Husserlian phenomenology and discusses how this immersion then gave rise to a deep investigation of the mathematical foundations of both geometry and the new space-time physics under the rubric of what Weyl called "pure infinitesimal geometry" (reine infinitesimale Geometrie)which, in turn, is intended to reflect the philosophical primacy of that which is directly and immediately presented to the Husserlian transcendental ego in the phenomenological here-and-now. On this basis, in particular, Weyl developed what he himself conceived as a constructive mathematical "essential analysis" (Wesensanalyse) of the nature of space (and of space-time), and he put this analysis to physical use in articulating, for the first time, the idea of a gauge transformation. 254 There is no doubt, therefore, that Weyl's ambitious combination of mathematical, physical, and philosophical thinking provides a striking illustration of the wide-ranging intellectual fruitfulness of transcendental phenomenology. I do wonder, however, whether Weyl's mathematical construction can provide a complete philosophical answer to the problem addressed, in rather different ways, by Schlick, Reichenbach, and Cassirer (and, in an earlier and less complicated time, by the original transcendental idealism of Kant)-the
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problem, namely, of how modern mathematical physics obtains a concrete empirical meaning in terms of the sensibly given objects of our perceptual experience. 255 So what I propose to do, in the remainder of this section, is to step back from the details of Weyl's construction and to consider the relationship between Hussedian phenomenology and the prospects for a propedy "transcendental" solution to this problem from a more general point of view. It would appear, to begin with, that Hussedian phenomenology is in a much better position than early logical empiricism to do justice to the empirical application of modern mathematical physics, since Hussed does not take his starting point from a stark opposition between abstract conceptual thought and concrete (but purely subjective and ultimately ineffable) intuitive experience. As Hussed explains in the first chapter of his Ideas for a Pure Phenomenology and Phenomenological Philosophy (1913), phenomenological intuition or "essential insight" (Wesenserschauung) inextricably involves both intuitive and conceptual elements, in so far as individual concrete realities are experienced as instances of universal kinds or "essences." In doing geometry, for example, we intuitively apprehend particular geometrical figures as instances of the kinds "circle in general," "triangle in general," and so on; and we thereby arrive at judgments of "eidetic generality" (1913, §5) expressing both strict universality and necessity (§6). Contingent empirical judgments, in turn, are then necessarily subject to judgments of eidetic generality, in so far as they are governed by both formal "mathesis universalis"-including, arithmetic, pure analysis, and the theory of manifolds (§8)-and the various "material eidetic sciences" (§10), such as geometry, rational mechanics, and so on. Thus, whereas Hussed largely follows Kant in his conception of how empirical knowledge is framed, and made possible, by necessary a priori knowledge,256 he emphatically does not embrace the Kantian distinction between concepts, on the one side, and intuitions, on the other: all of our knowledge, both a priori and empirical, both universal and particular, is fundamentally intuitive. 257 In his last work, The Crisis of European Sciences and Transcendental Phenomenology, written in the years 1934-37, Hussed makes his critique of Kant on this point more explicit, and, at the same time, he thereby clarifies the sense in which all knowledge, for him, rests on an "intuitive" or "evident" basis. Hussed begins (§25) by emphasizing that Kant's starting point is the rationalism or intellectualism of Leibniz and Christian Wolff, based on "a systematic universe of 'logical laws', the theoretical totality of truths called upon to function as norms for all judgments supposed to be capable of objective truth" (1954,95; 1970,92). This conception, Husser! implies, supplies Kant with his model for the faculty of the intellect-"pure understanding" or "pure reason"-but Kant also wants to explain how the
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deliverances of the distinct faculty of "sensibility" (the world as it "appears" to our senses). are nevertheless ?ecessarily subje~t t~ the a priori normative claims of logic and mathematics. Kant's solutiOn is a "regressi"e" or "hypothetical" account of how the two underlying faculties of "pure intuition" and "pure reason" interact with one another so as to produce the world as it ordinarily appears to us (1954, 97; 1970,94): "Kant now undertakes to show, in an in fact regressive procedure, that, if ordinary experience is actually supposed to be experience of obJects of natureof objects which, in accordance with being and not-being, being constituted so or otherwise, are supposed to be capable of being known in objective truth, and thus scientifically-then the intuitively appearing world must already be a product [Gebilde] of the faculties of 'pure intuition' and 'pure reason', the same faculties which express themselves in mathematics and logic in explicit thought." Because of his explicit rejection of Kant's "merely regressive" method, together with an implicit rejection ofI
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mirage, an hallucination, or a dream. Within the life-world itself we have inductive regularities ("anticipations") that guide us in this process of continual correction ("real" or "actual" physical objects are perceived as stable in certain ways as we move around them, for example); and objective or positive science necessarily arises from this process as we then atten1pt continually to correct and refine such regularities by reference to more accurate and comprehensive scientific laws of nature. For Husserl, Galileo's original mathematization of nature in the seventeenth century is paradigmatic of this procedure (§9), and its goal is a precise geometrical description of both space and physical objects arrived at by successively idealizing the rough and approximate shapes, bodies, and motions of ordinary experience in terms of perfectly exact geometrical figures and trajectories. 258 Thus, the task of all objective science, more generally, is to begin from the practical and theoretical problems intrinsic to the life-world itself, and then, on this basis, to pursue an infinite, never to be completed project of systematic inquiry and continual refinement analogous to the indefinite process of successive idealization that first gave rise to the science of geometry. By contrast, the task of a phenomenological science of the life-world itself is oriented in precisely the opposite direction. Rather than leaving the immediately and intuitively given realm behind in an indefinite process of continual idealization and abstraction, our aim is to remain entirely within this realm in order to understand as clearly as possible that and how the original intentional n1eaning of all the properly objective sciences derives solely from their relationship to the life-world. 259 In particular, whereas all possible objective sciences-mathematics, physics, history, psychology, and so on-must have their own forms of "evidence" or "insight" (their own forms of Wesenserschauung and Wesensanalyse), these forms of evidence are by no means ultimately or originally intuitive in the sense of the life-world (§34d): The life-world is a realm ofwhat is originally evident. That which is evidently given is, respectively, immediately present as experienced in "itself" in perception or as itself remembered in memory. Every other kind of intuition is a procedure of making present [Vergegenwartigen J. Every mediate cognition belonging to this sphere-broadly speaking, every kind of induction-has the sense of an induction on what is intuitable, something possibly itself perceivable or remembered as having been perceived, and so on. All thinkable verification [BewahrungJ leads back to these modes of evidence, because the "thing itself" (in whatever mode) lies in these intuitions as that which is intersubjectively actually experienceable and verifiable, without being a substruction of thought. Whereas such a substruction, by contrast, in so far as it has a claim to truth at all, can only have actual truth by being related back to
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such evidences.... It must be completely clarified, and thus brought to ultimate evidence, how all the evidence of objective-logical accomplishments, in which objective theory (mathematical, natural-scientific) is grounded in accordance with form and content, has its hidden sources of grounding in ultimately accomplishing life, in which the evident givenness of the life-world continually has, has acquired, and acquires anew its pre-scientific sense of being. From objective-logical evidence (mathematical "insight," natural-scientific, positive-scientific "insight," as it is being attained by the inquiring-grounding mathematician, and so on), the way leads here back to primal evidence [Urevidenz], in which the life-world is continually pre-given. (1954,
130; 1970, 127-28) The life-world is itself intuitively evident in so far as it is immediately given-perceptually, indexically, and ostensively-to an I-here-now, and there is nothing more evident "behind" or "beneath" this world on the basis of which its claim to "reality" and "truth" could either be questioned or justified. On the contrary, all further claims to "reality" and "truth," as pursued by the objective sciences, can themselves be raised and justified only on the basis ofwhat is immediately and intuitively evident in the originally given life-world experience from which they necessarily arise. 260 It is important to see, however, that what Husserl primarily means by "evidence" here is direct and immediate presence-perceptually, indexically, and ostensively-to an I -here-now; and he is not particularly concerned, accordingly, with the justificatory notion of certainty (with what is often called "self-evidence").261 Husserl is not primarily interested in epistemic certainty at all, but in the characteristically transcendental task of explaining, or making comprehensible to ourselves, the ultimate grounds on which the very possibility of objective knowledge finally rests
(§55): In accordance with this, one also understands the sense of the demand for an apodicticity of the ego and of all transcendental cognitions obtained on this transcendental basis. Having arrived at the ego, one is aware that one stands within a sphere of evidence behind which it is absurd to attempt to inquire-back [zuriickfragen]. By contrast, every ordinary appeal to evidence, in so far as a further inquiry-back [Riickfragen) is thereby supposed to be cut off, is theoretically no better that an appeal to an oracle in which a God reveals itself. All natural evidences, those of all objective sciences (not excepting those of logic and mathematics), belong in the realm of "things which are obvious" [Selbstverstiindlichkeiten) , which in truth have their background of incomprehensibility [Unverstiindlichkeit]' Every evidence is the name
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of a problem except for phenomenological evidence, after it has reflectively clarified itself and shown itself to be ultimate. It is naturally a ridiculous, although an unfortunately common misunderstanding to combat transcendental philosophy as "Cartesianism," as if its "ego cogito" were a premise or sphere of premises in order to deduce from it knowledge in the usual sense (whereby one naively speaks only of objective knowledge) with absolute "security." The point is not to secure objectivity but to understand it.... [And] the only real explanation is to make [something] transcendentally understandable. (1954,
192-93; 1970, 188-89) Russerl's project is not epistemic foundationalism but, like Iunt's, one of fashioning a transcendental explanation of the underlying conditions in transcendental subjectivity of the possibility of objective knowledge. 262 The difference, from Husserl's point of view, is that IZant's explanation (improperly) appeals to "hypotheses" about the cognitive faculties of the transcendental subject, while Husserl finds his ultimate grounding in the transcendental phenomenological structure of the ordinary and always present life-world itself, as it is directly and immediately given to all of us. Whatever the merits of Husserl's complaint about Iunt's "merely regressive" form of transcendental argumentation,263 it is clear, from our present point of view, that IZant's transcendental explanation of the possibility of mathematical natural scientific knowledge (because of his Leibnizean model of the faculty of pure understanding and EuclideanNewtonian model of the faculty of pure sensibility) is not capable, in principle, of comprehending the further application of mathematics to our experience of nature in modern post-Newtonian mathematical physics. By contrast, Husserl's understanding of how the objective mathematical sciences necessarily emerge from the transcendental structure of the lifeworld has no problems on this score, because the indefinitely extended sequence of ever more comprehensive and accurate approximations and idealizations is in no way limited to Euclidean-Newtonian science. Russerl's picture of this infinite approach to a "limit-idea" (Grenzidee) is closely analogous, in this respect, to the genetic conception of knowledge of the Marburg School which, as we have seen, is also explicitly intended to embrace the further development of mathematical natural science beyond the seventeenth and eighteenth centuries (compare the paragraph to which note 242 above is appended). Indeed, in Cassirer's case, as we have also seen, it is even possible, on this conception, to develop a subtle and penetrating transcendental perspective on the general theory of relativity (compare note 245 above, together with the paragraph to which it is appended, and also note 253). The problem, from Russerl's point of view, is that Cassirer (and the Marburg School more generally) takes the actual
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empirical object of scientific knowledge to be a "limit-idea" reached only at the end of scientific inquiry (aufgegeben) and has no room for anything immediately given (gegeben) prior to and independently of the process of scientific inquiry itself (compare the paragraph to which note 251 above is appended). By basing the distinctive perspective of transcendental phenomenology in the originally pre-given life-world, therefore, Husserl is then in a better position to do justice to the sensible, perceptual, and experiential dimensions of modern mathematical scientific knowledge. 264 Such an account of the original sensible, perceptual, and experiential il1tentional meaning of modern mathematical physics-and, in particular, general relativity-would parallel Husserl's account of the development of the matllematical science of geometry (and its first extensive application to nature by Galileo) in the Crisis. We begin from the life-world and thus from our initial perspective here-and-now at a given place on the surface of the earth. We begin, therefore, with ancient astronomy, which describes me perceptible motions of the sun, moon, stars, and planets in terms of the changes in the angles between lines of sight as seen from a point on the earth's surface. In order to obtain a description that comprehends all such points together (and, in particular, all possible changes in latitude), we tllen obtain the familiar two-sphere model, where the earth appears as a tiny sphere at the center, the outermost large sphere contains the fixed starts and rotates daily on its axis from east to west, and the sun, moon, and planets appear on further spheres rotating in the contrary direction (yearly, monthly, and so on). Modern science, of course, takes its starting point by replacing this geocentric model with a heliocentric model, but, to see the full force of this move, it is necessary to appreciate the way in which we are thereby able to integrate traditional mathematical astronomy with a new mathematical description of terrestrial motion. 265 This integration of course begins with Galileo, whose (idealized) mathematical descriptions of falling bodies, projectiles, pendula, and so on are supposed to be valid everywhere on the surface of the earth. Moreover, in addition to the idealizations already briefly described (see notes 47 and 258 above), we also abstract from the earth's curvature and imagine that all downward accelerations due to gravity are parallel to one another and orthogonal to the flat horizontal plane representing the surface of the earth. Our Galilean description, therefore, is supposed to be valid (approximately) in the "laboratory frames" of classical physics, and our problem now is to integrate the Galilean mathematical description of terrestrial motions with the best description of the celestial motions available in mathematical astronomy. It was precisely this problem that was first successfully solved by Newton, using a concept of force and general laws of motion applicable everywhere in the universe, together with a theory of universal gravitation (as a particular Newtonian force) governing both
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celestial and terrestrial phenomena. Yet this theory also gave rise to a new fundamental problem about the relationship between our mathematical description of nature and our perceptual experience, since Newtonian absolute space, on Newton's own account, is by no means presented to Our senses. IZant's original version of transcendental idealism was intended, among other things, to address precisely this probleln; and the solution Kant arrived at, in the Metaphysical Foundations of Natural Science, was to view absolute space as a limiting construction out of perceptible, physically definable relative spaces. Moreover, the modern solution, first proposed in the late nineteenth century, is closely analogous to 10nt's, in so far as we replace absolute space with a privileged class of relative spaces or frames of reference-the inertial fran1es of reference-which can then be concretely given (or at least concretely approximated) in our actual perceptual experience of the empirical world. 266 Thus, the "laboratory frames" attached to the surface of the earth (in which, as we have seen, the Galilean descriptions of various terrestrial motions are formulated) provide us with good approximations to an inertial frame for the phenomena standardly considered in them. Nevertheless, as Newton showed, for the purpose of astronomical science-and in order to integrate this science with terrestrial physics-it is necessary to move to a more exact frame of reference defined by the center of mass of the solar system. This astronomical frame of reference is a better approximation to an inertial frame than our initial "laboratory frames," and it is correspondingly more distant from our sensible perceptual experience here on earth (and thus more distant, as it were, from the Husserlian life-world); nevertheless, it is still reachable fron1 our initial experience here on earth by a well-defined sequence of empirically specifiable mathematical operations. Einstein's development of the general theory of relativity then involved a sequence of natural extensions of the concept of an inertial frame. As we have seen, the special theory of relativity, developed in 1905, gave a central place to this concept, and it proceeded, in particular, to adapt this concept to the new empirical discovery of the invariance of the velocity of light-which stood in apparent contradiction, as Einstein put it, with what he called the principle of relativity. Einstein resolved this apparent contradiction by revising the classical conception of (absolute) simultaneity and thereby revising the way in which inertial frames having different velocities are supposed to be related to one another (by the Lorentz rather than the Galilean transformations).267 Building on Einstein's work, Minkowski subsequently arrived in 1908 at a fundamentally new mathematical structure for describing space, time, and motion: the structure of what we now call Minkowski space-time. Nevertheless, this new space-time structure is still related to the empirically given phenomena observable in (approximately)
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inertial frames of reference, and these fi-ames, in turn, are related to our sensible perceptual experience of the life-world in a way precisely analogouS to the situation in Newtonian theory.268 Yet the empirical phenomenon of gravitation, as described by Newtonian theory, could not be easily reconciled, in turn, with the new spatia-temporal structure of Minkowski space-time. For Newtonian universal gravitation acts instantaneously across arbitrarily large spatial distances (across the solar system, for example), and it thus involves, at the . level of space-time structure, a relation of absolute simultaneity-precisely the relation now rejected by Einstein's new theory. Einstein therefore embarked on the project of developing a field theory of gravitation compatible with the new space-time structure, one in which gravitational interaction would propagate with the velocity of light. The theory of general relativity, published in 1915-16, is Einstein's solution; and, in this theory, the classical law of inertia-that free particles affected by no external forces traverse Euclidean straight lines with constant speed in any inertial frame (a law which remains equally valid in the special theory: compare note 267 above )-is no longer valid. It is replaced by a new principle, the principle of equivalence, according to which freely falling particles acted on only by gravitation traverse straightest possible four-dimensional trajectories or geodesics in a variably curved perturbation of Minkowski space-time. As a consequence, inertial reference frames (which are supposed to be themselves moving uniformly and rectilinearly) no longer exist in the new space-time structure of the general theory. Nevertheless, there are still local inertial frames which approximate, in small regions and appropriate conditions, the global inertial frames of our previous theories. 269 Hence, there is still an empirically well-defined mathematical route leading from our actual experience of the life-world here on earth to the empirical application (to such phenomena as light, gravitation, and so on) of our ever more abstract mathematical theoretical structures. This sketch of how the mathematical structure of general relativity successively acquires its empirical application and meaning is analogous, as I have suggested, to Husserl's account of the development of the mathematical science of geometry in the Crisis. For Husserl there portrays the development of geometry in terms of a twofold sequence of successive idealizations: the idealization or precisification of the inexact spatial forms given in ordinary perceptual experience by the ideal and exact figures studied in pure geometry, together with a parallel idealization or precisification of the empirical practice of spatial measurement (compare note 258 above, together with the paragraph to which it is appended). Moreover, Husserl's account of how all exact science, more generally, develops by an indefinitely extended sequence of idealizations or approximations is analogous, as we have seen, to the genetic conception of knowledge of the Marburg
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School (compare the paragraph to which note 263 above is appended). The problem, as we have also seen, is that the Marburg School focuses only on the infinitely distant "lin1it-idea" thereby successively generated, and it thereby neglects the perceptual, indexical, and ostensive features of OUf sensible experience originally given in the life-world. The solution, as we now see, is to add an account of how our mathematical theories of space, time, and motion are successively related to concrete, empirically given frames of reference, which are themselves perceptually, indexically, and ostensively centered on a phenomenological I -here-now. 270 And it is in this way, in particular, that transcendental phenomenology can indeed be viewed as a natural extension of Iunt's original transcendental solution to the problem of absolute space. But there is also a fundamental disanalogy. Kant takes the constitutive principles governing his sequence of more and more adequate approximations to the concept of absolute space (basically, the principles of Euclidean geometry and the Newtonian Laws of Motion) to be a priori given once and for all. The entire point of the approach just sketched, however, is to avoid this assumption, and to make room, instead, for the radical transformations in both geometry and mechanics that have characterized the actual historical development from Newton to Einstein. It follows, therefore, that our account of how the sequence of (what we take to be) more and more adequate approximations to a correct mathematical description of space, time, and motion successively acquires its empirical meaning cannot be developed purely a priori. On the contrary, it is an essentially historical account, depending on the concrete details of the actual development in question. It is by no means surprising, then, that Husserl, in the Crisis, explicitly builds historicity into the transcendental subjectivity underlying the originally given life-world: How does one have world-consciousness in the narrower sense, while one has self-apperception as this human being? Here we soon see, as another a priori, that self-consciousness and consciousness of others are inseparable. It is unthinkable, and not merely [contrary to] fact, that I should be a human being in a world, without being a human being. There does not need to be anyone in my field of perception, but fellow human beings are necessary as actual and as known, and as being possibly encountered in an open horizon. I am factually in an inter-human present and an inter-human open horizon; I know myself to be factually in a generative interconnection, in the unitary flow of an historicity, in which this present is an historical present of humankind (together with the world known to it), [with] an historical past and an historical future. (1954,256; 1970, 252-53)
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Thus, the original form of transcendental subjectivity is not simply I-herenoW, as it were, but we-(here-and-there)-(now-and-then). As Husserl himself puts it (1954, 258; 1970,255): "Just as each I-subject has an original perceptual field with an horizon to be opened by free activity leading to ever new perceptual fields, always pre-indicated, in turn, both determinately and indeterminately; so each such subject has its empathy-horizon [Einfiihlungshorizont] , that of its co-subjectivity, to be opened through direct and indirect traffic with the chain of others, each as another for another, who can always have yet further others, and so on.,,271 The essential structures of transcendental subjectivity (in a "reduced" form not assuming anything about the real existence of this or that subject or community of subjects) therefore include the structures of a universal, trans-historical human community: If! posit validity to empathy (in accordance with its intentional validity-correlates in "co-performance"), [these structures] become essential structures for every alter-ego thinkable by me, and the problems then arise of the universal community established through empathy and its essential special forms-precisely those which appear as objectivized in the natural world-view, namely, as family, people, community of peoples, and then as essential structures of human historicity. They here arise, however, in a reduced form, and result in the essential structures of absolute historicity, namely, that of a transcendental community of subjects, which, living through intentional community in these most universal and also special a priori forms, has the world as an intentional validity-correlate and continually creates it anew, in ever new forms and stages of a cultural world. (1954,262-63; 1970,259) It is an essential a priori feature of transcendental subjectivity, in other words, that the world it has as an intentional correlate not only necessarily includes an historical community comprehending all human subjects (together with its sub-communities), it is also necessarily viewed as being continually "co-created" (qua intentional object of knowledge) as an historical cultural product of this same community. It is in this sense that all scientific knowledge, in the Crisis, is viewed as necessarily historical, including the science of transcendental phenomenology itself. 272 However, although it might thereby be possible to articulate the necessary a priori structures governing all possible historical scientific developments, it is far from clear that we can then give a transcendental explanation of how certain specific scientific theories-at certain specific stages of scientific development-acquire their empirical meaning. Such an explanation, as we have seen, depends on concrete details of the actual historical process, and it would therefore appear to be
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entirely contingent. So how can such an explanation possibly be comprehended within a properly transcendental philosophy? Indeed, once we have given up on IZant's original ambition to delineate in advance the a priori structure of all possible scientific theories, it might easily seem that a properly transcendental explanation is hopeless. We have no way of anticipating a priori the specific constitutive principles of future theories, and so all we can do, it appears, is wait for the historical process to show us what emerges a posteriori as a matter of fact. How, then, can we develop a philosophical understanding of the evolution of modern science that is at once genuinely historical and properly transcendental? This question arises for any attempt to make transcendental philosophy essentially historical, not only for transcendental phenomenology. In Cassirer's version of the genetic conception of knowledge characteristic of the Marburg School, for example, scientific development is depicted as an historically given sequence of abstract mathematical structures converging on-but never actually arriving at-an ideal limit-structure. In this way, as we have seen, the development in question is framed by the regulative ideal of a "universal invariant theory of experience" containing all previous stages of the sequence as approximate special cases (compare the paragraph to which note 244 above is appended, together with the two following paragraphs). But this purely regulative ideal equally frames every stage of the historical sequence, and it tells us nothing specific, therefore, about any given stage. 273 Similarly, where Cassirer has characterized a necessary transcendental ideal or endpoint for the historical sequence, Husserl (here more explicitly following IZant himself) has characterized a necessary origin in transcendental subjectivity-in the immediately given "subjectiverelative" form of the life-world. Yet, once again, this can tell us nothing specific about any given stage in the development of the mathematicalphysical sciences, and, in particular, it would seem that it cannot transcendentally illuminate the historical process by which the empirical meaning of any given stage is actually constituted. One way of understanding my own project for an essentially historical form of transcendental philosophy-which I call the dynan1ics of reason-is as an attempt to resolve precisely this problem. 274
Synthetic History and the Dynamics of Reason What I call the dynamics of reason, then, is an atten1pt to describe the historical developments on which we have principally been focusing-the evolution of space-time physics from Newton to Einstein-within a reconceived version of I
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temporary post-Kuhnian philosophy of science remains focused on precisely these developments and, in particular, on what Kant took to be a priori constitutive principles of geometry and mechanics. Although Kant had rery good reasons to view the principles in question as having such a constitutively a priori role, we now know, of course, that they are not in fact a priori in the stronger sense of being fixed necessary conditions for all human experience in general, eternally valid once and for all. And it is for precisely this reason that Kant's original version of transcendental philosophy must now be radically reconceived. I begin, therefore, with the idea of relativizing Kantian constitutive principles of geometry and mechanics, as first proposed by Reichenbach in his 1920 book on The Theory ofRelativity and A Priori Knowledge. Such principles still function, throughout the development from Newton to Einstein, as necessary presuppositions for applying our (changing) conceptions of space, time, and motion to our sensible experience, but they are no longer eternally valid once and for all. 275 Instead of global necessary conditions for all human experience in general, we have merely local necessary conditions for the empirical application of a particular mathematical-physical theory at a given time and in a given historical context. For example, while Euclidean geometry and the Newtonian Laws of Motion are indeed necessary conditions for giving empirical meaning to the Newtonian theory of universal gravitation, the situation in Einstein's general theory of relativity is quite different. The crucial mediating role between abstract mathematical theory and concrete sensible experience is now played by the light principle and the principle of equivalence, which together ensure that Einstein's revolutionary new description of gravitation by a semi-Riemannian (infinitesimally Minkowskian) four-dimensional geometry of variable curvature in fact says something about concrete empirical phenomena: namely, the behavior of light and gravitationally interacting bodies. I have defended and further articulated this conception of what I call the relativized a priori in a number of places-most fully in Friedman (2001, Part Two, §§l, 2)-and I shall not repeat these arguments here. As suggested several times above, however, I now think that my earlier presentations were too closely tied to the problem, first formulated by Reichenbach and Schlick, of establishing a "coordination" (Zuordnung) between abstract mathematical structures and concrete empirical phenomena. 276 The difficulty arises when one accepts the sharp distinction, emphasized by Schlick, between an uninterpreted axiomatic system and intuitive perceptible experience, and one then views the constitutive principles in question (which, following Reichenbach, I called "coordinating principles" or "axioms of coordination") as characterizing an abstract function or mapping associating the former with the latter. This picture is
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deeply problematic, I now believe, in at least two important respects: it assumes an overly simplified "formalistic" account of modern abstract mathematics, and, even worse, it portrays such abstract mathematics as being directly attached to intuitive perceptible experience at one fell swoop. As we have seen, there is an important tradition in modern abstract mathematics, associated with Felix Klein's work in group-theory and projective geometry, which self-consciously resists such overly "formalistic" interpretations of Hilbert's axiomatic method and, accordingly, still maintains a balance between the modern axiomatic method and the more intuitive conception of mathematics originally associated with Kant. 277 Moreover, as Ryckman (2005) has emphasized, Hermann Weyl's deep contributions to the mathematical foundations of general relativity represent a natural continuation of this same tradition, and so it is by no means the case, for example, that a "formalistic" picture of modern abstract mathematics is necessarily forced upon us by this particular theory.278 Finally, as Scott Tanona has emphasized in his contribution to the present volume, it is a serious mistake to view our abstract mathematical representations (in either Newtonian mechanics, special relativity, or general relativity) as selfcontained theoretical structures directly and independently "coordinated" with perceptual experience. Rather, later theories draw their empirical meaning from earlier theories, viewed as necessary presuppositions for describing the concrete frames of reference within which alone actual empirical phenomena can then be exhibited. 279 Our problem, therefore, is not to characterize a purely abstract mapping between an uninterpreted formalism and sensory perceptions, but to understand the concrete historical process by which mathematical structures, physical theories of space, time, and motion, and mechanical constitutive principles organically evolve together so as to issue, successively, in increasingly sophisticated mathematical representations of experience. We have sketched such an historical account in the last section, appealing to a particular interpretation ofHusserl's conception of transcendental phenomenology in the Crisis. It is all the more urgent, therefore, that we now develop a response to the final question I raised there concerning how to reconcile the necessity and a priority demanded by transcendental philosophy with the unavoidable contingency of history. This problem, as I suggested, is not particularly salient for Husserl himself, for his aim, it appears, was only to delineate the transcendental conditions of historicity in general constitutive of the immediately given life-world, not to give a transcendental explanation of the particular concrete developments which then may occur against the background of these general conditions. Since we, however, are attempting to attribute some kind of transcendental significance to particular sets of constitutive principles produced at particular
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times and in particular historical contexts-we are proposing the "relativized a priori" as a necessary revision of Kant's original conception-it is definitely incumbent on us to respond to this problem. The response I wish to propose here is two-fold. First, the historical evolution from Newton to Einstein essentially involves an important set of developments in modern abstract mathematics, and pure mathematics itself is a central (and relatively uncontroversial) locus of a priority and necessity. Second, and more importantly, the historical evolution in question also essentially involves an extension or continuation of Kant's original attempt to comprehend the application of mathematics to our sensible experience within what he called transcendental philosophy, and Kant's original conception thereby continues actively to inform the development of new sets of constitutive principles (while, at the same time, undergoing a radical transformation in turn). With respect to the first point, then, pure mathematics itself (the very mathematics whose application to sensible experience we are attempting transcendentally to comprehend) has a necessary a priori deductive structure of incredible richness and complexity. It is for precisely this reason that rationalist philosophy from Plato to the present has always taken such mathematics to be the prime exemplar of the a priori rational necessity it then attempts to find much more generally. Beginning with Kant, however, this kind of rationalism took an unexpected transcendental turn: the special a priori status of pure mathematics consists in the circumstance that, although it is not strictly objective knowledge on its own, it nonetheless functions as a necessary presupposition of all empirical knowledge (compare note 236 above).280 Kant's transcendental explanation of applied mathematics thereby explains the necessary a priori character of pure mathematics as well, and this answer to the question "how is pure mathematics possible?" then inherits, by the "analytic" or "regressive" method of the Prolegomena, the necessary a priori character of pure mathematics itself. 281 Of course Kant's main example of this method talces traditional Euclidean geometry for granted, and Kant articulates his characteristic conception of space as the pure form of our outer sensible intuition as an explanation (among other things) for the astonishingly productive use of this geometry in Newtonian physics. In modern mathematics, as we have seen, geometry in particular has undergone a profound transformationso as to include projective geometries, non-Euclidean geometries of constant curvature, the theory of transformation groups, Riemann's theory of manifolds (including geometries of variable curvature of any dimension), the Helmholtz-Lie theorem, and so on-and these developments culminate, in a sense, in Einstein's application of a four-dimensional geometry of variable curvature to a real physical interaction in nature. Nevertheless,
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despite this profound transformation, it does not follow that geometry, as a purely mathematical subject, exhibits any less deductive necessity. On the contrary, we now have a much richer and more general deductive structure in which traditional Euclidean geometry takes its place among a vastly expanded set of geometrical possibilities, all woven tightly together with one another (and with other branches of modern mathematics) by a vastly expanded set of necessary deductive relations. There is still room at least to hope, therefore, that a transcendental explanation of the application of modern geometry to nature might still be possible. Whereas, for example, it is of course entirely contingent that Riemann developed his theory of manifolds in 1854 and Einstein then applied it to nature in 1915-16, the mathematical deductive relationships within this theory are not contingent at all. 282 But the real question, of course, concerns the nature and possibility of precisely such a transcendental explanation. What would such an explanation look like, and how, in particular, could it somehow "inherit" a priori necessity from the pure mathematics whose application to nature it seeks transcendentally to explain? How, for that matter, did Kant's own transcendental explanations "inherit" such a priori necessity? The answer to this last question requires a more careful discussion of the synthetic method of the first Critique, which is supposed to complement, as I have suggested, the analytic method of the Prolegomena (compare note 281 above, together with the paragraph to which it is appended). And, once we have become clearer about this question, it might then be possible to extend this synthetic method to our own post- Kantian philosophical situation. Kant's synthetic method, as we have seen, appeals to his conception of the two rational faculties of sensibility and understanding. The answer to the question "how is pure mathematics possible?" appeals to the necessary structure of our pure sensibility, as articulated, synthetically, in the Transcendental Aesthetic; the answer to the question "how is pure natural science possible?" appeals to the necessary structure of our pure understanding, as articulated, synthetically, in the Transcendental Analytic. Yet there is an obvious objection to this procedure: how can such proposed transcendental explanations inherit the (assumed) a priori necessity of the sciences whose possibility they purport to explain unless we can also somehow establish that they are the unique such explanations?283 From our present point of view, for example, it does not appear that Kant's explanation of the possibility of pure mathematics is uniquely singled out in any way; on the contrary, our greatly expanded conception of purely logical or analytic truth suggests that an appeal to the faculty of pure sensibility may, after all, be entirely superfluous. Indeed, from the point ofview of the antipsychological approach to such questions that dominated much of twenti-
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eth-century analytic philosophy, it appears that all consideration of our subjective cognitive faculties is similarly explanatorily superfluous. In Kant's own intellectual context, however, explanations of scientific knowledge in terms of our cognitive faculties were the norm-for empiricists, rationalists, and (of course) Aristotelians. Everyone agreed, in addicion, that the relevant faculties to consider were the senses and the intellect; what was then controversial was the precise nature and relative importance of the two. As we have seen, empiricist views, which denied the existence of the pure intellect or its importance for scientific knowledge, were, for Kant, simply out of the question, since they make a priori rational knowledge incomprehensible (see again note 281 above). Moreover, the conception of the pure intellect that was most salient for Kant, as we have also seen, was that of Leibniz, where the fundamental structure of this faculty is delineated, in effect, by the logical forms of traditional Aristotelian syllogistic. But this conception of the pure intellect, Kant rightly saw, is entirely inadequate for representing, say, the assumed infinite extendibility and divisibility of geometrical space, which had recently proven itself to be both indispensable and extremely fruitful in Newtonian mathematical physics. Nevertheless, Newton's own conception of space as the divine sensorium was also entirely unacceptable on theological and metaphysical grounds, and so the only live alternative left to Kant was the one he actually came up with: space is a pure form of our sensibility (as opposed to the divine sensibility), wherein both (infinitely iterable) geometrical construction and the perception of spatial objects in nature (like the heavenly bodies) then become first possible. 284 Kant's answer to the question "how is pure mathematics possible?"which, as we have seen, also necessarily involves an explanation of how such mathematics is empirically applied (compare note 280 above, together with the paragraph to which it is appended)-therefore operates against the background of an existing set of intellectual resources in a particular historical context. Geometry, for Kant, is limited to the classical system of Euclid; the pure understanding or pure intellect is delimited by the logical forms of Aristotle; the available conceptions of space and time are exhausted by the Leibnizean and Newtonian alternatives; and so on. Kant's construction of a synthetic description of our faculties of sensibility and understanding can only be understood against the background of precisely these resources-mathematical, logical, metaphysical, and theological-as Kant delicately navigates within them and eventually radically transforms them. The revolutionary and completely unexpected result, that space and time are pure forms of our (human) faculty of sensibility and that, considered independently of sensibility, our faculty of understanding yields no (theoretical) cognition at all, then emerges as the practically unique solution to the problem set by the existing intellectual
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resources: it is the only available conception of our rational faculties that does simultaneous justice to both Newtonian mathematical physics and Leibnizean (as opposed to Newtonian) natural theology and metaphysics. It is of course entirely contingent that I
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As we have seen, for example, Helmholtz's neo- Kantian scientific epistemology had deep roots in Kant's original conception. For, on the one hand, Helmholtz explicitly rejected the Naturphilosophie of Schelling and Hegel in favor of a return to Kant's original dynamical theory of matternOW understood, however, in a way that does justice to the new empirical discoveries in energetics, electricity and magnetism, chemistry, and biology which, in part, had inspired this Naturphilosophie. 286 And Helmholtz's dynamical theory of matter, like Kant's original theory, represented a particular realization of a more general principle of causality or lawlikeness in nature.2 87 On the other hand, however, Helmholtz also developed a distinctive conception of space as a "subjective" and "necessary form of our external intuition" in the sense of Kant (see the paragraph to which note 140 above is appended); and, while this conception was certainly developed within Helmholtz's empirical program in sensory psychology and psycho-physics, it nevertheless retained important "transcendental" elements. 288 In particular, space is "transcendental," for Helmholtz, in so far as the principle offree mobility is a necessary condition for the possibility of spatial measurement-and, indeed, for the very existence of space and spatial objects. This analysis, as we have seen, represents the heart of Helmholtz's characteristic theory of space-perception, and it culminates in his purely mathematical contribution to the "space problem" now known as the Helmholtz-Lie theorem (compare notes 141 and 142 above, together with the paragraph to which they are appended). Moreover, as we have also seen, the condition of free mobility-which permits arbitrary continuous motions of rigid bodies-represents a natural generalization of Kant's original (Euclidean) conception of geometrical construction, in the sense that Euclidean constructions with straight-edge and compass, carried out within Kant's form of spatial intuition, are generated by the group of specifically Euclidean rigid motions (compare note 39 above, together with the paragraph to which it is appended). The essential point, however, is that free mobility also holds for the classical non-Euclidean geometries of constant curvature (hyperbolic and elliptic), and so it is no longer a "transcendental" and "necessary" condition of our spatial intuition, for Helmholtz, that the space constructed from our perception of bodily motion obeys the specific laws of Euclidean geometry (compare note 148 above, together with the paragraph to which it is appended). Nevertheless, the Helmholtz-Lie theorem also shows, from this point of view, that Helmholtz's generalization of the Kantian conception of spatial intuition is, in an important sense, the minimal such generalization consistent with the nineteenth-century discovery of non-Euclidean geometries. 289 In sum, although Helmholtz's theory of space perception was much more clearly and explicitly anchored in empirical psychology than Kant's
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(see again note 288 above), his conception of both causality and geometry retained important elements of Kant's "transcendental" faculty psychology. The requirement of lawlikeness, in particular, represents the fundamental a priori (regulative) principle arising from our conceptual or intellectual faculty, and this principle, applied to our sensible or receptive faculty in the guise of the principle of free mobility, then results in a "necessary" and "transcendental" description of the structure of our spatial intuition: not specifically Euclidean geometry, but only the more general structure common to the three classical geometries of constant curvature, is now seen as a necessary presupposition for both spatial measurement and all perception of the objects of our spatial experience. 290 Poincare, as we know, then radically transformed Helmholtz's theory in turn. Although he, like virtually everyone else in the late nineteenth century, maintained a serious interest in the new biology, psychology, and psycho-physics, Poincare's own principal orientation, of course, was directed towards pure and applied mathematics (compare note 161 above, together with the paragraph to which it is appended). In particular, Poincare's use of group theory and the Helmholtz-Lie theorem-although clearly indebted to Helmholtz's work (see note 166 above, together with the paragraph to which it is appended)-was explicitly framed by a hierarchical conception of the mathematical sciences reflecting Poincare's distinctive approach to the philosophy of mathematics. This approach was developed within the Kleinian tradition, where (as we have said) one still maintained a balance between the purely abstract use of the modern axiomatic method and the more concrete and intuitive conception of mathematics associated with Kant, and, for Poincare, it followed that one should thereby explain the application of pure mathematics to our perceptual experience in terms of precisely the hierarchical conception in question. 291 Poincare, to begin with, views pure arithmetic as a synthetic a priori science involving the ineliminable use of an essentially non-logical principle of reasoning by recurrence or mathematical induction. This principle, for Poincare, rests on the fundamental intuition of indefinitely repeatable succession or iteration-a conception which, as I have said, is very close indeed to Kant's original philosophy of arithmetic (compare note 164 above, together with the paragraph to which it is appended). At the next lower level of the hierarchy is analysis or the theory of mathematical magnitude (also explained with an eye towards its intuitive meaning and perceptual application: see the same paragraph); and, at a crucial intermediate level, below the sciences of arithmetic and analysis but above the sciences of mechanics and empirical physics, is the science of geometry. In partieular, whereas the mathematical structure and empirical meaning of the science of geometry presupposes the existence of the two preceding sciences, it is presupposed, in turn, by the two succeeding ones.
