Causality, Method, and Modality: Essays in Honor of Jules Vuillemin (The Western Ontario Series in Philosophy of Science)

CAUSALITY, METHOD, AND MODALITY

THE UNIVERSITY OF WESTERN ONTARlO SERIES IN PHILOSOPHY OF SCIENCE A SERIES OF BOOKS IN PHILOSOPHY OF SCIENCE, METHODOLOGY, EPISTEMOLOG LOGIC, HISTORY OF SCIENCE, AND RELATED FIELDS

Managing Editor ROBERT E. BUTTS

Dept. of Philosophy, University of Western Ontario, Canal

Editorial Board JEFFREY BUB, University of Western Ontario L. JONATHAN COHEN, Queen's College, Oxford WILLIAM DEMOPOULOS, University of Western Ontario WILLIAM HARPER, University of Western Ontario JAAKKO HINTIKKA, Florida State University, Tallahassee CLIFFORD A. HOOKER, University of Newcastle HENRY E. KYBURG, JR., University ofRochester AUSONIO MARRAS, University of Western Ontario .rORGEN MITTELS1RASS, Universitiit Konstanz JOHN M. NICHOLAS, University of Western Ontario BAS C. VAN FRAASSEN, Princeton University

VOLUME 48

CAUSALITY, METHOD, AND MODALITY Essays in Honor of Jules Vuillemin With a Complete Bibliography of Jules Vuillemin

Edited by Gordon G. Brittan, Jr. Regents Professor of Philosophy, Montana State University

KLUWER ACADEMIC PUBLISHERS DORDRECHT I BOSTON I LONDON

Library of Congress Cataloging-in-Publication Data Causality. method. and modality: essays in honor of Jules Vui11emin I edited by Gordon G. Brittan. Jr. p. cm. -- (The University of Western Ontario series in philosophy of science; v. 48) Essays in English or French. Includes bibliographical references. ISBN 0-7923-1045-4 (a1k. paper) 1. Phi losophy. 2. Vui 11emin. Jules. I. Vui 11emin. Jules. II. Brittan. Gordon G. III. Series. B29.C336 1990 110--dc20 90-19840

ISBN 0-7923-1045-4

Published by Kluwer Academic Publishers. P.O. Box 17.3300 AA Dordrecht. The Netherlands. Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel. Martinus Nijhoff. Dr W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers. 101 Philip Drive. Norwell. MA 02061, U.S.A.

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Table of Contents

PREFACE

vii

Jules Vuillemin / "Ma vie en bref'

1

Patrick Suppes / "Indeterminism or Instability, Does It Matter ?"

5

Erhard Scheibe / "Covariance and the Non-Preference of Coordinate Systems"

23

Pierre Laberge / "Kant's 'Platonic' Argument in Behalf of the A Priori Character of the Representation of Space"

41

Duchesneau / "The Sense of the A Priori Method in Leibniz' s Dynamics"

53

Gilles Gaston Granger / "Methode axiomatique et idee de systeme dans l'oeuvre de Jules Vuillemin"

83

Gordon G. Brittan, Jr. / "Algebra, Constructibility, and the Indeterminate"

99

Fran~ois

Karel Lambert / "On Whether an Answer to a Why-Question Is an Explanation If and Only If It Yields Scientific Understanding"

125

Ruth Barcan Marcus / "Some Revisionary Proposals About Belief and Believing"

143

Brian Skyrms / "Quantification, Modality, and Semantic Ascent"

175

David Wiggins / "Temporal Necessity, Time and Ability: a philosophical commentary on Diodorus Cronus' Master Argument as given in the interpretation of Jules Vuillemin"

185

Jules Vuillemin / "Replies"

207

List of the Publications of Jules Vuillemin, 1947-1989

225

Preface Deservedly so, Jules Vuillemin is widely respected and greatly admired. It is not simply that he has produced a large body of outstanding work, in many different areas of philosophy. Or that he combines to an unusual degree rigorous standards with a very wide perspective. Or even that in his path-breaking accounts of algebra, of !)escartes, of Kant and of Russell, he showed in new and profound ways how the histories of science and philosophy could be used to illuminate each other. It is also that he has pursued the application of formal techniques and the defense of liberal institutions with a rare singlemindedness and courage. In a time and place where the former were generally ignored and the latter often attacked, he carried on, at some personal cost, embodying a traditional and ideal conception of the philosophical life, bridging national differences. Those who know him also treasure his friendship. Always curious, he delights in new facts and new experiences, and continually heightens the perception of those around him. Almost yearly, at the College de France he introduced brand new courses always with fresh and fruitful inSights. Exceptionally solicitous, he follows the lives of the families around him in great detail. The devotion of his students is legend. His personal energy is also legend. Many of us have followed him bounding up the stairs two at a time or through the gardens of the Luxembourg, his wit and irony apace. Some of us have been fortunate enough to ski with him along the ridges of the Jura dividing France and Switzerland, arriving back at his mountain home exhausted and famished, to enjoy the unending hospitality that he and his wife Gudrun, herself an established scholar, provide. The participation in this volume d'hommages of well-know philosophers from widely diverse areas of interest and a number of different countries testifies to the importance and influence of Jules Vuillemin's work. We hope that this volume, presented to him in his vii

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PREFACE

seventieth year and upon retirement from a distinguished career at the College de France, will underline the importance and extend the influence. In the Vie de Rance, Chateaubriand wrote: "Tout est fragile; apres avoir vecu quelque temps, on ne sa it plus si on a bien ou si on a mal vecu." Tout est fragile. But from a provisional point of view, who would not agree that Jules Vuillemin has tees bien vecu.

MA VIE EN BREF

I was born the 15th of February, 1920, at Pierrefontaine-IesVarans, a village in the Doubs. My father had entered the civil service, in the reconquered province of Lorraine, first at the prefecture of Metz, then at the sous-prefecture of Chateau-Salins, a small countrytown at that time both comfortable and prosperous, where I passed my childhood. Of my secondary and preparatory studies at the College de jesuites in Metz, then at the Lycee Louis-Ie-Grand in Paris, I have retained the memory of a liberal, but diSciplined instruction, isolated from the quarrels and noises of the outside world, solid in ancient languages, in French, in history, in mathematics - the ax had not yet been applied to separate letters and sciences - behind the times in physics: I heard the special theory of relativity mentioned as a curiosity in 1936. My baccalaureat in hand, I made a trip to Berlin in August, 1936. It opened my eyes to the imminence of hostilities. France, caught up in class struggles, strikes, local disorders, in the conflict in Spain, engaged in speeches instead of acting. A few weeks before the AnglO-French declaration of war, I entered the Ecole Norrnale Superieure for a period of four years. The circumstances seemed hardly favorable for philosophical studies. Nonetheless, I remember how it felt: the "dr6le de guerre," the mobilization of my class in May, 1940, the overwhelming defeat, several months in the "Camps de jeunesse" of Marshall Petain, the return to an occupied Paris, a failed attempt to get to England, far from undermining the scholastic background necessary for philosophical reflection, gave to it a kind of seriousness and urgency

G. G. Brittan (ed.), Causality, Method, and Modality © Kluwer Academic Publishers 1991

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which are ordinarily missing in peace time. At the Ecole Normale, Cuzin served as my mentor and introduced me to Kant. At the Sorbonne, I took the courses of Bachelard, Brehier, Gouhier, and Cavailles. Cuzin and Cavailles were shot for the part they played in the resistance. Having received the agregation in philosophy in 1943, I was named a teacher at the Lycee de Besan~on, not far from the village where my family had located. Teaching is the best way of learning, for the student stops at the point where the master has not understood. I did a survey, during my Besan~onian year, of the shadowy zones heretofore covered over by scholarly rhetoric. The war, however, pressed on both town and school. Four of my students were arrested. One of my friends was deported to Buchenwald. He was declared "disappeared" the day the camp was liberated by the American army. Guerillas organized. I rejoined my village group. The ebb-tide, with its dramas, began. Happily, the army, disembarked at Toulon, hurried on its way. It happened in a night. In the morning we were liberated. When armored cars and trucks left for the East, I thought the war was over. I was mistaken by more than six months. I re-entered Paris in a holiday mood and obtained, for five years, a position at half-salary at the Centre National de la Recherche Scientifique. I set myself the task of making clear what remained of rationalism when the Absolute was removed from its foundation. The history of Kantian interpretation showed the interpreters coming to grips with this question. The French philosophical world split at the time principally between the dogmas of Existentialism and Marxism: there an ontology of anguish and nausea, here an unscrupulous activism which, in Paris, continues to pervert thought even to the present day. In my doctoral thesis, I sought to examine

MAVIE

3

these dogmas critically, in order to take from them whatever finite reason was able to find worth discussing. I married and had two children, a girl and a boy. Maurice Merleau-Ponty, elected to the Sorbonne, proposed me as his successor at Lyon. I was not chosen. After a short stay at the Lycee, the Ministry named me, on condition that I reside there, to the University of Clermont-Ferrand, where I lived until 1962. I found old comrades there and made new friends. I was able to appoint assistants, all of whom have since made their way in the world. Up to this point, I had worked without method. I asked Descartes and Kant to reveal theirs to me. I noticed that their thought remained scarcely intelligible and sometimes impenetrable if one did not go back to the sciences which had inspired them or which they had created. Under the direction of Pierre Samuel, who taught in the science faculty, I deepened my mathematical knowledge a little. I was encouraged by Martial Gueroult, professor at the College de France, who honored me with his friendship. Gueroult, an historian of philosophy, insisted on the architectonic and proof methods by which philosophical systems characterize themselves. Around him we formed a kind of school - Ginette Dreyfus, Victor Goldschmidt, Louis Guillermit, and I - and worked together in concert. At the Sorbonne I had not received any education in logic. I learned it in the pages of Russell. Going from one idea to another, I discovered philosophy, or rather contemporary Anglo-Saxon philosophies, almost unknown in France at that time. In 1962, Gueroult sponsored me and brought about my election to the College de France, as successor to the suddenly deceased Merleau-Ponty. The retirement of Gueroult, my divorce, several visits abroad, in particular to Canada and to the United States, my remarriage, the explosion of 1968 at the College de France where I observed the revolutionary gesticulation, the death of Gueroult, of Goldschmidt, of

JULES VUILLEMIN

4

Guillermit, of Ginette Dreyfus, and of several others who are not inscribed in the philosophical register, a trip to Japan, such are the events which marked my life during this period. I had a difficult time at the beginning in Paris. Two auditor~ in my course on Russell, lost in an enormous room, permitted very few illusions. I did not give up. I continued to apply logical analysis, whose virtues I had just discovered, to the theologies of Aristotle and St. Anselm, to the constitution of the sensible world, from Russell and Whitehead to Carnap and Goodman. Making my way, however, a difference between me and the majority of Anglo-Saxon analysts emerged. There were those who, singlemindedly interested in chasing down grammatical errors in the talk of philosophers, forgot the existence of SCientific languages. But even those who applied the method of "rational reconstruction" to these latter more often imposed on them principles of their own choice. I resisted this violence done to history, and trusted in the sciences such as they are, and not such as they should be. Moreover, it is presumptuous to neglect the philosophical tradition. It therefore appeared to me necessary from the outset to put some order into this tradition by proposing a rational classification of possible types of philosophical systems. That done, it was necessary to ask whether contemporary science - and especially quantum mechanics - favors the selection of a given type of philosophical system or whether, taking the traditional classification apart, it suggests new concepts and new principles to philosophy. It is with the response to this question that I am now occupied. Jules Vuillemin

INDETERMINISM OR INSTABILITY, DOES IT MATTER?·

Patrick Suppes 1. Skepticism About Determinism

In my recent book, Probabilistic Metaphysics, I have argued at some length against determinism as a viable philosophical or scientific thesis. I want first to review those arguments and then go on to look at an alternative way of viewing phenomena. Instead of the dichotomy deterministic or indeterministic, perhaps the right one is stable or unstable. In expressing my skepticism about determinism I shall not linger over a technically precise definition. It seems to me that the intuitive notion that phenomena are deterministic when their past uniquely determines their future will serve quite adequately in the present context. The natural basis of skepticism is our remarkable inability to predict almost any complete phenomenon of interest, and even more, our inability to write down adequate difference or differential equations. Consider, for example, a gust of wind and its effect on

-The first draft of this paper was read at a symposium on indeterminism at the annual meeting of the Pacific Division of the American Philosophical Association, March 23, 1985. A later draft was the basis of a lecture at the College de France at the invitation of Jules Vuillemin on May 12, 1986. It is a pleasure to dedicate this paper to him. Our many conversations on a variety of philosophical topics have both enlightened and delighted me. 5

G. G. Brittan (ed.), Causality, Method, and Modality © Kluwer Academic Publishers 1991

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PATRICK SUPPES

leaves of grass, the branches of a tree, the particles of dust agitated in various ways. It seems utterly out of the question to predict these effects in any detail. Moreover, it seems hopeless even to think of writing down the equations, let alone solving them. It might be noted the particles of dust, at least, would be within the range of the phenomena of Brownian motion, and the hopelessness of actually predicting such motion has been recognized for a long time. Of course, this example of Brownian motion raises a problem that needs remarking. One standard view of classical physics is that all phenomena are deterministic,--it is just that we are unable to analyze some phenomena in adequate detail. But even here there is reason for skepticism. The standard result of the standard theory of Brownian motion is that because of the high incidence of collisions the path of a particle is continuous but differentiable almost nowhere (only on a set of measure zero). Given that the path is this kind of trajectory, it becomes obvious that determinism is out of the question just because of the many collisions. It is a familiar fact of classical mechanics that collisions in general cause great difficulty for deterministic theorems. The kind of result that we have in the case of Brownian motion is not just a matter of difficulty, it is a matter of principled hopelessness. So I take it that insofar as the phenomena I have just described fall within the purview of the theory of Brownian motion, determinism is ruled out. For many familiar human phenomena we do not even have the elements of schematic analysis given by the probabilistic theory of Brownian motion. Examples are easy to think of. A favorite of mine is the babble of speech. The idea of ever being able to determine the flow of talk even between just one set of persons, not to speak of a billion, given whatever knowledge you might hope to have seems ridiculous and absurd. There is no reason whatsoever to think we will ever have theories that lead to deterministic results. It is certainly true that in occasional high states of deliberation we formulate very

INDETERMINISM OR INSTABILITY

7

carefully the words we are going to utter, but this is not the standard condition of speech. Moreover, even in such states of high deliberation we do not and are not able consciously to control the prosodic contours of the utt~rance. In fact, as we descend from the abstract talk of grammarians and model theorists concerned with semantics to the intricate details of the actual sound-pressure waves emitted by speakers and received by listeners, the problem of having a deterministic theory of speech looms ever more hopeless. I have the same skepticism toward deterministic theories of vision. Such a theory for any serious level of detail seems out of the question. The reasons for thinking this are many in number. The long history of theories of vision and the difficulties we still have in giving detailed partial descriptions of what the visual system is sensing provide some evidence. Detailed physiological studies showing that the human eye is sensitive to even a single photon provide other kinds of evidence, as do quantitative studies of eye and head movements. The extraordinarily complicated nature of the transduction that takes place in the optical system in order to send messages to the central nervous systems is another case in point. Someone might want to claim that we could have a gross deterministic theory of vision, but such a theory would be superficial and uninteresting. The actual mechanisms seem intrinsically subtle and complex. Of course, there are some kinds of complex problems that we feel confident in tackling, but anyone who has taken a serious look at problems of vision will back away rather rapidly from optimistic claims about having within the framework of contemporary science, or science as we can foresee it to be in the future, a workable, detailed deterministic theory. What I have had to say about speech and vision applies also to the sense of smell. The evidence seems pretty good that this sense is sensitive down to the presence of a substance at the molecular level. Moreover, what theories there are of the activities of single

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recognition cells are probabilistic in character. As far as I know, no one has attempted to propose a serious deterministic theory of smell. These familiar phenomena I am using to buttress my reasons for skepticism about determinism are easily matched by a dozen others. Given the extraordinarily small number of phenomena about which we can have a deterministic theory, there is cause for psychological and philosophical speculation as to why the concept of determinism has ever achieved the importance it has in our thinking about the world around us. To adopt a broad deterministic view toward the world does require not quite the extreme faith of the early Christians, but at least that of such diverse eighteenth-century optimists as Kant and Laplace-. Surely one psychological root of the faith in determinism is its conflation with prediction. Hegel (1899, p. 278) reports that Napoleon in a conversation with Goethe remarked that the conceptual role of fate in the ancient world has been replaced by that of bureaucratic policy in modern times with the implication that uniform predictability of individual behavior subject to the bureaucracy is, in principle, what we can now have. The search for methods of prediction has ranged from zodiacs to chicken gizzards and is found in every land. The primitive urge to know the future has in no way been stilled by modern science, but only rechanneled into more austere forms. The new skepticism. so I am arguing, should be about the omnipresence of determinism, not the omniscience of God.

*Historically we probably need to think of Kant as a cryptodeterminist.

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2. How to Save Determinism

Before making some direct comparisons with indeterminism in terms of instability there are some preliminary points to be made about unstable systems. The intuitive idea of instability in mechanics is this. Wide divergence i.n the behavior of two systems identical except for initial conditions is observed even when the initial conditions are extremely close. There are two aspects of unstable systems that make prediction of their behavior difficult, and therefore make difficult the realization of the deterministic program, even if the systems are, in fact, deterministic. One source of difficulty is that the initial conditions can be measured only approximately. If a system is not stable in the appropriate sense--I omit a technical definition here but it is straightforward to give one--, it will be impossible to predict its behavior for any but short intervals of time with any accuracy. In this case, we attributed predictive failures to a possibly small uncertainty in the initial conditions. We shall leave aside in the present discussion whether this uncertainty should be treated epistemologically or ontologically. Some later remarks will have something more to say about this issue. A second aspect of an unstable system can be that the solutions are not given in closed form, and calculations based on various methods of series expansion, etc., will not give accurate predictions. In other words, we cannot count on numerical methods to give us a detailed result for periods of prediction of any length. If the system is unstable, the accumulation of small errors in numerical methods of approximation, which may be the only ones available, can lead to unavoidable problems of accuracy. This last problem is especially true of systems that are governed by nonlinear differential equations. What I have said thus far applies to very simple systems of differential equations as well as complex ones. The solutions of the equations may be unstable but they do not seem to exhibit the kind

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of behavior we so directly associate with indeterministic or probabilistic behavior. It might be argued that the simplest systems of linear differential equations that are unstable do not represent something comparable to indeterminism. Yet it is true that for such unstable linear systems the accuracy of predictions will be poor, given, as is always the case in real situations, any errors in the measurement of initial conditions. In other words, unstable deterministic linear systems capture an important aspect of indeterminism, namely, our inability to predict future behavior on the basis of knowledge of present behavior. There is another aspect also of such linear systems that needs to be noted. In most applications, the linearity of the real system that is being modeled by the linear differential equations is only approximate. Almost always, deviations from linearity in the real system--the fact that the linear differential equations are only approximations--J, will make our ability to predict actual phenomena even more limited.

3.