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This hierarchical conception of the mathematical sciences underlies poincare's fundamental disagreement with Helmholtz. For Helmholtz, as we have seen, the principle of fi:ee mobility expresses the necessary structure of our form of external intuition, and, following Kant, Helmholtz views all empirical investigation as necessarily taking place within this already given form. Helmholtz's conception is Kantian, that is, in so far as space has a "necessary form" expressed in the condition of free mobility, but it is also empiricist in so far as which of the three possible geometries of constant curvature obtains is then determined by experience (see the paragraph following the one to which note 166 above is appended). For Poincare, by contrast, although the principle of fi-ee mobility is still fundamental, our actual perceptual experience of bodily "displacements" arising in accordance with this principle is far too imprecise and indefinite to yield the empirical determination of a specific mathematical geometry: our only option, at this point, is to stipulate Euclidean geometry by convention, as the simplest and most convenient idealization of our actual perceptual experience (see note 167 above, together with the paragraph to which it is appended). For, as we have also seen, experiments with putatively rigid bodies, for Poincare, involve essentially physical processes at the level of mechanics and experimental physics, and these sciences, in turn, presuppose that the science of geometry is already firmly in place (compare note 168 above, together with the paragraph to which it is appended). In the context of Poincare's hierarchy, therefore, the principle of free mobility and the Helmholtz-Lie theorem precisely express our necessary freedom to choose-by a "convention or definition in disguise"-which of the three classical geometries of constant curvature is the most suitable idealization of physical space. One of the most striking and important applications of Poincare's hierarchical conception involves his characteristic perspective on the problem of absolute space and the relativity of motion expressed in his discussion of the next lower level in the hierarchy, (classical) mechanics. Ai> we have seen, Poincare's key idea is that what he calls the (physical) "law of relativity" rests squarely on the "relativity and passivity of space" and therefore reflects the circumstance, essential to free mobility and the Helmholtz-Lie theorem, that the space constructed from our experience of bodily displacements is both homogeneous and isotropic: all points in space, and all directions through any given point, are, necessarily, geometrically equivalent. Thus Poincare's conception of the relativity of motion depends entirely on his philosophy of geometry; and this is especially significant, from our present point of view, because Poincare's ideas on the relativity of motion were also inextricably entangled with the deep problems then afflicting the electrodynamics of moving bodies that were eventually solved (according to our current understanding) by Einstein's special theory of relativity.292
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I shall return to Einstein below, but I first want to emphasize that the connection Poincare makes between his philosophy of geometry and the relativity of motion represents a continuation of a problematic originally prominent in Kant. In particular, whereas Helmholtz, as we have seen transformed Kant's philosophy of space and geometry, and Mach, as w~ have also seen, participated in a parallel transformation of Kant's approach to the relativity of motion, neither Helmholtz nor Mach established a connection between the two (compare note 160 above, together with the paragraph to which it is appended and the three preceding paragraphs). Yet it was an especially central feature of Kant's original approach to transcendental philosophy that the two problems were in fact closely connected. While Kant's answer to the question "how is pure mathematics possible1" essentially involved his distinctive perspective on Euclidean constructive operations, his answer to the question "how is pure natural science possible?" involved the analogous constructive procedure by which Newton, from Kant's point of view, arrived at successive approximations to "absolute space" via a definite sequence of rule-governed operations starting with our parochial perspective here on earth (compare notes 45 and 46 above, together with the paragraphs to which they are appended and the two following paragraphs). Indeed, the way in which Kant thereby established a connection between the problem of space and geometry and the problem of the relativity of motion was intimately connected, in turn, with his overarching conception of the relationship between sensibility and understanding, his fundamental distinction between mathematical and dynamical principles of pure understanding, and his characteristic perspective, more generally, on the relationship between constitutive and regulative transcendental principles. 293 Now it was Mach, as I have suggested, who first forged a connection between Kant's original solution to the problem of "absolute space" and the late nineteenth-century solution based on the concept of an inertial frame ofreference. 294 And it is clear, moreover, that Poincare himself was familiar with this same late nineteenth-century solution (compare note 175 above, together with the paragraph to which it is appended). However, it is also clear that Poincare's attempt to base his discussion of the relativity of motion on his philosophy of geometry runs into serious difficulties at precisely this point; for, as we have seen, he is here forced to distinguish his "law of relativity" from the "principle of relative motion." The latter applies only to inertial frames of reference, moving uniformly and rectilinearly with respect to one another, while the latter applies, as well, to noninertial frames of reference in a state of uniform rotation: it follows from the "relativity and passivity" of space, for Poincare, that uniform rotations of our coordinate axes should be just as irrelevant to the motions of a physical system as uniform translations. Therefore, the full "law of relativity," as
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poincare says, "ought to impose itself upon us with the same force" as does the more restricted "principle of relative motion." Poincare must also admit, however, that the more extended "law of relativity" does not appear to be in accordance with our experiments (e.g., Newton's famous rotating . ) .295 bucket expenment It is for this reason, as we have seen, that Einstein's appeal to what he calls the "principle of relativity" in 1905 is entirely independent of poincare's "law of relativity," and it is also independent, accordingly, of Poincare's "conventionalist" philosophy of geometry. Einstein's principle is limited, from the beginning, to inertial frames of reference (moving relative to one another with constant velocity and no rotation), and his concern is rather to apply this (limited) principle of relativity to both electromagnetic and mechanical phenomena. Thus, in particular, whereas Poincare's "law of relativity" involves very strong a priori motivations deriving from his philosophy of geometry (based on the "relativity and passivity of space"), Einstein's "principle of relativity" rests on the emerging experimental evidence suggesting that electromagnetic and optical phenomena do not in fact distinguish one inertial frame from another. Einstein "conjectures" that this experimentally suggested law holds rigorously (and for all orders), and he proposes to "elevate" it to the status of a presupposition or postulate upon which a consistent electrodynamics of moving bodies may then be erected. Hence, Einstein's understanding of the principle of relativity is also entirely independent of Poincare's carefully constructed hierarchy of the mathematical sciences, and it is for precisely this reason, I have suggested, that Poincare himself could never accept Einstein's theory.296 Nevertheless, as I have also suggested, it appears overwhelmingly likely that, although Einstein did not embrace Poincare's "conventionalist" philosophy of geometry, Einstein's use of the principle of relativity was explicitly inspired by Poincare's more general methodology described in Science and Hypothesis-according to which the fundamental principles of mechanics, in particular, are "conventions or definitions in disguise" arising from "experimental laws" that "have been elevated into principles to which our mind attributes an absolute value" (see notes 186 and 187 above, together with the paragraphs to which they are appended). In Einstein's case, the experimental law in question comprises the recent results in electrodynamics and optics, and Einstein now proposes to "elevate" both the principle of relativity and the light principle (which together imply that the velocity of light is invariant in all inertial frames) to the status of "presuppositions" or "postulates." These two postulates together then allow us to "stipulate" a new "definition of simultaneity" (based on the assumed invariance of the velocity oflight) implying a radical revision of the classical kinematics of space, time, and motion. In particular,
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whereas the fundamental kinematical structure of an inertial frame of ref erence in classical mechanics is defined by the three Newtonian Laws of Motion, (a revised version of) this same structure in Einstein's theory is rather defined by his two postulates. 297 A central contention of lZant's original version of transcendental philosophy, as we know, is that the three Newtonian Laws of Motion are not mere empirical laws but a priori constitutive principles on the basis of which alone the Newtonian concepts of space, time, and motion can then have empirical application and meaning. What we have just seen is that Einstein's two fundamental "presuppositions" or "postulates" playa precisely parallel role in the context of special relativity (compare again note 297 above). But we have also seen significantly more. For Poincare's conception of how a mere empirical law can be "elevated" to the status of a "convention or definition in disguise" is a continuation, in turn, of I
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Yet Einstein's creation of the general theory of relativity, as we have seen, involved an even more striking engagement with Poincare's "conventionalist" methodology (compare note 207 above, together with the paragraph to which it is appended), which, I contend, makes the transcendentally constitutive role of this theory's fundamental postulates (the light principle and the principle of equivalence) even more evident. As we have also seen, however, the point is a rather subtle one, for Einstein had to reject precisely Poincare's "conventionalist" philosophy of geometry in this context, and he had explicitly to reject it, moreover, in favor of Helmholtz'S opposing "empiricist" conception (compare note 209 above, together with the paragraph to which it is appended). Indeed, the circumstance that Einstein explicitly embraces "empiricism" here can easily suggest-as it did for some of the early logical empiricists and continues to do for some contemporary Einstein interpreters-that the general theory, in particular, entails the final elimination of all Kantian and neo-Kantian conceptions of the a priori. 301 What I now want to argue, on the contrary, is that this very rejection of Poincare in favor of Helmholtz should rather be seen as an especially important vindication of the present neo- Kantian conception of a transcendentally constitutive relativized a priori. The first point to make is that the principle of equivalence (together with the light principle) plays the same role in the context of the general theory that Einstein's two fundamental "presuppositions" or "postulates" played in the context of the special theory: namely, they define a new inertial-kinematical structure for describing space, time, and motion. Because Newtonian gravitation theory involves an instantaneous action at a distance (and therefore absolute simultaneity), it was necessary after special relativity to develop a new theory of gravitation where the interactions in question propagate with the velocity of light. Einstein solved this problem, via the principle ofequivalence, by defining a new inertial-kinematical structure wherein the freely falling trajectories in a gravitational field replace the inertial trajectories described by free particles affected by no forces at all (the principle of equivalence thereby replaces the classical law of inertia). But, as we have seen, the principle of equivalence rests on a well-known empirical fact: that gravitational and inertial mass are equal, so that all bodies, regardless of their mass, fall with exactly the same acceleration in a gravitational field. In using the principle of equivalence to define a new inertial-kinematical structure, therefore, Einstein has "elevated" this merely empirical fact (recently verified to a quite high degree of approximation by Lorand von Eotvos) to the status of a "convention or definition in disguise"-just as he had earlier undertaken a parallel "elevation" in the case of the new concept of simultaneity introduced by the special theory. 302 Nevertheless, as I have emphasized, Einstein did not reach this understanding of the principle of equivalence all at once. He first operated,
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instead, within an essentially three-dimensional understanding of special relativity, and he proceeded to develop relativistically acceptable models of the gravitational field by considering the inertial forces (like centrifugal and Coriolis forces) arising in non-inertial frames of reference within this framework (see note 204 above, together with the paragraph to which it is appended). It was in precisely this context, in particular, that Einstein came upon the example of the uniformly rotating frame, and it is at this point (and only at this point) that he then arrived at the conclusion that the gravitational field may be represented by a non-Euclidean geometry. Moreover, as I have also emphasized, this use of non -Euclidean geometry was essentially three-dimensional, limited to purely spatial geometry, and Einstein did not arrive at the idea of a four-dimensional non-Euclidean geometry-where space-time geodesics represent freely falling trajectories affected only by gravitation-until he had generalized his conception to what we now call the four-dimensional (semi- )Riemannian geometries of variable curvature. 303 It was in precisely the context of this line of thought, finally, that Einstein found that he now had explicitly to reject Poincare's "conventionalist" philosophy of geometry (see again note 207 above, together with the paragraph to which it is appended). Yet this was far from a simple rejection of Poincare's methodology in favor of a straightforward "empiricism." For Einstein also explicitly says, in the same passage, that "sub specie aeterni" Poincare is actually correct-so that, in particular, Einstein's reliance on a Helmholtzian conception of "practically rigid bodies" is here merely provisional. I have suggested, therefore, that we can best understand Einstein's procedure as one of delicately situating himself between Helmholtz and Poincare. Whereas Einstein had earlier followed Poincare's general "conventionalist" methodology in "elevating" the principle of relativity (together with the light principle) to the status of a "presupposition" or "postulate," he here follows Helmholtz's "empiricism" in rejecting Poincare's more specific philosophy of geometry in favor of "practically rigid bodies." It does not follow, however, that Einstein is also rejecting his earlier embrace of Poincare's general "conventionalist" methodology. Indeed, Einstein had already opposed Poincare's philosophy of geometry in the case of special relativity, and for essentially the same reason he does so here: Poincare's rigid hierarchy of the sciences, in both cases, stands in the way of the radical new innovations Einstein himself proposes to introduce. 304 But why was it necessary, after all, for Einstein to engage in this delicate dance between Helmholtz and Poincare? The crucial point is that Einstein thereby arrived at a radically new conception of the relationship between the foundations of (physical) geometry and the relativity of space and motion. These two problems, as we have seen, were closely connected
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in Kant, but they then split apart and were pursued independently in Helmholtz and Mach (compare the paragraph to which note 293 above is appended). In Poincare, as we have also seen, the two were perceptively reconnected once again, in so far as Poincare's hierarchical conception of the mathematical sciences incorporated both a modification of Helmholtz's philosophy of geometry and a serious engagement with the late nineteenth-century concept of inertial frame (see the paragraph to which note 292 above is appended, together with the paragraph to which note 294 is appended). Indeed, it is for precisely this reason, as we now see, that Poincare's scientific epistemology was so important to Einstein. Einstein could not simply rest content with Helmholtz's "empiricist" conception of geometry, because the most important problem with which he was now faced was to connect the foundations of geometry with the relativity of motion. But Einstein could not rest content with Poincare's conception either, because his new models of gravitation had suggested that geometry has genuine physical content. Einstein's radically new way of reconfiguring the relationship between the foundations of geometry and the relativity of motion therefore represents a natural (but also entirely unexpected) extension or continuation of the same conception of dynamical and relativized constitutive a priori principles he had first instantiated in the creation of special relativity (compare the paragraph to which note 299 above is appended, together with the two preceding paragraphs). Just as he had earlier shown how an extension or continuation of Kant's original conception could accommodate new and surprising empirical facts (the discovery of the invariance of the velocity of light), Einstein here shows how a further extension of this same tradition can do something very similar in facilitating, for the first time, the application of a non-Euclidean geometry to nature. In this case, however, it is not the relevant empirical fact (the well-known equality of gravitational and inertial mass) that is surprising, but the entirely unforeseen connection between this fact and the new geometry. And what makes this connection itself possible, for Einstein, is precisely the principle of equivalence-which thereby constitutively frames the resulting physical space-time geometry of general relativity in just the same sense that Einstein's two fundamental "presuppositions" or "postulates" had earlier constitutively framed his mathematical description of the electrodynamics of moving bodies in special relativity (what we now call the geometry of Minkowski space-time).305 Whereas the particular geometry in a given general relativistic space-time is now determined entirely empirically (by the distribution of mass and energy in accordance with Einstein's field equation), the principle of equivalence itself is not empirical in this sense. This principle is presupposed-as a transcendentally constitutive condition-for any such geometrical description of space-time to have genuine empirical meaning in the first place. 306
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Post-Kuhn ian philosophy ofscience As I indicated at the beginning, the approach to transcendental philosophy I call the dynamics of reason is intended not only to make sense of the historical progression from Newton to Einstein, but also to have implications for contemporary post-Kuhnian philosophy ofscience, and to contain a response, in particular, to Kuhn's conception of scientific revolutions. The relationship between the two projects, as I have suggested before, is that Kuhn's central notion of a temporarily agreed upon paradigm definitive of a given episode of normal science is analogous, in important respects, to the notion of the relativized constitutive a priori first proposed by the early Reichenbach. Indeed, Kuhn himself, very late in his career, explicitly acknowledged the analogy, and he then characterized his view, accordingly, as "Kantianism with movable categories. "307 Finally, one of the central examples Kuhn uses to articulate and defend his conception is precisely the transition between Newtonian mechanics and Einsteinian relativity theory-the very example motivating both Reichenbach's introduction of the relativized constitutive a priori and my own conception of the dynamics of reason. 308 It is in the course of considering this example, in particular, that Kuhn invokes the notion of incommensurability to make a devastating point against the prevalent "development-by-accumulation" model of scientific progress he is most concerned to reject. Kuhn argues that the meanings of the fundamental terms of mechanics-terms for space, time, and masshave profoundly changed in the transition from Newton to Einstein. Therefore, the laws that are derived as approximate special cases (involving small velocities, for example) from Einstein's theory are not really Newton's laws, since the fundamental concepts of space, time, and mass have radically different meanings in the two theories. Hence, the relationship between them is not one of straightforward logical comparability; rather, what Einstein says about the world is logically incomparable or incommensurable with what Newton says.309 Kuhn's concept of incommensurability dismayed professional philosophers of science, just as it encouraged and emboldened the emerging postmodern criticisms of the Enlightenment ideal of scientific rationality. If succeeding theories in a scientific revolution are not logically comparable with one another, it seems that there is no longer a clear sense in which the transition from earlier to later is a rational and progressive one. It seems that scientific revolutions, in Kuhn's sense, present us with a radically "relativistic" dilemma: Newton is enclosed within his conceptual framework, Einstein within his, and there is no objective ground for rationally comparing them. Moreover, it is clear, I think, that the revolutionary transition from Newton to Einstein raises a deep and acute challenge to the Enlightenment ideal of rationality as originally championed by Kant quite
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independently of the details of Kuhn's characteristic approach. For it follows from Einstein's new theories, of course, that what Kant took to be the universal constitutive framework governing all human thought and experience-the geometrical-mechanical framework comprising Euclidean aeometry and the Newtonian Laws of Motion-is not fixed and universal ;fter all. We instead have merely local geometrical-mechanical frameworks maintained at particular times and within particular contingent historical circumstances; and, in this sense, the original Kantian ideal of a truly universal and trans-historical mathematical framework governing our fundamental concepts of space, time, matter, and motion was entirely impossible after Einstein's work. Hence, from this point of view, a profound challenge to the original Enlightenment ideal was also now inevitable; and Kuhn's theory of scientific revolutions then embodied this challenge in a particularlv clear and effective way. 'One of the most striking features of Kuhn's approach is its essentially historical character. Kuhn aimed to replace the older philosophical images ofscience with nothing more nor less than a new historiography of science. The history of science, for Kuhn, is not merely a source of examples illustrative of the new philosophical image he wants to promote; this image is itself irreducibly historical-science, for Kuhn, just is an historical process of a certain kind. But what is missing from Kuhn's approach, from my point of view, is a complementary consideration of the parallel developments in the history of mathematics and scientific philosophy that have been intimately connected, in turn, with the developments in the properly empirical sciences he primarily considers. In particular, the transition from Newton to Einstein was intimately connected, as we have seen, with an especially rich set of developments in mathematics and scientific philosophy, beginning with Kant and continuing with Naturphilosophie, Helmholtz, Mach, Poincare, and Einstein himself; and it is precisely here, in my view, that a properly philosophical response to Kuhn can now best be found. 310 My strategy, then, is to present an alternative historical narrative describing the transition from Newton to Einstein that includes the parallel developments in both mathematics and scientific philosophy from Kant through the logical empiricists. 311 I have presented this narrative in an earlier section, and I have given a neo- Kantian philosophical gloss on it in the first part of the present section. I have argued, on the one hand, that the transition from Newton to Einstein centrally involves a succession of relativized constitutively a priori principles (the Newtonian Laws of Motion in the Context of the Principia, the light principle and the principle of relativity in the context of special relativity, the light principle and the principle of equivalence in the context of general relativity), and the existence of such diverse constitutively a priori principles, on my view, captures the
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essence of Kuhnian incommensurability.312 But I have also suggested, on the other hand, that the detailed historical route from earlier to later constitutive principles exhibits the latter as natural transforn1ations of the former-arising as a sequence of "minimal extensions" of our KantianNewtonian starting point in a succession of new mathematical, empirical, and philosophical situations. 313 It is precisely this that shows the later principles to be still transcendentally constitutive in a natural transformation and extension of I
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the best example to consider when studying the structure and evolution of science. Koertge then suggests, quite reasonably, that my Kantian apparatUS of relativized constitutively a priori principles may not apply very directly to the case of chemistry, for example, and, accordingly, that other post-Kuhnian approaches to scientific change, such as Imre Lakatos's, may, in this respect, have a significantly more general domain of application. I want to emphasize, however, that what I call the dynamics of reason is not intended to be a general theory of scientific change at all-rather, it is a particular historical narrative accompanied by a particular philosophical gloss. The point, on my view, is that the transition from Newton to Einstein is the most important challenge to the Enlightenment ideal of scientific rationality championed by Kant (as Kuhn himself eventually came to realize, at least implicitly: see again notes 308 and 310 above), and I am attempting to respond to this challenge, accordingly, by reexamining this particular transition in considerably more historical detaiL317 Most importantly, I give equal emphasis to the complementary developments in both mathematics and scientific philosophy that accompany this transition, and thereby malce it possible, on my telling, to generalize and extend the original Kantian conception in the way just indicated. The special case of geometry and mechanics (our fundamental concepts of space, time, matter, and motion) is centrally important to me, therefore, because it was centrally important to Kant; and the transition from Newton to Einstein, in particular, then reveals both the full force of the resulting challenge to Kant and the means finally to overcome it. 318 This brings me to Alan Richardson's contribution, which raises the question of a Kantian or Hegelian dynamics of reason. Richardson sees a lingering commitment to a problematically Kantian conception of this dynamics in my move from relativized constitutive a priori principles to universal regulative a priori principles-as modeled, in particular, on Cassirer's notion of the ideal limit-theory towards which our historical sequence of mathematical-physical theories is continuously converging. 319 Richardson is dissatisfied, in particular, with the clear suggestion in Friedman (2001, 64-65) that the idea of a continuous approximation to such an ideal limit represents the key element in a post-Kuhnian defense of scientific rationality, and he then proposes a more Hegelian conception where the way in which new theories grow out of preceding ones (regardless of their possible approach to a final limit) is instead given pride of place. I now think that Richardson is basically correct about this. Indeed, especially in Part Two of Friedman (2001), one can already see the beginnings of a shift in emphasis from what I there call retrospective inter-paradigmatic (communicative) rationality (of an earlier conceptual framework seen as an approximate special case of a later one) to what I call prospective inter-paradigmatic (comrnunicative) rationality (of a later conceptual framework seen as growing by
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a series of natural conceptual transformations from an earlier one). My present account of the dynamics of reason is clearly dominated by the latter point of view, and, in this sense, I now think that Richardson was being especially prescient in urging me to emphasize the Hegelian elements in my own approach. These elements are especially clear and explicit, of course, in my present claim that it is precisely the historical development from Kant and Newton to Einstein and the logical empiricists which vindicates my continuing use of Kantian terminology. 320 Nevertheless, it is equally important on my approach to retain this Kantian terminology and, in particular, the fundamental Kantian distinction between regulative and constitutive a priori principles. 321 Although Richardson is certainly correct, in addition, to emphasize the Hegelian elements in Cassirer's version of the genetic conception of knowledge (especially, and very explicitly, in the third volume of the Philosophy of Symbolic Forms), what is missing from Cassirer's approach, as I understand it, is a proper appreciation of Kantian constitutively a priori principles-now relativized, a la Reichenbach, to particular times and particular historical circumstances. For, as I have been arguing, such principles are still needed to explain the (evolving) application of pure mathematics to our perceptual experience in our (evolving) geometrical-mechanical theories of space, time, matter, and motion. 322 Moreover, it is precisely this conception of relativized constitutively a priori principles which then allows us, as I have also argued, both to appreciate the full force of the resulting challenge to Kant's original conception and to find the means finally to overcome it. These means, as it turns out, essentially involve an historicized (and in this sense Hegelian) version of transcendental philosophy, but it is still a version of (Kantian) transcendental philosophy nonetheless. 323 In particular, in my account of the transition from Newton to Einstein we find both constitutive principles determining the empirical meanings of the fundamental concepts of space, time, matter, and motion (the Newtonian Laws of Motion, the light principle and the principle of relativity, the principle of equivalence) and regulative ideas at the level of scientific philosophy mediating and guiding the transition from one set of constitutive principles to another. Such regulative ideas include Helmholtz's scientific epistemology of space and geometry, Mach's critique of absolute space and motion in connection with the emerging new conception of inertial frames, and Poincare's transformation and extension of Helmholtz's epistemology in the light of newer developments in pure mathematics, the concept of an inertial frame, and the apparently paradoxical situation arising in the electrodynamics of moving bodies. For, as we have seen, it is precisely by engaging with these ideas at the level of scientific philosophy that Einstein was then able to craft radically new constitutive principles governing space, time, matter, and motion in first special
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and then general relativity. Such ideas are regulative rather than constituti"e however, because they do not themselves secure the empirical meaning; of the fundamental concepts in question, and they do not, accordingly, define a new space of real or empirical possibilities in relation to these concepts. As Kuhn would describe it, such ideas, at the philosophical or meta-scientific level, do not define even a local consensus J.mong all (or almost all) practitioners of the discipline or subfield in question (in this case, scientific philosophy). Nevertheless, they are still able constructively to guide the process of moving from one set of constitutive principles (one space of real or empirical possibilities) to another without constitutively constraining the outcome. 324 In this case, therefore, the scientific developments in question depend on a complex system of mutual interactions between three distinct levels: constitutive principles determining the empirical meanings of an evolving set of geometrical-mechanical concepts and principles, regulative ideas at the philosophical or meta-scientific level mediating the transition from one such constitutive framework to another, and advances in empirical or experimental inquiry issuing in both new constitutive principles by a process of "elevation" (the Michelson-Morley and related experiments, the Ebtvbs experiments) and empirical tests of the resulting finished theories (the deflection of light in a gravitational field, the anomaly in the perihelion of Mercury). However, it is also worth noting, before we leave this topic, that there are important cases where we find mutual interactions between the regulative or meta-scientific level and the empirical or experimental level with no developed mathematical constitutive framework yet in place. One example we have already considered is the influence of Schelling's Naturphilosophie on Oersted's experimental discovery of electromagnetism, where a perceptive extension of Kant's transcendental phi10sophy in a new philosophical and scientific context played a crucial mediating role in guiding and making possible the experimental discovery in question. 325 Another important example is discussed by William Newman-in his insightful contribution to the present volume and his remarkable recent book, Newman (2006)-where Robert Boyle's creation of the mechanical atomic theory dominating much of early modern chemistry is shown to have arisen from his appropriation of the experimental work of the Aristotelian atomistic "chymical" natural philosopher Daniel Sennert. 326 Experimental investigation can fruitfully interact with the process of scientific theorizing, therefore, not only in the context of precisely formulated mathematical theories, but also at the level of the more loosely formulated conceptions belonging to what we might still appropriately call natural philosophy-where regulative ideas of reason at the meta-scientific level mediate and guide this process towards the eventual construction and articulation of precise mathematical theories. 327
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I now turn, finally, to Robert DiSalle's contribution to the present volume, and to his outstanding new book, DiSalle (2006). DiSalle is centrally concerned, as I am, with "the philosophical development of physics from Newton to Einstein," and he is centrally concerned, as well, with the essential role of the philosophical work of Kant, Helmholtz, Mach, and Poincare in this development. Moreover, he also sees special philosophical significance in this particular transition because certain characteristically philosophical questions about the empirical meanings of fundamental theoretical concepts are crucially, and intrinsically, involved. At the same time, however, DiSalle gives a rather different interpretation of the "inner logic" of the transition in question, one in which what DiSalle calls critical philosophical (or conceptual) analysis plays an internal role in the very construction of the new physical theories. Paradigmatic of this procedure is Einstein's conceptual analysis of simultaneity at the beginning of his 1905 paper-which, for DiSalle, shows critical philosophical reflection essentially at work in the internal practice of science itself. Such reflection not only functions, therefore, at what I call the meta-scientific level, as a kind of external source of rational motivation for making the transition from Newtonian mechanics to special relativity; rather, it is an essential part of the content of Einstein's new theory. 328 DiSalle takes his approach to be complementary rather than opposed to my own: he is adding something to my version of the dynamics of reason rather than contradicting it. I view his approach, similarly, as complementary rather than opposed to mine, and I have no doubt, in particular, that critical conceptual analyses of the kind DiSalle describes in fact playa central role in the development from Newton to Einstein. 329 The difference between us is therefore largely one of emphasis. But perhaps this difference of emphasis is, after all, an important one, for DiSalle (especially in his contribution to the present volume) also takes his approach to exemplifY properly "transcendental" argumentation (and even to vindicate, in a sense, the concept of the synthetic a priori). So the best understanding of the prospects for a properly "transcendental" contemporary philosophy of science may, in the end, be at stake here. And, in any case, my own version of the dynamics of reason can be substantially clarified by carefully considering its relation to DiSalle's. I shall conclude, therefore, by considering the three central cases of critical conceptual analysis described by DiSalleNewton's discussion of space, time, and motion in the Scholium to the Definitions of the Principia, Einstein's analysis of simultaneity in his 1905 paper, and Einstein's use of the principle of equivalence in the foundations of general relativity-where, in each case, I shall focus, more specifically, on the differences between my interpretations and DiSalle's. DiSalle argues that Newton's famous discussion in the Scholium shows, from a philosophical point of view, that Newton is following the same crit-
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ical and empirical method later employed by Einstein, and, accordingly, [here is much more continuity between the two than has been traditionally supposed. Newt~n ~s reacting, in particular, to Descartes's treat~ent of motion in the Prznczples of Phzlosophy (1644), where, on the one SIde, Descartes articulates a vortex theory of planetary motion (and of light) and, on the other, maintains that the proper "philosophical" definition of motion is change of situation relative to the immediately contiguous parts of matter-so that Descartes is then able to say, for example, that the earth is truly at rest because it is not moving relative to the immediately contiguouS parts of the vortex carrying around the moon. 330 This definition, Newton realizes, is dynamically incoherent: Descartes needs the concept of true or absolute rotation in his vortex theory-as the source of centrifugal forces-but he then has no room for this concept in his "philosophical" definition. Newton's well-known discussion of absolute rotation in the Scholium is then an alternative definition of true or absolute motion, precisely in terms of the centrifugal forces thereby resulting. Such forces, in other words, provide us with a well-defined empirical measure of the (true) motions in question. 331 For DiSalle, therefore, Newton is here executing a critical conceptual analysis of the concept of absolute motion based on what DiSalle characterizes as a dialectical engagement with pre-existing mechanical practice and its philosophical interpretation. 332 Existing mechanical practice is founded on the law of inertia, and it is an unavoidable consequence of this law that rotating bodies, in particular, are subject to centrifugal forces arising from the inertial tendency to continue in a straight line tangent to circular motion. Moreover, the accepted principle of the conservation of momentum-or, equivalently, the equality of action and reaction-allows us, in turn, empirically to identity states of absolute rotation, because the centrifugal forces in question are not counterbalanced in any action-reaction pair. It is by proceeding in precisely this way, in fact, that Newton is then able fully to resolve the question of the true motions in the solar system in Book III of the Principia, where the true center of orbital motion is now identified as its center of mass. Newton's new definitions of space, time, and motion thereby make it possible, for the first time, to turn the question of the true motions in the solar system into an empirically answerable one. Newton is not "postulating" absolute space, time, and motion as unobservable "theoretical entities" in order to explain these motions, he is rather crafting empirically applicable definitions of space, time, and motion (in the Scholium), together with corresponding "interpretive principles" (the Laws of Motion), in order to make it possible to describe them. 333 I believe that DiSalle's analysis, as I have suggested, is both correct and very illuminating as far as it goes. But an important piece of the story, from
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my point of view, is still missing. For Newton's application of the Laws of Motion in first describing (and indeed defining) the true motions in the solar system essentially involves applying the conservation of momentum_ or, equivalently, the equality of action and reaction-to precisely the force of universal gravitation, whereby one body exchanges momentum with another body immediately and at a distance. 334 But it was a central tenet of the then dominant mechanical philosophy, of course, that no such action at a distance is possible, and, accordingly, exchanges of momentum on this view, were restricted to interactions by impact. DiSalle (2006: 40-42) suggests that Newton, in developing and applying his Laws of Motion, was simply making explicit the coherent core of the concept of motion already employed by the mechanical philosophers themselves, and Newton thereby demonstrated, in particular, that their restriction to interactions by impact was in fact entirely arbitrary.335 The problem, however, is that, whereas later Newtonians (including Kant) were quite happy to conceive gravitation as an immediate action at a distance across empty space, Newton himself was seriously troubled by this idea. He appeared deliberately to leave it open in the first (1687) edition of the Principia that gravity may ultimately be explained by mechanical impact; and he also speculated in Query 21 of the Optics about an interplanetary aetherial medium as the cause of gravity.336 Moreover, Newton famously declared that the idea of action at a distance is an "absurdity" in his well-known letter to Richard Bentley of February 5,1693: It is inconceivable that inanimate brute matter should, without the
mediation of something else, which is not material, operate upon and affect other matter without mutual contact, as it must be, if gravitation in the sense of Epicurus, be essential and inherent in it. And this is one reason why I desired you would not ascribe innate gravity to me. That gravity should be innate, inherent, and essential to matter, so that one body may act upon another at a distance through a vacuum without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity, that I believe that no man who has in philosophical matters a competent faculty of thinking can ever fall into it. Gravity must be caused by an agent acting constantly according to certain laws; but whether this agent be material or immaterial, I have left to the consideration of my readers. (Janiak 2004,102-103) And what is most striking, from my point of view, is the suggestion that the true cause of gravity may be an immaterial agent-perhaps even God himself. I believe that Newton's neo-Platonic metaphysics of space and divine activity played a crucial role here. For Newton, as we have seen, (absolute)
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empty space is metaph~sically fun.damental, for it is "an affectio? of every kind of being," lllcludlllg God himself. God, through his ommpresence, can thereby create matter or body by endowing certain determined re l1ions of space with the conditions of mobility, impenetrability, and obedi~nce to the laws of motion; and God can do this anywhere in space by his immediate thought and will, just as our souls can move our bodies by our immediate thought and wil1. 337 Thus, Newton views even God's creative activity as subject to the condition oflocal presence which all natural philosophers of the time (both mechanical philosophers and Aristotelians) imposed on the efficient causality by which one created material substance could act on another. It seems clear, therefore, that Newton shared this conception of efficient causality with his contemporaries, and he also shared the widespread view, of course, that the clearest and most fundamental example of causal agency is the creative activity of God. Newton's metaphysics of space then made it possible for him to maintain that universal gravitation involves an immediate exchange of momentum across empty space (as his physics requires: compare note 334 above) while, at the same time, preserving the more traditional ideas of causality and agency he shared with his contemporaries (for an immaterial or massless agent certainly does not participate in the exchange of momentum). Indeed, from Newton's own point of view, his conception of the creation of matter by God makes maximal room for divine creative activity and thereby avoids the threat of atheism opened up by the Cartesian conception of material substance (compare note 7 above, together with the paragraph to which it is appended).338 By neglecting or downplaying the role of Newton's metaphysics of space, therefore, we miss the opportunity to explain how Newton was then actually able-despite his (at the time) well-grounded qualms about action at a distance-to expand the space of intellectual possibilities available to the mechanical philosophy so as to include the case of instantaneous momentum exchange at a distance opened up by his own formulation of the Laws of Motion. It then fell to Kant, as we have seen, to redefine the concept of causal action so that, in the phenomenal world, what it now means for one material substance to act on another is, in effect, simply for it to exchange momentum with another: action at a distance, in Kant's reinterpretation of the metaphysical tradition he inherited, is now perfectly possible and intelligible (compare note 101 above, together with the paragraph to which it is appended and the preceding paragraph). ArId whereas Kant, as we have also seen, made essential use of both Newtonian physics and Newtonian metaphysics, he eventually had to undertake a quite radical transformation of the entire preceding tradition (compare note 105 above, together with the paragraph to which it is appended and the preceding paragraph). By neglecting the central role of these metaphysical and
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theological issues we miss the full force of Kant's characteristically "transcendental" replacement for traditional metaphysics. 339 DiSalle's central example of the use of internal philosophical reflection in the practice of science, as already noted, is Einstein's celebrated analysis of simultaneity at the beginning of his 1905 paper. (This is also the example on which DiSalle primarily focuses in his contribution to the present volume.) DiSalle argues, however, that the precise nature of Einstein's conceptual analysis has not been properly understood. The logical empiricists, for example, took it as a model for their verificationist theory of meaning, and Bridgman (1927) understood it as a paradigmatic operational definition. For DiSalle, by contrast, the true nature of Einstein's analysis is considerably more subtle. Einstein realized, on the one hand, that the invariance of the velocity of light following from Maxwell's equations and the principle of relativity allowed one to introduce a new concept of simultaneity that dissolved the apparent contradiction between the principle of relativity and the light principle (compare the paragraph to which note 192 above is appended). But Einstein also realized, on the other hand, that the concept of simultaneity implicit in classical mechanics did not have an empirically well-defined meaning after all. The reason for this, however, was not that the classical concept of simultaneity failed to have a verification procedure or operational definition; rather, the precise way in which it was embedded within the classical mechanical framework involved a tacit presupposition about its empirical meaning which turned out, eventually, to be false. Newtonian gravitational interaction-operating instantaneously at a distance-gave a clear physical meaning, at least in principle, to the classical relation of absolute sin1ultaneity. Yet there was no direct way to use "gravitational signaling" empirically to determine this relation, and so one relied, instead, on the light signals by which one empirically determined relations of gravitational interaction (as in Newton's analysis of the solar systen1 in Book III of the Principia, which begins from the observationally established "phenomena" described by IZepler's laws). It was of course well-known at the time that light travels with finite velocity, but this fact was not yet integrated theoretically with Newtonian mechanics. One simply assumed-tacitly-that light, like all mechanical processes, obeys the classical velocity addition law, and one supposed that its velocity could thereby be successively corrected for so as to approximate, in principle, the instantaneous causal propagation of gravitational interaction. What the Michelson-Morley and related experiments showed, however, is that this assumption is actually false; and, as Einstein then realized, Maxwell's theory of electromagnetism and optics yields a new concept of simultaneity, internal to electrodynamics, according to which light has the same constant velocity c in every inertial frame. Einstein's analysis does not rest,
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therefore, on a simple-minded application of verificationism or operationalism, but on a critical dialectical engagement with both pre-existing physical practice and the unexpected empirical facts which then rendered this practice conceptually problematic. I believe, once again, that DiSalle's reading of Einstein's analysis is both correct and very illuminating-indeed, it is by far the best overall interpretation of this analysis I have seen. Nevertheless, I believe that other important features of the transition from Newton to Einstein in this case may still be insufficiently emphasized. In particular, the precise role and significance of the crucial empirical facts revealed by the Michelson-Morley and related experiments may not be fully captured by DiSalle's reading. For, in the first place, there was no real incoherence in the pre-existing Newtonian conception of (absolute) simultaneity before the striking new discoveries in the electrodynamics of moving bodies to which Einstein was responding. It is true, as DiSalle suggests, that "gravitational signaling" did not provide an empirically applicable criterion for the classical concept of (absolute) simultaneity, and one was therefore forced to rely on the transmission of information from the heavenly bodies by light-which, of course, took place with a finite rather than an infinite velocity. Yet this situation was already relatively well understood within the Newtonian tradition, and it remained relatively unproblematic, from a conceptual point of view, as long as one could assume that the finite velocity of light obeyed the classical velocity addition law. 340 This is why it was in fact so surprising when the Michelson-Morley and related experiments eventually indicated that light did not obey the classical velocity addition law late in the nineteenth century. Accordingly, and in the second place, the classical conception of (absolute) simultaneity, based on the Newtonian law of the equality of action and reaction and the resulting possibility of instantaneous action at a distance, is still present, of course, in the late nineteenth-century concept of an inertial reference frame. It is for precisely this reason (as DiSalle rightly emphasizes) that Einstein then needed subtly (and "dialectically") to transform this concept when surprising new experiments suggested that light has the same constant velocity c in all such frames: in particular, Einstein then substitutes what he calls the principle of relativity and the light principle for the Newtonian Laws of Motion (compare note 297 above, together with the paragraph to which it is appended). But what I have been arguing, in this connection, is that Einstein also needed the idea that simultaneity could be empirically defined by a free "stipulation" of our Own in order to complete his reconceptualization of the classical notion of an inertial frame and that he appropriated this idea from Poincare. Einstein appealed to Poincare's notion of "elevating" a mere empirical fact to the radically new status of a "convention or definition in disguise" in charac-
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tenzmg his own procedure for taking the new empirical discoveries in question-now in the guise of two fundamental "presuppositions" Or "postulates"-to furnish precisely the new "definition of simultaneity" he then arrives at. 341 By neglecting these additional elements of Einstein's procedure we miss Poincare's pivotal role here, and, as I have also argued, we thereby miss the important sense in which Einstein's new definition of simultaneity represents the very first instantiation of the relativized yet still transcendentally constitutive a priori (see again note 299 above, together with the paragraph to which it is appended and the preceding paragraph). A third central example, for DiSalle, is Einstein's use of the principle of equivalence in the foundations of general relativity. DiSalle argues that we can find an important strand in Einstein's motivations for this principle, consisting of a critical dialectical engagement with the Newtonian concept of absolute acceleration, which is analogous to Einstein's earlier critique of the Newtonian concept of absolute simultaneity. Consider a privileged center of mass frame arising in Newtonian theory (like the center of mass frame of the solar system described by the law of universal gravitation)where, more specifically, all true accelerations are counterbalanced in action-reaction pairs. By Corollary VI to the Laws of Motion, however, there is no way empirically to determine whether this frame is accelerating in turn, provided that all bodies accelerate at the same rate and in the same direction (compare note 333 above). But gravitational force, as Newton well understood, produces quantitatively identical accelerations in all bodies, and, if the source of this force is sufficiently distant from the bodies in question, the directions of these acceleration are also practically the same. Indeed, Newton appealed to precisely this fact in arguing that the system of Jupiter and its moons, for example, could be treated practically as an inertial frame-that its acceleration towards the sun could be locally ignored. We therefore have a paradoxical situation arising within Newtonian gravitation theory itself: different frames of reference can locally satisfY all the empirical criteria for being an inertial frame, but these same frames can then be accelerating relative to one another. The solution is to accept the principle of equivalence and to acknowledge that the (classical) inertial frames and (classical) inertial trajectories are not empirically well defined after all. The true privileged trajectories (affine geodesics) are instead freely falling in a gravitational field, and the privileged frames of reference, accordingly, are the local inertial frames corresponding to such freely falling trajectories-each of which represents a partial (and local) perspective on the underlying four-dimensional geometry of variable curvature described by the finished general theory.342 In this sense, the relationship between the space-time of general relativity and the Newtonian decomposition of orbital motion into inertial and gravitational components is anal-
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agaus to the ~elati?nship be~een Mink?wski .space-time and a dec~mpo sitian, in any merual frame, mto three-dlmenslOnal space and one-dImensional time. And, in both cases, for DiSalle, we arrive at the appropriate new space-time structure by critical philosophical analysis of the empirical meanings of the fundamental concepts of the earlier (Newtonian) theories. 343 In the case of DiSalle's (undoubtedly very insightful) treatment of the principle of equivalence, however, we find a particularly clear and direct conu'ast benveen his approach and my own. For my approach gives a quite central place to Einstein's example of the uniformly rotating frame (and the resulting non-Euclidean spatial geometry), while DiSalle suggests that Einstein's use of this example is at best heuristic since it cannot actually warrant the four-dimensional geometrical structure employed in the finished theory.344 This is perfectly correct-and, indeed, from the point of view of the finished theory, Einstein's example of the uniformly rotating frame reveals no true space-time curvature in any case, for it arises in precisely the context of a flat Minkowskian space-time. 345 Yet what DiSalle's account does not quite satisfactorily explain, in my view, is how the idea of four-dimensional space-time geometry became a real physical possibility in the first place, and, more generally, it seems to me that DiSalle does not quite capture how difficult it actually was to arrive at this idea. The question whether a genuinely physical use of four-dimensional space-time geomeu'y is even possible is prior, in my view, to the question of its warranted correctness, and I claim that it was Einstein's use of the uniformly rotating frame, in particular, which first made such a four-dimensional geometry physically possible. So it is no accident, on this view, that the now accepted interpretation of the principle of equivalence in terms of four-dimensional affine curvature was only properly understood soon after Einstein's formulation of the final theory (compare note 343 above), and its application to Newtonian gravitation theory then followed several years later (compare note 342 above). DiSalle (2006, 129) acknowledges that "it was probably impossible to express this connection between gravitation and inertia [on his account] until it could be represented in the framework of space-time." In his view, however, Minkowski had already executed a critical conceptual analysis of special relativity in 1908, from which it follows that (2006, 115) "[t]he physically objective quantities must be expressed as the invariants of a fourdimensional structure" and "Einstein's 1905 paper only states the true spatio-temporal relations in a form bound by the limits of spatial intuition." However, it is noteworthy, as we have seen, that Einstein himself at first dismissed Minkowski's formulation as a mere mathematical trick, and he did not begin to appreciate its value until his own work on the principle of equivalence eventually led him-via the example of the uniformly
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rotating frame-to a four-dimensional space-time geometry in 1912 (compare notes 197 and 201 above). I believe that Einstein was basically correct that the real physical meaning of four-dimensional space-time geometry emerged only with his own work on the principle of equivalence in the years 1907-12. 346 .And I do not see, in particular, why it follows from Minkowski's work that the physically objective quantities must be expressed as four-dimensional invariants. To be sure, once the four-dimensional framework is already accepted, we can see that there is no separate three-dimensional space left invariant by the Lorentz group-the only geometrical invariant is the four-dimensional Minkowski metric itself. 347 But I see absolutely nothing defective, at this stage, in Einstein's original three-plus-one-dimensional formulation, where the Lorentz group transforms inertial frames onto one another, and each such frame has its own decomposition into three-dimensional (Euclidean) space and one-dimensional time (compare note 346 above). What does become clearly problematic in the traditional conception ofspatial geometry even in 1905, however, is the idea of rigid motion underlying the principle of free mobility in Helmholtz and Poincare; and it was Einstein-and Einstein alone-who then saw how delicately to exploit this situation, with precisely the example of the uniformly rotating frame, in order to turn space-time geometry into a real piece of physics for the very first time. 348 It therefore becomes perfectly understandable, on my view, why Einstein's own most developed account of the philosophical background and significance of general relativity focuses on the problem of physical geometry and, more specifically, on the debate concerning the empirical foundations of geometry between Helmholtz and Poincare. For it was Einstein's critical engagement with this debate that made it possible for hin1 eventually to arrive at the conclusion (after a long and arduous struggle) that gravitation may be represented by a four-dimensional nonEuclidean geometry of variable curvature. 349 Moreover, Einstein's engagement with Poincare was especially important here, since it was Poincare (unlike Helmholtz) who was then making important contributions to the electrodynamics of moving bodies, and it was Poincare (again unlike Helmholtz) who had also re-established an intimate connection between the problem of physical geometry and the problem of the relativity of motion in the context of precisely this new physical situation. As we have seen, Einstein's ongoing philosophical and scientific engagement with Poincare, therefore, allowed him not only to create radically new physical representational possibilities in both special and general relativity but also, at the same time, to give radically new meaning to the original I
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DiSalle, in both his contribution to the present volume and DiSalle (2006), emphasizes that critical conceptual analysis, in his sense, results in new empirical definitions of fundamental physical concepts (true rotational motion, simultaneity, inertia) which, in an important sense, are uniquely determined in the context in question (Newton in the seventeenth century, Einstein in 1905, Einstein in 1915-16). This points to the most basic error in the logical empiricists' notion of an "arbitrary convention," and it is the primary reason, for DiSalle, that the analyses he describes result in a distinctively (relativized) transcendental necessity. 351 I fundamentally agree with DiSalle on this point, and this is why I now reject the logical empiricists' notion of "coordinating principle" in favor of a truly transcendental constitutive a priori,352 and I also insist, accordingly, on the transcendental necessity of the principles in question-relative to a given intellectual situation. 353 However, in insisting on precisely such (relativized) transcendental necessity, I want to make a stronger point as wel1. 354 The relevant intellectual situation, on my account, essentially includes specifically philosophical elements that directly derive, by a series of what I call minimal (and in this sense unique) generalizations and extensions, from Kant's original transcendental analysis of the fundamental spatio-temporal structure framing Newton's physical theory. It is not merely that we can now discern, in Einstein's theories, principles which are analogous, in important respects, to what Kant took to be the (absolutely) transcendentally necessary principles of Newtonian theory. Rather, the principles now claimed to be (relatively) transcendentally necessary in Einstein's theories are themselves derived, in the context of successive new developments in both mathematics and the empirical sciences, from Kant's conception. Kant's original insight into the structure of Euclidean geometry, Newtonian physics, and, more generally, the state of both mathematics and the empirical sciences in the late eighteenth century, was so deep, and so systematic, that changes successively forced in one or another part of the Kantian framework in one or another new intellectual context (Naturphilosophie and turn of the century electro-chemistry, the discovery of nonEuclidean geometries in the early part of the nineteenth century, the development of the concept of inertial frame, the discovery of the invariance of the velocity oflight in the context of Maxwellian- Lorentzian electrodynamics, the rise of group-theory and the mathematical theory of invariance, and so on) then reverberated throughout this framework in extraordinarily productive ways. Extensions or continuations of Kant's original conception thereby decisively informed the new innovations in scientific philosophy by Helmholtz, Mach, and (especially) Poincare that eventually resulted in Einstein's theories. The very intellectual developments which, from our present point of view, decisively show that Kant's
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theory, in its original form, is no longer philosophically tenable, also prove, from an historical point of view, its remarkable enduring fruitfulness and power. But how can such a generalization of I(ant's original transcendental philosophy possibly be extended to the rest of modern physics and, in particular, to quantum mechanics (compare note 318 above)? DiSalle briefly considers the implications of his account of the transition from Newton to Einstein for a proper philosophical understanding of quantum mechanics at the conclusion of both his contribution to the present volume and DiSalle (2006). He suggests, in particular, that such an account of the philosophical significance of space-time physics might help us to reorient the problem of "interpreting" quantum mechanics towards philosophical questions, posed in a broadly Kantian spirit, about the empirical meanings of the fundamental concepts of the theory. We must look, in short, for constitutive principles, analogous to those that have successively emerged in the philosophical history of space-time physics, in the development of quantum theory as well. DiSalle suggests that some of the founders of quantum theory (such as Heisenberg and Pauli) saw then1selves as explicitly following Einstein's example in this respect; and, in his contribution to the present volume, he also suggests that we might conceive the uncertainty principle as establishing a new understanding of "causality" in the physical world and quantum entanglement as imposing a new understanding of "community."355 I propose a related reorientation of the "interpretation" problem in Friedman (2001, 120-23), and I very much agree with DiSalle, more generally, that what we most clearly see in quantum theory is the emergence of a new kind of fundamental theoretical structure-which we might call algebraic or logical structure-as a competitor and possible replacement for the geometrical-mechanical spatio-temporal structure that continues to frame our classical mechanical theories from Newton to Einstein. 356 I would only add, in light of the present discussion, that it was the lZantian transcendental analysis of such spatio-temporal structure which, above all, made a truly philosophical understanding of the development from Newton to Einstein first possible, and that it is now quite uncertain, accordingly, whether and how a similar transcendental understanding of the development of quantum theory is possible in turn. Nevertheless, I believe that considerable progress has been made in understanding this situation, and I would cite, in particular, recent work on Bohr's notion of con1plementarity by Beller (1992), Tanona (2004), and Dickson (2004),357 the articulation of logical interpretations (as an alternative understanding of complementarity) following the original work of Garrett Birkhoff and John von Neumann,358 and the development of modal and other algebraic interpretations following upon this develop-
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ment. 359 Also very intriguing, from the present point of view, are the prospects for attaining a transcendental understanding of quantum field theory, following on Weyl's phenomenological understanding of spacetime geometry, in the context ofWeyl's own extension of the concept of a gauge u'ansformation to quantum theory.360 Of course none of these approaches has yet resulted in a fully satisfactory philosophical understanding of the radically new structure of quantum theory-nor of the relationship between this structure, more specifically, and space-time structure (compare note 356 above). But the conclusion of DiSalle (2006) is undoubtedly correct that such an understanding of the structure of quantum mechanics is possible only on the basis of a prior understanding of the role of space-time structure throughout the development from Newton to Einstein. 361 And this gives us at least grounds for hope that Kantian transcendental philosophy may here have an opportunity, once again, to leave an unexpected scientific legacy of lasting value.