Chaos and Symbolic Dynamics

We now get down to essentials. Those special unstable solutions of differential equations that exhibit chaotic behavior provide the intended alternative to indeterminism. It would have been more accurate in certain ways to entitle this paper 'Indeterminism or Chaos, Does it Matter?', but the meaning of chaos is too special, and so it is the central concept of instability that should be kept to the fore. So, what do we mean by chaos? A brief but not quite technically correct definition is the following. A solution of a deterministic system of differential equations is chaotic if and only if it exhibits some aspect of randornness--or, as an alternative, sufficient complexity. To some, this definition would seem to embody a contradiction, and therefore no solutions would satisfy it. On the left-

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hand side we refer to a deterministic system of equations and on the right-hand side to the random character of its solution. How can a deterministic system have a random solution? This is what chaos is all about, and the discovery of the new phenomena of chaos is certainly a watershed change in the history of determinism. Before turning to the recent discussions of chaos, it will be useful to go back over the earlier history of developing the theory of random processes within classical mechanics. The origin of the approach, usually called the method of arbitrary functions for a reason to be explained in a moment, originates with Poincare, but has been developed in detail by a number of mathematicians in the first half of this century. Already a rather short qualitative sketch of the ideas in very accessible form is given by Poincare in Science and Hypothesis (1913). (The history of developments since Poincare has been chronicled in some detail by von Plato (1983).) Here I shall just give a sketch of the analysis of coin flipping, one of the most natural cases to consider. To a large extent I shall follow the recent treatment due to Keller (1986), but as somewhat modified in Suppes (1987). Without going into details, we shall assume a circular coin that is symmetric in all the ways you would imagine; second, dissipating forces of friction are entirely neglected; third, it is assumed that the coin does not bounce but on its initial point of impact flattens out to a horizontal position. In other words, from the initial point of impact the face up does not change. With this idealized model, the physical analysis is simple. Newton's ordinary law of gravity governs the vertical motion of the particle--we assume there is no horizontal motion. Second, we assume that the rotational motion is that of constant angular velocity so there is no angular acceleration to the rotation. Now with this situation, if we knew the exact initial conditions, we could predict exactly how the coin would land, with either heads or tails face up. In fact, the classical analysis of this case assumes rightly enough that we do not know the exact value of the

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initial conditions. The method of arbitrary functions refers to the fact that we assume an arbitrary probability distribution of initial vertical velocity and initial rotational velocity. Then as the initial velocity tends to infinity, whatever the arbitrary distribution we begin with, the probability of a head will be one-half. In other words, the symmetry in the mechanical behavior of the system dominates completely as we approach the asymptotic solution. Of course, in real coin-flipping situations we are not imparting an arbitrarily large vertical velocity to the coin, but the variation in the way that we flip will lead to a very good approximation to one-half. The point is that in this typical analysis, the randomness enters only through the absence of knowledge of initial conditions. It is an important example of randomness in mechanical systems, one that has only recently begun to be recognized again as an important example, but it is not the kind of example on which I want to concentrate here. To show that the conventional philosophical dichotomy between determinism and randomness is mistaken, I consider two important and much discussed examples. The first is a special case of the three-body problem, certainly the most extensively studied problem in the history of mechanics. Our special case is this. There are two particles of equal mass moving according to Newton's inverse-square law of gravitation in an elliptic orbit relative to their common center of mass which is at rest. The third particle has a nearly negligible mass, so it does not affect the motion of the other two particles, but they affect its motion. This third particle is moving along a line perpendicular to the plane of motion of the first two particles and intersecting the plane at the center of their mass. From symmetry considerations, we can see that the third particle will not move off the line. The restricted problem is to describe the motion of the third particle. The analysis of this easily described situation is quite complicated and technical, but some of the results are simple to state in informal terms and directly

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relevant to my focus on determinism and randomness. Near the escape velocity for the third particle--the velocity at which it leaves and does not periodically return, the periodic motion is very irregular. In particular, the following remarkable theorem can be proved. Let t l' t2,... be the times at which the particle intersects the plane of motion of the other two particles. Let sk be the largest integer equal to or less than the difference between tk+1 and tk times a fixed constant-. Variation in the sks obviously measures the irregularity in the periodic motion. The theorem, due to the Russian mathematicians Sitnikov (1960) and Alekseev (1968a,b; 1969a,b), in the version given by Moser (1975) is this.

Given that the eccentricity of the elliptic orbit is positive but not too large, there exists an integer, say such that any infinite sequence of terms sk with sk::2 a corresponds to a solution of the deterministic differential equation governing the motion of the third particle··. mEOREM.

a

A corollary about random sequences immediately follows. Let s be any random sequence of heads and tails--for this purpose we can use any of the several variant definitions--Church, Kolmogorov, Martin-LOf, etc.. We pick two integers greater than to represent the random sequence--the lesser of the two representing heads, say, and the other tails. We then have:

a

-The constant is the reciprocal of the period of the motion of the two particles in the plane. **The correspondence between a solution of the differential equation and a sequence of integers is the source of the term symbolic dynamics. The idea of such a correspondence originated with G.D. Birkhoff in the 1930's.

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Corollary. Any random sequence of heads and tails corresponds to a solution of the deterministic differential equation governing the motion of the third particle. In other words, for each random sequence there exists a set of initial conditions that determines the corresponding solution. Notice that in essential ways the motion of the particle is completely unpredictable even though deterministic. This is a consequence at once of the associated sequence being random. It is important to notice the difference from the earlier coin flipping case, for no distribution over initial conditions and thus no uncertainty about them is present in this tree-body problem. No single trajectory in the coin-flipping case exhibits in itself such random behavior. This example demonstrates the startling fact that the same phenomena can be both deterministic and random. The underlying explanation is the extraordinary instability of the deterministic phenomena. Before remarking further on the significance of this result, I turn to the second example which is an abstract discrete model of period doubling. Because the mathematics is more manageable it is a simple example of a type much studied now in the theory of chaos. The example also illustrates how a really simple case can still go a long way toward illustrating the basic ideas. Let f be the doubling function mapping the unit interval into itself. (1) xn+l = f(x n) = 2xn(mod 1), where mod 1 means taking away the integer part so that xn+llies in the unit interval. So if Xl = 2/3, Xz = 1/3, x3 = 2/3, x4 = 1/3 and so on periodically. The explicit solution of equation (1) is immediate: (2) xn+l + 2nx l (mod 1). With random sequences in mind, let us represent Xl in binary decimal notation, Le., as a sequence of l's and O's. Equation (1) now can be expressed as the rule: for each iteration from n to n+ 1 move the decimal point one position to the right, and drop whatever is to

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the left of the decimal point: .1011...-+ .011 .... We think of each xn as a point in the discrete trajectory of this apparently simple system. The remarks just made show immediately that the distance between successive discrete points of the trajectory cannot be predicted in general without complete knowledge of x l' If xl is a random number, i.e., a number between 0 and 1 whose binary decimal expansion is a random sequence, then such prediction will be out of the question unless xl is known. Moreover, any error in knowing xl spreads exponentially--the doubling system defined by equation (1) is highly unstable. Finally, it is a well-known result that almost all numbers are random numbers in the sense defined. Although the exact technical details are rather complicated for almost all chaotic systems, the first example of a restricted threebody problem was meant to illustrate orbital complexity and the second complexity of initial conditions. In any case, randomness can be an essential part of the behavior of what would seem to be quite simple deterministic systems.

4. The Troublesome Case of Quantum Mechanics. From what I have just said, the dements of a rejoinder to my earlier skepticism about determinism are apparent. The phenomena cited as examples of indeterminism are in fact just examples of highly complex, unstable deterministic systems whose future behavior cannot be predicted. The strongest argument against such view comes from quantum mechanics. Beginning in the 1930's there has been a series of proofs that deterministic theories are in principle inconsistent with quantum mechanics. The first proof of the impossibility of deterministic hidden variables was by von Neumann. The latest arguments have centered on the inequalities first formulated in 1964 by John Bell.

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Moreover, the associated experiments that have been performed have almost uniformly favored quantum mechanics over any deterministic theory satisfying the Bell inequalities. To those who accept the standard formulation of quantum mechanics, the various proofs about the nonexistence of hidden variables answer decisively the question in the title of this lecture. Indeterminism or instability, does it matter? For these folk the answer is affirmative. The negative results show chaotic unstable deterministic mechanical systems cannot be constructed to be consistent with standard quantum mechanics. The conclusion of this line of argument is that standard quantum mechanics is the most outstanding example of an intrinsically indeterministic theory. There is, however, a still live option for those of us who are not entirely happy with the orthodox theory of quantum mechanics and its many peculiar features. The option left open is to account for quantum phenomena in terms of something like the theory of Brownian motion, which is, of course, part of classical mechanics broadly construed. Nelson 0%7, 1985) has provided thus far the best defense of this approach. He has, for example, derived the Schroedinger equation, the most important equation of nonrelativistic quantum mechanics, from the assumptions of Newtonian mechanics. However, his recent analysis (985) ends up with Bell's theorem and the relevant experiments as a serious problem. The most feasible way out seems to be to develop a non-Markovian stochastic mechanics, which in itself represents a departure from classical nonlocality. The central problems of current physics are not much concerned with this alternative, but mathematicians and philosophers will continue to puzzle over the foundations of this century's most successful scientific theory. As long as the stochastic view in the sense of Brownian motion remains a viable option, the question posed in the title can be answered by a skeptical "Perhaps not". Consistent with this view, Laplace's concept of probability and thus

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17

of indeterminism also remains a viable option--probability is the expression of ignorance of deterministic causes.

5. Randomness as a Limiting Case of Unstable Determinism The existence of deep-seated randomness inside deterministic systems can be attributed to their great instability, and this suggests the road of rapprochement between determinism and randomness. A striking feature of randomness and instability is complexity. Moreover, recent definitions of randomness are in terms of complexity. The complexity of a sequence of finite symbols is measured by the length of a minimal computer program that will generate the sequence. (For asymptotic purposes, the particular computer or computer language does not matter.) A simple alternating sequence of l's and O's can be generated by a very short program. More intricate sequences require longer programs and are therefore more complex. Where this argument is going should be apparent. Random sequences are of maximal complexity. In fact, the programs required to generate them would have to be infinitely long. So what are random sequences? They are the limiting case of increasingly complex deterministic sequences. Randomness is just a feature of the most complex deterministic systems. And what of particular importance follows from this? The separation of determinism and predictability. The most complex deterministic systems are completely unpredictable in their behavior. Laplace's "higher intelligence" must be transfinite. He must be able to do arbitrarily complex computations arbitrarily fast. To give a modern ring to Laplace's basic idea, I propose this. Randomness is tbe expression of maximally complex deterministic causes.

PATRICK SUPPES

18

6. Does It Matter? Setting aside, for the moment, the -problem of hidden variables in quantum mechanics, we may argue that the philosophically most interesting conclusion to be drawn from the analysis outlined in this paper is that we cannot distinguish between determinism and indeterminism. The true-blue determinist can hold, without fear of contradiction, that all processes are determined. Confronted with the myriad examples of natural phenomena that cannot be predicted and that seem hopeless to try to predict, he can reply with serenity that even these processes are deterministic, but they are also unstable. The determinist can agree amiably enough that there are processes yet to be analyzed and that his belief that they too will turn out to be deterministic is only based on past experience. This last remark is meant to ring a Bayesian bell. Pure Bayesians are natural true-blue determinists. After all, de Finetti begins his two-volume treatise on probability by printing in capital letters: PROBABILIlY DOES NOT EXIST, a thesis Laplace would have heartily endorsed. The indeterminist, for his part, can just as firmly hold on to his beliefs, directly supported as they are by the phenomenological data in so many areas of experience. Moreover, with the possible exception of quantum mechaniCS, there seems to be no current possibility of giving a knockdown argument for either determinism or indeterminism. Under either theoretical view of the world, most natural phenomena cannot be analyzed in detail, and even less can be predicted. How drastic and serious these limitations are is not sufficiently appreciated. I gave a number of obvious examples in the first section, but even in that presumed citadel of mathematically developed science, classical mechaniCS, it is beyond our current capabilities to analyze a general

INDETERMINISM OR INSTABILITY

19

system of one particle having a potential with just two degrees of freedom·. Whichever philosophical view of the world is adopted, the impact on theoretical or experimental science will be slight. Probability has a fundamental role no matter what, and statistical practice is complacently consistent with either determinism or indeterminism. (The assumption of determinism plays no systematic role in Bayesian statistics, for example.) There remains the question of whether proofs of no hidden variables in quantum mechanics make a decisive argument against classical determinism. I have mentioned already some reasons for not accepting these results as the last word. I want to conclude with a more general argument. The essential point is the exceedingly thin probabilistic character of quantum mechanics. Roughly speaking, no correlations or other interactive measures can be computed in quantum mechanics. Perhaps most important, if we are examining the trajectory of a particle, no autocorrelations can be computed, Le., correlations of position at different times, but such a statistic is a most natural measure of probabilistic fluctuation in the temporal behavior of a particle. The probabilistic gruel dished out by the wave function of a quantum-mechanical system is too thin to nourish any really hearty indeterminist. Paradoxically enough, the reconstruction carried out so far of quantum phenomena within classical mechanics

•A system of one particle with two degrees of freedom is a system defined by the differential equations ··x - f(x), where x is a vector in the plane and f is a vector field on the plane. The system has a potential if there is a function U from the plane to the real numbers such that f - -dUJax.

20

PATRICK SUPPES

is probabilistically much richer. It would be ironical indeed if the deepest probabilistic analyses of natural phenomena turn out to be within a deterministic rather than indeterministic framework. Stern Professor of Philosophy, Stanford University

INDETERMINISM OR INSTABILITY

21

References Alekseev, V. M. 096Ba), Quasirandom dynamical systems. I. Quasirandom diffeomorphisms. Mathematicheskie USSR Sbornik 5, 73-128. Alekseev, V. M. o 96Bb), Quasirandom dynamical systems. II. One-dimensional nonlinear oscillations in a field with periodic perturbation. Mathematicheskie USSR Sbornik 6, 505-560. Alekseev, V. M. 0969a), Quasirandom dynamical systems. III. Quasirandom oscillations of one-dimensional oscillators. Mathematicheskie USSR Sbornik 7, 1-43. Alekseev, V. M. o %9b) , Quasi-random dynamical systems. Doctoral Dissertation. Mathematicheskie Zametki 6(4), 489-498. Translation in Mathematical Notes, Academy of Sciences, USSR 6(4), 749-753. Bell,]. (964), On the Einstein Podolsky Rosen paradox. Physics 1, 195-200. de Finetti, B. (1984), Theory of Probability, Vol. 1, New York: Wiley. Hegel, G. W. F. (1899), The Philosophy of History 0. Sibree, trans.) Colonial Press. Reprinted by Dover Publications, New York, 1956. Keller,]. B. (986), The probability of heads. American Mathematical Monthly 93, 191-197. Moser,]. (1973), Stable and Random Motions in Dynamical Systems With Special Emphasis on Celestial Mechanics. Hermann Weyl Lectures, the Institute for Advance Study. Princeton, N.].: Princeton University Press. Nelson, E. 0%7), Dynamical Theories of Brownian Motion. Princeton, N.].: Princeton University Press. Nelson, E. (985), Quantum Fluctuations. Princeton, N.J.: Princeton University Press.

22

PATRICK SUPPES

Poincare H. (913), Science and Hypothesis. Lancaster, Pa.: The Science Press Sitnikov, K. 0%0), Existence of oscillating motions for the threebody problem. Doklady Akademii Nauk, USSR 133 (2), 303-306. Suppes, P. (984), Probabilistic Metaphysics, Oxford:Blackwell. Suppes, P. (984), Propensity representations of probability.

Erkenntnis, 26, 335-358. von Plato, J. (1983) The method of arbitrary functions. British Journal for the Philosophy of Science 34, 37-47.

COVARIANCE AND TIlE NON-PREFERENCE OF COORDINATE SYSTEMS

Erhard Scheibe I

In his famous paper of 1916 on the foundation of general relativity Einstein has formulated the folloWing prinCiple that he himself calls "the postulate of general covariance,,1: (C) The general laws of nature are to be expressed by equations that are valid for all coordinate systems, Le. that are covariant under arbitrary substitutions. This principle Einstein has viewed as a strengthening of what he in the same paper calls "the postulate of general relativity". This postulate is given the following formulation : (R) The laws of physics have to be such that they are valid for arbitrarily moved reference systems. Einstein argues that the relativity postulate follows from the covariance postulate by saying3: For among aD substitutions we at any rate find those which correspond to all relative motions of the (three-dimensional) coordinate systems. In arguing thus Einstein obviously assumes that the reference systems mentioned in the relativity postulate can be described by certain coordinate systems mentioned in the covariance postulate such that the relative motions of the former are described by certain transformations (= substitutions) of the latter. Under this assumption the relativity postulate says the same of every element of a certain set as the covariance postulate says of every element of a much larger set. Obviously, the former would then follow from the latter. 23

G. G. Brittan (ed.), Causality, Method, and Modality © Kluwer Academic Publishers 1991

24

ERHARD SCHEIBE

Only one year after Einstein's basic paper Kretschmann has objected against the above argumentation "that every physical theory, by a mere mathematical ... modification of the equations representing the theory and without changing its content, can be forced to satisfy even the most general relativity postulate" 4. In his rejoinder Einstein immediately admitted this objection and also the general opinion of the physicists - if there is such a thing-seems to have become "that in Einstein's words 5-, of necessity, every empirical law can be given a general covariant formulation". In spite of this giving in and in spite of this general tendency Einstein has continued to argue that physics is in need of a general principle of relativity and that this can be achieved by founding it on a principle of general covariance. Thus, to give but one example, in the widely known book "The Evolution of Physics" written by Infeld but authorized by Einstein, in passing from special to general relativity we read 6: "Can we formulate physical laws so that they are valid for all coordinate systems, not only those moving uniformly, but also those moving quite arbitrarily, relative to each other? If this can be done, our difficulties will be over." Now, whatever is meant by the difficulties to be overcome, it is suggested that we get rid of them merely by satisfying the principle of general covariance (which here is lumped together with the principle of general relativity). And the theory of general relativity then is introduced precisely by this step. In view of Einstein's adherence to his original standpoint in spite of this admission one can hardly escape the conclusion - believe it or not - that in formulating the principles quoted above Einstein did not succeed in fully expressing the idea that was really important to him. This idea must have been of a sort that would not allow admitting for it what Einstein did admit for its seeming equivalent. Already in the paper of 1916 it is spelled out with respect to both principles. In the sentence immediately preceding the quoted covariance postulate (C) Einstein summarizes a foregOing argument by saying7:

COVARIANCE AND NON-PREFERENCE

25

There is no alternative but viewing all possible coordinate systems as to be placed on the same footing in principal (prinzipiell gleichberechtigt). Similarly, with respect to the relativity principle (R) Einstein concludes a corresponding argument by sayingS: (R') Of all possible spaces in relative motion to each other no one must be preferred a priori. And immediately after this sentence there follows the above quoted postulate of general relativity (R). These facts about the text, I think, leave no doubt that in formulating (C) Einstein wanted to express and thought to ~ expressed what he had said more informally and provisionally by (C'), and Similarly with respect to (R) and (R') . But not only the actual wordings of (C) and (C'), as well as (R) and (R'), but also the arguments preceding these sentences show that something different from (C) and (R) was meant by (C') and (R') respectively. Newton's gravitational theory can easily be given a generally covariant (equivalent) formulation 9. But this does not do away with the fact that there are preferred (galilean) coordinate systems. Covariance with respect to a set of coordinate systems may be necessary in order that none of those systems is preferred. But, as the newtonian example clearly shows, it is by no means sufficient. Therefore, since Einstein wanted to avoid preferred coordinate systems (or, for that matter, reference systems) the question still was which was stronger, for the democracy among the systems as not only necessary but also sufficient principle distinguished the theory of general relativity from its predecessor Up to this point my argument concerned covariance as well as relativity. This was because the text of the paper in question shows that Einstein made his mistake with respect to both, and this may (C')