NOTES * I am especially indebted to the editors, Mary Domski and Michael Dickson, for very helpfll1 comments on the entire manuscript. And I am grateful to the Center for Advanced Study in the Behavioral Sciences for a Fellowship in the academic year 2006-2007, during which most of the work on this essay was completed. 1. See Menn (1998). I have developed this approach with special emphasis on Descartes's relation to Galileo and the issue of Copernicanism in Friedman (2008). My reading of Descartes's "metaphysical foundations of physics," in this context, culminates in an interpretation of the argument for the existence of matter in the Sixth Meditation first developed in Friedman (1997a). 2. Domski cites Howard Stein (1967) in this regard, and we can also cite Stein (2002) for an authoritative account of Newton's metaphysics. De Gravitatione first appeared, together with an English translation, in Hall and Hall (1962). An improved translation by Christian Johnson, made with the assistance of Andrew Janiak, and consulting an earlier unpublished translation by Stein, appears in Janialc (2004): my parenthetical page references are to this edition. 3. Some of the most important writings of the Cambridge Platonists are collected in Patrides (1980). For discussion of the idea that space is an emanative effect of God see the exchange between J .E. McGuire and John Carriero in Bricker and Hughes (1990). See also the very careful discussion in Stein (2002, 266-72). In the course of his discussion Stein is led to claim (p. 269) that "the grounds for thinking that Newton's theory of emanation is neo-Platonic, or 'Cambridge Platonic,' are very weak." Whatever one may think of Stein's particular reasons for this claim, it seems to me very hard to deny, in any case, that
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Newton is appropriating neo-Platonic (and, indeed, 'Can1bridge Platonic') ideas for his own purposes here. 4. This, in a nutshell, is how I read the argument for the existence of matter of the Sixth Meditation: see Friedman (1997a; 2008). 5. Indeed, because of his more advanced conceptions of both mathematics and mathematical physics, there is even less room for such a gap in Newton than there is in Descartes. Newton's more tolerant conception of the class of curves admitted as the objects of geometry in the context of an intimate relation between the subject-matters of geometry and mechanics (as emphasized by Domski) means, in contemporary terms, that Newton adnuts transcendental as well as algebraic curves and, more generally, all curves generated by any kind of continuous motion which might be describable, in principle, by his new physics. 6. The centrality of absolute space, time, and motion in Newton is related to a central difference between the Leibnizean and the Newtonian calculus. Whereas Leibniz's approach is essentially algebraic, Newton's is essentially kinematical (involving absolute time as the ultimate independent variable) and therefore (especially in the Principia) essentially geometrical as well. See Bertoloni Meli (1993), De Gandt (1995). 7. As we observed (see note 2 above, together with the paragraph to which it is appended), Newton's principal target in De Gravitatione is Descartes rather than Leibniz, and it is certainly true, as we have also observed, that Leibniz's concept of substance is more Aristotelian (and, in particular, more closely linked to Aristotelian logic) than Descartes's. Nevertheless, the substance-attribute ontology is fundamental to Descartes as well, especially in his conception of space (= matter) as extended substance, whereas Newton's metaphysics of space is predicated on a rejection of this ontology. 8. For such a discussion, in the context of the Cartesian, Leibnizean, and Newtonian background, see Friedman (2006c). 9. IZant makes this connection in the New Exposition of the First Principles of Metaphysical Cognition and the Universal Natural History and Theory of the Heavens, both appearing in 1755. For discussion, and references, see Friedman (1992a, 5-14). As I point out there, an echo of the Newtonian doctrine of divine omnipresence occurs as late as the Scholium to §22 of the Inaugural Dissertation (1770). (Kant of course had no knowledge of Newton's unpublished De Gravitatione, but, as observed above, essentially the same metaphysics of space surfaces in such well-known published writings as the General Scholium to the Principia and Query 31 of the Optics.) For further recent discussions ofIZant's precritical metaphysics see Laywine (1993), Schonfeld (2000), Watkins (2005). 10. For extended discussion of Kant's "silent decade" see Carl (1989). This is also the subject of some of Alison Laywine's most recent work, for example, Laywine (2003). 11. See note 9 above, together with the paragraph to which it is appended. As I observed, there is an echo of the pre-critical theory of divine omnipresence even in the Inaugural Dissertation, where I(ant has already drawn a fundamental distinction between understanding and sensibility. The question Kant raises there (in the Scholium to §22) concerns precisely the causes of our sensible intuitions, and, in particular, the relationship between our sensible intuitions and the assumed ulti-
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mate substances constituting the intelligible world. The answer Kant (tentatively) suggests is that, si~ce both our mind an~ these "external things" are sustained by ,1 single infinite bemg, space, as the "sensibly cogmzed ul1lversal and necessary condition for the co-presence of all things," can thus be characterized as (God's) pheI/omenal omnipresence. In light of §22 itself, it appears that Kant is thereby invoking a pre-established harmony (instituted by God) between the purely intellectual reality of ultimate substances and our spatio-temporal sensibility to explain the necessary connection between this reality as it appears to us and this same reality as it is in itself. In §27 of the second edition transcendental deduction, Kant explicitly rejects such an explanation of the agreement between experience and its objects (which he calls a "preformation-system of pure reason") in favor of his new, critical explanation (which he calls an "epigenesis of pure reason")-where, as I understand it, the understanding rather creates the a priori order of sensible experience by injecting itself into the pure forms of sensibility. 12. In the Preface Kant self-consciously juxtaposes metaphysics in his new sense with what is clearly the tradition of Newtonian physics (4, 472): "Hence all natural philosophers who have wished to proceed mathematically in their occupation have always, and must have always, made use of metaphysical principles (albeit unconsciously), even if they themselves solemnly guarded against all claims of metaphysics upon their science. Undoubtedly they have understood by the latter the folly of contriving possibilities at will and playing with concepts, which can perhaps not be presented in intuition at all, and have no other certification of their objective reality than that they merely do not contradict themselves. All true metaphysics is drawn from the essence of the faculty of thinking itself, and is in no way fictitiously invented on account of not being borrowed from experience. Rather, it contains the pure actions of thought, and thus a priori concepts and principles, which first bring the manifold of empirical representations into the law governed connection through which it can become empirical cognition, that is, experience." All translations from this text are taken (occasionally wi th small emendations) from Friedman (2004), which contains the page numbers from volume 4 of the Akademie edition of [(ants gesammelte Schriften in the margins. In general, my citations to Kant's texts refer to the corresponding volume and page number of J(ants gesammelte Schriften, with the exception of references to the first Critique which are offered in standard AlB form. All translations from the German here are my own. 13. See Friedman (1992a, Chapter 4). A crucial issue arising in §38 is the relationship between the laws of nature and the geometry of space. And, while it at first sight appears that Kant is envisioning a purely geometrical derivation of the inverse-square law from the relations of concentric spherical surfaces around a common center, I argue that what is much more fundamental, in this very section, is Newton's derivation of the inverse-square law from the laws of motion and the geometry of the ellipse. 14. For more details of this reading see Friedman (1992a). For the issue of necessity, in particular, see Friedman (1992b). 15. This conception of absolute space as an idea of reason is further discussed in the references cited in notes 12 and 13 above. See the Introduction to Friedman (2004), in particular, for a relatively concise sketch of the entire argu-
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ment. As argued in Friedman (1992a, Chapter 4), this is how, more precisely, the understanding is able to "determine" the otherwise "indeterminate" space of pure sensibility. 16. These ideas, and the passage from Religion Within the Limits of Reason Alone, are discussed somewhat more fully in Friedman (1992a, 47-51). Compare also the more recent discussion, in the context of the earlier metaphysical investigations of Descartes, Newton, and Leibniz, in Friedman (2006c). 17. See note 12 above. Several pages later Kant explains that "the schema for completeness of a metaphysical system, whether it be of nature in general, or of Corporeal nature in particular, is the table of categories" (4, 473-74), and he attaches a lengthy footnote to this sentence on the transcendental deduction. 18. The construction and proof is distinctive of Euclidean geometry because we are making essential use of the parallel postulate in Proposition 1.29, the very first time it is applied in the Elements. 19. This proof, since it precedes Proposition 1.29, does not depend on the parallel postulate. Thus it is equally valid in hyperbolic geometry (constant negative curvature). However, since it (tacitly) assumes the indefinite extendibility of any given straight line segment to a longer such segment (under this particular interpretation of Postulate 2), it is not valid in elliptic geometry (constant positive curvature). 20. This passage makes it especially clear that Kant does not envision a direct process of quasi-perceptual acquaintance with (or inspection of) a given particular figure as the ground of such geometrical truths: "A new light dawned on the first man (whether he may be Thales or whoever) who demonstrated the isosceles triangle; for he found that he must not inspect what he saw in the figure, or even in the mere concept of it, and, as it were, thereby learn its properties. He must rather bring forth what he himself has injected in thought [hineindachte J and presented (through construction) in accordance with concepts, and, in order to know something a priori, he must attribute nothing to the thing except that which follows necessarily from what he himself has placed in it in accordance with his concept." The reference to Proposition 1.5 is made explicit in a letter to Christian Schiitz of June 25,1787 (where Kant also changes "equilateral [gleichseitigerJ" in the printed text to "isosceles [gleichschenkligter]"). 21. It is by means of precisely the distinction between schema and image, therefore, that Kant solves the Berkeleyean generality problem which initially stimulated Beth (1956-57): see Friedman (1992a, note 59 on 90-91, 124-27). 22. Corresponding to the concept of line (segment) in Euclid are two such constructive operations: Postulate 1, which takes two arbitrary points as input and yields the line segment connecting them as output, and Postulate 2, which takes two arbitrary line segments as input and yields the extension of the first by the second as output. 23. Kant discusses this principle in The False Subtlety of the Four Syllogistic Figures (1762), as well as in the Lectures on Logic compiled by Jasche (1800). 24. This explanation of the relationship between geometrical reasoning and the more general calculative manipulation of magnitudes in algebra and arithmetic was first presented in Friedman (1985): compare Friedman (1992a, 83-89). It is discussed more fully, in the context of Kant's conception of magnitude and the Euclidean-Eudoxian theory of proportion, in Friedman (1990b): compare
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Friedman (1992a, 104-22). More generally, it is clear that there is a very significant connection between Kant's conception of arithmetic and the categories of l uantity or magnitude [Griifle] from Kant's remarks in the Schematism (\142--43/B182): :'The pure schema. of magnitude .(quantitatis), as a concept of the understanding, IS number, which IS a representatIOn compnsmg the successIve .lddition of unit to (homogeneous) unit. Thus number is nothing else but the unity of the synthesis of the manifold of a homogeneous intuition in general, in so far as I generate time itself in the apprehension of the intuition." Daniel Sutherland 17004a 2004b, 2006) has recently developed a detailed and penetrating account ~f Kan~'s conception of magnitude in relation to both the Euclidean-Eudoxian theory of proportion and early modern discussions of algebra and arithmetic. Indeed, Sutherland suggests a new central role for intuition here, in so far as Kant must assume that the magnitudes in question are what Sutherland calls strictly homogeneous. This means, in particular, that completely "similar and equal" magnitudes-such as two identical cubic feet of space, for example, or two identical intervals of time-must, when added together, result in a larger such magnitude. We thereby obtain a further important differentiation between conceptual and intuitive thinking, since two identical concepts added together (conjoined) result in the same concept: A + A = A (conceptual addition, unlike addition of magnitudes, is necessarily idempotent). The iterative addition of units underlying arithmetic and the concept of number, for Kant, therefore presupposes that the units so added are strictly homogeneous in Sutherland's sense-a situation which I, and all previous interpreters prior to Sutherland, had overlooked. I now agree with Sutherland on the importance of strict homogeneity and, in particular, that the pure intuitions of space and time perform an additional important role in Kant's philosophy of mathematics by providing us with examples of strictly homogeneous manifolds of intuition. However, it is still important to emphasize (and I believe Sutherland agrees) that arithmetic and algebra concern homogeneous manifolds of "an intuition in general" (rather than exclusively spatio-temporal such manifolds), and, in this way, the concept of magnitude, together with its schema, is more abstract (and more intellectual) than the particular spatio-temporal homogeneous manifolds of our (specifically human) forms of intui tion. 25. This point was first made in Zeuthen (1896). 26. More precisely, we can represent all the points constructible by straightedge and compass construction in the Euclidean plane by the Cartesian product of a square-root extension field of the rationals with itself, whereas the full set of points generated by a true (second order) continuity axiom is of course represented by R2, where R is the real numbers. An important intermediate case, studied by Tarski, uses a (first order) continuity schema, and is represented by a Cartesian product over any real closed field: see Tarski (1959). 27. See especially Kant's discussion at B39-40: "Space is represented as an infinite given magnitude. Now one must certainly think every concept as a representation which is contained in an infinite aggregate of different possible repreSentations (as their common characteristic), and it therefore contains these under itself. But no concept, as such, can be so thought as if it were to contain an infinite aggregate of representations within itself. Space is thought in precisely this way, however (for all parts of space in infinitum exist simultaneously). Therefore,
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the original representation of space is an a priori intuition, and not a concept." As I point out in Friedman (1992a, 67), following Henry Allison and Manley Thomson, the distinction between "containing within" and "containing under" refers to the traditional logical notions of the intension and extension of a concept-where the intension of a concept includes all the concepts comprising its definition, and the extension of a concept includes all concepts subordinate to it (as lower species to a genus). There is thus an inverse relationship between extension and intension (where the intension of a concept includes all superordinate concepts). Since Kant holds that there are no lowest species, the extension of a concept is always potentially infinite, but the intension of a concept (since Kant rejects infinite conjunctions and Leibnizean complete concepts) can never be infinite (not even potentially). The whole-part structure of concepts is therefore con1pletely different, in this in1portant respect, from the whole-part structure of intuition. 28. The connection between monadic logic, in the modern sense, and the traditionallogic of concepts (note 27 above) is that the latter, represented purely syntactically, considers only finite conjunctions consisting of primitive one-place predicates and their negations. We thereby obtain a connection, as well, between the necessary limitation of monadic logic to finite n10dels and I
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priori sensible intuiti?,n under which alone the schema of a pure concept of outer appearance can anse. 33. This point is closely related, therefore, to the important (and difficult) distinction Kant makes between space as form of intuition and as formal intuition (or "intuition itself') in §26 of the second edition transcendental deduction. But here is not the place to discuss this matter further, and I shall simply observe that Kant makes a particular point of illustrating that in geometry we are actually required to represent space as an object (B160n). For further discussion see Friedman (2000a, 197-99), (2003b). 34. See again note 20 above, together with the paragraph to which it is appended and the two succeeding paragraphs. 35. As many authors have observed, such whole-part spatial relations (together with Common Notion 5) take the place, in Euclid, of an explicit axiomatic theory of order such as we find in Hilbert. And this is then an additional role for spatial intuition in Euclidean proof, going beyond the purely existential role on which I have focused previously (see note 25 above, together with the paragraph to which it is appended). Lisa Shabel has recently given special emphasis to this feature of the Elements in relation to Kant. In particular, Shabel (2003a, reprinted from her doctoral dissertation) illuminatingly describes how the Euclidean Definitions and Common Notions articulate what she calls the "mereotopological" structure of Euclidean space (on the basis of which, for example, all spatial figures are seen as having an inside and an outside, separated by a boundary) and how this structure then allows us to derive ordinal relations (of greater than or less than) from mereological containment relations generated in the construction of a figure. Moreover, Shabel (2003b, §2) explains how this "mereotopological" structure is related to Kant's emphasis on the unity and infinity of space in the third and fourth arguments of the Metaphysical Exposition (see the two paragraphs following the one to which note 29 above is appended). Finally, Shabel (2003a, §§3.1, 3.2) discusses how the traditional generality problem for Kant's notion of pure intuition can be solved if we take Euclidean constructions as a model for what Kant calls the schemata of geometrical concepts (see note 21 above, together with the paragraph to which it is appended and the preceding paragraph). Unfortunately, however, Shabel fails to acknowledge my earlier discussion of this last point in Friedman (1992a) (see again note 21 above). 36. Compare the very beginning of the Transcendental Aesthetic at A19-20/B33-34: "In whatever manner and by whatever means a cognition may relate to objects, that by which it is related to them immediately, and towards which all thinking as a means is directed, is intuition. But this takes place only in so far as the object is given to us-and this, in turn, at least for us humans, is only possible in so far as the mind is affected in a certain way. The capacity (receptivity) to obtain representations through the manner in which we are affected by objects, is sensibility. ... The effect of an object on the faculty of representation, in so far as we are affected by them, is sensation. That intuition which relates to the object through sensation is empirical. The undetermined object of an empirical intuition is appearance." 37. Parsons (1992, 69) offers this as an "obvious" idea, although he does not embrace it unreservedly.
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38. See again the paragraph following the one to which note 29 above is appended. This point also allows us to answer a well-known objection to the first argument first raised by J,G. Maa~ (a colleague of Eberhard's), according to which it does not follow from the fact that one representation presupposes another that the latter representation is a priori: it order to recognize red objects, for example, one must first have the concept of red (and, more generally, color), but it of course does not follow from this that red (or color) is an a priori rather than empirical concept: see, e.g., the discussion of this objection in Allison (1983, 82-86). The crucial difference, for Kant, is that we have a necessary a priori science of space (geometry), whereas we do not have such an a priori science in other cases (like color). I am here indebted to discussions with Graciela De Pierris concerning the first two arguments of the Metaphysical Exposition; for her own discussion see De Pierris (2001). 39. See again the passage to which note 31 above is appended, together with note 32. Straight lines and circles thereby appear as what we call the orbits (confined to any two dimensional plane) of the Euclidean group of rigid motions in space. As explained in Friedman (2000a), an advantage of this reading is that it then allows us to connect Kant's theory of pure geometrical intuition with the later discussions of Hermann von Helmholtz and Henri Poincare (who were self-consciously influenced by Kant)-although there can of course be no question of attributing an explicit understanding of the group-theoretical approach to geometry via Helmholtz's principle of "free mobility" to Kant himself. I shall return to this topic when discussing Helmholtz and Poincare below. 40. See note 6 above. Of course Kant's prominent emphasis on Euclidean constructions marks an important difference with Newton, whose more modern (more tolerant) conception of the scope of geometry signals a significant break, as Domski explains, between Newton and Descartes (see note 5 above). In particular, Newton's willingness eventually to include all curves generated by any kind of continuous motion within the subject matter of geometry is associated with a breakdown of the distinction between "geometrical" and "mechanical" curves. I touch on Kant's position in relation to these issues in Friedman (2000a, 193-97). Briefly, although Kant is certainly familiar with basic applications of Newtonian-style geometry in the Principia using limits and the method of ultimate ratios (such as in the derivation of the inverse-square law from the geometry of the ellipse, for example: see note 13 above), his own view of what counts as specifically geometrical construction appears to be entirely traditional-going no further, as far as the examples Kant actually considers, than "plane" (i.e., straight-edge and compass) and "solid" (i.e., conic section) constructions in the sense of Pappus. 41. It is true, as noted above, that Leibniz appears to have envisioned some kind of extension of the traditional logic of concepts which would be more adequate for mathematical reasoning. Not only did he explicitly include infinite conjunctions or complete concepts (see note 27 above, together with the paragraph to which it is appended), but he also recognized the importance of relational reasoning (such as "IfJesus Christ is God, then the mother ofJesus Christ is the mother of God"), and, more generally, he wanted to include algebraic reasoning, beginning with the Euclidean theory of proportion, within the scope oflogic (see New Essays on Human Understanding, Book IV, chaps. xvii and ii, respectively).
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:-Ieverthel ess , Leibniz never explicitly developed such an extended conception of logic, and his official definition of necessary truth (including all of pure mathematicS) in terms of finitary conceptual containment remained squarely within the framework of the traditional logic of concepts. 42. More precisely, Kant's philosophy of mathematics involves an interplay between the pure intellect and pure sensibility, because the fi.mdamental activity of the pure productive imagination is the construction of concepts, resulting in their corresponding schemata (see again the paragraph to which note 22 above is appended, togetller with the preceding paragraph). It therefore involves the schematization of the pure intellect in terms of sensibility which is the outcome of the transcendental deduction. (As I suggested above-see note 11, together with the paragraph to which it is appended-precisely this injection of the pure understanding into tlle forms of pure sensibility takes the place, in the critical Kant, of God's creative activity within me space of his omnipresence.) Indeed, in §24 of me second edition deduction, Kant first illustrates the "transcendental action of the imagination," as "an action of the understanding on sensibility," precisely by the fundamental Euclidean constructions of drawing a line and describing a circle. For further discussion, see Friedman (2000a, 190-93), and, in connection with me idea that geometry thereby considers space as an object, see note 33 above and tlle pages cited there. 43. Of course it is true, as Parsons observes, that "[m]amematicians beginning with Cauchy and Bolzano did not wait for logicians to develop a polyadic logic in order to exploit the capacity of ordinary language to express such notions" (p. 146). Indeed, it is obvious that ordinary language (both inside and outside mamematics) has operated with what we now call polyadic quantificational reasoning beginning long before Frege first made me logic of this reasoning fully explicit in 1879; and this has made some of my claims-e.g., that certain polyadic quantificationallogical forms "did not exist" before Frege-appear shocking to a number of a readers: a particularly clear example is provided by Rusnock (2004), which Parsons cites in a footnote to me sentence just quoted. Perhaps my language was over-stated, but my point (of course) was simply that Frege first made me logic of these forms fully explicit-a task which was by no means easy or trivial. Moreover, when Kant speaks of logical or conceptual or discursive reasoning, he, too, has in mind an explicit logical theory of precisely what this amounts to, namely, the traditional logic of concepts Leibniz uses as his model of the pure intellect. When Kant then (rightly) rejects this model as inadequate for mamematical reasoning in favor of a faculty of pure sensibility, he is making me perfectly reasonable-and, in Kant's particular historical context, quite important-point that a Newtonian approach to pure mathematics is, in this respect, clearly preferable to the Leibnizean. Nevertheless, Rusnock (2004) is correct that the Leibnizean algebraic approach quickly became dominant in the eighteenm century and that such early nineteenm-century mathematicians as Cauchy and Bolzano made meir foundational contributions squarely on the basis of this earlier tradition: in this sense, Kant was already on the "wrong" side of history even in me eighteenm century. What I most want to emphasize here, however, is that (especially in tllis case) me "wrong" side of history can still be a deeply important one. For the fact remains that no explicit logical meory of purely conceptual or discursive reasoning adequate for mamematics (including ele-
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mentary geometry and arithmetic) was actually produced before Frege, and so when anti-Kantians like Bolzano claimed to have shown that mathematics is purel; "conceptual" after all, we were still owed a replacement for the traditional logic of concepts in order fully to vindicate this claim. Bolzano's great mathematical work no doubt contributed very significantly to the eventual development of this replacement (by leading to extremely complicated forms of quantificational reasoning which strained the capacities of ordinary language and thereby cried out for an explicit logical theory), but his philosophical opposition to Kant on the matter of "concepts" versus "intuitions" was (as Parsons suggests) equally important as well. Thus, despite being on the "wrong" side of history, Kant's very sharp focus on the relationship between logic and mathematics-and on the contrast between "concepts" and "intuitions" arising in his particular intellectual context-proved to be a decisive stimulus indeed to Frege's eventual breakthrough. 44. See 4, 301-302: "If one analyzes all one's synthetic judgments, in so far as they are objectively valid, one finds that they never consist of mere intuitions, which, as is commonly held, have been simply connected in a judgment through comparison; rather, they would be impossible if a pure concept of the understanding were not added over and above the concepts derived from intuition, under which the latter concepts have been subsumed and thus first connected in an objectively valid judgment. Even the judgments of pure mathematics in its simplest axioms are not exempt from this condition. The principle that the straight line is the shortest between two points presupposes that the line is subsumed under the concept of magnitude [Grlij.?e], which is certainly not a pure intuition, but has its seat solely in the understanding, and it serves to determine the intuition (the line) on behalf of the judgments that may be made about it with respect to their quantity, namely, [with respect to] plurality (as Judicia plurativa), in so far as it is thereby understood that a plurality of homogeneous [elements] are contained in a given intuition." As we saw in note 24 above, the resulting "unity of the synthesis of the manifold of a homogeneous intuition in general" is the schema of the concept of magnitude. 45. See pp. 127-28 of Laywine's contribution: "Isn't it possible that our power of understanding represents empirically given things a priori in a way somehow analogous to the mathematical examples we have been discussing? In other words, isn't it possible that the understanding has certain resources that are to experience what Euclid's constructive procedures are to circles and line segments? If so, might it not be the case that we can represent nature to ourselves a priori, because somehow we construct a model of it using these resources-whatever precisely they might be?" 46. Thus, the passage from A234/B287 quoted in note 32 above is part of Kant's explanation for why he calls these principles "postulates." In the immediately preceding sentence Kant says: "The postulates of modality thus ascribe nothing more to a concept than the action of the cognitive faculty by which it is generated." 47. In the first stage, considering Galileo's law of fall, we are dealing with a rectilinear acceleration of all bodies towards the earth's center. However, since we have not yet applied the equality of action and reaction, we are not taking account of the fact that both the earth and any falling body are mutually accelerating towards their common center of mass (which, of course, is very close to the center
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of the earth). It is in this sense that the motions considered at this stage are merely possible. At the next stage, however, by provisionally assuming the rectilinear acceleration of fall to be actual, we can determine (by small deviations from this acceleration due to centrifugal and Coriolis forces) the true rotation of the earth (as opposed to a corresponding opposite rotation of the starry heavens). In the last stage, finally, we establish Newtonian universal gravitation as a generalization of the Galilean law of fall, and we find rectilinear accelerations (in accordance with the inverse-square law) between any two bodies in the solar system (including between the earth and any falling body). These mutual accelerations are always equal and opposite, and, when described from the center of mass of the solar system, they precisely counterbalance the centrifugal accelerations arising from the corresponding states of true rotation describing the orbital motions in the solar system from the point of view of precisely this privileged center (just as, in the second stage, we find the true rotation of the earth as the cause of the small deviations-Coriolis accelerations-from the rectilinear acceleration of fall). 48. As I explain in Friedman (1992a, chapter 3, §IV), this central disanalogy between the two types of construction is closely related to the important contrast Kant draws between mathematical principles of pure understanding (the Axioms of Intuition and the Anticipations of Perception) and dynamical principles (the Analogies of Experience and the Postulates of Empirical Thought). Unlike mathematical principles, which are "constitutive with respect to intuition" (I can, for example, instantiate the categories of quantity a priori in pure intuition by means of geometrical construction), dynamical principles are "merely regulative principles of intuition" (A664/B692). Nevertheless, dynamical principles of pure understanding (unlike the regulative ideas of reason) are still "constitutive of experience" (ibid.), in the sense that they can be instantiated in a perceptual experiencewhich, in turn, is itself only possible in accordance with these very same principles. For more on the distinction between mathematical and dynamical principles, in the context of Kant's argument in the Metaphysical Foundations, see Friedman (1992a, chapter 4, §II). 49. In the second (1787) edition version of the Axioms of Intuition, Kant characterizes the concept of magnitude in general in terms of "the composition of the homogeneous [Zusammensetzung des Gleichartigen] and the consciousness of the synthetic unity of this (homogeneous) manifold" (B202-203). Compare note 44 above, and, for further discussion of Kant's concept of magnitude, see the references cited in note 24. 50. For the traditional Greek theory of (continuous) magnitude see Stein (1990); for its relation to Kant's philosophy of mathematics see Sutherland (2004b; 2006). Sutherland (2005) explores the metaphysical background to Kant's theory of magnitude in the thought of Leibniz and Wolff, and shows, in particular, that Kant accepted the purely philosophical definitions of similarity (identity in quality) and congruence or equality (identity in quantity) offered by Leibniz. In his contribution to the present volume, Sutherland explains how Kant rejected the Leibniz-Wolffian program for a philosophical reform of mathematics, despite accepting their philosophical definitions, precisely because of his fundamental insight into the limitations of traditional logic and the consequent need for pure intuition: compare again note 24 above.
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51. The basic point of this proposition is that two given motions can be Composed a priori in pure intuition (as opposed to a merely "mechanical" composition in accordance with the parallelogram of forces) only if the motions are represented in two different reference frames: what we would now call the Galilean transformation taking one such frame onto another then yields the desired operation of composition (the classical velocity addition law). 52. For example, in the case of the force of (Newtonian) gravitational attraction (which is paradigmatic, for Kant, throughout the Metaphysical Foundations), its magnitude is precisely determined by reference to the accelerations (changes of velocity) it produces between bodies with given masses. (How the quantity of mass is itself determined shall be discussed below.) Kant suggests that there is a close conceptual connection between force and motion already in the Second Analogy of the first Critique, where he speaks of acquaintance "with moving forces, or, what is the same, with certain successive appearances, which (as motions) indicate [anzeigen] such forces" (A207/B252). A footnote to this paragraph then makes it clear that, in accordance with the law of inertia, the motions that are relevant here are changes ofstate of motion or accelerations. 53. Compare Kant's remark to the fifth explication (489): "In phoronomy, since I am acquainted with matter through no other property but its movability, and may thus consider it only as a point, motion can only be considered as the describing of a space-in such a way, however, that I attend not solely, as in geometry, to the space described, but also to the time in which, and thus to the speed with which, a point describes the space. Phoronomy is thus the pure theory of magnitude (Mathesis) of motions." 54. Kant makes this clear in his remark to the official definition of what it means to construct the concept of a composite motion in the fourth explication (486-87): "It is required for the construction of concepts that the conditions of their presentation not be borrowed from experience, and thus not presuppose certain forces whose existence can only be derived from experience; or, in general, that the condition of the construction must not itself be a concept that can by no means be given a priori in intuition, such as, for example, the concept of cause and effect, action and resistance, etc. Now here it is above all to be noted that phoronomy has first to determine the construction of motions in general as magnitudes, and, since it has matter merely as something movable as its object, in which no attention at all is therefore paid to its quantity, [it has to determine] these motions a priori solely as magnitudes, with respect to both their speed and direction, and, indeed, with respect to their composition. For so much must be constituted wholly a priori, and indeed intuitively, on behalf of applied mathematics. For the rules for the connection of motions by means of physical causes, that is, forces, can never be rigorously expounded, until the principles of their composition in general have been previously laid down, purely mathematically, as basis." This is why the parallelogram of forces, in particular, cannot be appealed to here: compare note 51 above. 55. Kant's proof of the principle of the Anticipations (in the second edition) begins as follows (B207-208): "Perception is empirical consciousness, that is, one in which there is also sensation. Appearances, as objects of perception, are not pure (merely formal) intuitions, like space and time (for these, in themselves, can in no way be perceived). They therefore contain, beyond the [pure] intuition, also the
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matter for some or another object in general (whereby something existing in space or time is represented), that is, [they contain] the real of sensation, and thus a merely subjective representation, of which one can only be conscious that the sub'ect is affected, and which one refers to an object in general." J 56. Kant's rejection of the mechanical natural philosophy on this issue is the most explicit point of contact between the dynamical theory of matter of the Metaphysical Foundations and the argument of the Anticipations of Perception. In particular, Kant there illustrates the sense in which the filling of space or time is purely intensive, as opposed to extensive, as follows (A172-73/B214): "[E]ven if the whole intuition of a given space or time is real through and through, that is, no part of it is empty, there must still be infinitely [many] different degrees with which space or time is filled-because every reality has a degree, which can decrease to nothing (to the empty) through an infinite number of steps while the extensive magnitude [i.e., the volume] is unchanged, and the intensive magnitude in different appearances can be greater or smaller, although the extensive magnitude of intuition [i.e., the volume] is equal." Kant then provides an illustration of the view he wants to avoid (A173/B215): "Almost all students of nature, since they observe a great difference in the quantity of matter at the same volume (partly through the moment of gravity or weight, partly through the resistance to other moved mattel's), unanimously conclude that this volume (extensive magnitude of the appearance) must in all matters be [partly] empty, although in different measures. But who among these mostly mathematical and mechanical investigators of nature has been able to realize that tlley have based this conclusion simply on a metaphysical presupposition, which they nevertheless pretend to avoid? They assume, namely, that the real in space (I may not here call it impenetrability or weight, because these are empirical concepts) is everywhere of the same kind, and can only differ according to extensive magnitude, that is, according to [its] aggregate." On the contrary, Kant concludes (A174/B216): "[W]e see that, although equal spaces may be completely filled with different matters, in such a way that there is no point in either in which its presence would not be found, each reality could still have its own qualitative degree (of resistance or of weight), which, without diminution of extensive magnitude or aggregate, can be smaller and smaller to infinity, before it is transformed into the empty and disappears." 57. The Physical Monadology of 1756 is explicitly articulated within the more general metaphysical framework of the New Exposition of the Principles of Metaphysical Cognition of 1755. These two works, in the pre-critical period, are merefore related to one another approximately as the Metaphysical Foundations and the first Critique are related in the critical period. Again, for further discussion of the pre-critical period see the works cited in note 9 above. 58. As many authors have observed, the theory of the Physical Monadology bears important similarities to that developed by Roger Boscovich in his Theory of Natural Philosophy of 1758 (although it appears that the works of Boscovich and Kant were entirely independent of one another). For discussion of the development and influence of eighteenth-century dynamical theories of matter see McMullin (1978), Harman (1982a, 1982b). 59. More specifically, according to Proposition X of tile Physical Monadology and its accompanying Scholium, the fundamental force of attraction (representing
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Newtonian gravitation) diminishes as the inverse-square of the distance from the center, while the fundamental force of repulsion (representing solidity or impenetrability) diminishes as the inverse-cube of this distance. The two fundamental forces thereby determine a limit or boundary beyond which repulsion (and thus impenetrability) is no longer effective and attraction takes over unhindered. It is this boundary that defines the (small but finite) volume or "sphere of activity" of the physical monad, and Kant concludes, in the following Corollary, that all physical monads have the same volume: see 1,483-85. 60. The proof of the infinite divisibility of matter is articulated in the fourth proposition of the Dynamics. The following first remark explicitly targets the contrary view of "the monadist," according to which "the sphere of activity of this [ultimate simple] substance [would be divided] by the division of space, but not the acting movable subject itself" (504). 61. Compare note 59 above. A crucial difference between the two conceptions, as explained in Kant's first remark to the eighth proposition, is that Kant now holds that, whereas the fundamental force of attraction (representing Newtonian uillversal gravitation) still diminishes as the inverse-square of the (finite) distance from any attracting point, the fundamental force of repulsion diminishes as the inverse-cube of the infinitesimal or infinitely small distance. No finite "sphere of activity" is thereby determined, and each point in a region filled with matter therefore exerts the fundamental force of repulsion only on its infinitely close neighbors. This representation, as Kant himself emphasizes in the following second remark, gives rise to deep mathematical difficulties. 62. This argument is made in the second remark to the fourth proposition of Dynamics (compare note 60 above), where, in effect, Kant repeats the argument of the Second Antinomy. 63. See Euler (1755, §8). In his comments on this paper Truesdell (1954, lxxxi) calls it "a turning point in the history of physics," where, for the first time, "the continuum view of matter is put forward as a basic principle." In particular, "Euler had come to realize that the error of his predecessors lay not in their adherence to mechanics ... but in the wrong notion that mechanics is bound up with little physical particles .... In §§5-8 is Euler's final and rather disgusted rejection of corpuscles. Henceforth the principles of mechanics themselves are to be applied directly to the bodies of physical experience, and 'particle' is to mean only a mathematical point in a continuum model of matter." 64. As a passage on the following page (considered immediately below) makes clear, "hydrodynamics" is a misprint for "hydrostatics." To say that the property of fluidity is "original" is to say that the material in question is fluid all the way down, as it were, so that every part, no matter how small, is characterized by exactly the same fundamental property. Compare Kant's consideration of Euler's wave theory of light (as waves in an elastic aether) in his footnote to the first remark to the eighth proposition (519-20). 65. Kant's point about the composition of motions appears simply to amount to the claim that a given finite agglomeration of solid particles (touching one another, therefore, at a finite number of points) will not exert pressures in all directions. In Euler's analogous argument, however, we also consider the process of allowing the sizes of the particles to shrink-and their number to
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increase-to infinity. It is not entirely clear whether Kant was acquainted with Euler's paper. 66. The first definition states that a fluid allows (frictionless) displacement I\~thout separation of its parts (526-27). This definition corresponds to what we now call an inviscid fluid, which offers no resistance to shear stress, and it belongs, from a modern point of view, to hydrodynamics rather than hydrostatics. Indeed, in a state of hydrostatic equilibrium there are necessarily no shear stresses (arising fi·om fluid flow) in any case, and the distinction between viscous and inviscid fluids is d1erefore irrelevant. 67. When a monad is "compressed" by the pressures exerted by external monads, the repulsive sphere of activity of the original monad is not actually changed at all (since, in the terminology of note 69 below, this field of repulsion is a static rather man a dynamical field). Rather, the change of "shape" experienced by a monad in this way is entirely due to the superposition of its own (static) fields of attraction and repulsion with those of the external monads. It is in this sense that each monad's fields are rigidly attached to it. 68. Indeed, the Physical Monadology breaks off at precisely this point, and the remaining parts of the larger projected work were never completed. (The Physical Monadology represents only the first part-"Specimen 1. "-of a larger project with me general title, "The Use in Natural Philosophy of Metaphysics Combined with Geometry." ) 69. The crucial distinction, from a modern point of view, is between a system wim a finite number of degrees of freedom and a system with an infinite number. Thus, although the fields of attractive and repulsive force of the Physical Monadologyare indeed spread out over a continuum of spatial points, they are still static rather than dynamical fields, in the sense that tl1ey are capable of no changes not rigidly determined by the motions of their (finite number of) point-center sources. (This is a consequence of the fact that they represent instantaneous actionat-a-distance forces.) By contrast, the fields of matter represented in continuum mechanics are dynamical fields, wherein genuine dynamical actions can be continuously propagated over space and time. Typical modern fields of force, such as the electromagnetic field, for example, are also dynamical in this sense, and they, too, possess an infinite number of degrees of freedom. Indeed, the mathematics used to describe such fields arose directly out of continuum models of the optical and electromagnetic aether. It is therefore a serious mistake to view the force-center atomisms of Boscovich and the early Kant (see note 58 above) as progenitors of the field-theoretic conception of matter developed in the nineteenth century by such minkers as Michael Faraday and James Clerk Maxwell. Adickes (1924-25, vol. 1, 196, n. 1), for example, entirely misses this point when he equates the "monadological-dynamical" conception of matter of Boscovich and the early Kant with the later (field-theoretic) view of matter developed by "Ampere, Faraday, etc."-and, accordingly, expresses a very clear preference for Kant's 1756 theory of matter over that of 1786. 70. Compare Kant's illustration of the expansive force of matter by the elasticity or pressure exerted by the compressed air in an air pump in the following third explication (500). The idea that the fundamental force of repulsion paired with Newtonian universal attraction is most clearly manifested empirically by the
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expansive pressure exerted by a gas goes back to Kant's first introduction of the two fundamental forces in the Theory of the Heavens: see 1, 264-65. Here Kant articulates a cosmogenic conception according to which all other forms of matter then result from this state-of "complete chaos" or universal "dissipation [Zerstreung]" or "dissolution [Auflosung]" (225-27)-in a process where solid matter gradually precipitates out of the original gaseous "dissolution" under the action of the two fundamental forces. In the General Remark to Dynamics of the Metaphysical Foundations (526-29), the liquid and solid states are characterized by successively placing additional constraints (cohesion, internal friction, and so on) on the original state of permanent elasticity. In sharp contrast to the "mechanical" conception he is most concerned to reject, Kant takes the nature of solid or rigid matter to be the least well understood of all (529): "But why certain matters, even though they may have no greater, and perhaps even a lesser force of cohesion than other matters that are fluid, nevertheless resist the displacement of their parts so strongly, and hence can be separated in no other way than by destroying the cohesion of all parts in a given surface at once, which then yields the semblance of a superior cohesion-how, that is, rigid bodies are possible-is still an unsolved problem, no matter how easily the common doctrine of nature presumes to have settled it." 71. Again, this form of balancing argument contrasts sharply with the analogous argument presented in the Physical Monadology, where the balancing in question involves an atomism of discrete point-centers, and, accordingly, it is meant to explain the (finite) volumes of the resulting physical monads: see note 61 above, together with the paragraph to which it is appended. 72. This model of the atmosphere is developed, for example, in Euler (1755), as the first application of a more general theory of the equilibrium of fluids. Again, although we do not know for certain whether Kant was acquainted with this particular work, the "Vorerinnerung" to his remarkable Theory of the Winds of 1756 (1,491-92) shows that Kant was perfectly familiar with the model. 73. A proper understanding of Kant's critical version of the "balancing" argument therefore requires that we place it in the context of continuum mechanics. By contrast, if we consider it within classical particle mechanics, the result is nonsense, as Daniel Warren shows in detail in §§I-4b of his contribution to the present volume. In the final section of his paper (§4c), however, Warren sketches a more sympathetic reconstruction in the context of statics (including hydrostatics). It should be noted, however, that general formulations of modern continuum mechanics-including both statics and dynamics-essentially involve a principle of equilibrium between "body forces" (like gravitation) and "contact forces" (like pressure): see, e.g., Truesdell (1977, chapter 3, §1). In this sense, the distinction Kant sets up in the seventh explication of the Dynamics between "surface forces" (repulsion and expansive pressure) and "penetrating forces" (gravitational attraction and weight) can also be seen as entirely appropriate in this more general context. 74. See the paragraph above to which note 71 is appended. The very first occurrence of "quantity of matter" is found several pages earlier, in the first remark to the fourth proposition on infinite divisibility (where Kant explicitly opposes his earlier Physical Monadology): in the case of a true continuum, Kant says, "a greater
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or smaller space [can] be represented as completely filled by one and the same quantity of matter, that is, one and the same quantum of repulsive forces" (505). 75. This, of course, IS the standard Newtoman concept of quanuty of matter defined in the first definition of the Principia (p. 403): "Quantity of matter is a measure of matter that arises from its density and volume jointly." All references to the Principia are to Cohen and Whitman (1999). 76. Kant is tacitly assuming, in this scenario, that the density at every point in such an expanding continuum decreases uniformly (something that would not be true in the atomism of discrete force-centers articulated in the Physical Monadology). In the conu'ary case of compression considered in the remark to the third proposition, Kant explicitly makes a similar assumption, which "can be postulated in matter, in so far as essential elasticity belongs to it, as matter in general filling a space" (501). In particular, since "expansive force, exerted from every point, and in every direction, actually constitutes this concept [of matter]," "the same quantum of extending forces, when brought into a smaller space, must repel all the more strongly at every point, the smaller the space in which this quantum diffuses its activity" (ibid.). (The "quantum of extending [i.e., expansive] forces" here is thus the same as the "quantum of repulsive forces" of note 74 above.) 77. Compare note 73 above. The reason, according to Kant, that "true attraction," as a penetrating force, thus yields a measure of the quantity of matter is that "through true attraction all parts of matter act immediately on every part of another, whereas through expansive force only those at the surface of contact act, so that it is all the same whether much or little of this matter is found behind that surface" (524). I return to this point immediately below. 78. The particular example Kant considers is Newton's determination oftlle relative quantities of matter (masses) of Jupiter and Saturn by the accelerations produced on their moons. For a detailed discussion of this case see Friedman (1992a, chapter 3, 152-59). The general point Kant is making here is that Newton must take universal gravitation as a "true attraction" acting immediately at a distance in order to conclude that it is in fact proportional to quantity of matter in the relevant sense. In any case, it is important to remirtd ourselves that it is precisely this property of universal gravitation that enables Newton to determine the masses of the primary bodies ill the solar system-and thus implement the constructive procedure for determining the true motions ill the solar system that is central to the overall argument of the Metaphysical Foundations culminating in the Phenomenology: see note 47 above, together with the paragraph to which it is appended. 79. Compare 525: "A body, ill the physical sense, is a matter between determinate boundaries (which tllerefore has a figure). The space betlVeen these boundaries) considered in accordance lVith its magnitude, is the volume [of the body]." 80. Kant is obviously assumirtg here-unlike in the parallel passage from the Dynamics (526)-that the two matters of the same kind are to be considered in the same conditions of external compression, so that they thereby necessarily have the same density. The general point Kant is making ill the Dynamics is that, in his system of "relative" or "dynamical" density based on an intensive filling of space, "it is inappropriate to think of matters as related with respect to their density, if we do not imagine them as specifically of the same kind, so that one can be generated from the
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other by mere compression." Therefore, "since the latter [condition] by no means appears to be necessary to the nature of all matter in itself [unlike in the system of "absolute" or "mathematical" density favored by the mechanical natural philosophy], no comparison with regard to density can properly take place between matters of different kinds, between water and mercury, for example, even though it is Customary." Nevertheless, once we have a universally valid mechanical comparison of matters with respect to quantity of matter (in the first proposition of the Mechanics), we can then mechanically compare densities in terms of mass divided by volume. 81. Although here is not the place to give a detailed analysis, what Kant means by the exertion of "moving force" is precisely an exchange of momentum: here, in mechanics, "the force of a matter set in motion is considered as communicating this motion to another" (536). The first proposition states that "the quantity of matter, in comparison with every other matter, can be estimated only by the quantity of motion at a given space" (537). And the remark to this proposition then responds to the apparent circularity involved in the idea that quantity of matter can only be estimated in terms of quantity of motion, but quantity of motion, in turn, is "estimated by the quantity of the moved matter and its speed together," by distinguishing between the "explication of a concept, on the one hand, and that of its application to experience, on the other" (540): "The quantity of the movable in space is the quantity of matter; but this quantity of matter (the aggregate of the movable) manifests itself in experience only by the quantity of motion at equal speed (for example, by equilibrium [Gleichgewicht])." I return to this condition of "equal speed" immediately below. 82. Newton explains the dependence of all static forces involving machines (including the balance or lever, of course) on his Third Law of Motion (from which the conservation of momentum is derived in Corollary 3) in the Scholium to the Laws of Motion (Principia 428-30). 83. This point is well emphasized, and discussed in considerable detail, in Carrier (2001). 84. Compare a parallel passage in the essay on Negative Magnitudes of 1763 (2,199): "the weights on the two arms of the balance are at rest, if they are placed on the lever in accordance with the laws of equilibrium [Gleichgewicht]." 85. As Carrier (2001, §3) also rightly emphasizes, this means that Kant's reference to equal "velocities" can, in this context, be read in terms of equal infinitesimal velocities or accelerations. Indeed, Kant himself has already made this clear in the note to the first proposition, where, in characterizing the "moving force" uf a body, he says that "it may be that the velocity of its motion is finite or infinitely small (mere striving towards motion)" (539). In the General Remark to Mechanics Kant characterizes such an infinitely small velocity as "the moment of acceleration" (551). 86. This is essentially Newton's reasoning in the passage cited in note 82 above. 87. Thus, when we subsume Galileo's law of fall under the Newtonian theory of universal gravitation, the former emerges as a simplification or idealization of the true situation in two different respects: we ignore both the dependence of gravitational force on the distance from the center of the earth (in accordance with the inverse-square law) and the gravitational interaction between the falling body and the earth itself (in accordance with the equality of action and reaction). Ignoring