ERHARD SCHEIBE

26

make my claim that there ~ something wrong more plausible. From now on I shall concentrate on covariance, thereby deliberately leaving out the physically more interesting but, at the same time, much more difficult part of the matter. II

To obtain precise explications of the postulates (C) and (C') we have to get a clear idea of that part of a physical theory with respect to which those postulates can be given a precise reformulation. It turns out that the part usually called the "formalism" of a theory is sufficient for this purpose. A concept of formalism recently successfully applied to foundational studies is the concept of species of structures in the sense of Bourbaki lO . Roughly, the axioms of a theory then are statements (1)

L

[Xi 8]

about structures < XjS > where X is a finite system of principal base sets and s a finite system of elements of scale sets over the X, i.e. of sets generated by successively applying the operation of power set of a cartesian product, starting with the X. In this way the base sets X are structured by the typified sets s. It will be important for us that "concrete" sets such as the set of real numbers may, as so called auxiliary base sets, also take part in the process. Species of structures, well known from mathematics, are the algebraic ones, e.g. groups, rings, vector spaces etc., as well as the topological species of topological spaces, differentiable manifolds etc., but also mixed forms such as Hilbert spaces, Lie groups, etc. That also the formal part of a physical theory can be reconstructed as a species of structures will become plausible in particular through the species of geometrical structures that we are going to study.

COVARIANCE AND NON-PREFERENCE

27

Before doing so we have to introduce a second concept on the general level: equivalence of species of structures. For already Einstein's statement, connected with (C), "that every empirical law can be given a generally covariant formulation" signalizes that general covariance is not the invariable concomitant of a theory. The most we can expect in general is the existence of a covariant version of a theory which then, of course, must be equivalent to the original one. Similarly, the idea of a class of preferred coordinate systems, related to (CO), essentially means that we could have a version of our theory, equivalent to the original one, in which we can get along with that restricted class. In both cases equivalence includes the actual change of the internal concepts of the theory in question, i.e. a change of the typified sets s. We therefore have to provide for equivalence transformations (2)

t

= q[X, 81

, 8 = q- 1 [X, t1

which have to satisfy certain obvious conditions. A familiar equivalence of this type is, for instance, the one between boolean algebras and boolean lattices. In our further investigations we shall be concerned with certain species of geometrical structures only . But it was important to define the concept of equivalence on a most general level. Approaching geometrical theories we begin with the rather special but fundamental case of coordinate geometries. A coordinate geometry is about a space X structured by a set F of (local) coordinate systems and by nothing else. The coordinate systems, relating subsets of X to subsets of Rn , are transformed by (local) coordinate transformations of a given pseudo-group G of Rn 12. The best known coordinate geometry is the species of C00-differentiable manifolds. In this case G is a very large pseudo-group Goo' But we can choose also a relatively small group as, for instance, the Galileo group or the Lorentz group

28

ERHARD SCHEIBE

or the euclidean group (of Rn). With them are associated well known classical geometries underlying various physical theories. It is true that these geometries usually are not formulated as coordinate geometries 13. But they are equivalent (in the sense of (2)) to coordinate geometries. The matter is closely related to Felix Klein's Erlangen program14, and we may therefore call species of structures equivalent to coordinate geometries Klein geometries. Evidently not every geometry in the traditional sense, e.g. riemannian geometry, is a Klein geometry, - let alone physical theories in general. Let us, therefore, generalize the concept of coordinate geometry by the concept of analytical geometry. The structures that are the subject of these geometries are species structured not only by a set F of coordinate systems but also by further typified sets s, - geometrical objects as they are sometimes called. Following the somewhat old-fashioned way of presenting geometry15 we assume the s not only to have but just to be coordinate representations: they are partial (!) functions

(3a)

8 :

Fl.

~ D.

assigning to every coordinate system cP of a subset F of F an element s ( ) in the representation space Os of s. Os is a subset of a scale seconstructed from R, and the underlying pseudogroup G is represented on Os such that in the intersection of any two y, cP , 'II sF,

(3b)

v. (W . cP- 1) . 8(cp) = 8(W)

where rs is the representation of G. Well known examples of geometrical objects in the sense of (3) are fields. For them

(4)

Fl.

= F , D. ~ Pow (Rn

X

R(n H1))

COVARIANCE AND NON-PREFERENCE

29

and the elements of Ds in fact are partial functions on Rn with values in

R(nk+l).

Riemannian metrics, Lorentzian metrics, affine

connections are special cases of fields. But also curves and submanifolds can be viewed as geometrical objects covered by (3). It is, of course, true that such objects as well as fields can also be described by an intrinsic method not directly making use of coordinate systems. In fact, the intrinsic representation of geometrical objects has become fashionable of late 16. However, although this method may be more adequate in some cases, it has to be emphasized that the problems connected with covariance and the non-preference of coordinate systems would become meaningless if we were to follow that method throughout. This may be evident already from the wordings of the principles (C) and (C'): the term "coordinate system" explicitly occurs in them, and it is not to be seen how it could be eliminated without depriving the principles of their essential content. Part of this content is already contained in an assumption that we have to make if we now complete this outline of the concept of analytical geometry. Up to this point we were concerned mainly with the typification of the sets s. If it now comes to the axioms proper it is evident that a condition of invariance has to be imposed on them: Since our geometrical objects are represented by coordinate representations and since these representations are in general different from coordinate system to coordinate system, what we want to say about those objects s in terms of their representations seq»~ must be invariant under coordinate transformations. If, for instance, in differential geometry (G - G ) we were to establish the relation between a tensor field g ~ ~v and an affine connection rlCA.6 that the covariant derivation of the former with respect to the latter vanishes we would express this as usual by

ERHARD SCHEIBE

30

(5)

using any coordinate system. Now the essential thing about a statement like (5) is that it is invariant under coordinate transformations (3b): As a consequence of the special transformation rules for the 8"tv and rA.~, if (5) holds in one coordinate system it holds in any other. Obviously, this is what we have to require in general: It is on pain of inconsistency that we have to require that something said about s(
From now on we will restrict our considerations to the case G ~ Goo. The basic phenomenon to be studied is that there are equivalent analytical geometries 1; and 1;1 where

(6a)

G

cG

l

COVARIANCE AND NON-PREFERENCE

31

(non-trivial inclusion!) and where, correspondingly,

(6b)

Fe q[X; F] (= F 1 )

for the equivalence transformation q that defines the set Fl of coordinate systems of a space < Xi FI,SI > belonging to I:I from the corresponding set F of < Xi F,S> belonging to I:aff . The best known examples of such equivalences are the differential geometric reformulations of the classical geometries. Let us, for instance, define (locally) affine geometry I: as the coordinate geometry belonging to the pseudo-group Gaff of all (locally) affine transformations of Rn. Then I:aff is equivalent to the differential geometry, i.e. G I - Goo ' I:aff of a flat affine connection. Clearly (6a) holds for this case, and (6b) is a consequence of the natural definition of F I as being just the set of coordinate systems on X generated by F and Goo' Similar situations occur by reformulating (locally) euclidean geometry as a special case of riemannian geometry and (locally) minkowskian geometry as the species of flat lorentzian manifolds. It is worthwhile to pause for a moment and ask how the phenomenon described by (6) is possible. In the examples given so far one of the geometries was supposed to be a coordinate geometry. A case more typical for the general situation is the following. We consider a field theory governed by the simple relativistic wave equation (7a)

based on minkowskian geometry as a coordinate geometry. The amazing thing about a differential geometric formulation of this

ERHARD SCHEmE

32

theory is that the equation (7 a) , though invariant under the Lorentz group, simply is not invariant under arbitrary coordinate transformations of G 00 . On the other hand, the wanted formulation certainly has to include an equation that is invariant under the transformations of G00 . How does this come about? The answer is that what is at work in (7a) not only is the wave function f but also the minkowskian metric g which, however, is disguised since it enters the stage only through special coordinate systems for which

for k = 1,2,3 for IL f. v

(Sa)

Thus in fact our dramatis personae are f and g, and there is the wave equation

(7b)

v IA

(glA"!.L) ax"

= 0

evlJ. the covariant differentiation with respect to g) relating f

and an arbitrary lorentzian metric g and invariant under Goo in precisely the same sense as (7a) is invariant under the Lorentz group. Of course, (7b) is still much too general. But if we require g to be flat by the equation (Bb)

RIA "IA>'

=0

(R IJ.vd.' the curvature tensor), likewise invariant under Goo we are led back to the original equation (7a) through the existence of special coordinate systems with (8a).