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the interaction between the falling body and the earth, and thus taking the earth be at rest, corresponds to the first stage in Kant's constructive procedure for determining the true motions in the solar system: see note 47 above, together with the paragraph to which it is appended. 88. The gravitational force acting on Jupiter is given by mJa J = Gm JmM/r 2 , where mM is the mass of the moon in question. Holding mM and r constant, then, it follows that mJa J is proportional to mJ' This is precisely analogous, in the context of universal gravitation, to the way in which the gravitational force or weight of a falling body near the earth's surface is proportional to its mass. A tricky issue of interpretation/translation arises in Kant's passage, according to which the attracting body "imparts to itself a velocity of its own inherent motion ... , which, in like external circumstances, is exactly proportional to the aggregate of its parts [i.e., its mass]." Grammatically, "which" could refer to either velocity or (quantity of) motion, but only the latter makes conceptual sense-since the attracting body's (infinitesimal) velocity (acceleration) is certainly not proportional to its mass. 89. Ignoring the acceleration (and thus change of momentum) experienced by Jupiter is thus analogous to ignoring the acceleration (and thus change of momentum) produced in the earth by a falling body. Note that there is no effective way to determine the change of momentum experienced by the attracting body in either case: we (in the context of the Principia) can determine neither the velocity (i.e., acceleration) produced in Jupiter by the resistance of its moon nor that produced in the earth by the resistance of a falling body. In comparing the masses of Jupiter and Saturn by the accelerations produced in their respective moons, however, the exchange of momentum between these two primary bodies is of paramount importance, for it is by applying the equality of action and reaction to this interaction that Newton actually establishes that the accelerations produced in their satellites yields a measure of their relative masses (where the resulting exchange of momentum is then determinable, at least in principle, by the perturbations in their orbits thereby produced): for details, see again my discussion cited in note 78 above. 90. In the essay on Negative Magnitudes of 1763 Kant states a general metaphysical conservation principle governing positive and negative "real grounds" or "positings" (2, 194): "In all natural changes ofthe world the sum ofthe positive [real grounds], in so far as it is estimated in such a lvay that the agreeing (not opposed) positings are added and really opposed [i.e., negative real grounds] are subtracted from one another, is neither increased nor diminished." He then illustrates this general metaphysical principle by a more specific "mechanical rule" modeled closely on Newton's formulation of the conservation of momentum in Corollary 3 to the Laws of Motion (2,195). 91. It is in this way that the conception of quantity of material substance in the Mechanics is explicitly linked to the fourth proposition of the Dynamics stating the infinite divisibility of material substance. The latter depends on the fifth explication of the Dynamics, according to which (502): "Material substance is that in space which is movable in itself, that is, in isolation from everything else existing external to it in space." The proof of the second proposition of the Mechanics then turns on this same idea (541-42): "In every matter the movable in space is the ultimate subject of all accidents inhering in matter, and the aggregate of these movto
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ables, external to one another, is the quantity of substance. Hence the quantity of matter, with respect to its substance, is nothing else but the aggregate of substances of which it consists. Therefore, the quantity of matter cannot be increased or diminished except in such a way that new substance thereof arises or perishes." 92. In the limit, when the matter in question expands to infinity, we have zero density at every point as the same initial (finite) quantity of matter has now become continuously distributed over infinite space: see note 76 above, together with the paragraph to which it is appended. 93. By contrast, these two demands are not jointly satisfied by intensive magnitudes in general, and it is for precisely this reason that Kant now finds the theory of the Physical Monadology deficient (see the paragraph to which note 91 above is appended). In particular, according to the Physical Monadology the quantity of mass is entirely independent of the balancing of the two fundamental forces of attraction and repulsion: it is instead given by a third fundamental force-the force of inertia (vis inertiae)-which attaches to each point-like force-center entirely independently of the small finite volumes (spheres of activity) thereby determined (see 2, 485-86). From the point of view of the critical Kant, therefore, the intensive magnitude of vis inertiae must necessarily be capable of (continuously) diminishing to zero independently of whatever else may be happening in the region of space surrounding any given force-center, and so physical monads, from this point of view, cannot be material substances. 94. It is only in this way, too, that Kant can link the traditional concept of quantity of matter he adopts from Newton (see note 75 above) to the concept of quantity of motion (momentum) and its possible exchanges. In Newton, by contrast, the Second Law of Motion simply equates the dynamical concept of quantity of matter as the product of density and volume with the mechanical concept of mass (linked to the concept of inertia) by fiat. This is the basis for Mach's famous criticism of the Newtonian definition of quantity of matter in The Science of Mechanics; and it is for precisely this reason that Mach himself rejects the concept of quantity of matter as "metaphysical" and instead prefers to characterize the mechanical concept of mass implicitly by means of the Laws of Motion themselves. More precisely, Mach (1883, chapter 2, §5) appeals to the equality of action and reaction in stipulating that "we call bodies of equal mass those which impart equal and opposite accelerations acting on one another." Here, therefore, the mechanical concept of mass or inertia has been completely separated from any notion of quantity of matter involving the volumes and densities of the interacting bodies in question-and it could be fully instantiated, at least from a mathematical point of view, by a mere system of interacting point-masses. 95. See, in particular, Kant's well-known criticism of the Leibnizean monadology in the Amphiboly of Pure Reason (A274/B330): "[T]he Leibnizean monadology has no other basis at all than [the circumstance] that this philosopher represented the distinction between the inner and the outer merely in relation to the understanding. Substances in general must have something inner, which is therefore free from all outer relations and thus all composition. The simple is therefore the basis for the inner in things in themselves. But what is inner in its state cannot consist in place, figure, contact, or motion (which determinations are all outer), and we can therefore ascribe no other inner state to the substances than
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that whereby we ourselves determine our own sense inwardly, namely, the state of representation. " 96. Essentially the same argument appears in the Paralogisms of Pure Reason in the second edition of the Critique, as a "refutation of Mendelssohn's proof of the permanence of the soul" (B413-14). Mendelssohn had argued, following tradition, that the soul, as absolutely simple, must be permanent, precisely because it has no parts which could then be divided from one another. In particular, Mendelssohn had attempted to repair a gap he saw in this traditional argument by appealing to the law of continuity: such an absolutely simple soul could not just disappear (without any division of parts), because this would involve a discontinuous jump from being to non-being. Kant uses his critical conception of intensive magnitudes to reply that an absolutely simple soul could nonetheless continuously vanish into nothingness by a diminution in its purely intensive "degree of reality." 97. Kant first insists on the necessary spatiality of all substance in his marginal notes to the first edition of the Critique. In a marginal note to A182, for example, Kant adds (23, 30): "Here the proof must be so developed that it applies only to substances as phenomena of outer sense, and therefore from space-which, together with its determinations, exists at all times." In a further note to A183 Kant adds (23, 31): "In the soul there is no quantum of substance possible. Therefore also nothing which one could determine by any predicate and call permanent." These passages, in turn, are clearly related to some of the central changes Kant made in the second edition involving the addition of the Refutation of Idealism and the General Remark to the System of Principles, together with corresponding changes in the Paralogisms of Pure Reason such as the refiltation of Mendelssohn considered in the immediately preceding note. All of these changes reflect the very fundamental way in which Kant's work on the Metaphysical Foundations is inextricably implicated in the ongoing development of the critical philosophy. 98. See the paragraph to which note 90 above is appended. As we see from note 90 itself, the fundamental conserved quantity in the essay on Negative Magnitudes of 1763 is precisely the total quantity of "reality"-conceived as a "real ground" or source of causal action. 99. The concept of action plays a central role in the discussion of quantity of matter and quantity of motion in the Mechanics as well, since, according to the first explication, the parts of the "aggregate of the movable" in question "are considered as acting (moving) together" (537) and, according to the following remark, "[a]s the quantity of motion in a body relates to that of another, so also does the magnitude of their action" (539 )-where, as the context makes clear, the "action" exerted by one body on another in the guise of "moving force" is expressed by the corresponding exchange of momentum. 100. See A205/B250: "Action already signifies the relation of the subject of causality [i.e., substance] to the effect." 101. These considerations help to answer a question raised in Andrew Janiak's insightful contribution to the present volume: namely, if Kant's discussion in the first Critique must abstract from all empirical concepts like mass, how can the properly "transcendental" perspective of the first Critique (in contrast to the more
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empirical perspective of the Metaphysical Foundations) possibly provide an adequate metaphysical foundation for the Newtonian concept offorce? (Janiak concludes, in particular, that only the spatial distances between substances can characterize Newtonian forces from the "transcendental" perspective.) The answer to this worry, I believe, is that, although Janiak is certainly correct that no empirical concept like mass can be considered from the properly "transcendental" perspective, the critical concepts of reality, quantum of reality, and the real in space, together (most importantly) with the concepts of substance (and its quantity), causality, force, and action still remain. Moreover, as we have just seen, the concepts of quantum of substance and action are related to one another in the first Critique in a way that is precisely parallel to the relationship between the concepts of quantity of matter (mass) and quantity of motion (momentum) in the Metaphysical Foundations. In this sense, there is an abstract counterpart, in the first Critique, of the principle of the conservation of momentum that plays an absolutely central role (as Kant was one of the first to emphasize) in the Newtonian concept of force. Indeed, it is not too much to say, I think, that Kant now simply defines the exertion of a force (a causal action of one substance on another) in terms of the exchange of such a conserved quantity, so that "action at a distance," in particular, is therefore perfectly intelligible: if such an exchange occurs between spatially distant bodies (as, for example, a momentum exchange occurs between Jupiter and Saturn), then they causally act on one another regardless of what is happening in the intervening a space. 102. Compare note 95 above, together with the paragraph to which it is appended. The reference to the forces of attraction and repulsion in the present passage is to the critical version of the dynamical theory of matter, for, in the Physical Monadology, the concept of mass involves a third fundamental force or force of inertia (see note 93 above). By contrast, Kant's discussion of inertia in the third proposition of the Mechanics (543-44) equates this concept with the mere "lifelessness" of matter (which, in particular, can have no "internal" principles modeled on mental states or activities), and, in his remarks to the fourth proposition (549-51), Kant completely rejects the concept of a force of inertia. See also Kant's discussion of the Leibnizean monadology in his second remark to the fourth proposition of the Dynamics (507-508): "The ground for this aberration [denying the infinite divisibility of space] lies in a poorly understood monadology, which has nothing at all to do with the explanation of natural appearances, but is rather an intrinsically correct platonic concept of the world devised by Leibniz, in so far as it is considered, not at all as object of the senses, but as thing in itself, and is merely an object of the understanding-which, however, does indeed underlie the appearances of the senses. Now the composite ofthings in themselves must certainly consist of the simple, for the parts must here be given prior to all composition. But the composite in the appearance does not consist of the simple, because in the appearance, which can never be given otherwise than as composed (extended), the parts can only be given through division, and thus not prior to the composite, but only in it." 103. Compare note 16 above, together with the paragraph to which it is appended. In the Dialectic of the Critique of Pure Reason, starting with the first edition, Kant lists the three highest ideas of reason as God, soul, and world. In the context of the further development of his moral philosophy, however, Kant changes
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rJlis list to God, freedom, and immortality-where freedom (as required by moralitv) corresponds to the discussion of the idea of the world (and, in particular, to the Third Antinomy), and immortality corresponds to the discussion of the idea of rJle soul (and its presumed substantial permanence). The crucial step is taken in the Critiq1te ofPractical Reason (1788), where Kant views freedom as a "fact [Faktum] of reason," given in our personal experience of the moral law, and declares that it is precisely this "fact" which ultimately gives objective reality (from a practical point of view) to supersensible causality. The (practical) objective reality of God and immortality then follows from our experience of the moral law, in so far as they are key elements of "rational faith" (Vernunftglaube) functioning as necessary presuppositions (from a practical point of view) for approximating the Highest Good in nature. As we shall see in note 105 below, this change is also reflected in the Preface to the second edition of the Critique. 104. As we saw in my discussion of the Newtonian era, Kant's parallel transformation of the Newtonian metaphysics of absolute space is equally important. This transformation corresponds to Kant's argument for transcendental idealism in the First Antinomy (concerning the extent of the material universe in successively larger and larger regions of space), just as the critical conception of the dynamical theory of matter and material substance corresponds to the argument of the Second Antinomy (concerning the divisibility of matter into smaller and smaller material parts). For further discussion of this parallel see the Inu'oduction to Friedman (2004) and Friedman (2006c). 105. Compare note 103 above. The main point of this famous discussion, of course, is to indicate how the freedom required by morality can be reconciled with the "mechanism of nature" on the basis of "our critical distinction between two modes of representation (sensible and intellectual) and the resulting limitations of the pure concepts of the understanding" (Bxxviii). Kant concludes by (very briefly) incorporating the ideas of God and immortality as well (Bxxix-xxx): "Thus morality maintains its place, and the doctrine of nature does so as well, which could not have been achieved if critique had not first taught us our unavoidable ignorance with respect to things in themselves, and had not limited everything we can theoretically cognize to mere appearances. This same discussion of the positive use of the critical principles of pure reason can be given in relation to the concepts of God and ofthe simple nature of our soul, which I shall omit for the sake of brevity. Thus, I can also not even assume God, freedom, and immortality on behalf of the necessary practical use of my reason, if I do not simultaneously deprive speculative reason of its pretension to extravagant insights; because, in order to attain these [insights], it has to employ principles which-in so far as they in fact extend merely to objects of possible experience, even if they are applied to that which cannot be an object of experience-actually always transform this into an appearance, and [thereby] declare all practical extension of pure reason to be impossible. I therefore had to deny knowledge, in order to secure a place for faith; and the dogmatism of metaphysics-that is, the prejudice that reason can progress [in metaphysics] without critique-is the true source of all unbelief in conflict with morality, which is certainly always very dogmatic." 106. For a good introduction to the nineteenth-century philosophical situation afrer Hegel see Cassirer (1950). For an extended discussion of nineteenth-century
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academic neo-IZantianism see IZohnke (1991). More generally, the essays collected in Friedman and Nordmann (2006) provide an overview of recent work on "the IZantian legacy in nineteenth-century science" by both historians and philosophers. 107. I shall discuss the Marburg School ofneo-IZantianism founded by Cohen, and its relationship to both logical empiricism and phenomenology, in the follo vv ing section. 108. This is the lesson of Beiser's contribution to the present volume as well as Beiser (2006); see Beiser (2002), (2003) for fuller discussions. My o\vn recent work on Schelling and Naturphilosophie, upon which I am drawing here, appears as Friedman (2006a), (2006b). 109. As I pointed out in note 48 above, there is also a distinction between "constitutive" (mathematical) and "regulative" (dynamical) principles even within the principles of pure understanding-the latter, however, are still "constitutive of experience" (but not of intuition). Note, in addition, that there is an intimate connection, as well, between the distinction between constitutive principles of pure understanding and regulative principles of reason, on the one side, and the fundamental distinction between the faculty of pure intellect or understanding and the faculty of pure sensibility, on the other: constitutive principles of pure understanding must be applied to or schematized in terms of sensibility, whereas regulative principles of pure reason necessarily jail to have such schemata: see A664-67jB692-95. 110. See 518: "The action of the universal attraction immediately exerted by each matter on all matters, and at all distances, is called gravitation; the tendency to move in the direction of greater gravitation is weight. The action of the general repulsive force of the parts of every given matter is called its original elasticity. Hence this property and weight constitute the sole universal characteristics of matter, which are comprehensible a priori, the former internally, and the latter in external relations. For the possibility of matter itself rests on these two properties. Cohesion, if this is explicated as the mutual attraction of matter limited solely to the condition of contact, does not belong to the possibility of matter in general, and cannot therefore be cognized a priori as bound up with this. This property would therefore not be metaphysical but rather physical, and so would not belong to our present considerations." Ill. Compare 470-71: "So long, therefore, as there is still for chemical actions of matters on one another no concept to be discovered that can be constructed, that is, no law of the approach or withdrawal of the parts of matter can be specified according to which, perhaps in proportion to their density or the like, their motions and all the consequences thereof can be made intuitive and presented a priori in space (a demand that will only with great difficulty ever be fulfilled), then chemistry can be nothing more than a systematic art or experimental doctrine, but never a proper science, because its principles are merely empirical, and allow of no a priori presentation in intuition. Consequently, they do not in the least make the principles of chemical appearances conceivable with respect to their possibility, for they are not receptive to the application of mathematics." 112. It is no accident, therefore, that examples from contemporary (phlogistic) chemistry playa central role in Kant's discussion of the regulative use of reason in the Appendix to the Transcendental Dialectic of the flfSt Critique: see, e.g.,
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A645-66/B673-74, A562-63/B680-81. For further discussion of Kant's conception of chemisu'y in this period see Friedman (1992a, chapter 5). Kant appears to be despairing of the lack of precisely such a grounding in the fundamental forces of matter in the passage quoted in note III above. 113. As we pointed out in note 102 above, Kant's remark to his statement of the law of inertia in the Mechanics emphasizes precisely its connection with lifelessness-viz., the non-existence of any internal principle of change. See, in particular, the conclusion of this remark (544): "The possibility of a proper natural science rests entirely and completely on dle law of inertia (along with that of the persistence of substance). The opposite of this, and thus also me deam of all natural philosophy, would be hylozoism. From this very same concept of inertia, as mere lifelessness, it follows at once that it does not mean a positive striving to conserve its state. Only living beings are called inert in this latter sense, because they have a representation of anomer state, which they abhor, and against which they exert their power." 114. My formulation of this skeptical problem is indebted to the discussion of Salomon Maimon's post-Kantian skepticism in Franks (2003), almough Franks himself does not emphasize, as I do, me distinction between constitutive and regulative principles; he instead formulates what I take to be essentially the same problem by means of a distinction between scientific judgments and everyday or "ordinary" judgments. 115. These three forms thereby constitute what Schelling calls me "second potency" of the original dynamical process; and mey correspond, respectively, to the one-dimensionality of a line (between two magnetic poles), me two-dimensionality of a surface (in a distribution of charge manifesting static electricity), and the mreedimensionality of a volume (within which chemical or galvanic forces act): see Schelling (1803, supplement to book I, chapter 6). There is therefore a close connection, for Schelling, between me further articulation of the original dynamical process and a procedure of geometrical construction in pure intuition. Although Frederick Beiser does not discuss this particular connection between geometrical construction and Schelling's Naturphilosophie in his contribution to the present volume, he does discuss it in part N, chapter 4, §2 of Beiser (2002): compare note 108 above, together wim the paragraph to which it is appended. 116. For Schelling, this dialectical continuation takes a quite precise and specific form: corresponding to the magnetic, electrical, and chemical (or galvanic) forms, we then have, as the "mird potency" of the original dynamical process, the biological powers of reproduction, (nervous) irritability, and sensibility: see again Schelling (1803, supplement to book I, chapter 6). 117. See Schelling (1848,54; 1988,40): "Finally, if we comprehend nature as a single whole, then mechanism, i.e., a past-directed series of causes and effects, and purposiveness [Zweckmafl~keit], i.e., independence of mechanism, simultaneity of causes and effects, stand opposed to one another. In so far as we now unite these two extremes, an idea of a purposiveness of the whole arises in us-nature becomes a circle that returns into itself, a self-enclosed system." 118. See (1848, 46-47; 1988, 35): "This philosophy must admit, therefore, that mere is a graduated development [Stufenfolge] of life in nature. Even in mere organized matter there is life, but only life of a limited kind. This idea is so old,
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and has been preserved until now in the most varied forms up to the present day(already in the most ancient times the whole world was [regarded as] penetrated by a living principle, called the world-soul, and Leibniz's later period gave every plant its soul)-that one can well surmise in advance that some ground for this natural belief must lie in the human spirit itself. And it is in fact so. The entire mystery surrounding the problem of the origin of organized bodies rests on the circumstance that in these things necessity and contingency are united in the most intimate way. Necessity, because their existence is already purposive, not only (as in the case of the work of art) their form; contingency, because this purposiveness is nonetheless onlv actual for an intuiting and reflecting being." . 119. An especially important electro-chemical researcher of the time was Johann Ritter, who had encountered Schelling's Naturphilosophie at Jena and then was a close friend and collaborator of Hans Christian Oersted (see the following note): for further discussion and references see Friedman (2006a). 120. I argue in detail in Friedman (2006a) that Oersted's experimental discovery of electromagnetism-that an electric current or galvanic circuit is surrounded by a magnetic field-was in fact decisively influenced by Schelling's Naturphilosophie (to which he was first introduced by Ritter: see note 119 above). The key idea which inspired Oersted, as he himself explains, is that the forces acting in the galvanic circuit are intermediate between those active in magnetism and (static) electricity, respectively, in so far as they are less "bound" than the former but more "bound" than the latter: a direct magnetic effect of electrical forces should therefore be sought in the action of the galvanic circuit. 121. See (1848, 187; 1988, 149): "Therefore, already in the chemical properties of matter there actually lie the first, although still completely undeveloped seeds of a future system of nature, which can unfold into the most varied forms and structures, up to the point where creative nature appears to return back into itself. Thus, at the same time, fluther investigations are marked out, up to the point where the necessary and the contingent, the mechanical and the free, separate from one another. Chemical phenomena constitute the middle term between the two. It is this far, then, that the principles of attraction and repulsion actually lead, as soon as one considers them as principles of a universal system of nature." 122. Schelling is here self-consciously returning to precisely the "hylozoism" Kant explicitly rejects: compare notes 113, 116, and 118 above. This essentially biological or organic conception of nature then entails the overcoming of all skepticism in the sense that tlle closing of the circle implicated in notes 117 and 121 above (embracing both mechanism and teleology) means that transcendental philosophy and Naturphilosophie-spirit and nature-are ultimately identical, in so far as nature itself gives rise to both life in general and conscious or rational life in particular. See (1848,56; 1988,42): "Nature should be the visible spirit, spirit the invisible nature. It is here, therefore, in the absolute identity of spirit within us and nature outside us, that the problem of how a nature outside us is possible must be solved." 123. I can present only a brief outline here. For further details and references see Friedman (1992a, chapter 5), Friedman (2006a). 124 In the Opus postumum Kant goes so far as to say that, if the "transition" project he is now embarked on cannot be successfully completed, then there is a
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fundamental "gap" in the critical system. For the importance of this idea see Forster (1987); for further discussion see Friedman (1992a, chapter 5), Forster (2000). 125. Aside from the work of Ritter and Oersted mentioned in notes 119 and 120 above, this also included such seminal figures as Humphry Davy and Michael Faraday-both of whom were strongly influenced by Schelling's Naturphilosophie rhrough the mediating influence of Samuel Coleridge. For discussion of Davy and Faraday see Williams (1965; 1966). 126. See Helmholtz (1903, 88). For discussions of Helmholtz's place in the "back to Kant!" movement see again the references cited in note 106 above. See also Cahan (1992; 1993) for useful overviews of Helmholtz's many contributions. My own most detailed discussion of Helmholtz, on which I am drawing here, is Friedman (1997b). 127. I cite this work parenthetically in the text, using page numbers from both rhe reprinting in Helmholtz (1882) and the translation in Kahl (1971). 128. Compare Helmholtz's statement in that lecture (1903, 116): "Hence the investigation of sense perception leads us on also to that knowledge already found by Kant, that the proposition, 'no effect without a cause,' is a law of our thinking given prior to all experience." 129. In particular, Helmholtz gives an argument for the conclusion that all forces must be resolvable into fundamental forces of attraction and repulsion which is precisely parallel to Kant's argument for the same conclusion in the note to the second explication of the Dynamics of the Metaphysical Foundations. See "On the Conservation of Force" (1882, 15; 1971, 5): "[M]echanics takes the forces of material points, i.e., those of the points of a space filled with matter, to be basic. But points have no spatial relations relative to one another other than their distance, for the direction of the line between them can only be determined in relation to at least two other points. A moving force, which they exert relative to one another, can therefore only be the cause of an alteration in their distance, i.e., either an attractive or repulsive [moving force]." Compare Metaphysical Foundations (498-99): "Only these two moving forces of matter [i.e., attractive force and repulsive force-MF] can be thought. For all motion that one matter can impress on another, since in this regard each of them is considered only as a point, must always be viewed as imparted in the straight line between the two points. But in this straight line there are only two possible motions: the one through which the two points remove themselves from one another, the second through which they approach one another. But the force causing the first motion is called repulsive force, whereas the second is called attractive force. Therefore, only these two kinds of forces can be thought, as forces to which all moving forces in material nature must be reduced." 130. Thus Helmholtz, in this respect, entirely agrees with Kant's anti-hylozistic view that the law of causality, instantiated in matter, is ultimately equivalent to "mechanical" lifelessness: compare notes 113 and 122 above, together with the paragraphs to which they are appended. Helmholtz makes only a few brief remarks on organic nature at the end of his memoir on the conservation of energy. However, his later 1869 lecture on "The Aim and Progress of Physical Science" (translated in Kahl [1971]) includes a lengthy criticism of all attempts to inject a
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"life-soul" or "vital force" into nature and explicitly appeals to Darwin's theory of evolution to argue against intrinsic purposiveness or teleology in nature. 131. Helmholtz gradually came to realize that a reduction to (time-indepen_ dent) Newtonian central forces of attraction and repulsion does not succeed in the case of electricity and magnetism, and, accordingly, that his claim that conservation of energy implies an exclusive reduction to such Newtonian forces is incorrect (compare note 134 below). Helmholtz therefore abandoned his reliance on Wilhelm Weber's action-at-a-distance theory in his own work on electricity and magnetism, and instead became the leading advocate on the continent ofMaJl.owell's field theory. 132. Helmholtz's 1855 lecture generalizes this to the more basic inference to the existence of an external world (1903, 115-16): "[W]e never immediately perceive the objects of the external world, but we only perceive the effects of these objects on our nervous apparatus, and this has been so from the first moment of our life onwards. In what way, then, have we first reached across from the world of the sensations of our nerves into the world of actuality? Obviously, only through an inference; we must presuppose tlle presence of the external object as cause of our nerve-excitation; for there can be no effect without a cause." This passage is then immediately followed by the sentence quoted in note 128 above. 133. In Kahl (1971), however, the first sentence is grossly mistranslated so as to have Helmholtz saying that he still considers "Kant's epistemological insights" to be correct. 134. Helmholtz's note goes on to suggest that he was also earlier mistaken in separating matter too sharply from the forces or laws in accordance with which it acts-and so he here appears, more generally, to be rejecting the a priori theory of matter he had borrowed from Kant as well. It seems likely that Helmholtz's movement away from this position was closely connected with his realization that electromagnetic forces cannot be assimilated to "unalterable" central forces, and thus that the fundamental equivalence he had tried to set up between phenomenological energy conservation, on the one side, and a reduction to masses and central forces, on the other, does not in fact hold good (see notes 2-4 from 1881; and compare note 131 above). This particular route from phenomenological lawlikeness to "ultimate unalterable causes" lying behind the appearances was therefore closed. 135. I cite this work by page numbers of volume III of the third (1910) edition and the (1925) English translation of this edition, respectively The most extended defense of Helmholtz's "empiricist" theory is found in §26, "On Perceptions in General." The second (1896) edition drastically revises this section by incorporating much of the content of Helmholtz's famous 1878 lecture, "The Facts in Perception," to be discussed below. (The third edition is a posthumous reprinting of the first, and it gives the page numbers of the original edition in the margins.) 136. For the development and basic ideas of Helmholtz's theory I largely follow the lucid and balanced account in Hatfield (1990, chapter 5). Hatfield is particularly careful to distinguish the "empiricist"/"nativist" debate in the psycho-physiology of perception from the traditional "empiricist"/"rationalist" debate in modern epistemology: see Appendix A. I also agree with Hatfield in seeing a gradual "retreat" from tile early position of 1855: see pp. 208-14.
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137. Helmholtz (1910, 433; 1925, 533). An earlier passage makes the connection with the (conceptual) faculty of understanding more explicit (1910, 21; 1925,23): "The representation ofa spatially extended body, e.g., a table, includes a mass of individual observations. There lies comprised therein the entire series of images that this table would provide me if I were to consider it from various sides and from various distances, if I were to lay my hands successively on the various points of its surface. Such a representation of a particular individual body is thus actually already a concept, which comprehends under itself an infinite number of particular intuitions following one another in time, all of which can be derived from it, just as the generic concept 'table' in turn comprehends within itself all particular tables and expresses their common characters." 138. The connection with the perception of space, as mediated by bodily motion, is indicated shortly before the passage last quoted (1910, 26; 1925,29): "Of the greatest importance, finally, for the fixity of our conviction in the correctness of our sensory perceptions are the tests that we undertake by means of the optional motions of our body. There thereby arises the same kind of fixed conviction, relative to merely passive observation, that we gain in scientific investigations through the experimental method. The proper ultimate ground, through which all our consciously executed inductions receive the power of conviction, is the law of causality. " 139. Compare the earlier discussion, quoted in note 137 above, concerning the localization of a table in space by means of the (regular) "series of images that this table would provide me if I were to consider it from various sides and from various distances." If we compare the talk about "concept[s]" and "generic concept[s]" in this passage with the remarks about "general rules" and "universal generic concepts" in the passage from the introduction to the monograph on energy quoted in the paragraph to which note 127 above is appended, it seems clear that the inductive regularities in question lie wholly on the side of the "appearances." (And it also seems to follow, accordingly, that it is no longer a "condition for the complete conceptualizability of nature" that we find "ultimate causes" behind the "appearances"-for "general rules" and "universal generic concepts" are found precisely on the side of the "appearances.") 140. All quotations from "The Facts in Perception"-and from "On the Origin and Significance of the Axioms of Geometry" (see below)-are taken from Hertz and Schlick (1921), translated in Cohen and Elkana (1977): in this case, from (1921,117; 1977, 124). 141. This result, as originally sketched by Helmholtz, was later proved by Sophus Lie within his theory of continuous groups. For the work of Helmholtz and Lie see Torretti (1978, §3.1). For a philosophically and mathematically sophisticated discussion of Helmholtz and Riemann see Stein (1977, §§VI, VII). 142. For a discussion of Helmholtz's results in the context of his theory of space-perception see Richards (1977). Richards emphasizes, in particular, that Helmholtz had first hoped thereby to derive the specifically Euclidean character of perceptual space: overlooking the existence of spaces of constant negative curvature described by Bolyai- Lobachevsky geometry, he argued that the specific character of our perceptual space could be derived from three-dimensionality, free mobility, and infinity (where this last condition rules out the case of constant
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positive curvature). Only after he had become acquainted with Eugenio Beltrami's work on models for Bolyai-Lobachevsky geometry in 1868 was Helmholtz then able to correct this mistake, and this led him to his final transformation of the Kantian idea of space as a "necessary form of our external intuition" discussed immediately below. 143. See (1921, 117-21; 1977, 124-28). Helmholtz views the condition of free mobility, in particular, as a necessary condition for the possibility of spatial measurement, and thus for the application of geometry to experience. For further discussion see the works cited in notes 141 and 142 above, as well as Friedman (1997b). 144. This is the title of Appendix 2 to "The Facts in Perception": see (1921, 140-42; 1977, 149-52). For further discussion see again Friedman (1997, especially note 56). 145. Helmholtz formulates this "fundamental problem" at the beginning of "The Facts of Perception" as follows (1921, Ill; 1977, 117): "What is truth in our intuition and thought? In what sense do our representations correspond to actuality?" 146. Comparing the passage just quoted with the closely related passage quoted at the end of the paragraph to which note 145 above is appended, it follows that "the lawlilce ordering in the realm of the actual" is the same as "the lawlilce in the appearance." Moreover, both passages should be compared with the passage from 1881, to which note 133 above is appended, where Helmholtz formulates the principle of causality as "nothing other than the presupposition of the lawlikeness of all the appearances of nature." For further discussion of Helmholtz's "sign theory [Zeichentheorie]" of perception in this context see Friedman (1997b). 147. In this sense, there is also a counterpart of the Kantian conception of the schematization of the intellect in terms of sensibility in Helmholtz: by subjecting our sensations to the condition of free mobility (an especially fundamental kind of lawlikeness), we insure the application of geometry to our outer perceptions (see note 143 above). In Kant himself, however, tlle situation is considerably more complicated, because of his fundamental distinction between mathematical and dynamical principles (compare note 48 above): in particular, it is possible to instantiate the categories of quantity a priori in pure spatial intuition, and thereby apply geometry to the objects of geometrical construction, independently of the application of tlle principle of causality. Nevertheless, successfully to apply geometry to the objects of experience, I believe, it is also necessary to apply all the corresponding Analogies of Experience-including, of course, the principle of causality. For the application of geometry to these objects is inextricably entangled, in turn, with the more general, and essentially dynamical procedure Kant calls "time determination"; and this point, in turn, is closely connected with Kant's distinction between space (and time) as form of intuition and as formal intuition (see note 33 above): for further discussion see Friedman (1992a, 197-203), and again Friedman (2000a, 197-99), (2003b ). 148. As we pointed out above (see note 39, together with the paragraph to which it is appended), there is a presentiment of the condition of free mobility in Kant himself. What Kant did not (and could not) see, of course, is that we can then abstract this condition (the existence, in modern terms, of a group of rigid motions
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or isomen-ies) from the specific character of Euclidean geometry: for further diswss ion see again Friedman (2000a), where I explain, in particular, how we can thereby extend the procedure of straight-edge and compass construction to all three classical geomen-ies of constant curvature. 149. For a balanced discussion of the interpeneu'ation of "naturalistic" and "normative" (or "transcendental") elements in Helmholtz's thought see again Hatfield (1990): compare note 136 above. Kant makes clear the irrelevance of empirical psychology to the discipline he here calls "transcendental logic" (another name for "transcendental philosophy" in this context) in an important passage in [he Critique of Pure Reason (A54/B78): "As pure logic [this discipline] has no empirical principles, and hence borrows nothing (as one sometimes persuades oneself) from psychology, which thus has no influence at all on the canon of the understanding. It is a demonstrated doctrine, and everything in it must be certain entirely a priori." (Of course Kant does not mean by "psychology" the same thing Helmholtz does.) 150. Paul Pojman's contribution to the present volume details some of the confusions that result from this assimilation in connection with Mach's influence on Carnap's Aufbau in particular. 151. For a detailed discussion of Fechner's psycho-physics and philosophy see Heidelberger (1993/2004). 152. Mach's neutral monism is already clear in the first chapter (in all editions), and it is very explicitly stated in Mach (1886, 141; 1959, 310): "There is no gulf between psychical and physical, no inside and outside, no sensation to which an external thing different from it would correspond. There are only elements ofa single kind, from which the supposed inside and outside are composed-which are then precisely inside and outside only according to the way in which they are temporarily considered." Thus, Mach's motivations for taking the fundamental objects of all scientific study to be what he calls "elements" have very little in common with traditional empiricism (1900, 208; 1959, 312): "Anyone who has in mind the union of the sciences into a whole, however, must seek for a representation to which he can hold fast in all domains. Now if we resolve the whole material world into elements, which are simultaneously also elements of the psychical world (which, as such, are called sensations); and if, further, we regard it as the sole task of science to inquire into the combination, the interconnection, the mUUlal dependency of these homogeneous elements of all domains; we can then reasonably expect to build a unified, monistic structure on this representation and to be free of unfortunate confusing dualism. Indeed, if one regards matter as that which is absolutely constant and unchangeable, one then, in fact, destroys the interconnection between physics and physiology." In particular, one of Mach's central ideas is that his conception of the "elements" will thereby enable psychology and psychophysics to fit together smoothly with contemporaneous developments in the new physical science ofphenomenological thermodynamics. It is by no means surprising, then, that some of Mach's most important "historical-critical" sUldies, including his History and Root ofthe Principle ofthe Conservation ofEne1ZJY (1872) and Principles ofthe Theory ofHeat (l886), concern energetics and thermodynamics. Mach is concerned, above all, to defend the priority of phenomenological interpretations of these developments (in terms of differential equations governing directly measurable quantities
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such as temperature, pressure, and heat, for example), as opposed to the search for an underlying molecular model typified by the work of Ludwig Boltzmann. 153. Indeed, the similarities between Mach's views and those of American pragmatism are in no way accidental in view of the close interactions between Mach and William James to which Mach alludes (in all editions) in a footnote at the beginning of the centrally important chapter on "the influence of the preceding investigations on our conception of physics" in The Analysis of Sensations. 154. For discussion of this nineteenth-century debate see Turner (1994). Again, this debate should not be confused with the traditional philosophical debate between "empiricisn1" and "rationalism" (see note 136 above): indeed, from a philosophical point of view, Helmholtz is more "rationalist," Mach more "empiricist." 155. Here I am especially indebted to discussions with Paul Pojman on this topic, and to his Indiana University dissertation Pojman (2000). As Pojman points out, both Hering and Mach operated within the more explicitly teleological German understanding of evolution (as defended by Ernst Haeckel) and, in addition, were both Lamarckians. For Mach, in particular, this meant that cultural and historical evolution was entirely continuous with biological evolution. 156. For the relationship between Mach's work in The Science ofMechanics and the conten1poraneous development of the concept of an inertial frame of reference see the beautiful discussion in DiSalle (2002b). In what follows, I linl, my earlier discussion of Iunt with DiSalle's treatment of Mach. 157. This en1phasis on the importance of the equality of action and reaction in defining a privileged frame of reference mirrors Mach's parallel concern for giving empirical meaning to the Newtonian concept of mass: see note 94 above. Kant, as we have seen, was again doing something quite sinillar, in so far as he takes quantity of motion as the "empirical criterion" or condition for the "application to experience" of the concept of quantity of matter: see the paragraph to which note 98 above is appended. I do not know whether Mach had studied Kant's Metaphysical Foundations (he does not refer to it in The Science ofMechanics); nevertheless, the similarities between their respective discussions of both the Newtonian concept of quantity of matter and the Newtonian conception of absolute space, time, and n1otion are very striking, and, in any case, are certainly quite important from a conceptual point of view. 158. For discussion of this late nineteenth-century literature on the concept of an inertial fran1e see Torretti (1983, §1.5), DiSalle (1988), (1991), (2002c). 159. The advantage of more abstract formulations of the Newtonian Laws of Motion in terms of an equivalence class of inertial frames of reference is that the privileged cosmic fran1es assumed by Iunt and Mach exist only in very special conditions that are difficult-if not impossible-to realize. In Kant's case, for example, he is assuming a special cosmology of indefinitely nested concentric galactic structures converging, in some sense, on a single limit structure: this is why IZant assumes that a unique counterpart of "absolute space" thereby results, and, accordingly, he entirely overlooks the importance of Galilean relativity. 160. In Helmholtz, by contrast, although we certainly find a deeply insightful transformation ofIZant's conception of space as a form of intuition, we find nothing comparable in the case of IZant's analysis of absolute space, time, and motion; and, more generally, Helmholtz, unlike Mach, has nothing really new to contribute
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here. Kant's original theory of space thus had two central components-space figured as both an object of pure geomeu'y and a framework for physical motionwhere Helmholtz insightfully transformed the first component and Mach did something similar with the second. These two components, in Kant, are closely related to his distinction between space as form of intuition and as formal intuition: see note 147 above. 161. As we shall briefly note below, Poincare's scientific epistemology bears clear traces of the influence of both Fechner's psycho-physics and Darwinian evolutionary biology, but these influences pale in comparison with the importance of recent work in pure and applied mathematics for Poincare. 162. Thus, for example, one of Poincare's most famous mathematical discoveries was that certain groups of automorphisms in complex analysis are isomorphic to the group of isomeu'ies or rigid motions of Bolyai-Lobachevsky geomeu'y-a discovery which is vividly described in the chapter on "Mathematical Invention" in Poincare (1908). And it was this discovery which then led to Poincare's celebrated models of the hyperbolic plane: for a discussion of the latter see Torretti (1978, §2.3.7). 163. Thus, in the course of a discussion critical of those he takes to be excessively influenced by Hilbertian axiomatics, Klein (1909, 383-84; 1939, 187) expresses his characteristic standpoint as follows: "In the case of such people who are only interested in the logical side of the question, and not in the intuitive or general-epistemological side, one often finds the opinion nowadays that the axioms are only arbitrary propositions that we set up entirelyfreely, and the fundamental concepts, ultimately, are also only arbitrary signs for things with which we wish to operate. What is correct in such a view, of course, is that within pure logic no basis for these propositions and concepts is found, and that they therefore must be furnished or suggested from another side-precisely by the influence of intuition. However, the authors [in question] often express themselves much more one-sidedly, and so we are repeatedly forced nowadays, in connection with modern axiomatics, straightaway once again into that philosophical position which has been known since ancient times as nominalism. Here the interest in the things themselves and their properties is entirely lost; and one speaks only of how they are to be named and the logical schemata in accordance with which they are to be operated. One then says, for example, that we call a triple of coordinates a point 'without thereby thinking of anything,' and we stipulate 'arbitrarily' certain propositions that are to be valid of these points; one can set up arbitrary axioms in an entirely unlimited way, so long as one satisfies the laws of logic and talces care, above all, that there is no contradiction in the ensuing structure of theorems. I myself in no way share this standpoint, but take it to be the death of all science: the axioms of geometrrin my opinion-are not arbitrary but rather rational propositions, which are motivated, ingeneral, by spatial intuition and regulated, in their particular content, by considerations ofpurposiveness [Zweckmafligkeitsgriinde]." 164. I present a more detailed discussion of Poincare's conception of arithmetic-and of his hierarchy of sciences more generally-in Friedman (1999, chapter 4). For discussions of Poincare's views on arithmetic and logic see Parsons (1983, chapter 6), Goldfarb (1988), Folina (1992), (2006). 165. It is here that Poincare explicitly invokes Fechner's experimental work on the relationship between the intensities of sensations and their stimuli-and refers,
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in particular, to the concept of just noticeable differences (compare note 161 above). We know, as a consequence of Fechner's work, that, in general, there are neighboring elements in any sensory continuum (sensations of color, sound, heat, weight, and so on), such that two pairs of adjacent elements AB and BC are both indistinguishable, while A is nonetheless distinguishable from C. Transitivity of identity therefore forces us to conclude that B is distinguishable from A and C after all, and that it can be sensibly perceived as such under an appropriate "magnification." This process can then be iterated indefinitely on all new sensibly indistinguishable elements emerging at successively greater degrees of magnification, and it is in precisely this way, for Poincare, that our mind begins to create the true and exact mathematical continuum from the rough and approximate sensory or physical continuum. 166. Poincare calls this result "Lie's theorem," and, in general, he is quite ungenerous in acknowledging his debt to Helmholtz. Indeed, the only acknowledgement of Helmholtz's work on the foundations of geometry in Science and Hypothesis occurs right after Poincare mentions the original discovery of nonEuclidean geometry by Bolyai and Lobachevsky (1902,50-51; 1913b, 56): "The question was not exhausted; it soon made a great stride by the publication of the celebrated memoir by Riemann entitled: On the Hypotheses which Lie at the Basis of Geometry. This paper has inspired most of the recent works of which I shall speak further on and among which it is fitting to cite those of Beltrami and Helmholtz." For Helmholtz, Riemann, Lie, and Beltrami see notes 141 and 142 above, together with the paragraph to which they are appended. Perhaps the most important point, in this connection, is that Helmholtz himself did not have the concept of a group, and this is why, as I point out in note 141, it was left to Lie to formulate the proof of the relevant result. 167. Poincare expresses this simplicity group-theoretically: the group of Euclidean rigid motions or isometries is the only one containing a normal subgroup-i.e., a subgroup of transformations (in this case the translations) that is itself invariant under the action of the whole group. For further discussion, in relation to both Helmholtz and Kant, see Friedman (2000a). 168. For further discussion of Poincare's hierarchy, and the circularity he thereby finds in the idea of an empirical determination of spatial geometry, see Friedman (1999, chapter 4). 169. Poincare derives the law of this contraction from his own models of hyperbolic geometry, which are variants of the well-known Beltrami-Klein model: see note 162 above. 170. Such a group of displacements, for example, would not have a normal subgroup: see note 167 above. 171. It is here that Poincare appeals to the theory of evolution by natural selection (compare again note 161 above) and suggests, in particular, that his hierarchy of the sciences runs in parallel, as it were, with an historical and cultural evolution of the human species (1902,109; 1913b, 91): "It has often been said that ifindividual experience could not create geometry, it is not the same with ancestral experience. But what does this mean? Does one mean to say that we are not able experimentally to demonstrate Euclid's postulate, but that our ancestors were able to do it? Not in the least. One means that by natural selection our mind has adapted
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ro the conditions of the external world, that it has adopted the geometry most
advantageous to the species; or in other words the most convenient. This is entirely in conformity with our conclusions; geometry is not u'ue, it is advantageous." 172. For further discussion of the law of homogeneity and Poincare's response ro Riemann see Friedman (2000a, 204ff.) 173. Thus, Poincare, unlike Helmholtz, does not appear to take free mobility as a condition for the possibility of spatial measurement as such, for he explicitly recognizes that Riemann has a more general method for measuring length based on the infinitesimal line-element (compare note 143 above). 174. In emphasizing that Poincare's U'eatment of the relativity of motion rests squarely on his philosophy of space and geometry-here and in what follows-I am in very substantial agreement with the excellent discussion in DiSalle (2006, §3.7). 175. This is clear from Newton's original formulation of Corollary V (p. 423): "When bodies are enclosed in a given [relative] space, their motions in relation to one another are the same whether the space is at rest or whether it is moving uniformly straight forward without circular motion." 176. For such generally covariant formulations of all the classical "space-time theories," including general relativity, special relativity, and Newtonian mechanics, see Friedman (1983). The key point is that we formulate all the classical laws of motion using a four-dimensional affine connection rather than the ordinary (threedimensional) derivatives of spatial position with respect to time. 177. In accordance with note 176 above, these additional terms are the components of the four-dimensional affine connection-which vanish in inertial frames of reference but not in non-inertial frames. 178. As Poincare himself puts it (1902, 141; 1913b, 111): "[T]hese two propositions, 'the earth turns' and 'it is more convenient to suppose that tlle earth turns', have one and the same sense; there is no more in the one than in the other." 179. Several paragraphs earlier Poincare had described the general point of his book as follows (1902, 2; 1913b, 28): "We shall see that there are several kinds of hypotheses; that some are verifiable and once confirmed by experiment become fecund truths; that others, witllout being able to lead us into error, may' be useful to us in fixing our ideas; that others, finally, are hypotheses only in appearance and reduce to definitions or conventions in disguise." 180. See, for example, Miller (1981, chapter 1). 181. Poincare arrived at these results based on his reading of Lorentz's 1904 theory of the electron: see Miller (1981, chapter 1, §1.14). 182. This claim was first given prominence in Whittaker (1953), which describes Einstein's 1905 paper as setting forth "the relativity theory of Poincare and Lorentz." Most historians and philosophers of science have not accepted Whittaker's view: see, e.g., Torretti (1983, §3.8). Nevertheless, Poincare's claim to at least an "independent discovery" of special relativity has been revived and defended by Zahar (1989, chapter 5). For a recent and very balanced historical account see Darrigol (2004). 183. This is to the reprinting of his 1905 paper in the well-known collection of original papers on relativity, Lorentz, et al. (1913); the standard English version Occurs in the (1923) translation of me fourm (1922) edition of mis collection.