COVARIANCE AND NON-PREFERENCE

33

The study of such examples does, of course, mean little with respect to the question of general theorems related to our phenomenon. As regards theorems, the principles from which we started come to mind 17 . First, we have seen Einstein suggesting that the laws of nature should be expressed by equations covariant with respect to arbitrary coordinate transformations. Translated into the terminology developed so far this would mean that those laws have to be formulated as axioms of a differential geometry. One reaction to this proposal is that, since we do not yet know the laws of nature, only the future development of physics will tell us whether Einstein was right. But then there came the objection that the postulate might be vacuous after all, - that we can always satisfy it whatever the final laws of nature may be. In contrast to the intention that Einstein may have had with his original proposal, the intention connected with its analytical version can hardly be anything but to bring about a prooJof this version. If, however, we want to prove something we must give it a fairly precise formulation, replacing such expressions as "the laws of nature" by some well defined concept of physical theory. Let us take as such a concept the concept of geometry developed in II. Then the logico-analytical version of the principle of general covariance becomes (el) Every analytical geometry (with G :s; Goo) is equivalent to an (analytical) differential geometry, i.e. an analytical geometry having the pseudo-group Goo' Is this provable? For a proof we could proceed as follows. Let

(9a)

L

[X; F, s]

be the given analytical geometry with pseudo-group G. Then the conjunction

(9b)

CG[X; Foo ]

"L [X; F, s] "F ~ Foo

ERHARD SCHEmE

34

where the first member is the coordinate geometry belonging to G00 evidently is equivalent to (930) with (6) being satisfied. To establish the equivalence we only have to define F00 as being the set of coordinate systems generated by F and Goo. But (9b) is not yet an analytical geometry with respect to Goo. We would still have to bring about the situation described in II, especially by (3). It is far from clear whether this can be done in each and every case, and we will not go any further into this matter 18. It must suffice to make the reader feel that as soon as we try to be a bit more precise in this business as is usual we find ourselves in a situation not easy to control. IV

With respect to the equivalences of analytical geometries satisfying (6) there is complete symmetry between the two following questions (A) Given an analytical geometry 1:. with pseudo-group G. Is there an analytical geometry 1:.1 having a larger pseudo-group G1,Le. satisfying (6a), but still equivalent to 1:. in the sense of (6b)? (A') Given an analytical geometry 1:.1 with pseudo-group G1. Is there an analytical geometry 1:. having a smaller pseudo-group G, Le. satisfying (6a), but still equivalent to 1:.1 in the sense of (6b)? In the previous section we have discussed (A) for the extreme case that G 1 - Goo. A far reaching positive answer to (A) in this case was (e+). But we raised doubts as to its validity. The corresponding positive answer to (A') certainly is wrong: There is no logico-analytical version (e' +) of (e') as there may be one for (e). Rather we have (e~ There are differential geometries that are not equivalent in the sense of (6b) to any analytical geometry having a

COVARIANCE AND NON-PREFERENCE

35

smaller pseudo-group, cf. (6a) with G 1 = Goo' An uninteresting instance of (C') would be the coordinate geometry with pseudo-group Goo' i.e. the theory of infinitely open differentiable manifolds. But also Einstein's theory of general relativity, if it is given a suitable formulation, seems to be a candidate for (C') although a proof is still missing. However, pointing out (C') I do not pretend to have found an adequate explication of Einstein's original (C'). Taken literally it in fact is an explication. But it grants a theory its virtue of not distinguishing special coordinate systems simply by letting its axioms being sufficiently weak. And thiS, in turn, does not seem to be a virtue of a (metatheoretical) principle. It is here where our decision to concentrate on covariance and the nonpreference of coordinate systems leads to consequences showing that that viewpoint may be a bit too narrow. Nonetheless I shall conclude this paper by discussing some variations of the idea of non-preference of coordinate systems. To this end let me introduce two concepts related to the one in question. The essential concept entering (C') was: (B) The analytical geometry ~l with pseudo-group G 1 is not equivalent in the sense of (6b) to any analytical geometry ~l having a smaller pseudo-group G in the sense of (6a). Consider now the following concept (B 1) For any relevant condition on a coordinate system, if it can be proven from ~1 that there are coordinate systems satisfying that condition then it can also be proven that every coordinate system satisfies the condition. In other words: There is no condition for which it could be proven that some but not all coordinate systems satisfy it. This is perhaps the most direct explication of the idea that in the geometry ~1 no coordinate systems are preferred to others: In the field theory defined by (7b) and (Bb) there are privileged coordinate systems

ERHARD SCHEIBE

36

precisely in the sense that we can prove that in some coordinate systems (7a) (or (8a)) holds whereas in others it does not. The new concept (B 1) is stronger than (B). For by virtue of (6b) any reduction of the pseudo-group of 1:1 immediately leads to a condition distinguishing certain coordinate systems. On the other hand, (B l) would not hold for general. relativity because for this theory there are conditions distinguishing certain coordinate systems without reducing G00 . The condition on a coordinate system adapting it to the light cones at every point of its domain is a case in point. Besides (B 1) there is another concept (B 2) related to (B) but presumably weaker than it. This concept was suggested by]. Anderson 20 and made more precise by M. Friedman 21 . In the following I give my own version of the matter. Let 1:1 be an analytical geometry whose pseudo-group G 1 of coordinate transformations is a group acting on Rn22. It may then happen that G 1 is categorical in the following restricted sense: With respect to the arguments "Xl" "F l " and "sl" in

(10) any two models < Xl; F1". sl'" > and < Xl; F1,,· sl'" > are isomorphic. If this happens and < Xl; F1". sl'" > is a model of 1:1 ' then s 1 is called an absolute object in that structure. There are absolute objects occurring of necessity: Any two models of 1:1 necessarily are isomorphic with respect to their sets of coordinate systems F 1 and F 1'- If, therefore, s1 is definable in terms of F 1 then it will be an absolute object. Such is the case, for instance, if the coordinate geometry on which 1:1 based has the Lorentz group as its group of coordinate transformations and sl is the usual metric definable on this ground. But there are cases of absolute objects not definable in the coordinate geometry. If 1:1 is the differential geometric formulation of euclidean geometry we have categoricity

COVARIANCE AND NON-PREFERENCE

37

without the possibility of defining the metric in the coordinate geometry which, in this case, is the species of Goo - differentiable manifolds. The case of non-definable absolute objects leads to a stronger version of categoricity: 1:1 is strongly categorical with respect to "sl" if it is categorical and "sl" is not definable in terms of "F 1". Our third concept of irreducibility then is (B 2) The analytical geometry 1: 1, is a coordinate geometry or it is not strongly categorical with respect to any of its arguments "sl". One can easily see that (B2) follows from (B). For if (B2) does not hold 1:1 is not a coordinate geometry. Moreover, it is strongly categorical with respect to at least one of the arguments, say "sl". Given a model < Xl; F I'" S I'" > of 1: 1 we define a set of preferred coordinate systems F < F1 as follows: Because of the categoricity the model is isomorphic to a standard model < Rn; G 1". Sl'" > of 1: 1 , The isomorphism is effected by a coordinate system in Fl' The set of coordinate systems thus distinguished is smaller than F 1 because 1:1 was assumed to be strongly categorical. In this way equivalence to a theory with a smaller group G and the absolute object sl being eliminated can be shown. Of course, the general concept (B 2) does not do away with absolute objects altogether: If G is one of the classical groups we still are where we ever were. Consequently, just as in the case of general covariance the interesting case is the differential geometric one. Anderson wanted to avoid absolute objects under all circumstances - whether they are definable or not. In order to avoid the definable cases we have to make the group G as large as possible. And this nicely fits into the bunch of ideas originally introduced by Einstein. Professor of Philosophy. University of Heidelberg

ERHARD SCHEmE

38

Notes 1

A. Einstein: Die Grondlage der allgemeinen Reltitivitatstheorie. Ann. d. Phys.49 (1916) 769-822. Quoted from § 3.

2

ibid. § 2.

3

ibid. § 3.

4

E.

5

A. Einstein: Prinzipie//es zur a//gemeinen Relativittitstheorie. Ann. d. Phys. 55 (1918) 241-4. Quoted from p . 242 .

6

A. Einstein and L. Infeld: tbe Evolution of Physics. Cambridge 1938 . Quoted from p. 212 . As I learned from Don Howard the book was written entirely by Infeld,and Einstein only gave his name to fasten the sale.

7

A. Einstein: Die Grondlage der allgemeinen Reltitivitatstheorie. Ann. d. Phys, 49 (1916) 769-822. Quoted from § 2.

8

ibid. § 3.

9

E. Cartan: Sur Ies varietes a connexion affine et /a thoorie de la relativite generalisee. Ann. sci. Ecole Normale Super. 40 (1923) 326-412 and 41 (924) 1-25.

Kretschmann: Ober den physikalischen Sinn der Relativittitspostulate. A. Einsteins neue und seine ursprllngltche Relativitatstheorie. Ann. d. Phys. 53 (1917) 575-614. Quoted from p . 576.

COVARIANCE AND NON-PREFERENCE

39

10

N. Bourbaki: Elements of Mathematics. Theory of Sets. Reading Mass., 1%8 . CH. IV. For physical application see G. Ludwig: Die Grundstrukturen einer physikalischen Theone. Berlin 1978.

11

For the following see E. Scheibe: Invariance and Covariance. In: Scientific Philosophy Today, Essays in Honor of Mario Bunge. Ed. by]. Agassi and R. S. Cohen. Dordrecht 1982 . 311 - 31 .

12

For details see the Encyclopedic Dictionary of Mathematics. Ed. by S. Iyanaga and Y. Kawada. Cambridge, Mass., 1977.92 D and (a narrower concept) 108 Z.

13

A recent exception is W. G. Dixon: Special Relativity. CUP 1978. pp. 42 ff.

14

F. Klein: Elementarmathematik vom h6heren Standpunkt aus. Vol. II: Geometrie. Berlin 1925.

15

The standard monograph is]. A. Schouten: Ricci-Calculus. Berlin 1954.

16

See Misner, Ch. W., Thorne, K .S., and]. A. Wheeler: Gravitation. San Francisco 1973. What I am emphasizing is that , although the definition of, say, the concept of a vector field need not refer to coordinate systems, the definition is based on the concept of a differentiable manifold and this concept usually is defined by using coordinate systems.