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184. See Miller (1981, 129, together with endnote 10 thereto on p. 139): the Olympia Academy was an informal study group including Einstein and some of his friends, including Conrad Habicht and Maurice Solovine. 185. Thus it is hard to believe, in particular, that Einstein was not already aware, in 1905, of the second order experiments and Lorentz's appeal to the contraction hypothesis beginning in 1892. In any case, Poincare, at the same time, was appealing to an identical "principle [or "postulate"] of relativity" in his own work inspired by Lorentz's 1904 paper. In his 1904 St. Louis lecture, for example, Poincare states, as the "principle of relativity," that "the laws of physical phenomena must be the same for a stationary observer as for an observer carried along in a uniform motion of translation; so that we have not and cannot have any means of discerning whether or not we are carried along in such a motion" (this, of course, is his earlier "principle of relative motion," now extended from mechanics to electrodynamics as well); and, in his 1906 paper on the dynamics ofthe electron, Poincare explains the principle as follows: "This impossibility of experimentally demonstrating the absolute motion of the Earth appears to be a general law of Nature; it is reasonable to assume the existence of this law, which we shall call the postulate of relativity, and to assume that it is universally valid[; w]hether this postulate, which so far is in agreement with experiments, be later confirmed or disproved by more accurate tests, it is, in any case, of interest to see what consequences follow from it." Both of these quotations are taken from Miller (1981,79); as Miller points out, there is no reason to think that Einstein, in 1905, knew of Poincare's remarks in 1904, nor that Poincare, in 1906, was already acquainted with Einstein's paper. 186. Later, in 1921, Einstein explicitly uses the language of "elevation" [erheben] in connection with precisely Poincare's "conventionalism": see note 207 below. To the best of my knowledge, this striking language in Einstein's 1905 paper has not been previously noted in the literature on a possible line of influence from Poincare to Einstein (see note 182 above). 187. The case for the central importance of Science and Hypothesis, for Einstein, is further strengthened by the fact that Poincare had also emphasized the need for a free convention to establish the simultaneity of distant events at the beginning of Chapter VI on "Classical Mechanics" (1902, Ill; 1913b, 92-93): "Not only do we not have a direct intuition of the equality of two durations, but we do not even have such [intuition] of the simultaneity of two events produced in different locations; this is what I have explained in an article entitled La measure du temps." Indeed, in this 1898 article (here cited in a footnote), Poincare proposes a definition of distant simultaneity, based on light signaling, which is very similar to the one later proposed by Einstein in 1905. (The article is reprinted in The Value of Science in 1905.) Thus, whether or not Einstein was actually acquainted with Poincare's 1898 article, it would have been very hard to miss the clear statement in Science and Hypothesis. 188. Einstein does not explicitly use the terminology of "inertial" frames of reference in his 1905 paper. Nevertheless, he refers his results to "a system of coordinates in which the equations of Newtonian mechanics are valid" (1905, 892; 1923,38), and he is perfectly clear, as is already suggested by the second paragraph where the "principle of relativity" is introduced, that such "frames of reference"
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include all those moving with constant rectilinear velocity relative to one another. It appears, therefore, that Einstein is already quite familiar with the concept of inertial frame, which he could have easily learned about, for example, from the fourth (1901) edition of Mach's Science ofMechanics: see again note 158 above, together with the paragraph to which it is appended. 189. See, for example, Poincare's discussion of what he now calls "the principle of relativity" in Chapter VIII, "The Present Crisis of Mathematical Physics," of The Value ofScience (1905). After clearly explaining Lorentz's concept of local time using his own procedure for synchronizing clocks on the basis oflight signals (compare note 187 above)-which, from the point of view of special relativity, already implies the relativity of simultaneity-Poincare goes on to state that additional hypotheses, including the contraction hypothesis, are still needed. 190. This discussion makes the two components of the principle of relativitya priori and empirical-particularly clear. Poincare begins by saying that the "principle of relative motion ... imposes itself upon us for two reasons: first, the most common experience confirms it, and second, the contrary hypothesis is singularly repugnant to the mind." Then, in his discussion of uniform rotation, he states that this principle-despite its apparent conflict with experiment-"ought to impose itself upon us with the same force, if the motion is varied, or at least if it reduces to a uniform rotation." (See the paragraph to which note 175 above is appended.) Thus, for Poincare, the need to extend the principle of relative motion to cases of (uniform) rotation is entirely a priori. 191. The result is Poincare's own theory of the dynamics of the electron of 1905-6: see note 181 above, together with the paragraph to which it is appended. This theory, as Torretti (1983, 83) observes, is "experimentally indistinguishable from and mathematically equivalent to Einstein's electrodynamics of moving bodies" (compare note 182 above). 192. As we pointed out in note 188 above, Einstein does not use the terminology of "inertial" frames of reference in 1905. In the 1913 reprinting of his paper (note 183) Einstein adds a footnote to the passage quoted in note 188 with the qualification that "the equations of Newtonian mechanics" are here assumed to be valid only approximately. 193. The real problem, therefore, has nothing to do with an a priori commitment to the classical aether. Indeed, Poincare adopts a consistently cautious and skeptical attitude towards the "real existence" of the aether, and, in Chapter X of Science and Hypothesis, in a section on "physics and mechanism," he explicitly links this attitude to the negative results of the aether experiments and the emerging principle of relativity (1902, 196-202; 1913b, 144-48). Nor is the problem an antecedent commitment to what Einstein will later call "constructive" over "principle" theories. Indeed, Poincare, in Chapter VII of The Value of Science, makes a closely related distinction between "the physics of central forces" and "the physics of principles"-where the latter, in particular, abandons the attempt "to penetrate into the detail of the structure of the universe, to isolate the pieces of this vast mechanism" (1905, 175; 1913a, 299)-and he gives precisely the principle ofrelativity as a central example of the latter. The problem, rather, is that, when Poincare implements this principle of relativity in electrodynamics, he must then proceed in accordance with his "conventionalist" epistemology and hierarchy of the sciences,
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and he is thereby forced both to implement it at the level of a "constructive" theory (Lorentz's theory of the electron) and, in effect, to adopt the classical aether as well (as a privileged frame of reference arising from the intersection of the Galilean and Lorentz groups). Thus, although he does not emphasize the importance of Poincare's hierarchical conception of the sciences, I believe that Torretti (1983,87) comes closest to the truth when he blames Poincare's inability to accept Einstein's theory on his "conventionalism." 194. These are the formulations mentioned in note 176 above: for details see Friedman (1983), Torretti (1983), Earman (1989), DiSalle (2006). 195. This is the lesson of my more detailed discussion in Friedman (2002b), as well as the comprehensive treatment in DiSalle (2006)-see again §3.7 for DiSalle's particular discussion of Poincare. 196. This, in effect, is precisely what Poincare himself does in his 1912 lecture on "Space and Time" (see the paragraph to which note 189 above is appended): he expounds his "conventionalist" philosophy of space and his hierarchical conception of the sciences as an alternative to the "entirely new conception of space and time" proposed by "certain physicists"-an alternative which is still both tenable and desirable. And this confirms the point of note 193 above, according to which Poincare's "conventionalist" epistemology and hierarchy of the sciences lies at the root of his alternative interpretation (to Einstein's) of what we now call relativistic electrodynamics. 197. See Pais (1982, 152): Einstein dismissed Minkowski's formulation as "tiberfltissige Gelehrsamkeit" (a superfluous show of learning). 198. If the notion of absolute simultaneity has physical meaning then so does the Galilean group. But the intersection of the Galilean group and the Lorentz group (under which the Maxwell-Lorentz equations are covariant) is just the group of Euclidean rigid motions (translations and rotations), which contains no transformations to frames with varying velocities. 199. More precisely, the transverse and longitudinal (inertial) masses of the electron are now velocity dependent, due to the electron's contraction in the direction of its motion (compare the paragraph to which note 181 above is appended.) 200. The Newtonian gravitational force between two bodies is direcdy proportional to the product of their (gravitational) masses and inversely proportional to the square of the distance between them. Thus, if gravitational and inertial mass are equal (as Newton tacitly assumes), the acceleration produced on either body is entirely independent of this body's (inertial) mass. Indeed, it was already wellknown to Galileo that a falling body's acceleration is entirely independent of its weight. 201. See again Pais (1982, 152): Einstein acknowledges Minkowski in the introduction to a separate reprinting, in monograph form (Leipzig, 1916), of his fundamental 1916 paper on general relativity. 202. I discuss this paper, and its influence, in detail in Friedman (2002b). I shall touch on only the most important points in connection with our current topic here. 203. The reference to Moritz Schlick's theory of "implicit definitions" in his General Theory of Knowledge (1918) aligns Einstein with the more "formalistic" interpretations of Hilbert's axiomatic method. For the sense in which both Klein
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and Poincare resisted such ideas see note 163 above, together with the paragraph to which it is appended: the quoted passage from Einstein appears, in its very wording, to be aligning itself with the conception Klein criticizes. 204. Einstein first started using the terminology of "inertial" frames of reference, as far as I have been able to determine, in his "popular" exposition of the general theory in Einstein (1916b). In the fundamental published version (1916a) he rather uses the terminology of "Galilean" systems. 205. Norton (1985, §3) explains how rigorously to define a three-dimensional geometry in such cases-essentially, by inducing a spatial metric based on the planes of simultaneity orthogonal to the motion of the reference frame at each point. This procedure is well-defined, in Minkowski space-time, if and only if the frame is moving uniformly-with constant velocity (i.e., an inertial frame), uniform acceleration, or uniform rotation. Thus, in particular, the (non-Euclidean) spatial geometry of Einstein's uniformly rotating frame is itself rigorously well defined. 206. It was at this point, in particular, that Einstein turned to his friend the mathematician Marcel Grossmann for help and discovered the "absolute differential calculus"-i.e., the general method for differentiating by means of an affine connection (compare notes 176 and 177 above)-due to G. Ricci and T. LeviCivita. 207. See (1921, 8; 1923, 35): "Geometry (G) [according to Poincare's standpoint-MF] asserts nothing about the behavior of actual things, but only geometry together with the totality (P) of physical laws. We can say, symbolically, that only the sum (G) + (P) is subject to the control of experience. So (G) can be chosen arbitrarily, and also parts of (P); all of these laws are conventions. In order to avoid contradictions it is only necessary to choose the remainder of (P) in such a way that (G) and the total (P) together do justice to experience. On this conception axiomatic geometry and the part of the laws of nature that have been elevated [erhobene] to conventions appear as epistemologically of equal status." 208. As is well known, Einstein (1949, 51) explicitly credits his reading of Hume and Mach for helping him to appreciate the "optional" (JVillkiirlich) character of the classical notion of absolute simultaneity, and he does not there mention Poincare. John Norton's contribution to the present volume does an excellent job of explaining what Einstein probably had in mind: Einstein's crucial insight was that even our most basic concepts have their origin in experience; they may thus have unnoticed empirical presuppositions; and therefore, most importantly, they are then subject to (even radical) revision when such previously unnoticed presuppositions are uncovered and seen to be false. Thus, it was tlle skeptical or iconoclastic attitude towards our most basic concepts shared by both Hume and Mach that decisively influenced Einstein at this crucial juncture; and it is no wonder, accordingly, that Poincare (with his fundamentally conservative defense of our "ancient" ways of thinking: compare the paragraph to which note 189 above is appended) did not help Einstein here. Moreover, it is also very plausible, as Norton suggests, that Einstein was especially influenced by Hume's claim that certain basic concepts are "fictions"-which, in Einstein's mind, meant that we then have the freedom to revise them. However, as Graciela De Pierris emphasized in the discussion of Norton's paper at the South Carolina conference, Hume's own notion of "fiction" has nothing to do with such freedom-and rather involves the habitual
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operation of associative mechanisms. Yet Einstein would have found Poincare speaking of "the free activity of our mind" at the very beginning of Science and Hypothesis (see the paragraph preceding the one to which note 164 above is appended); he would have also found the idea that distant simultaneity is conventional (see note 187 above); and, most importantly, he appears to have definitely found the idea that certain empirical laws can be "elevated" to the radically new status of conventions or definitions in disguise. (As Norton points out in note 17 to his contribution, Einstein later explicitly mentions Poincare, along with Hume and Mach, as "of considerable influence" in precisely this early period.) I believe that Norton is perfectly correct, therefore, in emphasizing the entirely opportunistic character of Einstein's use of philosophical ideas; and I maintain that, even though Einstein may have been decisively stimulated by his own understanding of Hume and Mach at a particularly crucial point in his own thinking, Einstein's ongoing engagement with Poincare-in spite of, and even because of, the deep opposition between them on many fundamental issues-is much more interesting and important from both a philosophical and an historical point of view. 209. Einstein does not explicitly mention Helmholtz in "Geometry and Experience." However, in a closely related article on "Non-Euclidean Geometry and Physics" (1925), Einstein makes it perfectly clear that the opposition he has in mind is precisely that between Helmholtz and Poincare (pp. 18-19): "Either one accepts that the 'body' of geometry is realized in principle by the solid bodies of nature, if only certain prescriptions are maintained regarding temperature, mechanical stress, and so on; this is the standpoint of the practicing physicist. Then a natural object corresponds to the 'interval' of geometry, and all propositions of geometry thereby attain the character of assertions about real bodies. Tllis standpoint was represented especially clearly by Helmholtz, and one can add that without it establishing the [general-MF] theory of relativity would have been practically impossible. Or, one denies in principle the existence of objects that correspond to the fundamental concepts of geometry. Then geometry alone contains no assertions about objects of reality, but only geometry together with physics. This standpoint, which may be more perfect for the systematic presentation of a completed physics, was represented especially clearly by Poincare. On this standpoint the total content of geometry is conventional; which geometry is to be preferred depends on how 'simple' a physics can, by its use, be established in agreement with experience." Ryckman (2005, §3.3) emphasizes the importance of this passage in relation to the earlier argument of "Geometry and Experience" (1921). Ryckman suggests, in particular, that Einstein has a more balanced attitude here towards the two standpoints than in the earlier paper, and he connects this with Einstein's own evolving attempts to create a unified field theory in which all matter fields (including those constituting "practically rigid bodies") are to be explained in terms of space-time geometry. 210. Nevertheless, my discussion has made it clear, I hope, how important Einstein's engagement with Poincare was for the development of both special and general relativity. For, as we have just emphasized once again, Einstein arrived at special relativity in the first place by following Poincare's method of "elevation," and, as we have also seen, Poincare's work in the foundations of geometry, mechan-
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ics, and electrodynamics was very salient indeed in Einstein's creation of the general theory: compare the paragraph to which note 189 is appended, the paragraph to which note 201 is appended, and the intervening paragraphs. In particular, we might say that Poincare's work, in exploring and elaborating the classical spacetime framework as deeply and extensively as possible in the radically new situation with which both he and Einstein were now faced, first made it possible for Einstein to embark on the revolutionary new direction he (but not Poincare) then followed. (I am here indebted to James Mattingly's illuminating contribution to the present volume: see note 249 below.) 211. The upshot of general relativity is thus a conception of the relationship between geometry and physics that is much more "holistic" than Poincare's hierarchical conception, since geometry and empirical physics (i.e., the theory of gravitation) now constitute one indivisible whole. In this respect, as well, Poincare's conception is much more Kantian, in so far as geometry, for Poincare, is a presupposition of all empirical physics. Don Howard, in his contribution to the present volume and elsewhere, has called attention to Einstein's repeated use of "holistic" considerations to argue against Kantian and neo-Kantian philosophical ideas. In his contribution to the present volume, in particular, he confronts my own neoKantian views with Einstein's "holism," and he takes this "holism" to be exemplified in the passage quoted in note 207 above-which, Einstein says, is "sub specie aeterni" correct. I shall return to my own neo-Kantian views later, and I shall here simply make an historical point. Leaving aside the question of the appropriateness of ascribing such an "holistic" view to Poincare himself (as Einstein does: compare the paragraph to which note 207 is appended), the point (and Howard is perfectly clear about this) is that Einstein is here criticizing Poincare and explicitly rejecting such "holism" in favor of the more naively empiricist conception defended by Helmholtz. For, at just this crucial juncture in the discovery of general relativity, Einstein found it useful and fruitfil1 to adopt the Helmholtzian standpoint of the "practically rigid body." Thus, Einstein's relationship to "holistic" philosophical ideas is just as opportunistic as in all other cases (compare note 208 above): he defends them when it is advantageous to his ongoing physical thinking, rejects them when it is not. In the present case, as Einstein became more and more involved witll his own work on "constructive" matter theory (his program in unified field-tlleory: compare note 209 above), he became less and less willing to treat rigid rods and clocks as "irreducible elements in the conceptual framework of physics," and, accordingly, he became increasingly attracted to the "holistic" standpoint he (wrongly) attributed to Poincare. Nevertheless, Howard is certainly correct, more generally, that Einstein explicitly opposed Kantian and neo-Kantian ideas that he took to be incompatible with the freedom of the physicist to revise our most basic concepts; and this is why, in particular, he was earlier attracted to me empiricism of Hume and Mach, and later to the (explicitly anti-Kantian) empiricism of Moritz Schlick. 212. For details see Howard (1984), and compare note 211 above. 213. Schlick's invitation to Vienna was engineered by the mathematician Hans Hahn, then a professor at the University, probably with help from his friend Philipp Frank, Einstein's successor in theoretical physics at Prague. It also appears likely that Einstein himself may have aided in this effort-in any case, after visiting
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Schlick at Rostock in 1919, he wrote to Max Born about the need to find Schlick a professorship. For details see Stadler (1982, 117-20). 214. Einstein did this most prominently in his famous obituary address for Mach delivered in 1916 (which is discussed by Norton in §4.1 of his contribution to the present volume). 215. See Schlick (1918, §7, "Implicit Definitions"); the application to Kant's theory of the synthetic a priori is then made in chapter IlI.C. Einstein is referring to these parts of General Theory ofKnowledge in "Geometry and Experience" (see again note 203 above, together with the paragraph to which it is appended), and Einstein's well-known (albeit implicit) rejection of the Kantian synthetic a priori is given at the very beginning of his paper (1921, 3-4; 1923,28-29): "In so far as the propositions of mathematics refer to reality they are not certain; and in so far as they are certain they do not refer to reality. Full clarity about the situation appears to me be have been first obtained in general by that tendency in mathematics known under the name of 'axiomatics.' The advance achieved by axiomatics consists in having cleanly separated the formal-logical element from the material or intuitive content. According to axiomatics only the formal-logical element constitutes the object of mathematics, but not the intuitive or other content connected with the formal-logical element." 216. I have discussed the deep tensions arising from the logical empiricists' attempt to assimilate Poincare and Einstein in Friedman (1999, chapters 3 and 4), (2002b). These tensions surface most explicitly in an exchange between Schlick and Reichenbach in connection with the publication of Reichenbach's first (1920) book, The Theory ofRelativity and A Priori Knowledge. Reichenbach had there argued (correctly) that the general theory of relativity is incompatible with Poincare's "conventionalism," and, accordingly, he had defended a conception of the a priori in geometry and (relativistic) physics opposed to Poincare's: the content of those geometrical and mechanical principles that are properly characterized as a priori (as "constitutive of the concept of the object of [scientific] knowledge") varies from theory to theory (and is in this sense relativized to a particular stage of scientific theorizing), but, in the context of any given theory, it is not subject to conventional choice. In the context of general relativity, in particular, the metrical structure is now empirical (determined by the distribution of matter in accordance with Einstein's field equations), whereas the differentiabletopological structure is (constitutively) a priori-but there is no conventional choice (based on global simplicity or anything else) of what to include in each category. Schlick then criticized Reichenbach in his (1921) review of Ernst Cassirer's (1921) neo-Kantian interpretation of general relativity, on the grounds that Poincare's "conventionalism" (understood in Schlick's preferred "holistic" way as involving an ultimately conventional choice based on global simplicity) is now the only live alternative to the original Kantian view. The upshot of this exchange (which also involved a correspondence between the two) was that Reichenbach eventually gave up his conception of 1920 in favor of Poincare's "conventionalism" (largely as understood by Schlick), and Schlick added a new section on "Definitions, Conventions, Judgments of Experience" (§11) to the second (1925) edition of General Theory of Knowledge.
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217. For the results of this research see Reck (2004), (2007). An edition of Carnap's student notes on Frege's lectures at Jena has also recently appeared, together with an Introduction by the editors, as Reck and Awodey (2004). 218. Bruno Bauch was a leading member of the Southwest School of neoKantianism founded by Wilhelm Windelband. At Jena he was a close colleague and associate of Gottlob Frege's. Mter the Great War, for example, Frege joined Bauch's conservative Deutsche Philosophische Gesellschaftand published his last three papers ("Logical Investigations") in the official journal of this society, Beitriige des deutschen Idealismus. For Rauch and his relationship to Frege see Sluga (1980, 1993). For Carnap's relationship to Bauch and neo-Kantianism more generallyincluding discussions of Der Raum-see Richardson (1998), Friedman (2000b). 219. Carnap's exposition of modern mathematical logic here essentially follows Frege, and he reconciles Frege with modern axiomatics by conceiving Hilbertian "implicit deftnitions" as characterizations of what we would now call a structure (of Euclidean space, for example) within a Fregean higher-order logical system. For Carnap's ongoing attempts to synthesize Fregean higher-order logic with modern axiomatics see again Reck (2007). 220. Carnap describes this early influence ofKantian and neo-Kantian ideas on his thinking in his "Intellectual Autobiography" as follows (1963a, 4): "I studied Kant's philosophy with Bruno Bauch in Jena. In his seminar, the Critique of Pure Reason was discussed in detail for an entire year. I was strongly impressed by Kant's conception that the geometrical structure of space is determined by the form of our intuition. The after-effects of this influence were still noticeable in the chapter on the space of intuition in my dissertation, Der Raum." For further discussion of Der Raum, and the influence of geometry on Carnap's thinking more generally, see Mormann (2007). 221. For extended discussion of this "formal structuring" by modern mathematical logic in the Aujbau see Friedman (1987/1999), (1992b/1999), Richardson (1998); for its relationship to Kantian and neo-Kantian ideas see, in addition, Sauer (1985). Section §15 of the Aujbau reinterprets Schlick's "implicit definitions" as what Carnap calls "purely structural definitive descriptions": here various higher-order (set-theoretical) structures arising within the "constitutional system" of empirical concepts (starting with an initial non-logical relation referring to immediately given sense experience) are explicitly defined as the unique such structures satisfYing various conditions and constraints; and, in this way, Carnap incorporates the "coordinative definitions" or "coordinating principles" earlier discussed by Reichenbach and Schlick in their debate about "conventionalism" (note 216 above)-for details see Friedman (2004). For a different perspective on these issues within the Aujbau see Thomas Ricketts's contribution to the present volume, which focuses, in particular, on the logical problems surrounding Carnap's contemporaneous project in "General Axiomatics"-for the latter see Awodey and Carus (2001), Reck (2007). 222. As we shall see below (note 230, together with the paragraph to which it is appended) "Iogico-mathematical truth," for Carnap, does not include geometry (in its customary interpretation). 223. See Carnap's official formulation of the principle of tolerance in §17 of Logical Syntax (1934,45; 1937, 52]): "In logic there is no morality. Everyone may
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construct his own logic, i.e., his own form of language, as he wishes. Only, if he wants to discuss it with us, he must clearly indicate how he wishes to construct it ' [and he must] give syntactic rules instead of philosophical arguments." 224. See, for example, Friedman (1999, chapter 9, especially §§VI-VII). 225. This understanding of the choice between classical mathematics and intuitionism is suggested by Carnap's reply to Evert Beth in the Schilpp volume. Carnap first explains how he understands, in general, the use of a metalanguage, whether syntactic or (as in his post-Syntax period) semantic (1963b, 929): "Since the n1etalanguage ML serves as a means of communication between the author and the reader or among participants in a discussion, I always presupposed, both in syntax and in semantics, that a fixed interpretation of ML, which is shared by all participants, is given. This interpretation is usually not formulated explicitly; but since ML uses English words, it is assun1ed that these words are understood in their ordinary senses." And it is clear, from the context, that the "fixed interpretation" in question involves the standard interpretation of classical arithmetic. Moreover, Carnap then applies this point, in particular, to the choice between classical mathematics and intuitionism (pp. 929-30): "It seems to be obvious that, if two men wish to find out whether or not their views on certain objects agree, they must first of all use a common language to make sure that they are talking about the same objects. It may be the case that one of them can express in his own language certain convictions which he cannot translate into the con1mon language; in this case he cannot communicate these convictions to the other man. For example, a classical mathematician is in this situation with respect to an intuitionist or, to a still higher degree, with respect to a nominalist." 226. Thus, the point of the principle of tolerance is not to defend a completely level playing field for all possible views in the foundations of mathematics (classical, intuitionist, and so on), but to facilitate the elimination of fruitless philosophical disputes in favor of the project of language planning. Compare note 223 above, together with the paragraph to which it is appended. See also Carnap's discussion of "Language Planning" in §11 of his Autobiography. Carnap reports that this idea "did not immediately occur to [him]" when he first studied logic with Frege, but rather evolved gradually (1963a, 68): "Only later, when I became acquainted with the entirely different language of Principia Mathematica, the modal logic of C.I. Lewis, the intuitionistic logic of Brouwer and Heyting, and the typeless system of Quine and others, did I recognize the infinite variety of possible language forms. On the one hand, I became aware of the problems connected with the finding of language forms suitable for given purposes; on the other hand, I gained the insight that one cannot speak of 'the correct language form,' because various forms have different advantages in different respects. The latter insight led me to the principle of tolerance. Thus, in time, I came to recognize that our task is one of planning forms of languages. " 227. These attempts are most visible in "Two Dogmas of Empiricism" and (even more) "Epistemology Naturalized" (1969). See De Pierris (2003) for a discussion of this latter article and its inability, in particular, to entertain the possibility of any kind of epistemology other than foundationalism. Compare also Richardson (1997): see note 228 below.
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228. For discussion of this important paper see Richardson (1996). More genin messing the fundamental differences between Carnap and Quine in their understanding of both "epistemology" and "empiricism" my argument has considerable overlap with Richardson (1997). 229. This is Carnap's reason for consistently taking higher-order logic (or, equivalently, set theory) as part of the formal or inferential framework within which empirical testing proceeds-and it is this that most clearly and sharply distinguishes Carnap's conception of logic and mathematics from Quine's. In particular, Carnap cannot accept Quine's thoroughly holistic version of empiricist epistemology (compare the paragraph following the one to which note 226 above is appended), according to which (first-order) set theory, like physics, is adopted on empirical grounds-because a framework of real numbers, for Carnap, is presupposed by all such testing. For a different perspective on the relationship between Carnap's conception ofJogic and mathematics and Quine's see RickettS (2004). 230. This central difference, in Carnap's view, between arithmetic and analysis on the one side and geometry on the other has not been sufficiently appreciated. Thus, for example, Carnap's discussion of the analyticity of logic and mathematics in his "Intellectual Autobiography" (1963a, 46-49) explicitly excludes geometry (49): "In the foregoing, the term 'mathematics' is meant to include the theory of numbers of various kinds and their functions, furthermore abstract fields, e.g., abstract algebra, abstract group theory, and the like, but to exclude geometry"-where Carnap then goes on to emphasize the distinction between mathematical and physical geomeu-y. In thus taking geometry (in its customary interpretation) to be synthetic rather than analytic, Carnap retains an echo of the original Kantian conception, of Frege's conception, and of his own early position in Der Raum. Unlike all of these earlier views, however, Carnap now takes Einstein to have conclusively shown that the geometry of physical space is entirely a posteriori. 231. See Carnap (1942, xi), (1963b, 932), (1966, 257). 232. See Carnap (1966, 257). I am indebted to Awodey (2007) for emphasizing the importance of Carnap's appeal to Einstein in this context. 233. For further discussion see William Demopoulos's contribution to the present volume, and also Demopoulos (2007). Demopoulos illuminatingly emphasizes, in particular, the importance of the Hilbertian background for Carnap's late reconstruction of scientific theories. From this point of view, we might view Carnap's use of the (higher-order) Ramsey sentence here as his final attempt to synthesize an Hilbertian conception of the meaning of "abstract" scientific terms with a Fregean conception of logic: compare notes 219 and 221 above. 234. See, for example, Carnap (1950b; 1952). 235. See the detailed and elegant discussion in Zabell (2007). 236. This point plays a central role, for Kant, in §22 of the second edition Transcendental Deduction (BI46-47): "Sensible intuition is eitller pure intuition (space and time) or empirical intuition of that which is immediately represented in space and time as actual, through sensation. By determination of the former we can acquire a priori cognitions of objects (in mathematics), but only according to their form; whether there can be given things that must be intuited in this form remains eray'
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still unsettled thereby. Therefore, all mathematical concepts are in themselves not cognitions, except in so far as one presupposes that there are things which can only be presented to us in accordance with the form of this pure sensible intuition." For discussion see Friedman (1992a, chapter 2, §I). In Carnap's case, the idea tl1at mathematics, strictly speaking, "has no objects" of its own is expressed in his famous treatment of ontological questions in Carnap (1950a). 237. See Kant's explanation of the distinctive character of "transcendental" inquiry at A56/B80-81: "[N]ot every a priori cognition should be called transcendental, but only that through which we know that and how certain representations (intuitions or concepts) are applied wholly a priori, or are possible (that is, [through which we know] the possibility or the a priori employment of the cognition). Therefore, neither space nor any a priori geometrical determination thereof is a transcendental representation, but what can alone be called transcendental is the knowledge that these representations are not at all of empirical origin, and the possibility that they can nevertheless relate a priori to objects of experience." 238. For example, in "Carnap and Logical Truth," first appearing in 1963 in the Carnap volume of the Library of Living Philosophers, Quine begins with Kant's question how synthetic a priori judgments are possible, replaces this question (in light of the Frege-Russell reduction of mathematics to logic) with the question, "How is logical certainty possible?", and then portrays "Carnap's philosophy of logical truth" as an answer to this (Kantian question). Interestingly, however, Quine also inserts a striking disclaimer (1963, 385): "My dissent from Carnap's philosophy of logical truth is hard to state and argue in Carnap's terms. This circumstance perhaps counts in favor of Carnap's position. At any rate, a practical consequence is that, though the present essay was written entirely for this occasion, the specific mentions of Carnap are few and fleeting until well past the middle. It was only by providing thus elaborately a background of my own choosing that I was able to manage the more focused criticisms in the later pages. Actually, parts also of the earlier portions correspond to what I think to be Carnap's own orientation and reasoning; but such undocumented points are best left unattributed." 239. For further discussion see Friedman (2003b); and compare notes 33 and 148 above. See also note 236 above: pure mathematics (e.g., pure geometry) does not, strictly speaking, supply us with objects of cognition, which is lirnited to objects in nature ("appearances"), but only with what Kant calls the "forms" of such objects; that "there are things [i.e., appearances-MF] which can only be presented to us in accordance with the form of this pure sensible intuition" can only be established by the necessary schematization of the pure intellect in terms of pure sensibility-and therefore with respect to empirical sensibility as well. 240. See the paragraph to which note 45 above is appended, together with me three following paragraphs, for the analogy (and disanalogy) between geometrical construction and the Newtonian construction of the center of mass of me solar system on Kant's view. The remainder of this section then illustrates in detail how a schematization of the pure categories is involved in Kant's answer to the question how pure natural science is possible in the Metaphysical Foundations. 241. For Kant, by contrast, the very idea of transcendental philosophy is inseparable from his sharp distinction between understanding and sensibility. Thus, in
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discussing what he calls "u"anscendentallogic" (another name, in this context, for transcendental philosophy: compare note 149 above) in section II of the Introduction to the Transcendental Analytic, Kant explains (A55-56/B79-80) that transcendental logic (as opposed to entirely empty general or purely formal logic) is possible precisely because "there are pure as well as empirical intuitions," so that there can be an a priori logical discipline "in which one does not abstract from all content of intuition" but only from "empirical content." The main task of the Transcendental Analytic, in the metaphysical and u"anscendental deductions, is then to show that and how the two a priori faculties of pure understanding and pure sensibility are necessarily interrelated; and it is precisely this "synthetic" or progressive argument which then distinguishes the characteristic method of the first Critique from the "analytic" or regressive method of the Prolegomena. 242. This terminology is explicitly borrowed from Kant's "regulative" solution to the antinomies of pure reason in section VIII of the chapter on the antin0mies in the first Critique, and, more generally, it reflects the fact that the Marburg School, in accordance with another prominent trend in post- Kantian transcendental philosophy, is thereby reconceiving Kant's sharp distinction between regulative and constitutive principles as well. See note 109 above, together with the paragraph to which it is appended; as I point out there, there is an intimate connection, in Kant, between a sharp distinction between constitutive and regulative principles and an equally sharp distinction between sensibility and understanding. 243. Here see especially Cohen (1883); and, for further discussion, see Richardson (2006). In particular, beginning with the idea of a continuous series or function, Cohen interprets the differential to this function at a given point as infinitesimally capturing the rule of the series as a whole, and thus expressing, at a given moment, the general form of the series valid for all times. Cohen thereby reinterprets the Kantian Anticipations of Perception-where, for Kant, empirical intuitions (sensations) first enter the cognitive synthesis as "intensive magnitudes" defined at a given temporal instant-in terms of the a priori determination of our sensory experience by universal laws. 244. For further details about Cassirer, and his relation to Cohen, see Friedman (2000b), (2005), upon which I am drawing here. It is important to note, in particular, that Cassirer rejects the logicist critique of the Kantian synthetic a priori on the basis of his reinterpretation of the distinction between formal and transcendental logic (compare note 241 above): the logicist critique fails to grasp that pure formal logic is merely an abstraction from the essentially synthetic cognitive process by which the objects of empirical naulral scientific knowledge are successively constituted, and it is precisely this synthetic process, as described by transcendental rather than merely formal logic, which is philosophically fundamental. 245. Thus, we might conceive the sequence in question as beginning with the decomposition of the four-dimensional manifold of spatio-temporal events into Newtonian absolute (Euclidean) space plus absolute time (E3 x T), progressing from there to the structure of what we now call neo-Newtonian or Galilean spacetime corresponding to the late nineteenth-century concept of an inertial frame; then progressing to the structure of Minkowski space-time corresponding to the Lorentz (rather than Galilean) transformations between inertial frames characteris-
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tic of special relativity; and finally progressing to the new variably curved space-time structure of general relativity containing the (flat) structure of Minkowski spacetime as an approximate special case. For the different (affine) structures of Newtonian and relativistic space-times see note 194 above, together with the paragraph to which it is appended. For the nineteenth-century concept of an inertial frame and its relationship to neo-Newtonian or Galilean space-time see note 158 above, together with the paragraph to which it is appended. Finally, for the relationship between the space-time metric of general relativity and Minkowski spacetime see the paragraph to which note 206 above is appended. 246. Einstein explicitly applauded this anti-Kantian polemic in a well-known letter to ScWick: "I have not read anything so perceptive and true in a long time." For discussion see Coffa (1991, chapter 10), Howard (1984). 247. See ScWick (1921,100; 1979, 325): "[A]nyone who asserts the critical theory must, if we are to give him credence, actually indicate the a priori principles that must constitute the firm ground of all exact science.... We must therefore require a declaration of the cognitions, for example, whose source is space. The critical idealist must designate them with the same determinateness and clarity with which Kant could refer to the geometry and 'general doctrine of motion' that were alone known and recognized at his time." 248. For example, ScWick (1921,101; 1979, 326) quotes from the above passage of Cassirer's on "the function of 'spatiality in general' ... entirely without regard to its more particular determinations," and complains that this formulation is "hardly satisfactory[; f]or what complex of axioms is it that is supposed to be comprised in this assertion?" Note that this complaint does not apply to Reichenbach (1920), for Reichenbach there indicates specific "constitutive" principles which vary from theory to theory-and are thus in no way fixed and umevisable; ScWick's response to Reichenbach, accordingly, is that such relativized "constitutive" principles are better understood as conventions in the sense of Poincare (see again note 216 above). 249. James Mattingly, in his contribution to the present volume, has given special emphasis to this feature of Cassirer's conception of scientific method. In particular, according to what Mattingly calls "the Lorentzian strategy," it is ofren particularly fruitful to guide the direction of an imminent scientific revolution by exploring and extending our current theory as precisely and widely as possible-so as then to determine as clearly as possible which difficulties in our current theory can now be most profitably addressed. In view of the crucial role of Poincare in the case of Einstein's revolutionary creation of his theories of relativity, however, we might, in this context, more properly speak of a "Poincarean" rather than a "Lorentzian" strategy (compare note 210 above, and also note 350 below). 250. Here Schlick takes himself to be following Helmholtz's celebrated "sign theory" of perception (compare note 146 above, together with the paragraph to which it is appended), according to which our perceptions do not picture or resemble external reality but only correspond to it by a relation of "coordination" (Zuordnung). For a discussion of Helmholtz and ScWick on this matter see again Friedman (1997b). 251. Cassirer, like ScWick, appeals to Helmholtz's "sign theory" for the relation of "coordination" in question. However, the true significance of this theory,
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for Cassirer, consists in a denial of what he calls the "copy theory of knowledge" (correspondence with independently existing "things in themselves") in favor of (his version of) a "critical" theory. For further discussion see again Friedman (200 0b ), (2005), and also Ryckman (1991). 252. See again note 216 above, together with the paragraph to which it is appended. Despite his differences with ScWick on the status of "constitutive" principles, Reichenbach shares Schlick's conception of coordination in 1920. Empirical knowledge, according to Reichenbach (1920, chapter 4), entitled "Knowledge as Coordination" (Erkenntnis als Zuordnung) , is defined by an abstract mathematical function or mapping that relates an axiomatically defined mathematical structure to something else, concrete empirical reality, which is simply undefined prior to the coordination in question. The novelty is that Reichenbach (1920, chapter 6) also insists on the need for a special class of physical principles-"coordinating principles" or "axioms of coordination"-to insure that this correspondence is uniquely well defined. For further discussion see again Friedman (2005). 253. Ryckman focuses on the significance of Cassirer's work for a "transcendental" understanding of the condition of general covariance, which I have not explicitly discussed. In an appendix to his book Ryckman considers my recent revival of Reichenbach's (1920) version of the "relativized a priori," and, in his contribution to the present volume, he rightly criticizes me for relying too uncritically on the ScWickean and Reichenbachian notion of "coordination." The present discussion of the application of mathematics in modern physics is largely a response to Ryckman's work, and I am also indebted to a number of discussions with him about these issues. 254. A central part of Weyl's construction, as Ryckman (2005, §6.3.2) explains, is a purely infinitesimal solution to the Helmholtz-Lie "space problem" which, accordingly, is not limited to spaces of constant curvature (see note 141 above, together with the paragraph to which it is appended). In this way, in particular, Weyl intends to extend the group-theoretic connection between sense experience and mathematical geometry forged by Helmholtz and Poincare (see also note 166 above, together with the paragraph to which it is appended) to the variably curved space-time structure of general relativity. For Weyl, we have direct phenomenological acquaintance only with the infinitesimal tangent space to a manifold at any given point (at which the transcendental ego may be located); and our knowledge (articulated in an appropriate group of rotations) that the geometry of such a tangent space is Euclidean (or, in the case of space-time, pseudo-Euclidean) expresses the a priori "essence" (Wesen) of space (or, respectively, space-time). Weyl's construction then derives the existence of a Riemannian (or semiRiemannian) line-element or metric on the manifold from (1) the postulate that the mutual "orientations" of the Euclidean (or pseudo-Euclidean) tangent spaces at different points may vary arbitrarily (as long as such variation is appropriately continuous), and (2) the postulate that a unique affine connection on the manifold is thereby determined. The reason a gauge transformation arises is that tile metric tllUS constructed is determined up to a scale factor at any given point, but the scale factor in question can vary from point to point. In Weyl's unified field theory of gravitation and electromagnetism, the associated gauge transformation then governs the vector potential of the electromagnetic field.
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255. On the one hand, in order to capture the a priori "essence" of space (or space-time), it would appear that Weyl's "essential analysis" must take place in pure rather than empirical intuition. On the other hand, however, since one of Weyl's fundamental postulates (note 254 above) is the existence of a unique affine connection, it would appear that, in the case of space-time, we must assume a uniquely privileged state of motion represented by the affine geodesics in question; and it would then seem that the existence and uniqueness of such a state can only be given empirically. Indeed, in the case of space-time, it would appear that we must also rely on further empirical facts-here the privileged and invariant character of the velocity of light-in order to justify the claim that the infinitesimal geometry of the tangent space is pseudo-Euclidean (i.e., Minkowskian). Thus, in modern axiomatic presentations ofWeyl's construction, e.g., by Ehlers, Pirani, and Schild (1972), we assume two classes of privileged trajectories-light rays and freely falling trajectories-simply taken as given, empirically, at the outset. What I do not yet fully understand, therefore, is the precise role and character of the light principle and the principle of equivalence in Weyl's phenomenological approach. 256. Consider, for example, the discussion of the Kantian examples "all bodies are heavy" and "all material things are extended" in Husserl (1913, §7): the former has the mere empirical generality of a natural law, because it essentially refers to concrete existences in nature; the latter, by contrast, necessarily characterizes the material (or "regional") essence of a material things in general, entirely independently of what actually concretely exists in nature. 257. It is for this reason, above all, that Schlick (1918, §17) includes a polemical dismissal of just these early sections of the Ideas: because of Schlick's extremely stark opposition between conceptual thought and intuitive experience (see again the paragraph to which note 250 is appended), the notion of "intuitive knowledge" or "knowledge by acquaintance" is simply a "contradictio in adjecto" (1918, §ll). (This polemical dismissal of the Ideas was removed, as Schlick explains, from the corresponding discussion in the second edition-now §18.) By contrast, as we have seen, Carnap (1922) explicitly appeals to Husserlian Wesenserschauung in his account of the essential role of intuitive space in the foundations of general relativity: see the paragraph to which note 218 above is appended, and the references cited there; see also Friedman (1999, chapter 2) for a discussion of the relationship between Carnap and Weyl on this point. And, although Carnap abandoned intuitive space and the synthetic a priori in the Aufbau (compare note 221 above, together with the paragraph to which it is appended), there are still striking parallels between Carnap's "constitutional theory of the objects of experience" and Husserl's Ideas. For further discussion ofCarnap and Husserl see Ryckman (2007), together with the references cited therein: perhaps the most fundamental difference between the two, as Ryckman emphasizes, is that here, in the Aufbau, Carnap has given decisive philosophical priority to formal over transcendental logic, whereas the main point of Husserl's approach is to do precisely the opposite. 258. Geometry is only possible in the first place, according to Husserl, via successively idealizing the rough and approximate shapes of ordinary experience as we imagine (purely in thought) an indefinite extension of the continual increase in precision actually practically available in experience (straighter and straighter lines, flatter and flatter planes, and so on). Similarly, Galileo's mathematical science of
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motion is only possible by idealizing the u'ajectories of real projectiles, for example, as parabolas in a pu:e vacuum (which, we think, can in some sense be approximated in actual expenence), arnved at by consldenng only the mfluence of downward gravity and horizontal inertia (where the ideal uniform time governing inertially moving bodies as they n-averse equal spaces can also be approximated in experience by actual time-measuring devices). 259. Following his discussion of Galileo's mathematization of nature (§9), Husserl explains (§§10-27) how the characteristic problems of modern philosophy arise from a perverse (although in some sense natural and inevitable) misunderstanding of Galileo's accomplishment. In particular, when modern mathematical natural science and philosophy declares that only geomen-ically described nature is "objectively real" and that the "subjective-relative" domain of our actual intuitive experience is merely a misleading "appearance" of this "absolute" and "objective" realm, they forget that our new geometrical description of nature only has sense and meaning on the basis of precisely its necessary origin in the ordinary world of perception and experience-otherwise, it is a mere empty formalism. The actual intentional meaning and origin of geometry becomes hidden from us in an historical process of "sedimentation," and this leads to the most profound philosophical misunderstandings-the most fundamental of which is the creation of mind-body dualism by Descartes. These philosophical misunderstandings of the modern period are the source, for Husserl, of "the crisis of the European sciences" indicated in his title, and the task of transcendental phenomenology, accordingly, is to resolve this crisis by decisively clarifying the relationship between the positive sciences and the originally given life-world. 260. For Husserl there is, sU'ictly speaking, no intuitive "experience" of what is objective (1954, 131-32; 1970, 129): "The objective is precisely never experienceable as itself ... The question of the experienceability of the objective is no different from that of infinitely distant geomen-ical sn-uctures, and thus [no different] in general than that of all infinite 'ideas', for example, that of the experienceability of the infinity of the series of natural numbers." 261. In this sense, Husserl's use of "evidence" here parallels Kant's: see, for example, the discussion at A179-80jB221-23 of the distinction between mathematical and dynamical principles of pure understanding (see note 48 above), which "differ from one another not, to be sure, in certainty, which is established in both a priori, but in the kind of evidence, i.e., in their intuitiveness (and thus also in the manner of their demonstration)." For Kant, mathematical principles involve a priori constructions in pure intuition which are immediately (self- )presented to the subject-perceptually, indexically, and ostensively-entirely a priori in the pure forms of its (inner and outer) sensibility. 262. Compare Kant's remarks in §40 of the Prolegomena (4, 327): "Pure mathematics and pure natural science had no need, for the sake oftheir own security and certainty, of the kind of deduction we have just made of both. For the former rests on its own evidence, and the latter, although arising from pure sources of the understanding, nevertheless rests on experience and its thoroughgoing confirmation-experience being a witness [pure natural science] cannot entirely refuse and dispense with, because, despite all of its certainty, as philosophy it can never imitate mathematics. Therefore, the inquiry in question was not necessary for the salce
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of the two sciences themselves [i.e., pure mathematics and pure natural science_ MFJ, but for the sake of another science, namely metaphysics." Compare also note 105 above, together with the paragraph to which it is appended. 263. It should be observed, for example, that Kant's explanation of the possibility of synthetic a priori knowledge in the first Critique-on the basis of the interrelationship between the faculties of understanding and sensibility-is definitely not "regressive" or "analytic" in Kant's own understanding of these terms: see note 241 above, together with the paragraph to which it is appended. 264. It is noteworthy, in this connection, that Cassirer (1921) contains one of the earliest occurrences of the idea of a general "philosophy of symbolic forms" in Cassirer's writings. And it is especially noteworthy, in particular, that this philosophy of symbolic forms (as further developed throughout the 1920s) explicitly makes room for other layers of "symbolic meaning" (including what Cassirer calls the "representative meaning" exhibited in the "intuitive world" of ordinary sense perception) prior to and independently of the "purely significative meaning" of the abstract formal structures characteristic of modern mathematics, mathematical physics, and mathematical logic. See Friedman (2000b, chapter 6) and, for a more extensive and developed account, I
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266. For the modern concept of inertial frame and its relation to Kant see note 158 above, together with the paragraph to which it is appended and the three preceding paragraphs. As discussed in note 159 above, the concept of inertial frame represents a more abstract solution than Kant's in so far as such frames are not necessarily physically instantiated. Nevertheless, they can always be approximated by concrete frames of reference in our actual experience in the same sense that Kant's unattainable ideal "absolute space" can be so approximated. 267. As pointed out in note 188 above, Einstein, in 1905, characterizes the frames of reference in question as those "in which the equations of Newtonian mechanics are valid." As then explained in note 192, he qualifies this in 1913 to say that Newtonian mechanics is only approximately valid in such frames. The point is that the mechanical principles governing the u'ansfer of momentum must now be modified so that (velocity-dependent) relativistic mass now appears in place of Newtonian mass. These principles, the law of inertia, and the principle that light has a fixed constant velocity c now rigorously characterize the inertial frames of special relativity. 268. Indeed, as Scott Tanona emphasizes in his contribution to the present volume, the familiar "laboratory frames" of classical physics continue to play an essential role in relating the mathematical structure of Minkowski space-time to our sensible perceptual experience of nature: for example, the Michelson-Morley experiment, which can be taken as confirming that light has the fixed constant velocity c in every inertial frame (in accordance with the light principle and the principle of relativity), is itself carried out in precisely such a "laboratory frame"a frame which is approximately inertial according to both Newtonian and relativistic physics. Tanona rightly criticizes some of my earlier discussions, in this connection, for suggesting that the abstract mathematical structure of Minkowski space-time is somehow directly coordinated with our perceptual experience of light, entirely independently of prior physical theories. Tanona argues, on the contrary, that the empirical meaning of Minkowski space-time entirely depends on the empirical meaning that the classical concept of inertial frame already has in Newtonian physics. In Husserlian terms, no mathematical theory of space, time, and motion is directly coordinated with the life-world; there is rather an historical sequence of mathematical descriptions-beginning with basic geometry, chronometry, and astronomy and eventuating in general relativity-such that later descriptions in the sequence are coordinated with earlier ones. (Thus, I view Tanana's criticisms of my earlier discussions of coordination and the "relativized a priori" as closely related to Ryckman's criticisms mentioned in note 253: the present discussion is very much indebted to them both.) 269. The (semi-Riemannian) space-times described by general relativity are all infinitesimally Minkowskian, in the same sense that all Riemannian manifolds are infinitesimally Euclidean. For the concept of local inertial frame in general relativity see, e.g., Misner, Thorne, and Wheeler (1973,18-23,285-86), Torretti (1983, 136-37). As we have seen, Einstein himself did not first understand the principle of equivalence in this way, and he arrived at the variably curved four-dimensional geometry of what we would now call general relativistic space-time only as a result of his earlier investigations of the principle of equivalence within the framework of the (global) inertial frames of special relativity: see the paragraph to which note
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204 above is appended, together with the following paragraph. Thus, Einstein's actual historical route to general relativity was even more dependent on the earlier concept of inertial frame than is his finished theory. 270. Thus, if we consider the mathematical structures of note 245 above as elements of a typical Marburg sequence (as understood by Cassirer), what we then need to add is the above account of how the different structures in the sequence are related to concrete approximations to various types of inertial frames (including, in general relativity, local inertial frames). Similarly, whereas Cassirer views the relation of coordination as holding only between later mathematical structures in the sequence and earlier ones (compare the paragraph to which note 251 above is appended), we also consider the relations between these structures and concrete approximations to the appropriate kinds of inertial frame-where these latter, in turn, arise by idealization and precisification of our life-world empirical practices of spatio-temporal measurement. Thus, whereas Cassirer is right to insist (against ScWick) that it makes no sense to coordinate our mathematical theories with something undefmed and ineffable, he is wrong to suggest-at least in his pre-symbolicforms period (see note 264 above)-that we can also dispense with all relation to what is immediately and perceptually given, prior to and independently of our mathematical-physical theorizing. 271. Husserl had already (briefly) indicated the importance of intersubjectivity in §48 of the Ideas, by suggesting that "the factually separate worlds of experience [of different subjects-MF] fit together through the interconnections of actual experience into a single intersubjective world, the correlate of the unitary spiritual world (the universal extension of the human community)." But Husser! had not there discussed the resulting historicity-that, in the above words from the Crisis, "I know myself to be factually in a generative interconnection, in the unitary flow of an historicity, in which [the] present is an historical present of humankind (together with the world known to it), [with] an historical past and an historical future." This new emphasis on the importance of historicity is distinctive of the Crisis. 272. In the final section of the Crisis Husserl emphasizes the distinctive historical mission of transcendental phenomenology (1954, 268; 1970,264): "What was hidden from me in humanity I reveal in transcendental investigation. It is itself a world-historical process, in so far as it not only enriches the history of the constitution of the world with a new science, but also enriches the content of the world itself in every respect. Everything worldly has its transcendental correlate; for every new revelation for the investigator of human beings, the psychologist, there are new determinations of what is human in the world. No positive psychology, which does not have the transcendental psychology already at work at its disposal, can discover any such determinations of what is human and of the world." This historical mission, then, consists in precisely the discovery that the transcendental investigation of human subjectivity is entirely distinct from any supposed objective science of the soul (1954, 268-69; 1970, 265): "Phenomenology frees us from the old objectivist idea of the scientific system, from the theoretical form of mathematical natural science, and it thereby frees us from the idea of an ontology of the soul, which could be analogous to physics." It is in this way, in the end, that Husserl resolves the problem of mind-body dual-
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ism-which, as we have seen (note 259 above), he takes to be the ultimate source of the crisis in the European sciences (and in European culture) with which he was laced in the years 1936-37. 273. Thus, for example, when Cassirer, in 1921, proposes "spatiality in general" as our current best candidate for a geometrical "a priori" (see note 245 above, together with the paragraph to which it is appended), it would seem that this proposal essentially depends on entirely contingent circumstances of history: that the sequence depicted in note 245 has in fact (in 1921) terminated in general relativiry. This fact can be located relative to Cassirer's regulative ideal by the requirement that the next stage in the historical development should continuously emerge from it (compare the paragraph to which note 248 above is appended, together with the following paragraph). But this sheds no transcendental light on the actual historical process by which we arrived at general relativity in the first place. (In terminology developed in the next section, it contributes to a transcendental explanation of the retrospective rationality of general relativity in terms of an ideal limit-theory, but not of its prospective rationality in relation to earlier theories.) 274. Here, and throughout the remainder of this essay, I am especially indebted to Charles Parsons for raising this problem and stimulating me to talee it very seriously. 275. See note 216 above. The heart of Reichenbach's conception of what I call the relativized a priori is a sharp distinction between two different meanings of the original Kantian a priori: (i) necessary and unrevisable, fixed for all time, on the one hand; (ii) "constitutive of the concept of the object of[scientific] knowledge," on the other. Reichenbach's argument is that the second meaning can still be sustained in the context of the transition from Newton to Einstein, while only the first need now be dropped. 276. See notes 253 and 268 above. As I point out there, I am especially indebted to Thomas Ryckman and Scott Tanona in this connection. I am also indebted to discussions with Don Howard. 277. See note 163 above, together with the paragraph to which it is appended. Indeed, Hilbert's Foundations ofGeometry (1899) grew out of this same projective tradition, and Hilbert famously says in his brief Introduction that the task of axiomatizing geometry amounts to "the logical analysis of our spatial intuition." In this sense, the purely "formalistic" reading of Hilbert adopted by ScWick-and, following him, Einstein as well (see note 215 above)-represents a considerable oversimplification of Hilbert's own view. 278. See note 254 above, together with the paragraph to which it is appended and the preceding paragraph. From this point of view, my discussion of the problem of "coordination" in general relativity in Friedman (2001, 78-79) becomes problematic when it uncritically adopts the Schlick-Einstein picture of this theory (compare note 277 above) as if it were the only live alternative. 279. See note 268 above, together with the paragraph to which it is appended, the preceding paragraph, and the following paragraph. As the note points out, this means, in Husserlian terms, that no mathematical theory is directly coordinated with the immediately and perceptually given life-world itself. 280. See De Pierris (1992) for an extended discussion of Kant's revolutionary conception of the constitutive a priori against the background of traditional ratio-
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nalism-as represented, in particular, by the Cartesian notion of "clear and distinct ideas" of reason or the intellect. 281. An emphasis on the "regressive" or "analytic" character of Kant's transcendental method, as we have seen, is distinctive of the Marburg School of neoKantianism (compare note 241 above, together with the paragraph to which it is appended). And this emphasis is controversial, precisely because (as I point out there) the Prolegomena explicitly distinguishes its "analytic" method from the "synthetic" method of the first Critique. It seems clear, however, that the two methods are complementary rather than opposed to one another, even in the first Critique. Thus, especially in the second edition, Kant also addresses the questions how pure matllematics and pure natural science are possible, and, at several crucial junchlres, he appeals to the acmal existence of these sciences as a rock-bottom response to skeptical and empiricist alternatives. In §VI of the Introduction, for example, Kant (just as in the Prolegomena) interprets the general problem of pure reason, "how are synthetic a priori judgments possible?", as containing the two more specific subproblems, "how is pure mathematics possible?" and "how is pure namral science possible?", and he simply takes it for granted that the actual existence of these sciences puts the existence of synthetic a priori knowledge entirely beyond all doubt. In particular, in considering Bume's skepticism concerning the necessity of the causal relation-which then leads to skepticism about the possibility of any a priori metaphysics-Kant blames this result on Bume's insufficiently general understanding of the problem (B20): "[B]ume would never have arrived at this assertion, which destroys all pure philosophy, if he had kept our problem before his eyes in its [full] generality; for he would then have seen that, according to his argument, there could also be no pure mathematics (for it certainly contains synthetic a priori propositions), and his good sense would then surely have saved him from this assertion." Similarly, while considering (in §14) the circumstance that neither Locke nor Bume posed the problem of the transcendental deduction, and instead attempted a psychological or empirical derivation of the pure concepts of the understanding, Kant concludes (B127-28): "But the empirical derivation which both fell upon cannot be reconciled with the acmality of the a priori scientific cognition that we have-namely of pure mathematics and universal natural science-and is thus refuted by this fact [Faktum]." 282. From this point of view, it is by no means surprising that Busserl relies on geometrical examples when first introducing the basic notions of his phenomenological method in the Ideas, and he also presents a conception of the relationship between essential or "eidetic" truths and empirical truths explicitly modeled on Kant (see note 256 above, together with the paragraph to which it is appended): empirical knowledge is framed by both material eidetic sciences (such as geometry and rational mechanics) and formal eidetic sciences (including the theory of manifolds), and it is in precisely this way that we find both universality and necessity within our inmitive experience. Thus, although Busserl also makes a sharp distinction in the Ideas (§§71-75) between "exact" and "descriptive" eidetic sciences, and firmly places transcendental phenomenology in the latter category, I fmd it very hard to see how even a descriptive eidetic science can then proceed entirely independently of a complementary "analytic" or "regressive" approach oriented towards the exact sciences. In particular, I would understand Busserl's necessary or
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eidetic description of the transcendentally subjective structures of the life-world as complementary to his view that the objective mathematical sciences then necessarilv develop out of these structures: objective mathematical-physical space and time d~veloP out of subjective-relative spatiality and temporality, objective mathematical-physical laws out of subjective-relative sets of "anticipations," and so on (compare note 258 above, together with the paragraph to which it is appended and the preceding paragraph). 283. Kant often makes such claims to explanatory uniqueness, for example, in the Transcendental Exposition of the Concept of Space added to the second edition (B41): "Therefore, only our explanation makes the possibility ofgeometry as an a priori synthetic cognition comprehensible. Any mode of explanation that does not achieve this, even if it appeared to be similar to ours, can be most securely distinguished from ours by this criterion." I am indebted to Dagfinn F0llesdal for emphasizing to me the importance of the problem of uniqueness in this connection. 284. Compare note 43 above, together with the paragraph to which it is appended and the preceding paragraph. It is no accident, then, that Kant's discussion of the faculty of pure sensibility in the Transcendental Aesthetic prominently includes a discussion of the Newtonian and Leibnizean conceptions of space and time (§7), which alludes to the theological unacceptability of the Newtonian position at A40/B57: "The first party [of the mathematical investigators of nature] succeeds to such an extent that they make the field of appearances free for mathematical assertions. However, they confuse themselves very much by precisely these conditions when the understanding pretends to extend itself beyond this field." The theological point is then made explicit in the second edition (B71-72): "In natural theology, where one thinks an object that is not only no object of sensible intuition for us, but cannot even be an object of sensible intuition for itself, one takes care to remove the conditions of space and time from all of its intuition (for all of it cognition must be intuition and not thought, which is always a manifestation of limitations). But with what right can one do this, if one has previously made both into forms of things in themselvesand, indeed, into forms which, as a priori conditions of the existence of things, even remain when one has annihilated the things themselves? (For, as conditions of all existence in general, they must also be conditions for the existence of God.) There is therefore no alternative, if one does not pretend to make them into objective forms of all things, but to make them into subjective forms of our outer and inner mode of intuition." 285. As Mary Domski has emphasized to me, it is an important feature oflater developments that the specifically metaphysical and theological elements characteristic of the intellectual situation in which Kant originally found himself drop out, or at least become extremely attenuated. This is a consequence of Kant's fundamental transformation of the very meaning of metaphysics as practiced by his predecessors into what he then calls transcendental philosophy-as discussed in detail in the first two sections of the present essay. 286. See notes 129 and 130 above, together with the paragraph to which they are appended and the three preceding paragraphs. As I remark there, Helmholtz's blanket rejection of the N aturphilosophie of Schelling and Hegel as unscientific was certainly unfair, in so far as Schelling, for example, did in fact respond intelligently
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and insightfully to the scientific and philosophical situation in which he found himself (compare note 125 above, together with the paragraph to which it is appended). Yet Helmholtz, as I also remark (compare the paragraph to which note 127 is appended), had an important new weapon up his sleeve, namely, the discovery of the conservation of energy. Nevertheless, Helmholtz's attempts to comprehend specifically electromagnetic phenomena within his dynamical theory of matter (based on central forces of attraction and repulsion depending only on distance) eventually failed, revealing that electromagnetism involves fundamentally non-Newtonian forces after all (compare notes 131 and 134 above)-just as the Naturphilosophen had always suspected. The discovery of the invariance of the velocity of light (and, more generally, all forms of electromagnetic propagation) finally revealed, in a completely unexpected way, just how profounclly nonNewtonian electromagnetism really was. 287. See again the discussion referred to at the beginning of the preceding note. As I point out in notes 131 and 134 above, Helmholtz eventually gave up the strong link between the principle of causality and his dynamical theory of matter, and, accordingly, he then understood the former as expressing only the lawlikeness of the appearances: compare also note 146 above, together with the paragraph to which it is appended and the following paragraph. 288. See notes 143 and 144 above, together with the paragraph to which they are appended. For the relationship between Helmholtz's "naturalistic" (and "psychologistic") version of scientific epistemology and transcendental philosophy in Kant's original sense see note 149 above, together with the paragraph to which it is appended. 289. In particular, Helmholtz's conception represents the minimal such generalization that is consistent with the idea that our "subjective" and "necessary form of external intuition" may be described by a non-Euclidean geometry. As we know, Riemann's more general theory contains manifolds of variable curvature not subject to the Helmholtz-Lie theorem, and it is for precisely this reason that Weyl then attempted to generalize the "space problem" to comprehend the geometries arising in the general theory of relativity as well (compare note 254 above). Moreover, as I point out in Friedman (2000a, 209-11; including note 47 on p. 218), Weyl, too, conceived his work as a generalization of Kant's original theory of space as an (a priori) "form of appearance." I shall return to Weyl below, but the important point here is that Helmholtz is closer to Kant (in the sense that his generalization preserves the possibility of geometrical constructions analogous to Euclid's), whereas Weyl's work arises only as a further generalization, in turn, of Helmholtz's. 290. See note 147 above, together with the paragraph to which it is appended and the preceding paragraph. As I point out there, that Helmholtz's conception of the principle of causality was more regulative than constitutive did not represent as radical an innovation as it might seem, since a blurring of the sharp distinction between the two had already begun in the "post-critical" work of Kant himself (i.e., the Opus postumum) and was then characteristic of post-Kantian N aturphilosophie. 291. See note 163 above, together (again) with the paragraph to which it is appended. Poincare was thus a central representative of the approach to modern abstract mathematics characterized by a rejection of overly "formalistic" interpretations of Hilbert's axiomatic method-where the latter, by contrast, are definitive
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of the Schlick-Reichenbach understanding of the problem of "coordination" (compare note 277 above, together with the paragraph to which it is appended). 292. See note 174 above, together with the paragraph to which it is appended and the twO preceding paragraphs. The connection between Poincare's conception of the relativity of motion and Einstein's theory of special relativity becomes especially important when we also appreciate the extent to which Einstein's formulation of what he calls the "principle of relativity" was very likely influenced by Poincare-a point (already discussed above) to which I shall return below. 293. See note 48 above, together (again) with the paragraph to which it is appended. As the note points out, whereas mathematical principles of pure understanding are constitutive with respect to intuition, the dynamical principles are merely regulative with respect to inulition-but constitutive with respect to experience (compare note 147 above). Ideas of reason, by contrast, are not constitutive with respect to either intuition or experience (because they can never be £lIlly realized in perceptual experience) and are therefore purely regulative: "absolute space," as the forever unattainable "center of gravity of all matter," is just such a regulative idea (compare the paragraph following the one to which note 156 above is appended). 294. See notes 157 and 159 above, together (again) with the paragraphs to which they are appended. As note 157 remarks, although I do not know whether Mach actually studied Kant's Metaphysical Foundations, the similarities are striking indeed; and, whether or not he did so self-consciously, Mach in fact forged such a connection nonetheless. 295. See again note 175 above, together with the paragraph to which it is appended, the preceding paragraph, and the following paragraph. As I point out in the course of this discussion, Poincare here appeals to more general (and more complicated) formulations of the Newtonian theory of gravitation that also hold in uniformly rotating systems; and, as I also point out (notes 176 and 177 above), such a procedure is perfectly legitimate from the point of view of contemporary generally covariant formulations. The point remains, however, that the relativity expressed in Poincare's "principle of relative motion" (= Newton's Corollary V to tile Laws of motion) remains fundamentally different from the broader relativity expressed in his "law of relativity." Whereas the former represents a symmetry or invariance property of the Newtonian laws of nature (and, from a contemporary point of view, of Newtonian space-time), the latter most certainly does not. 296. See note 193 above, together with the paragraph to which it is appended and the four preceding paragraphs. As I point out in note 185 above, what Poincare, during this same period, came to call the "principle of relativity" was in fact limited to inertial frames-but, nevertheless, it continued to have both empirical and a priori motivations: see notes 190 and 191 above, together with the paragraphs to which they are appended. In terms of note 295 above, whereas Poincare's most fundamental "law of relativity" is a purely geometrical principle, expressing the necessary symmetries of three-dimensional (homogeneous) space, Einstein's "principle of relativity" expresses the symmetry or invariance properties of the laws of Maxwell-Lorentz electrodynamics-which we now take to be the symmetries of Minkowski space-time. The central problem with Poincare's hierarchy, from this
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point of view, is that it makes the three-dimensional geometry of space prior to the four-dimensional geometry of space-time: see note 195 above, together with the paragraph to which it is appended. 297. The crucial point, in tllls connection, is that Newton's tIllrd law-the equality of action and reaction-implicitly defines the relation of absolute simultaneity in a classical inertial frame, in so far as it allows us to coordinate action-reaction pairs related by the Newtonian law of (instantaneously propagated) gravitation: see note 157 above, together (again) with the paragraph to which it is appended, the preceding paragraph, and the following paragraph. Einstein's two postulates take over precisely tills role in the case of his new, relativized relation of simultaneity defined by (continuously propagated) electromagnetic processes. 298. Compare notes 170 and 171 above, together with the paragraph to which they are appended and the preceding paragraph: Euclidean geometry is singled out, for Poincare, in that it is both mathematically simplest and very naturally corresponds-rougWy and approximately-to our pre-scientific experience of bodily displacements. Just as Helmholtz's conception, as I have suggested, is the minimal extension of Kant's original conception consistent with the discovery of non-Euclidean geometries (compare note 289 above), Poincare's conception is the minimal extension of Helmholtz's consistent with the more sophisticated grouptheoretic version of the principle of free mobility due to Lie (compare note 166 above), the new perspective on the relativity of motion due to the modern concept of an inertial frame (compare again the paragraph to which note 175 is appended), and, most importantly, the apparently paradoxical new situation in electrodynamics arising in connection with precisely tills relativity of motion (compare note 191 above, together with the paragraph to which it is appended and the two preceding paragraphs). 299. Einstein's "elevation" of the principle of relativity and the light principle to the status of "presuppositions" or "postulates" definitive of a new geometricalmechanical constitutive framework is thereby shown to be possible, but not necessarily obligatory. Thus, as I emphasized above, Poincare, for example, had every right to resist tills move and to continue to maintain the Newtonian frameworkat least before the incorporation of gravitation into relativity around 1912 (see note 196 above, the paragraph to which it is appended, and the following paragraphs). Nevertheless, Poincare himself adnlits that the "entirely new conception of space and time" (due to Einstein and Minkowski) is a possible and legitimare "convention" in his 1912 lecture on "Space and Time" (see the paragraph to which note 189 is appended). 300. Compare the paragraph preceding the one to which note 286 above is appended, together with the two preceding paragraphs. In particular, that the "inner logic" of the successive intellectual situations in question proceeds against the background of, and explicitly in light of, Kant's original theory is what makes tills enterprise properly "transcendental." 301. See, in particular, note 211 above, where I discuss Don Howard's attempt, in his contribution to the present volume, to deploy Einstein's embrace of an explicitly anti-Kantian (and anti-neo-Kantian) epistemological "holism" against my own version of the relativized a priori. Einstein's stance, as Howard righdy points out, was then taken up by ScWick, who also associated such "holism" with
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the passages from "Geometry and Experience" now under consideration: see note 216 above, together with the paragraph to which it is appended. As I point out in both places, however, the situation is considerably complicated by the circumstance that, in the passages in question, Einstein himself (wrongly) associates "holism" with Poincare-and then proceeds to reject it in favor of his own understanding of (Helmholtzian) "empiricism." 302. See note 200 above, together with the paragraph to which it is appended, the twO preceding paragraphs, and the following paragraph. In Friedman (2001, 86-91), I develop more fully the parallel between these two cases of "elevating" a mere empirical fact to the status of a (relativized) a priori principle by first examining the relationship between the invariance of the velocity of light (as recently verified in the Michelson-Morley experiment) and Einstein's new definition ofsimultaneity, and then the relationship between the equality of gravitational and inertial mass (as recently verified in the Eotvos experiments) and the principle of equivalence. 303. See note 206 above, together with the paragraph to which it is appended. As I point out there, the crucial factor in motivating a truly four-dimensional understanding of tlle principle of equivalence is that the non-inertial (uniformly) rotating frame in question has not only a non- Euclidean spatial geometry (due to length contraction) but also a non-standard temporal metric (due to time dilation). As note 205 explains, rigorously defining the non-Euclidean spatial geometry in this case is actually rather delicate, requiring, in effect, the consideration of all the instantaneous inertial systems that are tangent to the non-inertial frame at every point. The situation is similarly delicate with tlle non-standard temporal metric in question, where, once again, the result is rigorously well defined only if the fi-ame is rotating uniformly. 304. See the paragraph to which note 211 above is appended; and compare note 296 above, together with the paragraph to which it is appended and the following paragraph. As I point out there, Einstein could not embrace Poincare's philosophy of geometry even in 1905, since it privileges a priori the three-dimensional geometry of space over the de facto symmetries of the laws of motion (which, on our current understanding, express the four-dimensional geometrical symmetries of space-time). Einstein's divergence from Poincare on this point is even stronger in general relativity; for, not only do we now use non-Euclidean geometries to describe both space and space-time, but we have also definitively given up (in both cases) the homogeneity and isotropy (constant curvature) of the underlying geometry. Einstein thereby ultimately arrived at a radically new conception of geometry envisioned by neither Helmholtz nor Poincare. 305. It is in precisely this way that contingent empirical principles-the light principle and the principle of equivalence-can then playa constitutive role and can tlms instantiate, in particular, the concept of the contingent or relativized a priori. And it is precisely this concept, in my view, which therefore needs to be added to Weyl's deep mathematical analysis of the "space problem" in the context of transcendental phenomenology: compare notes 254, 255, and 289 above. By contrast, it appears that Weyl himself had envisioned a purely a priori solution with no reliance on contingent empirical facts (1927, §18, 99-100; 1949, 137): "A way for understanding the Pythagorean nature of the metric expressed in the Euclidean rotation group precisely on the basis of the separation of a priori and a posteriori
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has. been given by ~e a~th~r: Only in the cas~ of this g~oup does the intrinsically aCCidental quantltauve distnbuuon of the metric field uruquely determine in all circumstances (however it may have been formed in the context of its a priori fixed nature) the infinitesimal parallel displacement: the non-rotational progression from a point into the world. This assertion involves a deep mathematical theorem of group theory which I have proved. I believe that this solution of the space-problem plays the same role in the context of the Riemann-Einstein theory that the Helmholtz-Lie solution (section 14) plays for rigid Euclidean space. Perhaps the postulate ofthe unique determination of 'straight-progression' can be also justified from the requirements of the phenomenological constitution of space. "-What I remain unsure of is this last idea. 306. The historicized version of transcendental philosophy I am attempting to exemplify therefore sheds striking new light, I believe, on the truly remarkable depth and fruitfulness of Kant's original version. Kant's particular way of establishing a connection between the foundations of geometry and the relativity of motion-which, as we have seen, lies at the heart of his transcendental method (compare note 293 above, together with the paragraph to which it is appended)-has not only led, through the intervening philosophical and scientific work of Helmholtz, Mach, and Poincare, to a new conception of the relativized a priori first instantiated in Einstein's theories, it has also led, through this same tradition, to a radically new reconfiguration of the connection between geometry and physics in the general theory of relativity itself. There can be no question, of course, of Kant having "anticipated" this theory in any way. On the contrary, as we have also seen, Kant was not even aware of the new discoveries in electro-chemistry which first precipitated the discovery of electromagnetism in the early years of the nineteenth century (compare the paragraph following the one to which note 124 above is appended, together with the two preceding paragraphs), let alone the apparently paradoxical situation in late nineteenth- and early twentieth-century electrodynamics facing both Einstein and Poincare. The point, rather, is that Kant's own conception of the relationship between geometry and physics (which was limited, of course, to Euclidean geometry and Newtonian physics) then set in motion a remarkable series of successive reconceptualizations of this relationship (in light of profound discoveries in both pure mathematics and the empirical basis of mathematical physics) that finally eventuated in Einstein's theory. 307. See the first foomote of Richard Creath's contribution to the present volume. 308. These parallels between Kuhn's conception of scientific revolutions and the relativized a priori are developed in Friedman (2001). See also Friedman (2002a; 2003a). 309. This argument is made in Kuhn (1962, Chapter IX). As I discuss in the references cited in note 308 above, there is considerable irony in the circumstance that Kuhn here names "early logical empiricism" as the main example of the prevalent interpretation of scientific progress he aims to refute. See also Reisch (1991) for an important seminal discussion of this irony. 310. Again, Kuhn himself became clearly aware of the Kantian and neoKantian background to his conception of scientific revolutions only relatively late
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in his career: compare notes 308 and 309 above. As I explain in Friedman (2003a), Kuhn's historiography also has deep roots in the neo-Kantian tradition, mecliated by the in~uence of such philosophically oriented historians of science as Ernst Cassirer, Emile Meyerson, Helene Metzger, Leon Brunschvicg, Anneliese Maier, and (especially) Alexandre Koyre. 311. The logical empiricists are of particular significance here for at least three reasons: (i) their assimilation of Einstein's theories had a decisive influence on twentieth-century philosophy of science; (ii) Kuhn himself at first took them as the main representatives of the prevalent philosophical image of science he wished to reject; (iii) Kuhn, relatively late in his career, also recognized the analogy between the Kantian and neo-Kantian elements of their philosophical approach and the corresponcling elements in his own approach. See again notes 308 and 309 above, together with the references cited there. 312. This point is discussed in more detail in Friedman (2001, Part Two, §3)-see especially pp. 98-99. The basic idea is that, although a later theory in the sequence (e.g., general relativity) does indeed contain the laws of an earlier theory (e.g., Newtonian gravitation theory) as an approximate special case, the constitutive principles of the earlier theory have now entirely lost their constitutive (possibility-defining) role. To move to a new set of constitutive principles (e.g., the principle of equivalence) is precisely to abandon the constitutive role of the older principles (the Newtonian Laws of Motion), so that what counted as constitutive principles in the older theory now hold only as mere empirical laws valid (approximately) in special empirical circumstances (small velocities, relatively isolated gravitating systems, and so on): they are not "elevated" to the status of more than merely empirical laws in the new theory. 313. For the sequence of such "minimal extensions," beginning with Kant and leacling from Newtonian mechanics through special and then general relativity, see note 289 above, together with the paragraph to which it is appended and the prececling paragraph, note 298, together with the paragraph to which it is appended and the prececling paragraph, and note 306, together with the paragraph to which it is appended. 314. See Friedman (2001, 83-86) for further cliscussion of the relationship between the notion of real (or empirical) possibility, the "space of reason" then governing relations of empirical justification or testing, and the resulting asymmetry between mere empirical laws formulated against the background of possibilitydefining constitutive principles and these constitutive principles themselves. The basic idea is that the mathematical and mechanical relativized constitutive principles (e.g., the Newtonian Laws of Motion, the principle of equivalence) must first be in place before particular empirical laws (the Newtonian law of universal gravitation, Einstein's gravitational field equations) can then be tested: we need to define what is even really (physically) possible before we can determine what is actually (empirically) true. 315. Again, see Friedman (2001,99-115) for further cliscussion of this kind of expansion of the space of available intellectual possibilities. The basic idea is that the most fundamental problem posed by Kuhn's concept of incommensurability is to explain how it becomes rational and reasonable to expand the space of intellectual possibilities from one subject-defining paracligm to another and that
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concurrent developments in scientific philosophy playa crucial mediating role here. (The meta-scientific reflections of Helmholtz, Mach, and Poincare, for example, play precisely this role in the sequence of "minimal extensions" involved in the transition from Newtonian mechanics through special and then general relativity referred to in note 313 above.) After the resulting new space of intellectual possibilities is in place-once it is recognized, for example, that gravitation may be represented by a variably curved, four-dimensional space-time manifold-it is then (and only then) possible empirically to test Einstein's theory (his field equations) by the advance in the perihelion of Mercury or the deflection of light in a gravitational field. 316. In the terms of note 299 above, the arguments leading to a new geometrical-mechanical constitutive framework show only that it is possible, not that it is obligatory. Nevertheless, as I point out there, Poincare, in effect, recognized the new constitutive framework of special relativity as possible in 1912, and he would therefore have had good reason (especially in view of Einstein's own very serious engagement with Poincare's meta-scientific ideas) to accept the framework of general relativity as possible as well (if, of course, he had lived). 317. Richard Creath puts the point very aptly in his contribution by remarking that the historical narrative I present is not an illustration of my argument-it is the argument itself. In this sense, the particular examples on which I focus are not intended to support a general thesis as part of an inductive argument. (If so, as Koertge rightly suggests, it would not be a very strong inductive argument in any case.) Rather, the historical narrative I present is an historical response to a particular historical challenge. Nevertheless, I still intend it to leave us with a viable contemporary (generalized) Kantian conception of scientific rationality: compare note 318 below. 318. Of course it would be disappointing, nonetheless, if we learned nothing interesting or useful about other central cases of scientific revolutions from the particular historical events on which I focus. My hope, in this connection, is that the developments in question are sufficiently rich, from both a scientific and a philosophical point of view, that other such cases will also be thereby illuminated. In Friedman (2001, Part Two, §5), I briefly attempt to illustrate.how this might work for chemistry and biology (or, at least, the life sciences generally conceived). Just as Kant's original conception of scientific rationality, although primarily focused on the geometry and mechanics of space, time, matter, and motion, still had interesting and important implications for chemistry and biology, I hope that my generalized Kantianism may be able to do something similar by considering both the subsequent history of these sciences and that of post-Kantian philosophical reflection upon them. But perhaps the biggest challenge to my approach is quantum mechanics, where we have a new revolutionary theory of fundamental physics whose position in what I call the dynamics of reason still remains quite obscure. I make some preliminary suggestions in the section of Friedman (2001) cited above, and I shall return to this case briefly below. 319. This, of course, is Cassirer's version of the genetic conception of knowledge characteristic of the Marburg School: see notes 242 and 243 above, together with the paragraphs to which they are appended and the two following paragraphs. (As I point out in note 243, this is a topic on which Richardson himself has made
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important contributions.) For the application of this conception to the transition Ii-om Newton to Einstein see note 245, together with the paragraph to which it is appended. 320. For this claim, and the idea that a fundamentally historical-and in this sense Hegelian-approach represents my justification for the continuing use of Kantian terminology, I am especially indebted to discussion with Thomas Ricketts. 321. See again note 242 above, and compare note 109, together with the paragraph to which it is appended. As we have seen, a rejection of Kant's sharp distinction between regulative and constitutive principles pervades post-Kantian German philosophy-including both the absolute idealism of Schelling and Hegel and the neo-Kantianism of Cohen, Cassirer, and the Marburg School (compare note 106, together with the paragraph to which it is appended and the preceding paragraph)-and this rejection is intimately connected, in turn, with a parallel rejection of the sharp distinction between understanding and sensibility. Although I do not propose to defend Kant's original version of this latter distinction, the distinction between pure mathematics and its application to our perceptual experience remains centrally important in my conception of (relativized) constitutively a priori principles (compare note 322 immediately below). 322. This is the main theme of the preceding section on logical empiricism, neo-Kantianism, and phenomenology. See especially note 273 above, together with the paragraph to which it is appended and the preceding paragraph, where I compare Husserl and Cassirer in this regard. The point is that, whereas Husser! has specified a transcendentally constitutive origin for the historical process by which the empirical meanings of our mathematical-physical theories are successively constituted, and Cassirer has specified a transcendentally regulative endpoint, neither has described the intervening historical developments by which the empirical meanings of the various intermediate theories are actually constituted-neither has articulated an account of what I call the relativized constitutive a priori. 323. Richardson (note 9 of his contribution) suggests that, following Hegel, "the transcendental must ultimately disappear in favor of a logic of objective knowledge." What I have been urging, on the contrary, is that a properly historicized Kantianism appropriate to the development from Newton to Einstein essentially involves a (similarly historicized) version of precisely transcendental philosophy: see again note 299 above, together with the paragraph to which note 306 is appended. Although Mark Wilson's preference for local over global principles, expressed at the conclusion of his very generous and insightful contribution to the present volume, is associated with a rather different philosophical point of view than Richardson's, it is along these lines, too, that I would reply to him. We still need what I am calling relativized constitutively a priori principles even in the thorougWy local context of general relativity (the assumption of an infinitesimally Lorentzian space-time metric, the principle of equivalence), despite the fact that we are well aware that these principles remain subject to unpredictable revisions as our empirical knowledge progresses. 324. See Friedman (2001, Part Two, §4) for a fuller discussion of these two levels (although without invoking the terminology of regulative ideas to describe the philosophical or meta-scientific level). In the case of Einstein's appropriation of
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the debate on the foundations of geometry between Helmholtz and Poincare, for example, Einstein was able to arrive at a new possibility-defining constitutive principle, the principle of equivalence, without resolving the foundational debate itself. Rather, all he needed was to take this debate very seriously (in the context of special relativity, the equality of gravitational and inertial mass, and the example of the uniformly rotating frame of reference) in order to generate the radically new empirical possibility of representing the gravitational field by a non-Euclidean geometry. In particular, anyone who took the foundational debate seriously (including Poincare himself) would (in the context in question) have ample grounds for taking Einstein's expansion of the space of empirical possibilities seriously as well-and would thereby have ample grounds for taking Einstein's new representation to be really (empirically) possible. 325. See note 120 above, together with the paragraph to which it is appended, and note 125, together with the paragraph to which it is appended. A comprehensive mathematical theory of electromagnetism was only achieved considerably later in the work of Maxwell. 326. Again, no truly quantitative version of chemical atomism was achieved until John Dalton's work on combining weights in the early years of the nineteenth century, and no comprehensive mathematical theory was achieved until the development of electro-chemistry throughout the nineteenth century finally resulted (against the background of Lorentz's theory of the electron) in the quantummechanical theory of chemical valence developed in the twentieth century. This is one important reason, as Koertge observes, that the apparatus of relativized a priori constitutive principles does not apply very directly to the history of chemistry. Nevertheless, as I suggest in note 318 above, my version of the dynamics of reason may still apply indirectly to this case-here in the guise of what I am now calling regulative ideas. Mary Domski has emphasized to me that it is well worth reminding ourselves, at this point, that the regulative use of reason played a distinctive role in Kant's own conception of chemistry, both in the critical period (see note Il2 above, together with the paragraph to which it is appended) and in the Opus postumum (see the paragraph to which note 122 above is appended to, together with the following paragraph). I cannot develop the idea here, but I believe that chemistry remains the locus for a distinctive kind of interaction between regulative and constitutive principles throughout its history-and, indeed, even today. 327. Recent work on the history and philosophy of experimental practice has emphasized its essentially open-ended character-as a trial-and-error procedure for stabilizing the interactions between our contrived experimental devices and our theoretical frameworks rather than a method for producing certain (or relatively certain) "observation sentences": see, e.g., Cartwright (1997), Galison (1997), Hacking (1993), Pickering (1992), (1995). My own account of how the construction and articulation of precise theoretical frameworks is conditioned by a looser web of interactions between regulative, constitutive, and experimental levels is intended to complement this work. Domenico Bertoloni Meli's innovative and perceptive contribution to the present volume makes a related point in so far as he shows that, although the axiomatic (Archimedean) tradition in seventeenth-century mechanics was self-consciously motivated by the search for self-evident principles "to which the mind naturally consents," it actually functioned in a considerably
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more open-ended and this sense "experimental" fashion. It is especially appropriate, therefore, that Bertoloni Meli urges historians of seventeenth-century science to devote as much attention to the development of this tradition as to the development of the experimental method. Bertoloni Meli (2006) extends his innovative approach to the entire development of mechanics in the seventeenth-century, showing how both experimental and engineering practices were fruitfully intertwined with mathematical axiomatic thinking (often via thought experiments). 328. See DiSalle's remarks in the introductory section of his contribution to the present volume (pp. 527-28): "The role of philosophy in conceptual transformation, it seems to me, goes beyond its contributions to enlarging the common conceptual space, and providing for intelligible discourse across theoretical divides. In the most important historical cases-those cases in which, as in that of relativity, the most fundamental concepts are most radically redefined-we find that philosophical argument is essential, not only to the rational evaluation of a new theory, but to the very construction of the theory; the conceptual transformation is precisely the product of a conceptual analysis. In such a case, in other words, philosophical analysis is more than a means of articulating and defending the new theory against its predecessor; the philosophical analysis is the new theory itself. Deeper philosophical understanding is then not only the accessory to revolutionary scientific understanding; the new theory itself represents a revolution in philosophical understanding." 329. In the Preface to DiSalle (2006, xii), he generously remarks that his work has deep roots in a philosophical engagement ,"vith mine; I should add that Friedman (2001) has equally deep roots in a philosophical engagement with hisespecially with DiSalle (1991; 1995 )-and, in the present essay, I am also especially indebted to DiSalle (2002b): see note 156 above, together with the paragraph to which it is appended and the two following paragraphs. Finally, my present discussion of the precise sense in which what I call relativized a priori principles are still transcendentally constitutive in (a generalization and extension of) the original Kantian sense owes much to my reading of DiSalle (2006) in connection with Friedman (2007), portions of which I am drawing upon here. 330. As DiSalle points out, that Newton's principal target in the Scholium is Descartes was first made clear in Stein (1967). As DiSalle (and Stein) also point out, this becomes perfectly explicit in Newton's unpublished De Gravitatione. DiSalle takes himself, more generally, to be reviving the original philosophical perspective of Stein (1967)-which, DiSalle suggests, has not been fully appreciated by the subsequent literature. 331. See also the earlier discussion in DiSalle (2002a), and compare Rynasiewicz (1995). 332. This kind of "dialectical engagement" is Hegelian, in so far as it involves what Hegelians call "immanent critique" of a preceding conception or practice. Thus DiSalle's approach, in this respect, emphasizes the elements Richardson finds missing (or under-emphasized) in my own-and, in particular, the idea that the primary motive force of revolutionary change is supplied by inconsistencies or incoherences at a given stage of knowledge which are then overcome by "specific negation" at a later stage: see pp. 286-89 of Richardson's contribution. However, while Richardson takes such an approach to entail the ultimate disappearance of the
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transcendental (compare note 323 above), DiSalle understands such dialectical engagement as paradigmatic of transcendental argumentation: see pp. 528-29 of his contribution to the present volume, and compare DiSalle (2006, 155-56) for an important qualification of the sense in which what he calls dialectical engagement is explicitly Hegelian. 333. Newton's conceptual analysis, according to DiSalle, is not fully successful, even on its own terms, because the Laws of Motion only give empirical meaning to the concepts of absolute rotation and acceleration, not the concepts of absolute velocity or rest. It is precisely this defect that is remedied in the twentiethcentury conception of neo- Newtonian or Galilean space-time (compare note 245 above). Indeed, as DiSalle points out, it was already remedied by the concept of an inertial frame of reference developed in the late nineteenth century, and Newton himself comes very close to anticipating this solution in Corollary V to the Laws of Motion. Finally, as we shall see below, there is an analogous-but more subtledefect in the Newtonian concept of absolute (rectilinear) acceleration as well, as revealed by Corollary VI. 334. See note 47 above, together with the paragraph to which it is appended, for Kant's gloss on Newton's argument, and see the references cited in note 15 for my interpretation of this Kantian gloss. A crucial case, for example, is that of the mutual (equal and opposite) momentum exchange between Jupiter and Saturn, which must take place entirely independently of any intermediate matter (such as an interplanetary aether) existing around and between them. 335. See DiSalle (2006, 41-42): "Compared with the mechanical philosophy, [Newton's] program is inherently more modest in its presuppositions-necessarily so, since its presuppositions are less restrictive than those of the mechanists concerning what form physical interaction may take. For in the Newtonian view, any interaction is physically intelligible as long as, and just to the extent that, it conforms to the laws of motion." 336. Thus, for example, in the Scholium to Section 11 of Book I of the Principia, after discussing the three-body problem at some length, Newton writes (p. 588; my emphasis): "I use the word 'attraction' here in a general sense for any endeavor whatever of bodies to approach one another, whether that endeavor occurs as a result of the action of the bodies either drawn toward one another or acting on one another by means of spirits emitted or whether it arises from the .. action of the aether or air or of any medium whatsoever-whether corporeal or incorporeal-in any way impelling toward one another the bodies floating therein." As is well known, Roger Cotes objected to Newton on this score in their correspondence and argued that Newton must, after all, assume that gravitational attraction acts immediately at a distance, entirely independently of any possible interactions (and resulting momentum exchanges) with an intervening mechanical aether: see Koyre (1965, chapter 7), and also Stein (1967). (It should be noted, in this connection, that the aether Newton speculatively invokes in Query 21 of the Optics is not mechanical, since its pressure arises from a short-range repulsive force; moreover, in the General Scholium added to the second (1713) edition of the Principia, Newton explicitly argues that the true cause of gravitational attractionwhatever it might be-cannot be mechanical.)
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337. See note 2 above, together with the paragraph to which it is appended and the following paragraph, and compare the paragraph to which note 6 is appended. In the paragraph to which note 8 is appended, the preceding paragraph, and the remainder of that section (up through the paragraph to which note 16 is appended), I explain how Kant transforms both Leibnizean metaphysics and the Newtonian metaphysics of space so as to avoid the theologically objectionable features of (literal) divine omnipresence. 338. Of course Newton's conception of divine agency is highly unorthodox, and, from a more traditional point of view, one would certainly not constrain God's creative activity by the requirement of local presence governing the interactions of material substances. From this point of view, it is Newton who opens up the threat of atheism (or rather pantheism) by seeming to materialize God. However, although Leibniz, for example, thus stands on firmer theological ground than Newton, he does not have a competing metaphysics adequate for natural philosophy and physics. Kant's problem (compare note 337 above) was precisely to construct such a competing metaphysics along broadly Leibnizean lines while simultaneously doing full justice to Newtonian physics. 339. DiSalle (2006, 56-66) illuminatingly describes Kant's radical transformation of the preceding metaphysical tradition (especially the Leibnizean tradition) in light of Newtonian physics. As DiSalle puts it (pp. 57-58): "Rather than being a source of alternative views from outside metaphysics-to be refuted, absorbed, or derived from metaphysical first principles-Newtonian physics is in itself a philosophical critique of metaphysics as traditionally practiced. This general viewpoint was already suggested by Newton himself, as we saw in the previous chapter. But it was Kant who developed and applied it in detail, and who made it the means of transforming philosophy as definitively as Newton had transformed physics." The reference to the previous chapter is to such passages as that quoted in note 335 above; and the difference in emphasis between my approach and DiSalle's is manifested, in this case, by my focus on precisely the metaphysical and theological preoccupations which continued to frame the discussion of absolute space and gravitational attraction at a distance in both Newton and Kant. 340. In particular, Kant emphasizes the importance of light for our perception of the heavenly bodies-and the relationship, in particular, between light and universal gravitation-in the Third Analogy of Experience. Mter appealing (in the second edition of the Critique) to a mutual interaction between the earth and the moon as the ground for our perception of their simultaneity (B257-58), Kant remarks that such dynamical community can be either immediate or mediate (A212-13/B259), and he then gives the following example of a mediate dynamical community (A213/B260): "It is easy to notice in our experience that only the continuous influences in all positions of space can guide our sense from one object to another, that the light, which plays between our eyes and the heavenly bodies, effects a mediate community between us and them and thereby proves their simultaneity, [and finally] that we cannot empirically change our place ([i.e.,] perceive this change) without matter everywhere making it possible for us to perceive our position, and only this [matter], by means of its mutual influence, can verify their [the heavenly bodies'] simultaneity and therefore, all the way to the most distance objects, their coexistence." Thus, whereas the universal gravitational attraction
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between the heavenly bodies (including the earth and the moon, of course) effects an immediate dynamical community between them, it is the mediate community effected by light propagated through the aether that puts us into perceptual contact with the heavenly bodies in the first place. Once again, however, I in no way wish to imply that Kant somehow "anticipated" Einstein's critical analysis of simultaneity (compare note 306 above), for Kant, like everyone else at the time, simply assumed that the classical velocity addition law would also apply to light. (Kant takes light to be a form of motion through an aether, and he derives a version of the classical velocity addition law for motion in general in the Phoronomy of the Metaphysical Foundations: see note 51 above, together with the paragraph to which it is appended.) 341. See again note 186 above, together with the paragraph to which it is appended and the following paragraph, and compare note 208, together with the paragraph to which it is appended. As I discuss in note 208 in connection with John Norton's contribution, Einstein later tended to name Bume and Mach, rather than Poincare, as central sources of philosophical inspiration for his realization that the classical notion of absolute simultaneity is "optional." I suggest, however, that it was Poincare's relatively conservative stance towards the basic concepts of classical mechanics (in contrast to the more skeptical and iconoclastic attitudes of Bume and Mach) which was responsible for this. In any case, it was Poincare, rather than either Bume or Mach, who, despite his conservatism, was intimately engaged with the very same scientific and philosophical situation faced by Einstein himself. 342. Compare note 269 above, together with the paragraph to which it is appended, for the concept of local inertial frame. Note that this solution can be implemented equally well in the context ofNewtonian gravitation theory (where the problem in fact originally arises), and the result is the variably curved reformulation of that theory first articulated in Cartan (1923-24; compare note 194 above), in which freely falling trajectories in a Newtonian space-time define a new class of (non-flat) affine geodesics. DiSalle (2006, §4.6) is perfectly clear about this-and he suggests, in particular, that the new situation in electrodynamics addressed by special relativity gave this solution, at the time, a particular force and urgency (resulting, in this case, in the variably curved space-time of general relativity). 343. DiSalle does not claim that the argument he sketches in the case of general relativity was explicitly made by Einstein; rather, he makes a strong case that it can be seen as implicit in Einstein's philosophical motivations by explaining (2006, §4.6) how the theory was understood in a manner quite close to this very soon after its first publication-namely, by Weyl (1918) and Eddington (1918; 1920). 344. See the discussion in DiSalie (2006, §§4.4-4.5), and compare the earlier treatment in DiSalle (2002b, 186-88). In particular, DiSalle's own analysis of the principle of equivalence leads to a definite construction (2006, §4.5) of variably curved four-dimensional affine geometry in terms of "geodesic deviation." 345. Einstein, in the period 1907-12, was, in effect, using the non-vanishing components of the affine connection in certain non-inertial coordinate systems to represent the gravitational field; intrinsic space-time curvature, however, is indicated by the non-vanishing of these components in every coordinate system (see note 176 above; and compare the paragraph to which note 204 is appended).
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346. See again notes 197 and 201 above, together with the paragraphs to \vhich they are appended and the intervening paragraphs. What I argue there is that, if we are only considering the electrodynamics of moving bodies, then there is really nothing to choose, from a physical point of view, between Einstein's (relativistic) three-plus-one-dimensional formulation, Minkowski's four-dimensional fornlulation, and Poincare's (semi-classical) three-plus-one-dimensional formulation. It is only when we attempt to incorporate gravitation into this framework that the true superiority of the four-dimensional forn1ulation then emerges. 347. Compare DiSalle (2006, 115): "By recognizing the relativity of time, Einstein had raised the spatia-temporal displacements, the Lorentz transfonnations, to a privileged status that the spatial displacements could no longer legitimately claim-because, unlike the Galilean transformations, the Lorentz transformations implied that spatial geometry was no longer a fundamental invariant." 348. Thus, even though the three-dimensional geometry in any inertial frame remains Euclidean, the Lorentz contraction precludes the possibility of ison1etrically moving putatively rigid bodies. And it is this fact, as Einstein showed, which then leads to anon-Euclidean geometry in certain non-inertial fran1es-where inertial forces, according to the principle of equivalence, are equivalent to a gravitational field. Of course this is not quite the same thing as a true gravitational field according to the finished theory (compare note 345 above); but, in my view, we need to go through an intermediate stage that is not yet fully coherent from the later point of view in order to explain how the finished theory first caIne to be recognized as a genuinely live physical possibility. TIns is also why, on my view, it is wrong to dismiss examples like the uniforrrlly rotating frame as merely heuristic and, more generally, to maintain a sharp distinction, accordingly, between the context of discovery and the context of justification or warrant. For such apparently lllerely heuristic examples often play an essential role (as in this case) in making it reasonable or rational (and in this sense justifiable) to expand the space of intellectual possibilities in the first place. 349. See the paragraph to which note 202 above is appended, the paragraph to which note 211 is appended, and the intervening paragraphs, for n1Y reading of "Geometry and Experience" in connection with the history of general relativity; and compare also note 315. In terms of the point made at the end of note 348 above, it follows that a blurring of the distinction between context of discovery and context of justification is a necessary consequence, on my view, of a properapprcciation of ICuhn's conception of scientific revolutions: what are norn1ally called heuristic elements may be crucial in justifying the initial expansion of the space of representational possibilities, and what are normally called justificatory elements then single out (in the process of empirical testing) which of these new representational possibilities are actually realized. 350. See again note 306 above, together with the paragraph to which it is appended and the preceding paragraph, and compare note 324, together with the paragraph to which it is appended. Thus it is crucial, on this account, that Einstein appropriated both philosophical and scientific elements from Poincare. For Poincare was not only very actively engaged with the same scientific problems as was Einstein (the concept of an inertial frame, the electrodynanlics of moving bodies' the role of electromagnetic signaling in the determination of simultaneity, and
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the consequent need to revise the theory of universal gravitation), he was also very actively engaged with the parallel philosophical problems intertwined with these developments (the relativity of motion, the empirical foundations of physical geometry, and, above all, the need for a fundamentally new kind of physical principle arrived at by "elevating" an empirical law to the status of a "convention," "presupposition," or "postulate"). It is true that Poincare, in contrast to Einstein, took a basically conservative stance relative to the classical spatio-temporal framework, but the perceptive and innovative way in which he crafted this stance-mathematically, physically, and philosophically-then made it possible for Einstein himself to move, on this basis, in a more radical direction. It is in this sense, in the terminology of Mattingly's very illuminating contribution to the present volume, that I want to speak of a "Poincarean" rather than a "Lorentzian" strategy here: see note 249 above. 351. See again pp. 528-29 of DiSalle's contribution to the present volume (and compare note 332 above). DiSalle first writes (p. 528): "Indeed, such an argument deserves to be called a transcendental argument. If a certain way of defining a concept is shown to be a condition of the possibility of employing that concept at all, and thereby a condition of the possibility of the scientific reasoning on which the concept depends, then it can hardly be seen as an arbitrary coordination. Nor, therefore, can the argument for it be seen as an a priori appeal to convenience or simplicity of the framework that the definition constitutes. The argument, rather, reveals the new conception in its transcendental role, as uniquely making possible the synthesis of experience under formal scientific principles." He then adds (p. 529): "The transcendental necessity revealed in these arguments is of an inescapably relative kind: the conditions of possibility for theoretical understanding evolve with our theoretical knowledge, and so make different demands at different moments in the history of physics. And this is why the a priori principles are revisable. It is not because we are free to adopt whichever ones we wish, but, arguably, precisely because we are not free; a certain a priori frameworks cannot be maintained in the face of arbitrary new evidence, and so at certain points in the history of physics the transcendental principles must be revisited, and a new transcendental argument is required." 352. For DiSalle's rejection of logical empiricism on this point see especially (2006, §5.2), which begins (p. 158): "This book suggests an alternative, not only to the logical positivists' account of the history of physics, but also to their view of the nature of theories. It suggests that the picture of scientific theories as uninterpreted formal systems, linked to experience by arbitrary stipulations, involves some deep misunderstandings about the nature of interpretive principles-and, therefore, of the nature and evolution of scientific theories." For my rejection of the logical empiricists' notion (prominent in Schlick and Reichenbach) of a "coordination" between abstract mathematical structure and concrete sensible experience see note 252 above, together with the paragraph to which it is appended and the two preceding paragraphs (compare also notes 253 and 268). As I point out in note 216, although Reichenbach had not originally associated this notion of "coordination" with the idea of an "arbitrary convention" (derived from Poincare), he later, under Schick's influence, acquiesced in this idea. Beginning with the paragraph to which note 283 is appended, together with the four following paragraphs,
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I then develop my alternative account of both the uniqueness of the constitutive principles in question and their (relativized) transcendental necessity. It should also be observed, finally, that Poincare's conception of a "convention or definition in disguise" did not involve the idea of an arbitl'ary coordination in the sense of the logical empiricists: our choice of Euclidean geometry, for Poincare, is in fact essentially unique, given both its mathematical simplicity and its specific relationship with our rough and approximate pre-scientific experience of bodily displacements (see note 298, together with the paragraph to which it is appended). 353. As Michael Dickson has emphasized to me, the question of the uniqueness-and therefore necessity--ofthe (relativized) transcendental principles in question is actually rather delicate. Have I not claimed, for example, that Poincare was well within his rights to resist the new geometrical-mechanical framework of special relativity and to continue to maintain the classical Newtonian framework (see note 299 above)? From my point of view, therefore, DiSalle may be expressing the relevant kind of transcendental necessity a bit too strongly in such passages as those quoted in note 351 above. In particular, in the case of special relativity, in 1905, I would claim only that Einstein's new framework is the unique non-classical alternative (unique minimal extension of the classical framework) appropriate to the intellectual situation in which he found himself. Nevertheless, I would agree with DiSalle that, by the time we get to the creation of general relativity in the years 1912-15, this new framework is the only remaining live alternative (compare again note 346 above). 354. In articulating this stronger point I am especially indebted to conversations with Graciela De Pierris. 355. See, in particular, p. 549: "A compelling argument for an interpretation of quantum mechanics, then, would do more than show that certain principles are a plausible foundation from which the theory may be derived. It would also show that those principles, whatever they may be, play an indispensable constitutive role in our understanding of fundamental concepts." DiSalle does not elaborate further, but the uncertainty principle presumably involves a radically new understanding of causality because of the duality (complementarity, incompatibility) between the precise changes of momentum and precise spatial trajectory of a physical particle, while quantum entanglement involves a radically new understanding of community due to the presence of something like instantaneous interaction at a distance in the context of a relativistic understanding of simultaneity. 356. It is becoming increasingly clear that the deepest problems facing the foundations of quantum mechanics derive from precisely the difficulties in unifying it consistently with relativity theory: entanglement and non-locality in the context of special relativity (compare again note 355 above), the so far unresolved problem of quantum gravity in the context of general relativity (compare Mattingly's contribution to the present volume). It is therefore worth reminding ourselves that both relativity and quantum theory originated from deep and unexpected problems encountered in connecting the behavior oflight and other forms of electromagnetic radiation with the behavior of matter: problems encountered in late nineteenth and early twentieth-century electrodynamics of moving bodies, on the one side, and problems encountered in using electromagnetic radiation to probe the internal structure of the atom, on the other. It is also worth reminding
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ourselves that the indispensable background to (special) relativity is Lorentz's theory of the electron, which, among other things, explains interactions between electromagnetic radiation and matter in terms of resonant frequencies of electrons bound in an atom resulting from (as well as contributing to) such interactions, and that it is this theory, as well, which then initiates the properly quantum mechanical investigation of atomic structure beginning with Bohr (compare note 326 above). 357. Tanona's work, in particular, is especially relevant from our present point of view, for it is very much focused on the role of first the correspondence principle and then the principle of complementarity in Bohr's conception of the empirical meanings of the fundamental concepts of quantum theory: see Tanona (2002), (2004), and his contribution to the present volume. See Tanona (2002), (2004), especially, for detailed discussion of Bohr's initial investigations of atomic strucuire mentioned in note 356 above. 358. The original paper is Birkhoff and von Neumann (1936). Von Neumann proposed quantum logic as an alternative interpretation of (Bohrian) complementarity in a public discussion with Bohr in 1938: see his comments in Bohr (1939, 30-39). Later treatments of the quantum logical interpretation include Finkelstein (1962-63), Putnam (1969), Bub and Demopoulos (1974), Bub (1974), Demopoulos (1976), (1977), Friedman and Putnam (1978). 359. See Bub (1997) for a comprehensive discussion and references to the literature. Modal interpretations (and their relations to logic) have been an important focus of Michael Dickson's work: see, e.g., Dickson (1996), Dickson and Clifton (1998), Bacciagaluppi and Dickson (1999), Dickson (2000). Dickson's contribution to the present volume provides a more general illustration of the significance of algebraic structure for the understanding of quantum theory. 360. See note 254 above, together with the paragraph to which it is appended and the preceding paragraph. Ryckman (2005,171-76) discusses Weyt's extension and briefly comments on the prospects for such a phenomenological understanding of quantum field theory. Of course, if such a program were to succeed, then my own hesitations about Weyt's "purely infinitesimal" approach (see notes 255, 289, and 305 above) would have to be reconsidered. 361. See DiSalle (2006, 162): "If philosophers and physicists are to make philosophical sense of such a [quantum mechanical] structure, surely they will require a clear understanding-dearer, at any rate, than twentieth-century philosophy of science was able to achieve-of what the role of space-time structure really was, and how it functioned as a framework for other physical objects, interactions, and processes."