17

A different analysis of the principle of general covariance can be

ERHARD SCHEIBE

40

found in S. Weinberg: Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York 1972. pp. 91 ff. 18

For some further thoughts on the matter see the paper mentioned in n.11.

19

There are, of course, explications different from (C+). One possibility is to restrict the whole question to field theories in the sense of (4). Yet the problem of proving (C+) thus modified again is a matter not too easily settled.

20

J. L. Anderson: Principles of Relativity Physics. New York 1967. J. L. Anderson: Covariance, Invariance, and Equivalence: a Viewpoint. Gen. Rei. Grav. 2 (1971) 161-72.

21

M. Friedman: Relativity Principles, Absolute Objects, and Symmetry Groups. In: Space, Time, and Geometry. Ed. by P. Suppes. Dordrecht 1973. 2%-320.

22

This assumption simplifies the concept formation and the argument. But it seems not essential for the matter.

KANT'S "PLATONIC" ARGUMENT IN BEHALF OF TIlE A PRIORI CHARACTER OF TIlE REPRESENTATION OF SPACE

Pierre Laberge The transcendental exposition of the concept of space has perhaps received more attention than the metaphysical exposition. 1 This is undoubtedly because of the threat which non-Euclidean geometries pose to it and because of the interest which for this reason philosophies of mathematics sometimes have in it. Gottfried Martin has given a little more spice to the arguments of the metaphysical exposition in behalf of the a priori character of the representation of space by qualifying the first as "platonic" and the second as "aristotelian." 2 In what sense does the first deserve such dignity? On the other hand, are there any problems with it? The first argument (A23/B38) of the metaphysical exposition establishes the a priori character of the representation of space, as Martin summarizes it 3, by making the following case: "in order that it be possible to represent to myself certain things (etwas) as juxtaposed (als nebeneinander), it is necessary that the representation of the juxtaposition (des Nebeneinanders) be already presupposed." So, he continues, it has to do with the "Grundargument' 4 that Plato regularly uses to establish the Theory of Ideas and thanks to which we can prove, if proof there be, the a priori character of all representations a priori. Plato employs this argument, for example, in the Pbaedo (72e - 71) in order to prove the a priori character of the concept of equality, that is to say, to establish that equality is an Idea. "In order that I can recognize two things as equal," Martin again summarizes,5 ".. .1 must already have available the concept of 41

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equality." Only the reminiscence of the Idea of equality, of the "Equal in itself" (74a) makes possible recognition of the equality of these sticks of wood or of these stones (74b). The first argument of the metaphysical exposition is thus nothing but the celebrated reminiscence argument, demythologized and applied to the particular case of the representation of space. Martin's interpretation assumes that the reminiscence argument is intended to establish the Theory of Ideas. But it seems to me rather that the reminiscence argument presupposes the Theory of Ideas. Recognition of the equality of these sticks of wood or of these stones is reminiscence of the Idea of equality because it can only be reminiscence. It can only be reminiscence because (a) we must claim (and isn't this just to take the Theory of Ideas as given?) that the "Equal in itself.. .is something" (74a and b) and (b) much more is needed than the equality of sticks of wood or of stones "to match that which is Equal" (74d). Socrates intends to establish that the knowledge of this "Equal in itself," of which the recognition of the equality of sticks of wood or of stones can only be a reminiscence, must have been acquired before birth. To suppose that the reminiscence argument is intended to establish the Theory of Ideas renders it circular: there are Ideas; therefore we recall them when we perceive; since we recall them when we perceive, there are Ideas! To call the first Kantian argument "platonic" in this sense is equivalent to lending to Plato and to Kant a circular argument, circular in the same way, perhaps, as that imagined by D.P. Dryer: "In order to recognize objects as red, one must already have the concept of redness, "6 hence the concept of redness is not empirical. The conclusion would hold only on condition that one agreed from the outset either that the

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concept of redness is a Priori or that one cannot already possess the concept of redness without it being a Priori. If it corresponded to the argument imagined by Dryer, Kant's first argument would be no more successful: in order to recognize objects as spatial, one must already possess the concept of space, hence the concept of space is not empirical. That is perhaps the interpretation of P F. Strawson, for whom Kant's first argument, a banal tautology 7, does not even attain the majesty of a circle. If we have been able to loan Kant a platonic argument after having loaned Plato a circular argument, we might have been able to loan Kant a circular argument directly. Has this been done? VaihingerB put me on the track of Ueberweg. This latter criticizes Kant's argument for being a Zirkelschluss. After summarizing it as follows "Space is not an empirical concept drawn from external experiences; for the representation of space must already be at the foundation (schon zum Grunde liegen) of all concrete localization (Lokalisierung)," he adds in a footnote: "Was freilich ein Zirkelschluss ist." 9 Thus he defines a" Cirkelbewei1' in his System der Logik: 'when A is proved by Band B in its turn by A.10" In the Zirkelschluss for which Kant is criticized, the B can only be "already at the foundation," and the A the non-empirical or a Priori character of the representation of space. Kant would set out to prove the a priori character by the foundational character, but he would succeed in doing so only by tacitly basing the second on the first. It is in this way, moreover, that Hermann Cohen 11 interprets Ueberweg's objection since he tries to reply: if Kant intends to prove that the representation of space is not empirical, he does not undertake to prove at the same time that it is a priori. Whence Kant's merely

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negative formulation: space is not an empirical concept. Everything would transpire as if Kant had chosen this negative formulation in order to escape the circularity objection. For how could there be a circle if Kant is not trying to prove A? Is Kant's argument circular or "platonic" in Martin's sense? We can agree with Vaihinger 12 and Martin 13 in seeing in its first sentence the thesis to be established: "Space is not an empirical concept which has been derived from outer experiencesj" in the second sentence which begins with "Denn," the argument properly socalled: "for in order that certain sensations be referred to something outside me (that is, to something in another region of space from that in which I find myself), and Similarly in order that I may be able to represent them as outside and alongside one another, and accordingly not only as different but as in different places, the representation of space must be presupposedj" in the third sentence which begins with •Demnach," the restatement of the theSiS, whose truth now appears coupled with the soundness of the argument. The third sentence thus paraphrases the first two sentences in linking them: "The representation of space cannot, therefore, be empirically obtained from the relations of outer appearance. on the contrary, this outer experience is itself possible at all only through that representation." We may therefore forget it after having taken note of its antileibniziano-wolffian accent (A40, 856-57). To tell the truth, the argument divides into two sub-arguments: on the one hand, the representation of space, it is declared, is presupposed in order that I can refer impressions to something outside me (ausser mich), on the other hand, it is also presupposed in order that I can represent to myself things as outside and next to each

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other (ausser und nebeneinander). The first sub-argument is difficult to interpret. A tautology powerless to shake the empiricist? In fact, what empiricist would not subscribe to the impossibility of representing things outside of him without the representation of space? Unless the "ausser" in "ausser micb" is to be understood in a nonspatial sense, in the sense of praeter rather than extra? Whence the attempt of H. E. Allison to exonerate Kant of the tautology with which Straws on finds fault. Allison goes so far as to suggest that the second sub-argument must also be sheltered from Strawson's criticism. It would be tautological to assert that space must be presupposed for one to represent "things as outside and along side one another, and accordingly not only as different but as in different places." It would therefore be necessary to interpret the second sub-argument in the following manner: "in order to be aware of things as numerically distinct from one another, it is necessary to be aware, not only of their qualitative differences, but also of the fact that they are located in different places." 14 But, even if sound, 15 the second sub-argument thus understood would be advanced by Kant without proofl Moreover, the parallel with the argument in behalf of the a priori character of the concept of time: "For neither coexistence nor succession would ever come within our perception, if the representation of time were not presupposed as underlying them a priori' (A30/B46) begs for a literal interpretation: the representation of the coexistent or of the successive there, of the ausser und nebeneinander here, presupposes there the representation of time, here the representation of space. As regards the "ausser micb" of the first sub-argument, it is difficult to interpret it in a non-spatial way given Kant's parenthetical

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remark: "(that is, to something in another region of space from that in which I find myselO." In brief, eiJher the first argument deserves Strawson's criticism, or it is to be interpreted, although with difficulty, in Allison's way (in which case Kant advances it as gratuitously as the second sub-argument as interpreted by Allison), or it is simply a special case of the second sub argument. As well in consequence to discard it as at best superfluous, which leaves us with the single second sub-argument completely parallel to the argument (without sub-argument) in behalf of the a Priori character of the representation of time. It remains to ask if these two parallel arguments do not prove to be tautological or at bast circular. They would be tautological if they led back to the argument already mentioned in behalf of the a priori character of the concepts of redness and of space: "in order to recognize objects as red, one must already have the concept of redness;" in order to recognize objects as spatial, one must already haye the concept of space. But is this really the case? For Kant does not write that it is necessary to already have the concept of space to be able to recognize XiS as spatial, but that it is necessary to already have it in order to recognize them as ausser und nebeneinander. (One can, as Martin, blast away the "ausser' to save only the "neben".)16 In the same way, as regards time, Kant does not write that it is necessary to already possess the concept of time in order to recognize XiS as temporal, but that it is necessary to already have it in order to recognize them as co-existent or successive. But, it will be objected, don't we escape the tautology only to sink into circularity? How to prove that one cannot already possess concepts of space and time without them being a Priori and this