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Awodey, S. (2007). "Carnap's Quest for Analyticity: The Studies in Semantics." In Friedman and Creath, eds. (2007), 226-47. Awodey, S. and A.W. Carus (2001). "Carnap, Completeness, and Categoricity: The Gabelbal'keitssatz of 1928." Erkenntnis 54: 145-72. Bacciagalupi, G. and M. Dickson (1999). "Dynamics for Modal Interpretations." Foundations of Physics 29: 1165-1201. Beiser, F. (2002). German Idealism: The Struggle against Subjectivism, 1781-1801. Cambridge, MA: Harvard University Press. - - . (2003). The Romantic Imperative: The Concept of Early Gel'man Romanticism. Cambridge, MA: Harvard University Press. - - - . (2006). "Kant and Naturphilosophie." In Friedman and Nordmann, eds. (2006),7-26. Beller, M. (1992). "The Birth of Bohr's Complementarity: The Context and the Dialogues." Studies in History and Philosophy of Science 23: 147-80. Bertoloni Meli, N. (1993). Equivalence and Priority: Newton versus Leibniz. Oxford: Clarendon Press. - - - . (2006). Thinking with Objects: The Transformation of Mechanics in the Seventeenth Century. Baltimore: John Hopkins University Press. Beth, E. (1956-57). "Uber Lockes 'Allgemeines Dreieck'." Kant-Studien 49: 361-80. Birkhoff, G. and J. von Neumann (1936). "The Logic of Quantum Mechanics." Annals ofMathematics 37: 823-43. Bohr, N. (1939). "The Causality Problem in Atomic Physics" (with following discussion). In New Theories in Physics. Paris: International Institute ofIntellectual Co-operation. Bricker, P. and R.I.G. Hughes, eds. (1990). Philosophical Perspectives on Newtonian Science. Cambridge, MA: MIT Press. Bridgman, P.w. (1927). The Logic of Modern Physics. New York: Macmillan. Bub, J. (1974). The Interpretation ofQuantum Mechanics. Dordrecht: Reidel. - - . (1997). Interpreting the Quantum World. Cambridge: Cambridge University Press. Bub, J. and W. Demopoulos (1974). "The Interpretation of Quantum Mechanics." Boston Studies in the Philosophy of Science. Vol. XIII. Dordrecht: Reidel,92-122. Cahan, D., ed. (1992). Hermann von Helmholtz: Philosopher and Scientist. Berkeley and Los Angeles: University of California Press. - - - . (1993). Hermann von Helmholtz and the Foundations of NineteenthCentury Science. Berkeley and Los Angeles: University of California Press. Carl, W. (1989). Der schweigende Kant: Die Entwurft zu einer Deduktion der Kategorien vor 1781. Gottingen: Vandenhoeck & Ruprecht. Carnap, R. (1922). Der Raum: Ein Beitrag zur Wissenschaftslehre. Berlin: Reuther and Reichard. - - - . (1928). Der logische Aufbau der Welt. Berlin: Weltkreis. Translated by R. George (1967) as The Logical Structure ofthe World. Berkeley and Los Angeles: University of California Press. - - - . (1934). Logische Syntax der Sprache. Wien: Springer. Translated by A. Smeaton (1937) as The Logical Syntax ofLanguage. London: Kegan Paul.
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Biemel. The Hague: Martinus Nijhoff. Translated by D. Carr (1970) as The C1'isis of EU1'opean Sciences and Transcendental Phenomenology. An Introduction to Phenomenological Philosophy. Evanston: Northwestern University Press. Janiak, A., ed. (2004). Isaac Newton: Philosophical Wl'itings. Cambridge: Cambridge University Press. Kahl, R., ed. (1971). Selected Wl'itings ofHeI'mann von Helmholtz. Middleton, CT: Wesleyan University Press. Kant. 1. (1902-) Kantsgessammelte Schnften in 29 volumes. Berlin: Walter de Gruyter and predecessors. Klein, F. (1909). Elemental'mathematik vom hiihel'en Standpunlet aus. Teil II: Geometl'ie. Leipzig: Teubner. Translated by E.R. Hedrick and C.A. Noble (1939) as Elemental'Y Geometry from an Advanced Standpoint. Part II: Geometry. New York: Dover. Kohnke, K. (1991). The Rise of Neo-Kantianism: German Academic Philosophy between Idealism and Positivism. R. Hollingdale trans. Cambridge: Cambridge University Press. Koyre, A. (1965). Newtonian Studies. Cambridge, MA: Harvard University Press. Krois, J.M. (1987). Cassirer: Symbolic Forms and History. New Haven: Yale University Press. Kuhn, T (1957). The Copernican Revolution: Planetal'Y Astronomy in the Development of Western Thought. Cambridge, MA: Harvard University Press. - - - . (1962). The Structure of Scientific Revolutions. Chicago: University of Chicago Press. Laywine, A. (1993). KantJs Early Metaphysics and the Origins of the Critical Philosophy. Atascadero, Calif.: Ridgeview Press. - - - . (1998). "Problems and Postulates: Kant on Reason and Understanding." Journal of the History ofPhilosophy 36: 279-31l. - - - . (2003). "Kant on Sensibility and the Understanding in the 1770s." Canadian Journal ofPhilosophy 33: 443-82. Lorentz, A., et al. (1913). Das Relativitatsprinzip, eine Sammlung von Abhandlungen. Leipzig: Teubner. First English Translation 1923. Mach, E. (1883). Die Mechanik in ihrel' Entwicklung, historisch-kritisch da'l'lJestellt. Leipzig: Brockhaus. Translated by TJ. McCormak as The Science ofMechanics: A Critical and Historical Account ofits Development. La Salle, IL: Open Court. 1960. - - - . (1886). Beitriige zur Analyse der Empfindungen. Jena: Gustav Fischer. - - . (1890). Die Analyse der Empfindungen und das Verhiiltnis des Physischen zum Psychischen. Jena: Gustav Fisher. - - - . (1906). Space and Geometry. TJ. McCormack, trans., La Salle, IL: Open Court. - - - . (1959). The Analysis of Sensations and the Relation of the Physical to the Psychical. C.M. Williams, trans. (from the 1886 1st ed., revised and enlarged following the 1906 5th ed. by S. Waterlow). New York: Dover. McMullin, E. (1978). Newton on Matter and Activity. Notre Dame: University of Notre Dame Press.
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Menn, S. (1998). Descartes and Augustine. Cambridge: Cambridge University Press. Miller, R. (1981). Albert EinsteinJs Special Theory of Relativity: Eme1'lJence (1905) and Early Interpretation (1905-1911). Reading, MA: Addison-Wesley. Misner, C.W., K.S. Thorne, and J.A Wheeler (1973). Gravitation. San Francisco: W.H. Freeman. Lorentz, A, et al. (1923). The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theories of Relativity. W. Perrett and G.B. Jeffrey, trans. London: Methuen and Company. Mormann, T. (2007). "Geometrical Leitmotifs in Carnap's Early Philosophy." In Friedman and Creath, eds. (2007), 43-64. Mulder, H. and B. van de Veide-ScWick, eds. (1978). Moritz Schlick: Philosophical Papers. Vol. 1. Dordrecht: Reidel. Newman, W. (2006). Atoms and Alchemy: Chymistry and the Experimental Origins ofthe Scientific Revolution. Chicago: University of Chicago Press. Norton, J. (1985). "What Was Einstein's Principle of Equivalence?" Studies in History and Philosophy ofScience 16: 203-46. Pais, A (1982). (Subtle is the Lord . .. J The Science and the Life ofAlbert Einstein. Oxford: Oxford University Press. Parsons, C. (1983). Mathematics in Philosophy: Selected Essays. Ithaca: Cornell University Press. - - - . (1992). "The Transcendental Aesthetic." In P. Guyer, ed. The Cambridge Companion to Kant. Cambridge: Cambridge University Press, 62-100. Patrides, CA, ed. (1980). The Cambridge Platonists. Cambridge: Cambridge University Press. Pickering, A, ed. (1992). Science as Practice and Culture. Chicago: University of Chicago Press. Pickering, A (1995). The Mangle of Practice: Time, Agency, and Science. Chicago: University of Chicago Press. Poincare, H. (1902). La Science et rHypothese. Paris: Flammarion. Translated as Science and Hypothesis in Poincare (1913b). - - - . (1905). La Valeur de la Science. Paris: Flammarion. Translated as The Value of Science in Poincare (1913b). - - - . (1908). Science et Methode. Paris: Flammarion. Translated as Science and Method in Poincare (1913b). - - - . (1913a). Dernieres Pensees. Paris: Flammarion. Translated by J. Buldoc (1963) as Mathematics and Science: Last Essays. New York: Dover. - - - . (1913b). The Foundations ofScience. Trans. & ed. G. Halstead. Lancaster: Science Press. Pojman, P. (2000). Ernst MachJs Biological Theory of Knowledge. Doctoral Dissertation: Indiana University. Putnam, H. (1969). "Is Logic Empirical?" Boston Studies in the Philosophy of Science. Vol. V. Dordrecht: Reidel, 216-41. Quine, w.V. (1951). "Two Dogmas of Empiricism." Philosophical Review 60: 20-43. Reprinted (1953) in From a Logical Point of View. New York: Harper. - - . (1963). "Carnap and Logical Truth." In P. Schilpp, ed. The Philosophy of RudolfCarnap. La Salle, IL: Open Court, 385-406.
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_ . (1969). "Epistemology Naturalized." In Ontological Relativity and Other Essays. New York: Columbia University Press, 69-90. Reck, E. (2004). "From Frege and Russell to Carnap: Logic and Logicism in the 1920s." In S. Awodey and C. IZlein, eds. Carnap Brought Home: The View from Jena. Chicago: Open Court, 151-80. - - - . (2007). "Carnap and Modern Logic." In Friedman and Creath, eds. (2007), 176-99. Reck, E. and S. Awodey, eds. (2004). Frege)s Lectures on Logic. Carnap)s Student Notes, 1910-1914. Chicago: Open Court. Reichenbach, H. (1920). Relativitiitstheorie und Erkenntnis Apriori. Berlin: Springer. Translated by M. Reichenbach (1965) as The Theory ofRelativity and A Priori J(nowledge. Berkeley and Los Angeles: University of California Press. - - - . (1924). Axiomatik der relativistischen Raum-Zeit-Lehre. Braunschweig: Vieweg. Translated by M. Reichenbach (1969) as Axiomatization ofthe Theory ofRelativity. Berkeley and Los Angeles: University of California Press. - - . (1928). Philosophie der Raum-Zeit-Lehre. Berlin: de Gruyter. Translated by M. Reichenbach and J. Freund (1958) as The Philosophy of Space and Time. New York: Dover. Reisch, G. (1991). "Did IZuhn I
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Index
definition of, 419n10 of observables, 420n14-19 and Leibnizian calculus and logic, 578, 736n41, 737n43 and iterative reasoning in lZant, 590 Leibniz vs. Newton, 598, 730n6 and interpretation of quantum theor~ 728, 801n359 Carnap's attitude to, 772n230 analysis/analytic (cf. synthesis) ancient accounts of, 12-14, 19n13, 66 versus synthesis, 14-15, 84n20, 515-16,518 of matter, 46, 51,216, 515-16 in geometry, 66-67, 69-70, 84n21, 574,578, Barrow's attitude to, 78-80 analytic judgments and propositions, 98-99, 101, 108nl1, 147, 149n8, 151n33, 248,282,324,459,466, 479-82,484-85,487-88,498, 524-25,545,598,647, 668-77 mathematical analysis, 146, 474, 642,675, 687, 704, 761n162, 773n230 analysis situs, 157-58, 162, 164, 177-79, 188n27
1929 Davos disputation, 10 alchemy historian's attitude towards, 44-45, 57n3-4, 57n6-7 mixtures in, 48 and Paul of Taranto, 50 Boyle's and Sennert's connection to, 53, 55-56 Thomas Aquinas and, 58n14, 59n15 reversible processes in, 60n25 Newton's admiration for, 84n16 Alexander, Peter, 301 algebra and geometry in Descartes and Newton, 65-67,69-74, 76-79, 83n9, 84n15, 84n20, 85n21, 85n24, 574, 730n5 Newton's lectures on, 84n17 and geometry and arithmetic in lZant, 138-39, 141, 149n9, 151n26 as inspiration for Leibniz, 157, 159 use of signs in, 170-71, 189n44 and magnitudes, 190n48, 732n24 Babylonian understanding of, 270, 276n20 Heisenberg's use of, 407 as used to describe quantization, 411-13
823
824
Index
definitions, 169-70, 173, 178-79, 181 of concepts, 172-75, 178, 181-82, 189n47, 190n48,247, 373 dynamical, 227 of terms, 255 Husserlian essential, Wesenanalyse, 460-62, 467n4, 685-86, 688, 778n255 Russell's account of, 474 Quine on, 498, 508 conceptual, 464, 524, 528-29, 531,542,546,549,599,718, 725, 727, 795n328, 796n333 analytic philosophy, 10,280-81,308, 326,390,460,700-701 ancient mathematics, 66-74, 76-80, 83nl0,84n18-19,86n25,119, 129,270,574,588,602, 739n50 Andersen, Hans Christian, 341 Anderson, James L., 461-62 Apollonius, 66 approximation, 284, 287, 538, 541, 639-41,645,654,663, 680-83,687,690-95,701, 705, 708-709, 712, 715, 722, 741n57, 750nl03, 761n165, 765n192, 775n245, 778n258, 780n265-68, 781n270, 787n298,790n312,800n352 mathematical and physical, 33, 69,232, 336,403n5,606 method of successive, 430-31, 442-44, 467n2, 584-85, 623-24 approximate truth, 456, 464-65 Aquinas, Thomas, 44, 46-52, 55-56, 57nl0,58nll,n14-15, 59n16,60n24 Archimedes, 23-26, 28-29, 33-35, 37-38, 39nll, 66, 69, 72, 77, 83n13,574,794n327 Aristotle, 12-13, 19n13, 57n8-9, 58n12, 77, 86n31, 161, 184, 187n17-18, 271, 276n21, 285,409,528
Aristotelian tradition, 23, 26, 44-49, 51, 53, 55-56,60n25, 496,575,577-79,582,585, 701,717,721, 729n7, 780n265 arithmetic, 126-27, 131n8, 135, 137-42, 144-45, 147-48, 149n5,n8-9,150n18,n20-21, 151n36, 152n39, 167, 170, 175, 187n26, 188n31, 189n44, 190n48,247,318,459,474-75, 493-94,496,590,598,643, 646,668-70,674-75,686,704, 732n24, 738n43, 761n164, 772n225 vs. geometry, 70-71, 77, 84n18, 84n21, 132n22,249, 773n230 ars combinatorial, 159-63, 168, 172, 181, 184-85, 186nl0, 187n12, 188n37 astronomy, 40n32, 113,214,517, 520nl, 641, 650, 690-92, 781n268 Augustine, 86n28, 575 Avenarius, Richard, 305 Awodey, Steve, 292n2, 323, 326nl, n5, 328n18,n22-23, 329n25, 771n217,n221,773n232 axiom, 135, 149n8, 157-58, 164, 166, 173, 176, 230nll, 249, 252,264,359,371,455-58, 473-75,478,514-16,585-86, 588-92,598-99,601,639, 642-43,665-67,669,697-98, 704, 738n44,739n48, 739n49, 761n163, 766n203, 770n215, 771n219, 771n221, 776n248, 777n252, 786n291 in mechanics, 23-26, 28-31, 33, 35-38, 39n31, 150n21,n23, 151n36, 794n327 in geometry, 79-80, 345,458,487, 633-34,660-61,676,684, 702, 733n26, 734n32, 735n35, 767n206, 783n277 in philosophy, 121,247,249,252 in arithmetic, 137-38, 140-42, 147-48, 149n8,473-75,590
Index in logic and logicism, 323, 326n5, 598-99 of physical theory, 349-50, 393, 398,430-31,458,465, 467n16, 674-75, 677, 778n255
Bacon, Francis, 12, 54, 61n28 b~ance,23-29,33-34,37,612-13,
746n82,n84 balancing argument, 193-198, 200, 203-206,212-214,217-218, 220-221, 225-226, 229n8, 230n9,n12,233n23,n25, n27,n29,234n35,237n47, 608-610, 744n71,n73 Baliani, Gianbattista, 32-33, 39n21 Balmer, Johann Jakob, 433-34 Barcel6, Carlos, 392, 399-401 Barrow, Isaac, 66, 77-80, 82, 86n30-33, 87n34, 186n2 Bauch, Bruno, 666, 770n218, n220 Beck, Jakob Sigismund, 148 Becker, Oskar, 281 Beiser, Frederick, 149nl0, 1511134, 256n8-9, 257n23, 621, 75211108,75311115 Bellarmine, Roberto, 46 Beller, Mara, 352n6, 728 Berkeley, George, 81, 305, 637, 732n21 Bernays, Paul, 669 Bernoulli, Johann, 113-14, 116, 131n11 Besso, Michele, 367, 374, 376, 3831116-17, 3841120 Beth, Evert W., 135-136, 144, 1901155, 586, 732n21, 772n225 Birkhoff, Garrett, 728, 802n358 Boas, Marie, 19n14, 44-45, 82n2, 729n2 Bogen, James, 432, 452nl1 Bohm, David, 397 Bohr, Niels, 410, 433, 453n24, 728, 801n356, 802n357-58
825 Boltzma11n, Ludvvig, 559, 563, 665, 759n152 Bolzano, Bernard, 135, 142-148, 149n3, 151n28-29,n31-33, 598, 737n43 Boole, George, 271 Borgia, Cesare, 272 Born, Max, 341, 769n213 Bos, Henk, 68, 74, 83n9, 84n15, 85n21 Boscovitch, Roger Joseph, 556-57, 559, 7411158, 743n69 Bose, Satyendra Nath, 399 Bose-Einstei11 condensate, 399 Boyle, Robert, 23, 37, 43-45, 51, 53-56, 57n3, n6, 61n27-29, n31,717 Brahe, Tycho, 542 Bridgman, Percy Williams, 498, 721 Brouwer, Luitze11 Egbertus, Jan 668-69, 772n226 Biihler, IZarl, 265 Butterfield, Herbert, 44-45 Byrne, Patrick Hugh, 12, 19n13
Callende~ Cr~g, 396 Calzetta, Esteban, 392, 401 Campbell, Norman, 498 Caramuel, Jean, 33 Carnap, Rudolf, 16, 38, 283, 291, 292n3, 295-98, 300,333, 343,464, 473, 475-89, 489nl, n5,n9,493,495-97, 501, 521n2,554,566,621,664, 666-77, 771n217-22, 772n225, 77211229-32, 773n233-34, 774n236 vs. Heidegger and Cassirer, 10, 261-62,268,281-2 Carnap's Aujbau) 279-80, 295-96, 299,301,304,307-308, 311-26,326n2-3,n5, 327n6-13,328n14-18, n21-24,329n26,n28-30, 459-60,668,673, 759n150,
826
Index
771n221, 778n257 Mach's influence on, 295-96, 301-308 Principle of Tolerance, 460, 497, 670-71, 674, 771n223, 772n226 and Quine, 311-13, 315, 321, 326, 337,481,484,498-99,505, 507-508,671-73,675,678, 773n228, 774n238 Carson, Emily, 592, 597 Cartan, Elie, 466, 467n2, 798n342 Cartwright, Nancy, 465, 794n327 Carus, A.W., 323, 326nl, n5, 328n18,n22-23,771n221 Cassirer, Ernst, 9-11, 19n12, 256n7, 264-74, 274n5-7,n9, 275nl0-14, n16-19, 276n22, 279, 284-86,288-91, 292n9, 341, 346, 390-91, 393, 403nl,426, 462, 464, 554, 680-85,696, 715, 751nl06, 775n244,n248-49,776n251, n253, 781n270, 790n310, 792n319,n321-22 vs. Carnap, 10,261-62,268, 281-2 on cultural sciences, 262, 264-70 philosophy of symbolic forms, 264, 267-68,270,272,290,716, 780n264 on relativity theory, 333-34, 343, 394-95,402-403,682,690, 770n216, 782n273 Castelli, Benedetto, 32, 39n21 Cat, Jordi, 38nl, 408 Cato, 58n14, 272 Cauchy, Augustin Louis, 146, 558, 737n43 cause/causality, 13, 15, 86n31, 97-98, 233n26, 253, 272, 291,300,347,408,415,548, 549n3,681, 721-22, 728, 753nl17,801n355 physical/material/motive, 33, 35, 56,198-99,210-11,218,
220-22,224,226-27, 237n49, 238n51, 239n56, 579, 796n336 Humean account of, 81, 360, 375, 377, 381nl, 784n281 Kant's account of, 95-98, 101, 108nll, 230n13-14, 238n50, 524,543-46,582-85,604, 618-19, 622-23, 740n54, 749n98, 749nl00, 749-50nl0l, 755n128 in space-time theories, 450, 455, 463-64 in Helmholtz, 629-32, 635-36, 639, 703-04, 755n128, 755n129-130,757n138, 758n146, 758n147, 786n287, 786n290 certainty, 31, 81, 137, 189n45, 505, 564,689, 759n149, 774n238 mathematical, 67, 119, 124, 132n15,588 scientifi~74-78,301-304,296,
306, 532, 794n327 philosophical vs. mathematical, 155-56, 169-78, 180-83, 185, 189n39, 189n47, 524, 779n261-62 c~cle, 30, 68, 70-71, 73, 83nl0, 84n21,85n23-24,118-123, 125-126, 128, 131n12, 132n22, 157-158, 167, 188n34, 189n44, 229n8,231n20,233n25,270, 514,577,589-90,593,597, 599-600,662,685,719, 734n32, 736n39, 737n42, 738n45, 753nl17, 754n122, 763n175 Ciufolini, Ignazio, 350 clarity, 67-69, 75-76, 81, 575, 783n280 Clarke, Samuel, 96-97, 107n8, 574, 579-80, 598 Coffa, Jose Alberto, 426, 776n246 Cohen, Hermann, 108nl0, 261, 281, 341, 621, 678-79, 721nl07, 775n243,n244,793n321
Index Coimbrans, the, 46, 48-49, 58n12, n15 combination, see ars combinatoria conditions of experience, 5, 194, 196, 228, 234n35, 286, 529,596, 618,622, 696, 760n157 conductor and magnet experiment, 362-64, 366, 382n4, 533,653 congruency, 79-80, 87n37, 157, 161, 164, 186n4, 188n27, 739n50 constimtive a priori/principles, 267, 282-89, 291, 292n8-9, nIl, 335,343, 349, 351n4,423, 425,432,435,444,451, 455-57,461-64,466,494-99, 501-502,504,508,512-14, 516-20, 521n7, 524-25, 528, 537,573,622-24,627-28,635, 683,693,695-98,706-708, 711-17,723, 726-27, 738n48, 751nl09, 752nl14, 769n216, 774n242, 775n248, 776n252, 782n275, 783n280, 786n290, n293, 789n305, 790n312, 791n314,792n321-22, 793n323-24,n326,794n327, 795n329,800n352,n355 relativized, see relativized a priori contingent a priori, 333-38, 340, 342-43, 348-51, 351nl-2, 518, 789n305 convention/conventionalism, 32, 291,337,338,456,463-64, 512,527,542,545,548,559, 727, 771n221, 776n248 Poincare's conventionalism,S, 383n17, 467n3,487, 642-43, 645-46,650,654-55,661-62, 664, 666-67, 705-10, 763n179, 765n193, 766n196, 767n207, 770n216 Carnap's conventionalism, 318-19, 324, 327n13, 333,479,497 Einstein's use of, 341-42, 344-45, 347-48, 382n8-9, 531-32, 723, 764n186, 764n187, 768n208, 768n209,788n299,800n350
827 Reichenbach on, 351n4, 529, 800n352 convention vs. fact, 481, 484, coordination/coordinative definitions, 432-34,438-42,445-51, 451n3-4, 455, 457, 460-64, 466, 467n5,498, 507, 525-28,541-42,545,684, 698, 727, 776n250-51, 781n268, 782n270, 783n279, 800n351 Carnap on, 337, 13, 338n16, 771n221 Reichenbachian/Schlickean, 343, 351n4,430-31,443-44, 777n52-53, 787n291, 800n352 Einstein on, 344-345, 347, 349, 531,535 Friedman on, 424-27, 429, 456, 462, 697, 783n278 Copenhagen Interpretation, 397, 446, 453n24 Copernicus, Nicolaus, 499, 502, 542, 574-75, 729nl, 780n265 corpuscles, 43-46, 51-56, 57n3-4, 208,215,607-608, 742n63 correspondence, 30, 160-61, 301, 325,340,344-46,400,424-25, 438-39,445,447,449,477, 479, 489n7, 498, 547, 586,612, 634-35, 653, 661, 684, 738n47, 758n145, 768n209, 776n250-51, 777n252 Craig, William Lane, 483-84, 486 Crusius, Christian August, 180, 190n56 Cudworth, Ralph, 19n16, 77, 86n29 Cuomo, Serafina,74, 86n25 Cushing, James T., 408, 521nl Dalton, John, 518-19, 794n326 Darwin, Charles, 16, 502, 518, 755n130, 761n161 Daumer, Martin, 446 da Vinci, Leonardo, 272
828
Index
Dear, Peter, 25-26, 39n2, n7, n20, 45, 57n7, 85n22, 87n34 de Broglie, Louis, 397, 407 de Broglie-Bohm Theory, 397 Dedekind, Richard, 467n5, 474, 680 definition, 30-31, 114-18, 121, 126, 132n16, 140-41, 146, 151n33, 161-66, 168-85, 187n18, 188n29,188n35,361,367-68, 370-71,374,376-77,379,445, 473-75,478,518,535-36, 538-43,640-43, 727, 734n27, 744n75,799n351 in geometry, 80, 126-128, 131n7-10, 137, 155-58, 179 in science, 95-96 in philosophy, 155-56, 171, 185, 247-50,252, 255, 739n50 in Carnap's Aujbau) 311-17, 319-26, 326n3, 327nll, 328n17,329n29 implicit, 312, 326n3, 458, 661, 684, 766n203, 769n215, 771n219,771n221, 787n297 operational, 325, 427, 429, 539-40, 722 coordinating/coordinative, 337-38, 344-45, 351n4,429-30,443, 457-60,462-64,498,525-28, 531, 535, 545, 771n221, 776n252 conventional status of, 351n4, 382n8-9, 383n17, 531-32,645, 654-55, 707, 709, 723, 763n179,768n208,800n352 demonstration, 66, 78-79, 86n31, 107n5, 157-59, 162, 164-68, 172, 179-81, 183-85, 188n35, 247-48,266,495,588,594, 647,652, 732n20, 759n149, 762n171,764n185, 779n261 Demopoulos, William, 482, 489nl, n5-6, nl0, 490n12, 773n233, 802n358 de Omerique, Hugo, 66
De Pierris, Graciela, 494, 736n38, 767n208, 772n227, 783n280, 801n354 Descartes, Rene, 4, 8, 14, 65-77, 79-81, 82n2-3, 84n19, 86n26-28,243,494,500, 528-29,574-78,580,585,645, 689, 729nl, 730n7, 730n8, 732n16, 758n259, 779n259, 783n280, 795n330 on space and time, 13, 65 physics, laws of motion, 24-25, 33-34,36,65,208,214,558, 719 on mathematics and geometry, 66-74, 83n6,83n8-13, 84n14-15,84n21,85n23-24, 730n5, 736n40 Deschamps, Theodore, 33 determinism, 321, 388, 548 Dickson, Michael, 453n28, 728, 801n353,802n359 Dijksterhuis, E.J., 39n8-10, 44-45, 57n3 Dilthey, Wilhelm, 262-64 Dirac, Paul, 405-407, 409-15, 418, 418n2-5, 419n7, n9-10, n12-13, 420n14, n18, n20 DiSalle, Robert, 538, 542, 546n2, 641, 717-19, 721-28, 760n156, n158, 763n174, 766n194-95, 795n328-29, 795n330-32, 796n333,n335,797n339, 798n342-44, 799n347, 800n351-52,801n353,n355, 802n361 distinctness, 67-69, 75-76, 81, 155, 165-66,172,176,182,273, 575, 783n280 Domski, Mary, 85n22-23, 186n2, 574-75, 729n2, 730n5, 736n40, 785n285, 794n326 Drake, Stilln1an, 30, 39n4 Duhem, Pierre, 39n8, n27, 311, 313, 324,337-41,349,427-28,432, 442,465,513,558-59,671 Durr, Detlef, 446
Index dynamic/dynamical, 97, 104, 106, 107n7,234n33-34,408, 414-15, 424, 434, 453n21, 461-64, 529, 537, 549n3, 577, 579, 624-27, 630, 744n73, 753n115,753n116 Newtonia11, 4-6, 56, 92, 94-95, 106, 333-34, 748n94 in IZant, 109n18, 193-194, 198, 203,207-212,215-228, 230nI3,233n28,235n39, 236n41-42,237n46,n49-50, 238n51,239n55-56,240n57, n61, 602-10,613, 616, 620, 623, 739n48, 742n60, 742n62, 744n70, 744n73, 7451180, 747n91, 755n129, 779n261, 786n293, 797n340 dynamical space-time, 388-89, 394, 463 dynamics of reason, 16,279,281-83, 287,289-91,333,495,500, 504,511,520,553-54,573, 696, 708, 711-12, 714-15, 718, 792n318, 794n326 Earman, John, 364, 388, 453n21, 766n194 Eberhard, Johann Augustus, 149n5, 150n20,244, 592, 736n38 Eddington, Arthur Stanley, 464, 684, 798n343 Ehlers, Jiirgen, 349, 778n255 Einstein, Albert, 9, 289, 333-36, 361-68, 381nl, 382n4, 383n16, 384n18, 384n20, 387, 390-93, 395, 397, 399, 403n3, 405, 415-16,456,458,463-64,487, 496,502,506-507,512,519, 523-27, 529-35, 537-44, 546-48,565,573,621,640, 642,648,652-59,674-76,682, 684,692-93,696-99,705-18, 721-28, 763nI82,nI84-85, 764nI88,nI91,765nI92, nI96-97,766n201,n203-206,
829
767n209,768n210-11, 769n213,770n214,n216, 773n230,n232,776n249, 781n267, 781n269, 783n275, 783n277-78, 787n292, 787n296, 788n97, 789n2, 789n305-306,790n311, 791n314, 792nI5-16, 792n319, 793n323, 794n324, 797n340, 798n341,n343,11345, 799n346-48 philosophical views and influences, 337-51, 351n2, 352n6, 352n8, 359-61, 369-81, 527, 530-31, 642,652,660-64,707,710, 767n208, 770n214, 776n246, 788n301, 789n304, 799n350 on conventionality, esp. of simultaneity, 382n6, 382n8-9, 707, 723, 764n186, 765n193, 788n299 and Poincare, 383n17, 503, 654-55,658,660-64,707, 710-11, 723, 764nI86-87, 765n193, 767n208, 788n299, 789n301, 789n304 on quantum theory, 546 electrodynamics, 401, 408, 410, 543, 548,790n306,801n356 and relativity theory, 361-67, 369, 371, 374, 376-79, 391-92, 399, 529-30, 532-34, 539, 541, 546, 653-60, 705, 707, 711, 716, 722, 787n296, 787n298, 798n342, 799n346 and Poincare, 642, 648, 650-652, 726, 764n185, 765n191, 765n193, 766n196, 769n210, 799n350 demen~232n21,234n31, 399, 514-15,518-20,606,615 Aristotelian, 46-49, 51-53, 58nll-12,59nI5,nI8-19, 60n21,n23-24,181 Elsbach, Alfred, 345-46, 350 Empedocles, 46
830
Index
Engel, Friedrich, 115, 117, 121, 130n5,131n6,n9,nll,n15 engineering, 24, 32,498,671, 794n327 English, Jane, 483-84 E6tv6s, Lorand von, 709, 717, 789n302 Epicurus, 52, 209, 720 Eppley, Kenneth, 397-98 Erdmann, Johann Eduard, 114 essence/essential, 7, 15,43, 54, 80, 91-93, 101, 135, 144, 177, 196-97,199,201-202,204, 207-208,211-13,215,220, 228, 229n8, 230n14, 233n25, 234n31, 238n51, 254, 314, 394, 461,577,608,616,633-34, 777n254, 778n255, 778n256, 784n282 vs. existence, 248, 250 essential analysis, see analysis, Husserlian essential, Wesenanalyse essential idealization, 561-65, 567 divine, 576 Husserlian essential intuition, 667, 686 ether, 230, 362-65, 382n6, n9, 395, 401,435-36,530,534-35,538, 540,626-27,651-52,654,658, 720, 742n64, 743n69, 765n193, 796n334,n336,797n340 Euclid, 5-6,9, 13,23, 39n3,66, 77, 79-80, 85n24, 87n37-38, 114-26,128, 131n9,n13, 137, 139, 141, 147, 155, 157-58, 162-64,166,173,175,179-80, 183, 187n24, 189n44, 190n55, 327n13, 336, 343, 346,411, 423,425,437,440,443, 457-58,495-96,503,542,566, 583,586-94,597,599-601, 633-34,636,639,644-47,656, 661-63,665,667,679-83,690, 692-93,696,699,701-705, 707-709, 711-12, 724-27, 732n18,n22, n24, 733n26,
735n35,736n39,n40-41, 737n42, 738n45, 757n142, 758n148,762n166-67,n171, 766n198, 767n205, 768n209, 771n219, 775n245, 777n254, 778n255, 781n269, 786n289, 788n298, 789n303-305, 790n306, 793n324, 799n348, 800n352 Eudoxus of Cnidus, 175, 732n24, Euler, Leonard, 93, 107n3, 115,557, 560, 606-608, 742n63-65, 744n72 Everett, Hugh III, 397 experiment, 23-25, 26, 34-38, 45, 51-55,318,338-40,344,346, 378, 414, 420n20, 427, 429, 431-33,448,514,516,556, 620,623,628-29,632,634, 646-47,649-54,656-58,664, 674,676,705,707,717, 752nlll, 754n120, 757n138, 762n171, 763n179, 764n185, 765n190-91, 765n193, 794n327 Newton's bucket, 14, 373, 380 Galileo's, 29-32, 39n14 Newton's, 36-37, 40n32, 707 thought, 14,266,362-65,366, 382n4,396-98,646 Fizeau's, 378, 393-94 Michelson-Morley, 362, 393-94, 433, 435, 439-41, 452n15, 530-31,533,538-40,651,653, 717, 722-23, 781n268, 789n302 in quantum gravity, 396-97, 400-01 psycho-physical, 636, 638, 761n165 explanation, 14-15, 56, 61n31, 71, 94,99,207,213,215-17,220, 225-28, 234n34-35, 238n51, 253,281-82,349,408,415, 418n2, 429, 432, 452nl0-ll, 474-75, 515-17, 520nl,521n5, 524,529,548,644,651-52, 659,668,678,695,698,704,
Index 716, 719-20, 731nll, 750nl02 mechanical vs. dynamical, 209-11, 215, 229n8, 234n33 vs. understanding, 263 of the possibility, success, or failure of science, 283, 333, 512,673, 689-90, 699-701, 780n263, 783n273 mechanical vs. theoretical, 408-409, 415, Fabri, Honore, 33, 58n13 Fechner, Gustav, 303, 636, 638, 759n151,761n161,n165 Feyerabend, Paul, 1, 504, 512 Fichte, Johann Gottlieb, 244, 252, 621 Fizeau, Hippolyte, 378, 393-94 force, 5, 30, 37, 79, 272, 287-88, 336,338,344,349,380,388, 392,413,417,424,433,448, 463,524,530,537-38,542, 546, 556-57, 559-64, 568nl0, 577-79,581,583,602,623, 625-27,629-31,636,645, 647-48,659,661-62,664,709, 719-20, 739n47, 740n51, 741n59, 743n69, 748n93, 753nll5, 755n129, 755n130, 756n134, 765n193, 796n336, 799n348 in Newton's physics, general, 9, 36, 91-106, 107n5,n7, 109n15, n17-l8,423,514,534,549n3, 691, 746n82 gravitational, 13, 65, 87n35, 190n50, 336, 387,406, 535,658-59,720,724, 740n52, 746n87, 747n88, 766n200 I
831
237n44,n48-49,238n50-51, 239n53,n56,240n57,n61, 603-19, 740n52, 740n54, 742n61, 743n70, 744n73-74, 745n76-77, 746n81, 746n85, 749n99, 749nl0l, 750nl02, 752nll0,752nll2 elect:romagnetic, 340, 406, 427, 533-535, 656, 658, 754n120, 756n31, 786n286 form, formal, 145, 246-48, 254, 556, 626,645,667-68,671,678-80, 688,693, 695-96, 753nl16, 754n118,785n284 of bodies or substances, 44, 47-50, 52-53, 56, 58nll, 59n15, 60n21,n24,61n26, 535, 579 of judgment, 99, 108nll symbolic, in Cassirer, 264-72, 290, 716, 780n264, 275nll, 275n18, 276n19,290-91,780n264, 782n270 system, in Carnap, 297-98, 302, 772n226 of intuition, experience, sensibility, 341-43,423,425-26,435, 460-61,494,524-25,543, 581-82,585-86,591-98,602, 632-36,639,667,679,699, 701, 703, 708, 731nll, 733n24, 735n33, 737n42, 740n55, 758n142, 758n147, 760n160, 771n220, 774n239, 779n261, 785n284, 786n289 logical, 474, 591, 701, 737n43, 770n215,773n236, Foucault, Leon, 649 Frank, Manfred, 244, 256n3 Frege, Friedrich Ludwig Gottlob, 163,296, 314, 324, 326n3, 459-60, 473-75, 489n3, 589, 598, 666-68, 680, 737n43, 771n217-19, 772n226, 773n230,n233,774n238 Fresnel, Augustin-Jean, 651 Friedman, Michael, 18n7, 19n12, 19n17-18,56,61n31,86n28,
832
Index 136, 186n8, 290-91, 292n4, 292n6-7,292nl0-ll,429-30, 440, 443, 452n4, 452n5, 453n20,489n5,490nI4,493, 495-96,499-508,511-515, 517-20, 521n7,549, 565-67, 715, 728, 729nl, 730n4, 730n8, 752nl08, 754n119, 754n120, 754n123, 766n202, 772n224, 777n252, 786n289, 791n310, 791n312, 791n314, 791n315, 792n318, 793n324, 795n329, 802n358 methodology in history and philosophy of science, 3-11, 15-17, 18nll,43, 186n3,281, 526-27, 542, 545-47, 566-67 space, space-time, and accounts of them, 13-14,333-37, 349-51, 351n2, 351n3,453n21, 461-64, 466, 467n2,467n3, 517, 731n15, 759n148, 763nI76,766nI94,783n278 the relativized a priori, constitutive a priori, contingent a priori, 18n9-10, 287, 292n8, 333, 335,342, 351n2, 351n4, 423-26,431-32,450,455-56, 461-62,464,514,518,525, 527, 567, 697, 789n302, 790n308 Kant, 91-92, 107n3, 138, 140-41, 144, 146-47, 151n35, 183, 190n50, 190n55, 190n57, 193, 237n46,256n2,494, 549n4, 553-56, 567n8, 586, 592, 730n9, 731n13, 731n14, 732n16,732n21,732n24, 734n27, 734n28, 735n33, 735n35, 736n39, 736n40, 737n42, 739n48, 745n78, 751nl04, 753n112, 755n24, 758n147, 774n239 Cassirer, 261-62, 267, 270-72, 274n6, 282,284-89, 292n9, 390-91, 775n244,776n251, 780n264
on Carnap and logical empiricism generally, 279-89, 292n2, 295-97,313,319,325-26, 327n8, 328n11, 329n30,423, 498, 770n216, 771n218, 771n221, 774n236, 776n250, 778n257 on Helmholtz, 755n126, 758n143, 758n144, 758n146, 776n250 on Poincare, 761n164, 762n167, 762n168, 763n172, 766n195 Friedrichs, IZurt 0., 466, 467n2 Galilei, Galileo, 4-5, 14, 24-34, 37-38, 39n12-13,n15-21, 79, 265,365-66,393-95,436, 444-45,462,528-29,534,539, 574,583,600,612,641, 649-51,656,687,690-92, 729nl, 738n47, 740n51, 746n87, 760n159, 765n193, 766nI98,n200,767n204, 775n245, 778n258, 779n259, 780n265,796n333,799n347 Gassendi, Pierre, 52, 61n27 Geilker, Charles Don, 396 geometry, 4-6, 14, 25, 33, 77, 93, 119-20, 130n4, 131n12, 131n14, 131n22, 137, 148, 150n20, 171-72, 190n55,247, 249, 383n12, 463-65, 493-96, 526,542-44,573,639,685, 697-713, 727-29, 731n18, 731n19, 734n32, 738n43, 753n115, 761n163, 776n247, 777n254, 783n273, 790n306 and conventionalism, especially Poincare, 5, 9, 327n13, 344-45, 383n17, 467n3, 487-88, 642-649,655-57,662-64,666, 705-10, 762n168,763n171, 763 n174, 770n216, 788n298, 794n324,800n50,801n352 Einstein on, 9, 335, 344-47, 350, 351n2,659-666,676,707, 710-11, 767n207, 768n209,
Index 768n210, 770n215, 789n301, 789n304, 794n324 non-Euclidean, 9,411,518,665, 680, 710, 759n148, 761n162, 762n166, 762n169 Descartes on, 65-74, 76, 83n6, 83n8, 83n9,83nl0, 83nl1, 83nI3,84nI5,574-75,577-78, 736n40 in Newton and Newtonians, 70-74, 79, 84n18, 84n20, 84n21, 85n23-24, 86n31, 86n33, 87n36, 574-75,577,585, 598, 730n5, 730n6, 736n40 and Hi!bert, especially axiomatization, 80, 458, 475, 478, 487, 490n14, 666, 684, 783n277 I
833 724-26, 767n205, 769n211, 778n255, 781n269, 787n296, 789n303, 798n344, 799n347, 799n348 Helmholtz on, 628, 632-36, 645, 664, 708, 711, 716, 757n142, 758n143, 758n147, 786n289, 788n298, 794n324 Husser! on, 686, 688, 691, 693-94, 778n258, 779n59, 779n60, 781n268, 784n282, Gerlach, Walther, 447 Giere, Ron, 1-3, 18n3-4, n6 Glymour, Clark, 364 Godel, IZurt, 323-24, 326, 328n22, n24,496, 666, 669 Goldstein, Sheldon, 446, 567n5 Grafton, Anthony, 78 gravity, gravitational force, 13-15,25, 31, 35,218, 343,406, 418n3, 443,535,557,559-61,609, 612, 717, 741n56, 744n73, 752nl10, 777n254, 779n258, 800n350 in Newtonian mechanics, 36, 65, 87n35,92-97, 101-103, 106, 190n50, 194, 197, 230n14, 238n51, 388,462,502, 506, 514,517,537-38,540,546, 548,576,579,581,583-84, 610,613-14,622,641,658, 691, 720-25, 739n47, 740n52, 742n59, 742n61, 745n78, 746n87, 747n88, 766n200, 787n293, 787n295, 788n297, 796n336, 797n339, 797n340, 798n342 the transformation from Newtonian to Einsteinian, 333-36, 351n3, 466, 467n2, 659, 693,697, 709, 726, 788n299, 791n312, 794n324, 799n348 in relativity, 348-49, 387, 458, 506,660,662-63,710-11, 769n211, 789n302, 791n314, 792nI5,798n345, 799n346
834
Index
quantum and semi-classical, 387-92, 395-403, 403n5, 463, 566,801n356 Gunzig, Edgard, 392, 400-1 Haberlandt, Gotdieb, 307 Habermas, Jiirgen, 264, 284-85, 501-502, 514, 519, 521n7 Habicht, Conrad, 374, 383n16-17, 764n184 Hackel, Ernst, 303 Hagerstfom, Axel, 268 Hall, Marie, see Boas, Marie Hall, Rupert, 19n14, n16, 44-45, 57n2, 82n2, 729n2 Hamilton, William Rowan, 411, 413, 419n8,507 Hannah, Eric, 397-98 Hanson, Norwood Russell, 573 Heath, Thomas L., 80, 87n37-38, 186n6 Hegel, Georg Wilhelm Friedrich, 243-48, 252-56, 256n5,nll, 257n14,n20,n24,n28,279, 287-91, 292nl0-ll, 621, 629, 702, 715-16, 751nl06, 785n286,793n320-21,n323, 795n332 Heidegger, Martin, 10,261-64, 274n5, 281-82, 290 Heisenberg, Werner, 397,407,410-11, 414, 418n5, 546, 548,728 Helmholtz, Hermann von, 9, 342, 351n2,543-44,554,566,621, 628-39,642,644-45,647,649, 663,699,702-705,707-708, 710, 713, 716-17, 726, 736n39, 755n126-30, 756n131-36, 757n137,n141-42,758n143, n145-47, 759n149, 760n154, n160, 762n166-67, 763n173, 768n209,769n211, 776n250-51, 777n254, 785n286, 786n287-90, 788n298, n301, 789n304-305, 790n306, 791n315, 793n324
Hempel, Carl Gustav, 476, 482, 488 Henrich, Dieter, 244, 256n3 Hermite, Charles, 412-13 Herring, Ewald, 303, 307, 638-39, 760n155 Herz, Marcus, 125-28, 599 Hilbert, David, 80, 312, 326n3, 340, 456-59, 464, 467n8-11,473, 475, 478, 482, 487, 489n3, 490n14,496,590-91,598,642, 660,665-66,668-70,676,680, 683,697, 735n35, 761n163, 766n203, 771n219, 773n233, 783n277, 786n291 Hintikka, Jaakko, 135-36, 144, 190n55,489nl0,586 historiography, 3-4, 9-10, ,18nl0, 45,393,476,525,712, 790n310 Holden, Thomas, 93 holism, 298, 333, 337-38, 340-41, 344-47, 352n5, 466, 482, 506, 525, 665, 671, 673, 769n211, 770n716, 773n229, 788n301 Holton, Gerald, 39n7, 362, 376, 382n2, 383nl4-15 Howard, Don, 337, 340, 351n4, 352n5-8, 376, 379, 382n2, 769n211,n212,776n246, 788n301 Hubble, Edwin, 400, 500-501 Hu Bei-Lok, 392,401 Huggett, Nick, 396 Hull, David, 518 Hume, David, 81, 280, 341, 359-62, 372, 374-80, 381nl, 383n12, n16-17, 527,637, 767n208, 769n211, 784n281, 798n341 Husserl, Edmund, 10,268,281,290, 292nl0,343,460-61,465, 467n4,n13, 468n19, 666-67, 685-87,689-90,692-94,696, 698, 778n256-58, 779n259-61, 780n264, 781n268, 782n271-72, 783n279, 784n282, 793n322
Index Huygens, Christiaan, 24-25, 33-37, 39n24-27,n31,66,576 hylomorphism, 47, 56 idealization, 73, 131n12, 132n19, 222-23,272,285,311,315, 343-44, 346-47, 351nl,465, 467n16, 536, 560-65, 567,639, 644-45,654,687,690-91,693, 704-705, 746n87, 778n258, 781n266, 782n270 identity, 47, 98, 103, 108n14, 161, 165-66,178,184,248, 252-55, 303-304,413, 419nl0, 420n14, 444, 473-74, 494, 536, 681, 739n50, 754n122, 761n165 instrument, 68, 73, 97, 131n12, 283-84,289,304,344,428, 430,447-48,460,500,515, 634,645 intuition, 28-31, 34-35, 300, 341-43, 351nl, 460-61, 465, 467n7, 484, 487, 537, 540-41, 642-44,647,660-61,677,679, 684, 704-05, 725, 753n115, 757n137, 758n145, 761n163, 764n187, 770n215, 778n255, 778n257, 779n259, 779n260, 779n261, 783n277, 784n282, 786n289 I
835 758n142, 758n147, 760n160, 771n220, 773n236, 774n237, 774n239, 775n241, 775n243, 785n284, 787n293 Hegel and Schelling on, 245-49, 252-54, 256, 256n7,256nll, 257n12,257n14 Cassirer on, 265, 267, 288, 780n264 intuitionism, 669-70, 674, 772n225, 772n226, Isham, Christopher, 389 James, William, 303, 760n153 Janiak, Andrew, 14, 19n14, 108n9, 109n17, 720, 729n2, 749nl0l Jones, Alexander, 74, 86n25 Juncker, Johann, 515-16 IZaestner, Abraham Gotthelf, 115, 130n5, 131n6 IZant, Immanuel, 38, 81, 109n19, 113-14, 130n2, 132nI8-20, 149n7-8,nl0, 151n28, 151n34, 152n40,n42, 163, 189n40, 230nl0,240n62,285,334, 425-26, 467n4, 523, 529,544, 549n4,553-555,565-67, 696-700, 730nl0-ll, 732n21, 733n27, 738n46, 739n48, 740n55, 741n58, 749n96, 749nl0l, 750nl03, 751nl04-105,752nI12, 754n124, 773n236, 774n237, 774n239, 775n242, 779n262, 780n263, 783n280, 784n281, 785n283-84, 794n326, 797n337 intuition, see intuition, IZantian on space, see space, IZant on and mechanics, theory of matter, natural science, 91-106, 107n3-4,n7,108nl0-ll,nI3, 109n18, 190n50, 193-228, 229n6,229n8,230nI3-16,
836
Index
231n18,231n20,232n21,n23, 233n24,n28-30,234n31,n33, n35,n36, 235n39, n41, 236n42-43,237n44,n47-50, 238n51, 239n53, n55, 240n56-57, 543, 548, 556-64, 567n7-8, 580-85, 599-637, 640, 721, 730n9, 740n52-54, 741n56, 742n59, 742n61-62, 742n64-65, 743n69-70, 744n72-74, 745n76-80, 746n81,n746n85,747n87-88, 748n90, 748n93-95, 749n97, 749nlOl, 750nl02, 774n240, 796n334, 797n338, 797n340 on philosophy and especially metaphysics, 5-6, 121-30, 155-57, 169-85, 186n2, 189n47, 524,582,722, 731n12, 732n17, 774n241, 785n285, 797n339 later influence, formulations of, and reception, 7-11, 15-16, 19n18, 261-64, 266-67, 279-82, 285-89, 291, 292n3-4, n9,340-43,345-48,352n6, 390,423,455-59,496-97,525, 573-74,638-41,643-645, 665-69,676-81,683,685-87, 690,692,694,696,703-08, 711-18, 726-29, 752nl06-07, 754n122, 755n126, 755n128-30, 756n133-34, 758n142, 758n147-48, 760n157, 760n159-60, 762n167,769n211, 770n215-16, 771n218, 771n220-21, 773n230, 774n238, 775n243-44, 776n246-47, 778n256, 779n261, 781n266, 783n275, 784n282, 786n288, 786n289-90, 787n294, 788n300-01, 790n306, 790n310, 791n311, 791n313, 792n317-18, 793n320-23
and mathematics, geometry, and logic, 135-48, 149n3, 150n20, n26, 151n35-36, 155-57, 159, 169-85,493-95, 186n2-3, 189n44-45, 190n48, 190n55-57,243-45,247-52, 255-56, 256n2, 257n21-22, 271, 585-94, 597-99, 700-01, 731n13, 732n20, 732n23-24, 734n28-29, 735n33, 735n35, 736n38-40, 737n42, 737n43, 739n48, 739n49-50, 759n149, 775n241, IZepler, Johannes, 542, 583, 722 Klein, Felix, 642, 660, 697, 704, 761n163, 762n169, 766n203 Klugel, Georg Simon, 115, 130n5 Kneale, Martha, 261 IZneale, William, 261 IZoertge, Noretta, 513, 515, 521n3, n5, 714, 792n317, 794n326 IZoyre, Alexander, 107n8, 281, 790n310, 796n336 IZraft, Julius, 268 IZrois, John Michael, 264, 271, 274n7, 275n17, 780n264 IZuhn, Thomas, 3-4, 9, 16, 283-86, 291, 292nll, 333,337, 390-91, 393, 403n3, 428, 460, 466,493, 495,497,499-504,508,508nl, 511-13, 517, 520, 521n2, 525-27,573,696,711-16, 780n265,790n308-10, 791n311,n315,799n349 Lal
Index law, lawlikeness, 6-7, 15, 71, 107n7, 219,265,272,286-88,290-91, 303,335-39,344-46,348-50, 351n3, 369,394,462-464, 512, 517, 530, 537-40, 542-43, 583-84,607,629-32,634-36, 638-39,647-50,661-63,674, 681-82, 703-709, 731n12, 749n96, 751nl03, 756n134, 758n146, 768n208, 775n243, 785n282, 786n287, 791n312, 791n314, 799n350 of motion, 5,9,14,24-25,31, 33-34, 36, 39n30,93-95, 109nI7-18,201,210,222, 237n49,251,429,434,503, 530,557,559,576-80, 600-601,607,612,623, 640-41,648-50,660,691, 693-94,697,708-709,712-13, 716, 719-24, 731n13, 738n47, 740n51-52, 746n82, 747n90, 748n94, 753nl13, 760n159, 763n172,n176,781n267, 787n295-96, 788n297, 789n304,796n333,n335, 798n340 conservation, 24, 33, 517, 576-77, 583,614-17 force, especially gravitation, 94-95, 101, 190n50,205,212,232n23, 235n36, 333, 418n3,423, 514, 583-84, 626, 724, 736n40, 746n87, 788n297 causal, 96, 98, 272, 543, 629-31, 681, 755n128,n130, 756n134 mathematical, logical, 138, 141, 144-45, 150n25-26, 248, 252, 459-60,642,646,686,703, 731n13, 761n163 of nature, scientific, generally, 315-16,318,321,325, 327n13, 328n16, 409, 450, 506-507, 560,581-84,652-57,666,688, 752nlll, 764n185, 767n207, 778n256
837
Laywine, Alison, 132n18, n21, n23, 151n33-34,152n38,256n2, 599-600, 730n9-10, 738n45 Laz,Jacques, 142-143, 151n29-30 Le Cazre, Pierre, 33 Leibniz, Gottfried, 24, 38, 81, 187n15-19,188n27,n29,n37, 189n40, 574-75,578-582, 585, 591,597-98,619-20,686,690, 701, 730n8, 732n16, 734n27, 754nl18 matheo1atics, logic, 80, 84n14, 87n39, 144, 147, 150n26, 155-66, 168, 172, 177-85, 186nl,n4,n7-10, 187n20,n22, 243-44,252, 598, 678, 736n41, 737n43, 739n50 on space and time, mechanics, 82n3,218-19,224,235n41, 236n42, 555, 561, 577,579, 730n6-7, 785n284 metaphysics, 91-93, 96-101, 103-106, 108n13-14, 109n15, n19, 187n12, 524, 557,564, 578,579-581,604-605,616, 702, 748n95, 750nl02, 797n337-39 Le Roy, Edohard, 337-38 Le Tenneur, Jacques, 33 Lewis, C.I., 313, 772n226 Liberati, Stefano, 392, 399, 401 Lie, Sophus,411, 413, 419nl0, 632, 642,644-45,647,649,699, 703, 705, 757n141, 762n166, 777n254, 786n289, 788n298, 789n305 Liebert, Arthur, 261 lingua philosophica, 159-60, 162-63, 172, 181, 183 Locke, John, 81, 119, 132nI4,211, 637, 784n281 logic, 31, 78, 135-36, 146-47, 164, 255,261-68,271,279,283, 285, 292nl0-11, 296, 328n19, 328n22,342,456,489n7, 494-95,586,599,639,643, 680-82,686,689,700-702,
838
Index
704, 712, 728, 734n27-29, 737n43, 761n163, 761n164, 772n226,801n358-59 Kint,98-101,105-106, 108nl0-ll, 129, 137, 142-43, 149n7, 155-56, 172, 183-85, 190n55-57,496,591-92, 597-98,678, 732n23, 739n50, 759n149, 775n241 Leibniz, 157-63, 172, 186n8, 187nI2,n22,678,736n41 Wolff, 188n36 Aristotelian, 276n21, 494, 496, 578-79,582,585,701-702, 730n7 Carnap, 297, 299, 311-17, 319-26, 326n5,328nI5, 329n26,n28-29,479-86, 488-89,497,668-678, 771n219,n221,n222-23, 773n229-30,n233,778n257 Einstein, 346-48, 371, 376-77 and logical empiricism, 458-61, 473-76,479-86,488-89, 666-78, 770n215, 775n244, 779n264 Quine, 505, 512, 671-72, 774n238 logical positivism/empiricism, 1,6-7, 264, 279-83, 291, 292n2, nll, 298,303,306,308,325,333, 337, 352n5, 360,379,390, 393, 423,425,428,457-61,464, 473,476-78,487, 521n2, 523-27,529,531-32,535,541, 546,567,573,621,637,640, 642,664-66,673,678,682, 684-85,708,713,715,721, 726, 752nl07, 770n216, 790n309, 791n311, 793n322, 800n352 Lorentz, Hendrik, 366-67, 374, 378, 383n17, 389-92, 394-96, 399, 401, 403n3, n6, 436-38, 441-42,462,518,530,532-35, 538-39,547-48,651-58, 661-64,692,725,727,
763nI81-83, 764n185, 765n189, n193, 766n198, 775n245, 776n249, 787n296, 793n323, 794n326, 799n347-48,n350,801n356 MaaG, Johann Gebhard Ehrenreich, 244, 736n38 Mach, Ernst, 280, 295-97, 299-308, 342, 352n5, 359-62, 371-74, 376-381, 383nI6-17,384n20, 527,530,558-59,565,637-42, 665, 705-706, 710, 713, 716-17,727, 748n94, 759n150, nI52,760nI53-57,nI59-60, 764n188, 767n208, 769n211, 770n214, 787n294, 790n306, 791n315, 798n341 Machiavelli, Niccolo di Bernardo dei, 272 magnet and conductor thought experiment, 362-64, 366, 382n4,533,653 Mahoney, Michael, 24, 39n4-5, n24, n26,84n14 Maier, Anneliese, 46, 50, 57nl0, 58nl1, 60n24, 790n310 Maimon, Salomon, 244, 572n114 Makkreel, Rudolf A., 264 Mariotte, Edme, 23, 35, 37, 39n28 Martin, Gottfried, 137, 147-48, 149n4-5, 151n36, 152n39-42 Martin, Rosario, 392,401 mass, 37, 93, 95-97,101-104, 109nI8,203-204,211,221-23, 230n14, 231nI9-20, 234n32, 237n49, 238n51, 240n57, 336, 349, 380-81, 387-88, 396-98, 441,448,462,464,517,539, 542,548,556-57,561-63, 567nll,576-77,579,583,600, 602-603,611-14,618,622, 630,640,648,652,658-59, 692,709,711-12,719,721, 723, 738n47, 740n52, 745n78, n80, 747n88-89, 748n93-94,
Index 749nl0l,750nl02,756n134, 757n137,76011157, 766n199-200, 774n240, 780n265, 781n267, 789n302, 793n324 matter, 181, 248, 290, 306, 382n6, 516,518,524,546,585,596, 713, 716, 741n58, 759n152, 760n157 Early Modern views on, 13-14, 43-46,48,50-51,54-55,60, 65,193,575-77,579-80,718, 720-21, 729nl, 730n4, 730n7, 745n75 I
839 Maxwell, James Clerk, 363-67, 369, 374,377-79,392,394-95,401, 408,436,481,530,533, 539-40,651-53,656-58, 721-22, 727, 743n69, 756n131, 76611198, 78711296, 79411325 McMullan, Ernan, 1-2, 108n9, 511, 520111, 714, 741n58 111easurement, 31-34, 37, 69, 161-62,2361141,270,289, 344-47, 364, 368-69, 3831112, 397,402,403112,427-33, 435-37,439-42,445-50, 4521115,4531123,1125,463,498, 526,530-33,535-39,548,557, 563-64, 567nl0,602,613-14, 619,633-34,636,639-40, 645-46,649,654,662,673, 676,680-81,684,693, 702-703,719,7451175,1177, 7471189, 75811143, 75911152, 76311173,77811258, 78211270 mechanics/ mechanical/mechanism, 23, 85n24,93, 233n27, 333-35, 338-39,373,399,401,431, 443,511,515-17,520,520111, 530,535,542,564,573,575, 624-26,630,636,638,653, 682-83,686,694,711-19, 740n51, 742n63, 75311117, 754n121-22,75511130, 78411282, 788n299, 791n314, 792n316, 796n336 mechanical philosophy, 4, 43-45, 53-56,611131,216,577-78, 604,617, 720-21, 741n56, 744n70, 796n335 ancie11t, 23, 25 early modern, 23-26, 28, 35, 65, 583, 79411327 mecha11ical curves, 69, 72-73, 76, 831110,nI2-13, 574, 730n5, 7361140 Ka11t, 105, 1091118,203,207-12, 214-20,222,224,227-28, 230n8,2301113,2341134,
840
Index
235n39,236n41-42,238n51, 239n53,n55, 601-602, 609, 611-15, 617, 697-98, 746n81, 746n85, 747n90-91, 749n99, 750nl02, 751nl05, 753nl13, 792n318 Newtonian, classical, 194, 231n19, 336,371,390-92,410-11,455, 506-507,514,529,537,540, 557, 561, 567n5,631, 648, 651, 655-67,665,681,683,698, 708, 714, 728, 763n176, 764n188, 765n192, 781n267, 791n313, 792n315 mechanical explanation/ understanding, 209, 229n8, 407-08 quantum, 387-89, 394, 396-99, 402,403n5, 406-07, 410-12, 410-12,414, 420n14, 426-27, 435,446-51, 453n25, 526, 546-49,556,677,728-29, 792n318,794n326,801n355, n356,802n361 continuum, 555, 557-60, 565-66, 743n69, 744n73, 746n80 11ach, 373, 530,640-01,665, 748n94, 760n156-57, 764n188 Poincare, 643, 646-50, 654, 658, 664,704-07, 764n185,n187, 765n193, 770n216, 798n341, 11elanchthon, Philipp, 161 11endeleev, Dmitri, 518 11endelssohn, 11oses, 243, 749n96-97 11enn, Stephen, 575, 729nl 11ersenne, 11arin, 23, 31, 33, 37, 39n20,74 n1etaphysics/metaphysical, 5-6, 8-10, 87n39, 149nl0,495, 500,513, 521n3,545,547 Aristotle, 271 Carnap,299,326,671,673,676, 677,678 Descartes, 574-575, 729nl, 730n7, 732n16
Hume,375 I
Index Michelso11, Albert, 362, 393-94, 433, 435-36,439-40,530-34, 538-40,651,653,717,722-23, 78111268, 78911302 Michelso11-Morley experime11t, 362, 393-94,433,435-36,439-40, 530-34,538-40,651,653,717, 722-23, 781n268, 789n302 Mie, Gustav, 458 Mill, Joh11 Stewart, 665, 668 Mi11kowski, Herma11n, 424-25, 433, 435,438-39,442,444-45,447, 449, 451n3, 462, 547,658-59, 662-663,682,692,697,711, 724-25, 766n197,11201, 767n205, 77511245, 77811255, 781n268,n269,787n296, 788n299, 799n346 mixture, 43-50, 52-56, 56nl, 57n3, n8,nl0,58nll-12,n14, 59n17-19,60n21,1124-25,494, 516, 559 model, 128-29, 323, 349, 379, 393, 396-401,408,433,474,477, 479,483,485-86,489, 502-503,514,553,558,562, 565,604,606,628,662, 690-91, 709-10, 734n29, 738n45, 742n63, 743n69, 744n72, 759n152, 780n265 Molland, A.G., 74, 83nl0, 85n22 More, Henry, 14, 19n16, 77, 86n29, 575 Morley, Edward, 362, 393-94, 433, 435-36,439-40,530-34, 538-40,651,653,717,722-23, 781n268, 789n302 Muller, Johannes, 636 mysticism, 44-45, 245-247, 660 Natorp, Paul, 341, 346-47 Neurath, Otto, 518 Newcomb, Simon, 536 Newman, William, 39n2, 57n3, n6-7, 59n15, 60n22-24, 61n26, 717 Ne"rton, Isaac, 19n17, 38, 82n2-3,
841 83n7, 84n16-17, 86n30, 87n34, 109n18-19,177,226-27, 239n56,243,425,457,495, 519,557,560,619,628, 679-80,690,698,711-19, 727-29, 729112, 731n12-13, 732n16, 737n43, 746n86, 747n90, 760n157-58, 764n188, 765n192, 774n240, 780n265, 785n286, 788n299, 790n306, 791n312-13,n314-15, 792n319, 793n323, 795n330, 796n334, 797n338-39, 801n353 and absolute space, 13-15, 18n8, 371-73, 380, 444, 453n21, 533-35, 556, 562, 575-77, 579-81,584-85,598,640-42, 649,665,692,701,706,719, 730n6-7, 730n9, 751nl04, 760n157, 775n245, 783n275, 785n284, 795n330, 797n337 and action at a distance, 107n9, 197,558,581,605,622,709, 720-21, 749nl0l, 796n336 and attraction/attractive forces, 91-99, 102-103, 109n16-17, 197, 224, 238n51, 743n70, 756n131, 796n336 and calculus, 79-80, 84n14, 87n39 and geometry, 65-67, 70-74, 84n18-21, 85n22-24, 87n36, 186n2, 574-77, 597-98, 730n5, 736n40 and gravitation, 36-37, 871135, 101, 106, 1901150, 230n14, 333, 387-88, 418n3, 466;467112, 502,537,548,581,584,610, 622,691,693,697,709, 720-21, 722, 724-25, 738n47, 740n52, 741n59, 742n61, 745n78, 746n87, 76611200, 787n295, 788n297, 798n342 and mathematical physics/ mechanics, 4-5, 9, 39n30, 40n32, 107n5, 193-94, 198,
842
Index
208-10,221-22, 327n13, 333-37,365,391-92,394,423, 440,455,462,506-507,514, 517,526-29,531,536-44, 546-48, 549n3, 573-574, 583, 600-601,604-605,610, 612-13,631,648-50,656-58, 665,682,701,708,719-720, 723, 724-25, 745n75, 746n82, 747n89, 748n94, 763n175-76, 781n267-68, 787n295, 796n333,n335 and neo-Platonism, 19n16, 77, 86n29,575-577,720-21,729n3 and philosophy, 76-82 Newton's Bucket, 14, 373, 380-81, 707 Nobel, Alfred, 502 Norton, John, 348, 364, 366, 369, 377, 381, 382n3,n8,662, 767n205,n208,770n214, 798n341 noumenon,100,105,606,619-20 Nozick, Robert, 462 objectivity, 10, 129, 262-63, 268-70,272-74,279,281-82, 286, 292nl0, 296-97, 299, 302, 313,459-62,465,517,536-37, 540-41,543,593,624-25,677, 680-81,683,686-90,694,699, 712-14, 725, 731n12, 738n44, 750nl03,779n259-60, 780n264, 782n272, 784n282, 785n284,793n323 observation/observer/observable, 31, 40n32,48,86n33, 157-58,181, 316, 320,343, 351nl,440, 500-501,506, 509n2, 512,546, 558,620,631-32,652-53,661, 669,676,719,722, 757n137-38 role in theorizing about metaphysics, 197, 238n50 Kant's "General Observation on Dynamics," 207, 215, 217,
and intersubjectivity, 304 and theory, especially underdetermination of theory, 313,338-39,428-30,432-35, 443-44,455,457 observation language/sentences/ vocabulary, 325, 427, 433, 476-85, 487-88, 489n4-5, 490n13,498,503,672, 794n327 in space-time theories, 360-61, 363-65, 367-71, 378, 382n9, 384n19,388,393,400-402, 425-27,435,438,440-41, 445,447,450,531,534, 537,539-41,566,583, 600-601, 692, 764n185, 780n265 in quantum theory, 411, 413, 418n6,419n8,420n14-15,n19, 446-48 odd-number rule, 31-33 Oersted, Hans Christian, 622, 717, 754nl19-20,755n125 ontology, 2, 9, 80, 255, 322, 349, 438,514-15, 520nl, 524, 534, 547-48, 556, 730n7, 773n236, 782n272 ordinary language, 146, 248, 250, 265, 737n43 Page, Don N., 396 Paneth, Friedrich Adolf, 518 Pappus of Alexandria, 12-13, 74, 86n25, 574, 736n40 paradigm, 244, 247, 283-84, 286, 291,335,337,390,429,457, 466,499-501,503,511-12, 514,516-17,520,525-26, 554-55,565-67,577,711, 713-15, 791n315 parallel, 115-118, 130n5, 13111.8, nl0-ll, 132n13, 170,231,248, 459, 496, 549n3, 560-61, 587, 691, 732n18-19, 740n51, n54,
~~2~3~5n=3~9~~~~~~~~~~~~7~8~%=3~05
_
Index Parsons, Charles, 151n37, 585-86, 592, 597-98, 735n37, 737n43, 761n164, 783n274 Pauli, Wolfgang, 407, 418n5, 546, 728 Paul of Taranto, 50-53, 55, 60n22-23 Peano, Guiseppe, 271, 474 Pearson, IZarl, 562-63, 567n12 Peirce, Charles S., 271, 284 Pemberton, Henry, 65-66 Penrose, Roger, 402 perception/perceptual data, 247, 304, 426,434,440,537,562-63, 685,697-98,708,716, 761n165 Bolzano, theory of, 151n28 Carnap, 300,311-14, 316-22, 324 Cassirer, 266, 268, 273, 275n16, 276n23, 780n264 clear and distinct in Descartes, 68, 75 Helmholtz, 631-36, 703-704, 755n128, 756n132, n135-36, 757n138,n140,n142, 758n144-47, 776n250 Hume,375 Husserl, 687-94, 779n259 lZant,95, 136, 143, 173-74, 178, 239n56, 582, 586,591-98, 602-603,612,615,617-18, 627,692,701, 739n48, 740n55, 741n56, 758n147, 775n243, 797n340 Mach, theory of, 638-40 and theory-ladenness, 428-30, perihelion of Mercury, 658-59, 717, 791n315 pessimistic meta-induction, 2 Philoponus, John, 46,49, 57n8, 59n17 phlogiston, 515-16, 623, 626, 752nl12 Pirani, Felix, 349, 778n255 Planck, Max, 307 Plato, 12, 14, 19n16, 77, 86n28-29, 146, 246, 256n8, 264, 281,
843
328n21,379,408-409,494-95, 575, 577,670,698, 720, 729n3, 750nl02 Pliny, 48, 58n14 Plotinus, 575 Poincare, Henri, 9, 543-44, 566, 672, 713, 716, 718, 736n39, 761n161, n164-65, 763n172-73,n178-79,n181, 764n186, 765n191, 766n195, 786n291, 787n295, 790n306, 791n315,799n346,801n353 on conventions/conventionalism, 5, 327n13,456,467n3,487-88, 642-52,666-67,704-10, 764n187, 765n193, 766n196, 768n209-10, 770n216, 776n248,788n299,800n352 Duhem's critique of, 337-39 and Einstein, 342, 344-45, 351n2, 383n17,503,527,654-61, 663-664, 723, 726-27, 764n185-87, 765n91, 765n193, 766n203, 767n207-208, 768n209-10,769n211, 770n216, 776n249, 787n292, n296, 788n299, 788n301, 789n304, 792n316, 793n324, 798n341,799n350 on geometry, 5, 467n3, 487-88, 704, 706-10, 726, 761n162, 762n166-71, 767n207, 769n211, 777n254, 788n298, 789n304,800n352 on relativity/relative motion, 705-707, 726, 763n174, 763n182, 764n185, 765n189-90, 768n210, 787n292,n295-96,792n316 and structural realism, 2 and underdetermination, 313, 324 Poisson, Simeon-Denis, 411-14, 419n8,420n19 Popper, IZarl, 511-13, 519-20, 521n2-3,n8 Porphyry of Tyre, 161 possible world, 101, 578, 580
844
Index
Post, Heinz, 515, 517, 520, 521n6 postulate /postulates, 131n6, 131n13, 132n14, 132n22,394, 762n171, 777n254, 778n255 Carnap,325,477,479,481-82, 484, 489n7 Einstein, 361, 371,464,534,548, 653-54,657,707-11,723, 764n185, 788n297, n299, 799n350 Kant, 121-130, 132n18-20, 138, 147-48,589-92,594-95, 600-02, 734n32, 738n46, 739n48 Lambert, 113-121, 131nl1-12, 132n15-16, 147, 152n39, 599 of Euclidean geometry, 79-80, 85n24,496, 586, 589-92, 594-95,601-602,702, 732n18-19,n22 in mechanics, 24-26, 30-31, 34 Schultz, 137, 140-42, 147-48, 150n20-21 Poynting, John Henry, 553, 563 prediction, 231n19, 232n20, 344, 387-88,407,429,432-33,437, 440,442,447,500,502,517, 546,562-63,671 principle, 50-52, 54, 75-77, 86n28, 92, 123, 166, 231n16, 253, 265, 270,290,312,318,325,335, 338,345,347,392-95,424, 475,477-78,497,501-502, 505,513,534,536,554,632, 668, 682-84, 704, 719, 738n46, 738n48, 750nl02, 751nl05, 753nl13, 754nl18, 765n193, 789n302, 795n327, 795n329, 800n350-52,801n355 philosophical/epistemic/ metaphysical; see also specific principles in this entry, 6, 61n26, 95-96, 105, 107n7, 114, 125, 129-30, 132n15, 182,251-52, 257n21, 291, 292nl0, 322, 327n13,375,409,529,545-46,
566-67,578-79,582-84, 588-89,600-603,607,617, 623-24, 631,706, 731n12, 740n55, 747n90, 752nl09, 797n339,801n353 physical/empirical/scientific; see also specific principles in this entry, 23-26, 35, 38, 140, 177-78, 190n50, 196,207, 335, 337-38, 349, 382n6,423, 453n28,515-16, 518, 525-26, 528-29, 532, 541, 543-49, 557-58,578,607,617,623, 629-31,648,651,672,681, 707, 717, 726-27, 738n48, 740n54, 742n63, 744n73, 752nl09, 752nlll, 754n121, 759n149, 770n216, 779n261, 781n267, 787n292, 791n312, 800n350-51 symmetry, 33, 37 of relativity, 34, 362-69, 377, 379, 382n6, 393, 395, 529-30, 534, 539,649-60,664-65,692, 706-707, 710, 713, 716, 722-23, 764n185,n188, 765n189-90, 765n193, 781n268, 787n292, 787n295-96,788n299 conservation, 24, 36, 516-17, 583, 614,629-30,636,638,719, 747n90, 750nl0l mathematical/geometrical, 67, 76, 115-118, 126-27, 138, 138, 142, 147, 149n8, 164, 173, 175, 180,249,251-52,335,474, 507,578,581,591,694,697, 738n44, 738n48, 752nl09, 758n147, 770n216, 779n261, 787n292, 787n296 logical, 98, 188n38, 287, 292n11, 732n23 identity of indiscernibles, 108n14 of theology, 169, 189n39 of sufficient reason, 188n38, 579 regulative, 234n35, 284, 287-88, 292n9,602,622,627, 704, 706,
Index 715-16, 738n48, 752nl09, 753nl14, 775n242 constitutive, 281-88, 292n8, 292nll,335,425,466, 512-514, 517-18, 520, 521n7, 524-25,537,544,573,622-23, 627,684,694,696-99,706, 708, 713-17, 728, 752nl09, 753nl14,775n242, 776n247-48, 791n312, 791n314, 793n321, 793n323, 794n326, 795n329, 801n352 coordinating/bridge principles" 325,337-38, 343, 350, 351n4, 424-26,429-30,434,439-40, 449-51, 451n4, 455-57, 461-463,697, 727, 771n221, 777n252 of equivalence, 333-38, 348-50, 351n2,463-64,506, 518, 542, 659,662-62,693,697,709, 711, 716, 724-26, 778n255, 781n269, 789n302-303, 789n305, 791n312, 791n314, 793n323, 794n324, 798n344, 799n348 ligh~ 349,367-68,393-95, 424-25,430,435,439-40,450, 451n2,n4,452n14,n17, 463-64,534,548,653,664, 697, 707, 709-10, 713, 716, 722-23, 778n255, 781n267-68, 788n299, 789n305 Mach's, 359, 381 uncertainty, 397, 548, 728, 801n355 correspondence, 449 of causality, 455, 548, 583, 623, 629,635-36,639,703, 758n146-47, 786n287, 786n290 of tolerance, 460, 497,670-71, 674, 771n223, 772n226 General Correspondence, 515, 517 Euler's cut, 557-58, 560 free mobility, 566, 636, 639,
845 703-705, 708, 726, 736n39, 758n143,n147-48,788n298 complementarity, 802n357 principle and constructive theories, 348-49, 765n193 principle of equivalence, 333-37, 348-50, 351n2,463-64, 506, 518,542,659,662,692,697, 708-709, 711, 713, 716, 718, 723-25, 778n255, 781n269, 789n302-303,n305,791n312, n314, 793n323-24, 798n344, 799n348 Proclus, Lycaeus, 77 progress, 5, 7-8, 10, 16-17,44, 81-82, 84n18, 129, 190n58, 193,284-85,287-88,290, 292nll, 333, 371, 373, 391-93, 396,402-403,414-15,417, 421n23, 508n2, 511-13, 520, 625,638,673,675,677, 680-81, 711-12, 751nl05, 755n130, 775n245, 790n309, 793n323 pseudoscience, 45 Ptolemy, 517, 542, 575 Putnam, Hilary, 476, 489n4, 802n358 Pythagoras, 415, 475, 633, 789n305
quantum field theory, 388-89, 394, 401, 420n15, 676, 728, 802n360 quantum gravity, 387-88, 390-92, 395-96,398-403,463,565, 801n356 quantum mechanics, 4, 387-90, 392, 394, 396-99, 401-402, 403n5, 406-407, 410-15, 419nl0, n13, 420n14-15,n19,426-27,433, 435, 446-51, 453n25,n28, 526, 546-49,556,676,727-28, 792n318,793n326,800n355, 801n356,802n367-59, n361
846
Index
Quine, Willard van Orman, 16, 279, 311-13,324, 326, 329n27, 333, 337,457,466,481,484,493, 495-96,498-99,503-508,512, 518,525,671-73,675,678, 772n226, 773n228-29, 774n238 Ramsey, Frank P., 479-80, 482-84, 486, 488, 489n9, 490n13, 676, 773n233 rational reconstruction, 6, 297, 316, 322,334,350-51,390,457, 460-61,473-74,476-87,489, 489nl,505,507-508,676, 744n73, 773n233 reduction, 12, 19n13,43, 50-56, 60n23,65,94, 141, 165, 173, 175, 182, 184, 229n8, 240n62, 247,267-68,279,295-96,298, 301-302,304-307,312-14, 317-19, 323, 325-26, 327nl0, 345, 396,401, 403n2, 460, 467n5,476,531,630,638,643, 662, 666, 694-95, 755n129, 756n131,n134,763n179, 769n211,774n238 regulative a priori/ideal/principle, 9, 234n35, 262, 265, 267, 284-89, 291, 292n9, 348, 556, 562, 564-65,584,602,622-25, 627-28,635,680,683-84, 695-96, 703, 706, 715-17, 739n48,752nl09,nl12, 753nl14, 775n242, 783n273, 786n290, 787n293, 793n321-22,n324, 794n326-27 Reichenbach, Hans, 280, 282, 292nll-12, 333-34, 340-43, 350, 351n4, 382n8, 390, 423-24,426,430-31,442-44, 453n20, 455-57, 460-64, 466, 467n2,n16,498,524, 529, 531, 554,566,664,666,682, 684-85,696-97,711,715,
770n216, 771n221, 776n248, 777n252-53, 783n275, 786n291,800n352 Reinhold, I<arl Leonhard, 244, 252 relativity, 6, 9, 289, 327n13, 333-36, 345, 348-50, 351n2, 381, 382n6, n8,403n2, 424, 426-28, 430-31,433-47,449-51, 452n16,455-56,462-66, 467n3,n17, 503, 506-507, 544, 548, 549n2, 566, 573,648,662, 667,682, 724, 763n176, 776n245, 777n254, 778n257, 781n267-69, 782n270, 783n278, 786n289, 787n292, 788n299, 790n306, 791n312, 793n323,795n328,801n353 of motion, 34, 394, 655, 658, 705-706, 711, 763n174, 787n292, 788n298, 790n306, 800n350 philosophical and historical lessons or accounts of, 282, 341-43, 495-96,498,502,504,525-27, 532-33,546,665,675,683, 685,691,697-98,708-12,714, 717-18, 725-26, 769n211, 770n216, 789n304, 791n313, 792n315 axiomatization of, 349-50 development and early years of, 359-74, 376-79, 383n17, 500, 548,642,652,661,692-93, 763n182-83, 764n185, 764n188, 765n193, 766n201, 768n209-10, 769n211, 776n249, 782n269, 783n273, 792n316, 794n324, 798n342-43, 799n349 principle of, 34, 362-69, 377, 379, 382n6, 393, 395, 529-30, 534, 539,649-60,664-65,692, 706-707, 710, 713, 716, 722-23, 764n185,n188, 765n189-90, 765n193, 781n268, 787n292, 787n296, 788n299
Index of simultaneity, 366-70, 377-78, 383n17,395,403n2,654-55, 659, 765n189, 788n297, 799n347 and quantum theory, 387-91, 393-96, 399,401-402,403n5, 526, 546-47,677,801n356 problem of, 533, 536 and passivity of space, 648-49, 655, 657,660,706-707 Poincare's 'law of', 648-50,652, 655, 706-707, 787n295-96 relativized a priori, 7-8, 10, 16, 18nl0,281-84, 349, 390,423, 426,431,435,450,455-56, 461-62,466,512-13,517,520, 525,527,529,545,547, 566-67,697-98,708,711-16, 723, 726, 770n216, 776n248, 777n253, 781n268, 783n275, 788n297, n301 789n302, n305 790n306,n308,791n314, 793n321-23,794n326, 795n329,800n352,801n353 representation, 2, 160, 162, 328n21, 342,360,369,371,375,432, 446, 450, 451n4, 463-65, 563, 566,577,590,616,661,701, 725-26, 736n38, 740n51, 753nl13, 780n265, 781n266, 799n349 mathematical/geometric/algebraic, 69-72, 126, 189n44, 316-18, 321,336,424,463-65,474, 606-607,643,646,663,680, 698, 710, 725, 792n315, 794n324, 798n345 I
847
745n74, 749n95, 751nl05, 773n236, 774n237 symbolic/logical, 159, 339, 464, 479,673, 734n28 in Cassirer, 265-68, 271-73, 781n264 11ach, 303,639, 759n152 Helmholtz, 632-35, 639, 757n137, 758n145 Richardson, Alan, 284, 292n2 nll-12,295-97, 308, 313, 318-19, 326n4,328n14, 715, 771n218,n221,772n227, 773n228, 775n243, 792n319, 793n323, 795n332 Rickert, Heinrich, 261-64, 268, 272, 274n4-6 Riehl, Alois, 341 Riemann, Bernhard, 463-65, 503, 506-507,633,647,667,682, 697,699, 709, 757n141, 762n166, 763n172-73, 777n254, 781n269, 786n289, 789n305 Ritz, Walter, 366 Robertson, Howard Percy, 400 Roemer, Ole, 537 Romanus, Aegidius, 47-48, 58n12 Rovelli, Carlo, 389, 395-96, 398 Russell, Bertrand, 190n55, 271, 281, 296, 322-24, 328n19-20, 473-75,496,500,566,589, 637, 666-68, 680, 773n238 Ryckman, Thomas, 292n5-6, nl0, 351nl,n4,352n7,426,466, 467n4-5,n9,n14,n16, 468n17-18,684-85,697, 768n209, 776n251, 777n253-54, 781n268, 783n276,802n360 Rynasiewicz, Robert, 364, 795n331 j
Saa, Alberto, 392, 400-1 Salmon, Wesley, 292n2, 408 Sanzio, Raffael, 272
848 Sarpi, Paulo, 29-30, 39n15 Schaffer, Simon, 25, 39n7, 45 Schelling, Friedrich Wilhelm Joseph, 243-55, 256n4, n6-7, nIl, 257n12,n14-15,n20,621-22, 624-30,636,702,717, 752nl08,753nl15-17, 754nl19-20,n122,755n125, 785n286, 793n321 Scmld,AJfred, 349, 778n255 Scmlpp, Paul Arthur, 473, 772n225 ScWegcl,Friedrich, 247, 256nll ScWeiermacher, Friedrich Daniel Ernst, 247 Schlick, Moritz, 264, 274n9, 280, 340-41, 351n4,352n5, 360, 374, 377,426,456-60, 467n3, n5,490n14,660,664-67, 682-85, 697, 757n140, 766n203,769n211,n213, 770n215-16,771n221, 776n246-48,n250-51, 777n252-53, 778n257, 782n270, 783n277-78, 786n291,788n301,800n352 Schneider, Ilse, 341 Schroedinger, Erwin, 407 Schultz, Johann, 135-42, 144-48, 148nl,149n3,n5,n7-8, 150n20-26,151n32,n36, 152n42,184,598 Schwab, Johann, 244 scientific revolution, 2-7, 9-10, 16, 43-45, 55-56, 57n2,61n30, 284-86, 289, 292nll, 387, 390-91,393,466,495, 499-504,511-13,517,520, 523,526-28,531,546,554, 588,616,622,628,645,682, 697, 701, 711-12, 768n210, 776n249, 780n265, 783n280, 790n308,n310,792n318, 795n328,n332,799n349 Sellars, Wilfrid, 495, 504 Sellien, Ewald, 341 Sennert, Daniel, 51-55, 57n3-4, 60n24, 717
Index sense-datum language, 311, 326 Shabel, Lisa, 139, 149n9, 735n35 Shankland, Robert, 362 Shapin, Steven, 25, 39n7, 45, 55, 57n6,61n30 similarity, 155-58, 161-68, 172, 177-81, 183-85, 186n4, 187n18, 188n29,n35, 739n50 simplicity, 69, 71-72, 84n20-21, 85n24, 318-19, 322, 327n13, 347, 380, 406, 409, 418, 418n2, 420n20, 500,505, 507-508, 528,542,645,650,653-54, 661,667,671,674,704, 762n167, 768n209, 770n216, 788n298,800n351-52 simultaneity, 359-62, 366-72, 374, 376-79, 382n8-9, 383n17, 395, 397, 403n2, 431, 436-38, 445, 525,527-29,531-33,535-43, 547-48,654-55,657-58,663, 692,707,709,717-18,721-23, 726, 753nl17, 764n187, 765n189, 766n198, 767n205, n208, 788n297, 789n302, 797n340, 798n341, 799n350, 80111355 skepticism,S, 8, 53-54, 74, 86n27, 108nll, 120,341, 384n20,624, 626-27, 753nl14, 754n122, 765n193, 767n208, 784n281, 798n341 Slusius, Rene Fran~ois Walther, 66 Socrates, 98-99, 539 solipsism, 296, 299-300, 302, 304-305 Solovine, Maurice, 374, 383n16-17, 764n184 space/space-time, 31-32, 39n3, 80, 87n37,92-94,96-97,100-104, 106,225,346-47,388-89,394, 397, 400,403n2, 407, 424,427, 429-30,433-43,445,447-50, 451n3, 452n5, 453n22-23, n25, 455,457,459,465-66,514-15, 520,528,532,536-37,539, 545,547-49,559,566,574,
Index 662,687,696-97,701,712, 724, 727-28, 730nll, 731n15, 743n69, 744n74, 748n93, 749nl0l, 757n137-39, 758n143, 761n163, 762n168, 763n173, 763n176, 767n205, 768n209, 771n219, 776n248, 778n258, 781n268-69, 782n270,783n273,n277, 784n282, 789n303, 792n318, 793n323, 798n345, 799n347, n350,801n355,802n361 in Descartes, 13, 65, 575-78, 730n7 in Newton, in Newtonian mechanics, absolute, 13-15, 18n8,65,95, 336, 365, 371-73, 380,425,444, 453n21, 462, 526,529,533-34,542-43,546, 555-56,562,564,575-77, 579-81,584-85,598,601,619, 640-42,648-49,659,665, 679-80,692,701,706,708, 718-21, 729n3, 730n6-7,n9, 731n15, 751nl04, 760n157, n159, 763n175, 775n245, 776n245, 781n266, 787n293, n295, 796n333, 797n337,n339, 798n342 in lZant, 91, 95, 105, 107n7, 136-37, 144, 150n20, 173, 175, 179,195-97,199,201-04,207, 213-16, 218-21, 229n6-8, 231n18, 233n27, 234n36, 235n39, 237n49, 238n50, 239n56, 240n56-57, 247,282, 342-43,423,555-56,558,564, 581-86,589,591-601,603-11, 614-24,626-27,632,634-36, 639-41,679-80,692,694, 699-700, 703, 705-706, 715, 730n9, 731n13, 732n15, 733n24,n27,734n27,n29, 735n33,n35, 736n38, 737n42, 740n53,n55, 741n56, 742n60, 745n76,n79-80,746n81, 747n91, 749n97, 750nl02,
849
751nl04, 752nl11, 755n129, 758n147, 760n159-60, 761n160, 771n220, 773n236, 774n237, 781n266, 785n283-84, 786n289, 797n337,n339-40 in Cassirer, 290, 343, 462, 682-83 in Leibniz, 578-80,604, 730n6, 750nl02, 785n284, 797n337 in Carnap, 299-301,311-12, 316-18, 320-22, 325, 328nI5-16, 329n29, 343,665, 667,675, 771n220, 773b230, 778n257 in Mach, 304, 306 in Weyl, 343, 351nl, 463,658, 685,698,729, 777n254-55, 786n289, 789n305 in Hume, 360,375 in Einstein, relativity theory, 367, 369,371,377-79,387,503, 523,527,529-30,534-35, 541-43,546,573,642,653, 657-58,675-76,682,692,693, 709-11,716, 725-26, 773n230, 788n299, 789n304 in Poincare, 487,543-44,642, 644-50,655-58,660,663-64, 666, 705-10, 763n174, 766n196, 769n210, 787n296, 788n299, 789n304 in Helmholtz, 544, 631-36, 639, 644-45,647,664,703-706, 708, 716, 726, 736n39, 757n142, 760n160, 777n254, 786n289, 788n298, 789n304 in Schlick, 665-66, 683, 776n247 Speziali, Pierre, 376, 383n16-17 Spinoza, Baruch, 81, 130n4, 252, 621 Stachel, John, 362, 366, 382n2, n7, 662 Stackel, Paul, 115, 117, 121, 130n5, 131n6, n9, nIl, 132n15 Stadler, August, 340, 769n213 Stahl, Georg Ernst, 61n26, 514-16, 518,623
850
Index
Starkey, George, 45 Stein, Howard, 82n3, 102, 276n23, 729n2-3, 739n50, 757n141, 795n330, 796n336 Steiner, Mark, 416-17, 420n21 Stern, Otto, 447 Stern-Gerlach apparatus, 447 Stevin, Simon, 24, 26-27, 37, 39n11 Stewart, Ian, 77, 86n31 structural realism, 2, 18n5, 515 substance and alchemy, 44-48 in Aristotle, 45-46, 48, 161, 271, 579 in Cassirer, 394, in Hume, 375 in Leibniz, 92, 108n14, 109n15, 578, 730n7 in Kant, 91-92, 99-100, 108n11, 172, 196,228, 229n6, 230n9, n13,240n57,n62,524,543-45, 559,580-83,585,604-606, 609, 613-20, 622, 721, 730nll, 742n60,747n91,748n93,n95, 749n97, 749nl00-l0l, 750nl04,753n113 in Mach, 303 and matter, 172, 196,228, 229n6, 230n9,n13,240n57,n62,394, 518,562,605-606,613-20, 747n91, 748n93, 797n338 in More, 14 in Paul of Taranto, 50-52 and perception, 300, 562, 635 in Scholastic matter theory, 46-50, 56nl, 58n11, n15 and space, 14, 453n21-22, 576, 720-21, 747n91, 749n97 Substance and Function, 680, 682, 684 synthesis/synthetic and analysis, 14-15, 84n20 of appearances/experience, 528, 543-45,548,556,591,602, 738n44, 739n49, 775n243-44, 800n351 Aristotle on, 13
Carnap,282 chemical, 515-16 definitions, 169-70, 172, 178 in geometry and mathematics, 84n20, 123-24, 135, 138, 141-42, 145, 147, 189n40, 247-48, 599, 666, 668, 732n24, 734n29,785n283 Hegel, 287-88 of the imagination, 144-45, 582, 589, 734n32 judgments, 98-99, 107n4, 108n11, 123-24, 138, 175, 233n29, 478, 486, 738n44 in Kant's transcendental philosophy, 700-702, 704, 774n241, 784n281 in metaphysics, 182 in physics, 528 synthetic a priori,S, 93, 247-48, 251,341,475,496,512, 524-25,529,583,643,665-68, 676,678-81,683-85,718, 734n29, 770n215, 774n238, 778n257, 780n263, 785n283 synthetic/analytic distinction, 466, 481-82,498,599,671-78, 773n230 synthetic history/method, 1, 11, 13-17,43-44,573-74,696, 700, 702, 784n281
Tanona, Scott, 446, 452n12, 453n26, 508n2,697, 728, 781n268, 783n276,802n357 Tarski, Alfred, 666, 733n26 theory-ladenness, 427-30, 432, 443, 447,449-50 thermodynamics, 349, 382n6, 759n152 Thomson, James, 529, 536 thought experiment, 14,23, 36,266, 362-66, 369, 382n4, 397-98, 646, 794n327 time, 407, 411, 419n8, 423, 425, 429-30, 433ff, 450, 451n3,
Index 453n23, 514, 520, 536-38, 544-46,565,573,680,694, 697-98, 701, 712-13, 715-16, 743n69 in the scientific revolution, 30-32, 65, 79 in IZant, 107n7, 128-29, 144-45, 170,173,199-204,215,218, 224-26, 237n49, 341-42,423, 543-44, 581-82, 586, 591-92, 599,603,615,618-19,622, 640,679,708, 733n24, 740n53, n55, 741n56, 758n147, 760n160, 773n236, 785n282, 785n284, 792n318 in Bolzano, 144-45 reversal/reversibility, 231n19, 518 Schelling, Hegel on, 247 in Cassirer, 266, 269, 290 in Mach, 306, 360, 372, 378, 640-42,665, 760n157 in Einstein and relativity theory, 341-42, 347, 359-61, 367-69, 371, 379, 403n2, 434-42, 445, 523,527,531-32,535,539-41, 546,654,657-58,663,676, 707-709, 726, 788n299, 789n303, 799n347 in Hume, 360, 372-73, 375, 378, 383n12 in Lorentz, "local time," 378, 383n17, 651, 655, 765n189 in Newton, Newtonian mechanical, classical n1echanics, 379,443-45, 526,529,534,542,546,576, 579,640-42,648,665,680, 708, 718-19, 730n6, 760n157, n160, 775n245 Minkows~on, 547,692, 788n299 in Poincare, 648, 766n196, 788n299 in Helmholtz, 757n137, 760n160 in Hussed, 779n258, 781n268 Toletus, Franciscus, 46, 49, 59n17-19, 60n20-21, 61n26 Torricelli, Evangelista, 24, 31, 33, 35, 37-38, 79
851 Transcendental Analytic, 91-92, 129-30 Trautman, Andrzej, 463 triangle, 79, 120-22, 128-29, 131n8, 162, 166, 187n24, 543, 577, 587-90,594,645,685,701, 732n20 uncertainty principle, 397, 447, 548, 728,801n355 universal characteristic, 159-60, 162-63, 168, 181-85, 186nl0, 187n12, 188n37, 752nll0 universal language, 159, 181
Valerio, Luca, 29, 39n15 validity, 31, 67, 151n26, 172, 261-74,286,304,325,336, 348,369,388,401,412, 419n12, 436-37, 455, 466,494, 524,530,553,587,595,611, 641,652-54,657,660,668, 672,681,687,691-92,694-96, 732n19, 738n44, 745n80, 761n163,764n185,n188, 765nI92,775n243,780n264, 781n267, 791n312 values, 261-64, 267-69, 272, 283, 289, 314,403n3,417, 509n2, 513, 528,687, 764n187, 765n189,n193,795n328 van Rijn, Rembrandt Harmenszoon, 272 Verdaguer, Enric, 392, 401 verification, 55, 313, 317, 320, 327n12, 339, 344,411,480, 531,541,585,593,688,709, 721-22, 763n179, 789n302, 797n340 Vienna Circle, 261, 301, 307-308, 352n5,637,664-65,667-69 Visser, Matt, 392, 399, 40 I Viviani, Vincenzo, 31 von Holland, Baron Georg Jonathon, 117, 121, 132n15
852
Index
Von Neumann, John, 453n24, 728, 802n358 von Zedlitz, I(arl Abraham Freiherr, 114, 130n2
Wald, Robert M., 398, 463 Walker, Arthur Geoffrey, 400 Wallis, John, 77-79, 576 Warburg, Aby, 264, 269, 274n7-8, 275n17-18 Warburton, Willian1, 122 Warren, Daniel, 105, 109n19, 229n3, 239n51, 240n62, 744n73 Waclcins, John, 513, 730n9 vVertheimer, Max, 370, 382n4, nIl Westfall, Richard, 3, 9, 15, 39n30, 84n16, 108n9 Weyl, Hermann, 343-44, 351nl, 352n7, 460, 463-65, 467n16, 684-85, 697, 728, 777n254, 778n255,n257, 786n289, 789n305,798n343,802n360 Wheeler, John, 350, 781n269 Whitehead, Alfred North, 271, 322-23,496,667
Wigner, Eugene, 415-17, 420n21 Williams, L. Pearce, 197-198, 755n125 Wilson, Mark, 19n18, 416, 604, 793n323 Windelband, Wilhelm, 264, 771n218 Winrne, John, 473, 480-82, 489nl0 Winternitz, Josef, 345 Wittgenstein, Ludwig, 496, 553, 668 Wolff, Christian, 114-18, 121-22, 126, 130n4, 131n7-8, 132n16, ISln33, 155-57, 163-69, 177-85, 186nl, 187n23-2S, 188n27-28,n31-32,n34-36, n38,190n53,n57-58,686, 739n50 Woodward, James, 432, 452n11 Worrall, John, 2, 18n5, 490n13, 514-15 Wren, Christopher, 576
Zanghi, Nino, 446 Zeno, 563 Ziehen, Theodor, 281 Zschimmer, Eberhard 342