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without tacitly taking for granted either that these concepts are .a priori or that one cannot already possess them without their being .a priori? Perhaps the Dissertation of 1770 provides us with the response in paragraph 14 on time. I7 If it is necessary to already possess the concept of time, if the concept of time presupposed, it cannot be empirical. For if it is presupposed, one cannot define it therefore through that which presupposes it. One cannot define it by the order of succession I8 or "per seriem actualium post se invicem exsistentium." For quid significiet vocula post, non intelligo, nisi praevio jam temporis conceptu" (2, 399). This is to say that it cannot be defined empirically, since experience does not provide us with anything but this "series." Kant alludes to this difficulty when in paragraph 14.5, he castigates the circulus vitiosus into which "Leibnizius et asseclae' (2, 400-400 fall. Without doubt, he has in mind an analogous difficulty concerning space when he warns again against the danger of circularity in paragraph 15.d (2, 404). Nam, he could write, quid significent vocula "extra se invicem" non intelligo, nisi praevio jam spatii conceptu. The first argument in behalf of the a priori character of the representation of space thus escapes circularity, if it does escape it, to the extent that the definition of space to which its opponents would be driven presupposes that which it wishes to define and would thus itself be circular. 19 Professor of Philosophy. University of Ottawa

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Notes

lQuotations of the Critique of Pure Reason are from the translation by Norman Kemp Smith (Macmillan and Co., Ltd., 1933), following the A and/or B editions. All other texts of Kant's follow the Akademie Ausgabe. 2Gottfried Martin, Immanuel Kant (Berlin: Walter de Gruyter, 1%9), pp. 34-39. In a doctoral thesis defended in 1987 at the Universite de Provence, Aix-Marseille I, and which deserves publication, FrancoisXavier Chenet traces back to Johann Georg Feder the affirmation of a tie between the first argument of the metaphysical exposition and Plato. See Francois-Xavier Chenet, L' Esthetique transcendentale de la Critique de la raison pure. Etude analytique et critique~ p. 150. Martin, for his part, traces back the rediscovery of this tie to Natorp (Platos Ideenlehre, zweite Auflage, Leipzig, Felix Meiner, 1921, pp. 34ft). I have, however, not found a trace of this rediscovery in the passage indicated or elsewhere in Platos Ideenlehre.

3Ibid., p. 34. 4Ibid, p. 34. 5Ibid., p. 35. 60. p. Dryer, Kant's Solution for Verification in Metaphysics (London:

George Allen and Unwin, 1%6), p. 173.

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7p.F. Strawson, The Bounds 19(6), p. 58.

0/ Sense

(London: Methuen and Co.,

BHans Vaihinger, Kommentar zu Kants Kritik der reinen VernunJt, Band 2, zweite Auflage (Stuttgart: Union Deutsche Verlagsgesellschaft, 1922), p. 179 9F. Ueberweg, Grundriss der Geschichte der Philosophie~ Dritter Teil (Berlin: Ernst Siegfried Mittler und Sohn, 1907), p. 306. 1°F. Ueberweg, System der Logik, dritte Auflage (Bonn: Adolph Marcus, 1968), pp. 403-404. 11 Hermann Cohen, Kants Theorie der Er/ahrung, zweite Auflage (Berlin: Dumrnler, 1885),pp. 96-97. 120p. cit., pp. 156-157. BOp. cit., P. 33.

14Henry E. Allison, Kant's Transcendental Idealism (New Haven: Yale University Press, 19832, p. 83. 15Compare to Metaphysik Mrongovius, 29.1.2, p.830. 160p . cit., p. 35. Jules Vuillemin underlines the importance of attaching the "neben" to the "ausser" in the second edition of the

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Critique. See Jules Vuillemin, La philosophie de I'algebre. tome 1 (Paris: P.U.F., 1967), p. 457, note 3. 17Compare to 1) Metaphysik L',;, "Raum is die Ordnung der Dinge, sofern sie ausserhalb oder neben einander zugleich sind, Zeit ist die Ordnung der Dinge, sofern sie nach einander sind. Diese Definition ist aber tautologisch; will man sie verstehen, so muss schon der Begriff von Zeit und Raum vorhergehen. Denn nebeneinander bedeutet in verschiedenen Orten sein, also liegen die Begriffe von Zeit und Raum der Definition schon zum Grunde" (28.1, pp. 177-178). 2) Metaphysik Mrongovius: "Der autor erklaert Raum durch Ordo extra se positorum. Extra se positorum sind Dinge in verschiedenen Orten. Der Begriff des Orts sezt den Begriff des Raurns voraus, und der Begriff wird schon als bekannt angenommen: ordo plurium, quatenus post se existunt est tempus; nach einander seyn ist zu verschiedenen zeiten seyn, also ist idem per idem erklaert" (29.1.2, p.831).

18Baumgarten defines space and time as follows in paragraph 239 of his Metaphysica: "Ordo simultaneorum extra se invicem positorum est spatium, successivorim, tempus" (17, p. 79), Wolff, according to Kant, defines space as "ein bloss der empirischen Anschauung (Wahrnehmung) gegebenes Nebeneinandersein des Mannigfaltigen ausser einander" (6, p. 208). 19J3ut why, it will be asked, does Kant no longer invoke in the Critique the objection of the Dissertation? I have no response to this

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question. According to Vaihinger, op. cit., p. 369, Kant would have discovered between 1770 and 1781 that the defect attributed to the Leibniziano-wolffians was not a .. circulus vitiosus' but rather a "Cirkel in der genetischen Ableitung." But was this a reason to abandon the objection? Would it not have sufficed to label it differently?

TIlE SENSE OF TIlE A PRIORI MEmOD

IN LEIBNIZ'S DYNAMICS Fran~ois

Duchesneau

In the opuscule De primae philosophiae emendatione notione substantiae (1694), Leibniz mentions that he has devoted a special science, dynamics, to the notion of force, and that this methodological elaboration clarifies the metaphysics of substance. I To a certain extent this assertion causes problems. The scientific papers published by Leibniz up to that time do not support this claim to have founded a science of dynamics. Certainly, since the Brevis demonstratio erroris memorabibis cartesii (1686), Leibniz had been working to invalidate demonstratively the Cartesian principle of the conservation of quantity of motion. He proposes as an alternative a new measure of the motive force which would be conserved in mechanical exchanges, a measure based on the estimate of absolute force (potentia absoJuta) or living force (vis viva), expressed by the product mv 2. Since then, a stormy quarrel with the Cartesians concerning the possibility of deducing the new principle as foundation of the system of the laws of nature had broken out 2 . But on what basis was Leibniz able to claim that he had founded a science of dynamics? One factor in the answer to this question emerges from a more careful examination of the appeals to the a priori method of demonstration which appear progressively in his unpublished work and in certain exchanges of letters, in particular with De Voider, Johann Bernouilli, and Christian Wolff. Our ambition here will be limited to presenting some reasons which militate in favor of such a re-examination, then to initiate the process of analysis through a study of the first part of the Dynamica de potentia (1689-90) where recourse to the a priori method is first sketched. 53

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I

For Leibniz, there is no doubt that a genuine science assumes a structure of demonstrative arguments capable of giving way to a synthetic exposition. 3 The analytic procedures implemented with the aim of setting up the a posteriori proofs of the new conservation principle do not fail to suggest the possibility of establishing a system of demonstrations and conceptual connections to serve as a theoretical basis for the explanation of the phenomena. But this objective seemed to be achieved only if Leibniz showed that the analysis resulting in the principle of the conservation of living force was so structured that it could in turn beget a synthetic development capable of illustrating the value of the new principle as a keystone of theoretical physics. It was with this objective in mind that Leibniz without doubt wrote the Dynamica de potentia during his trip to Italy, in the period immediately prior to the declaration we have extracted from the De primae pbilosopbiae emendatione. The argumentative structure of this treatise, modeled on the chain of demonstrations in a system of geometry, provides material evidence of such a methodological orientation. And every more thorough study of it reveals the project of a synthetic deduction of the laws of mechanics founded on immanent force or vis insita. 4 The question, what can and must be a synthetic demonstration at the foundation of a special science, calls for interpreting how a priori and a posteriori grounds are juxtaposed and integrated in the structure of Leibnizian dynamics. In the main, commentators have tended to avoid this question by concentrating their attention on the essentially analytic and a posteriori steps evidenced in the texts of 1686 and in the initial controversy with the Cartesians, as well as later on, in the Specimen dynamicum (1695). Martial Gueroult is an exception to the extent that he devotes an entire chapter of his

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magisterial study, Leibniz: Dynamique et metaphysique, to the "a priori method (through motive action)." 5 Gueroult distinguishes two types of demonstrations at work in the dynamics. The demonstration a posteriori draws on the principle of equivalence between the whole effect and the full cause in order to infer from an empirical law, Galileo's law of falling boqies, a theorem about the conservation of living forces; this theorem requires a meta-empirical interpretation. In a parallel way, Leibniz infers from the combination of two equations of relational conservation (conservation of relative speed before and after impact, conservation of total directive progress of the system) an equation of absolute conservation, that of living force. To a certain extent, the justification appears to conform with the analogous truths of fact that experience can establish: thus the theorem conforms to the empirical laws of motion established by Huygens, Wallis, Wren and Mariotte, but at the same time the objective meaning that Leibniz attributes to it seems to depend on a system of rather metaphysical concepts. Does this representation of the a priori path make enough room for hypotheses which the understanding can construct in drawing on what is empirically given as interpreted according to mathematical models and by way of regulative principles? Can one otherwise consider that the inferential patterns involving combinations of equations, such as those set in place in the Dynamica de potentia and illustrated by the Essay de dynamique (c.1700), have nothing to do with an a priori step included within an analytic and generally a posteriori context? These combinations unquestionably enter into the ingredients of the a posteriori path as Gueroult does not hesitate to recognize. Consider now the a priori path such as it is presented to us. The living forces controversy aroused Leibniz to battle the Cartesians on their own ground, by adopting the strategy of a priori demonstration from geometrical phoronomy. Thus he tries to start from abstract considerations concerning space and time only. A

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scientific dead end, according to Gueroult: Leibniz would know how to avoid the paralogisms which stretch beneath the premises of his syllogisms only if the theoretical concepts were synthetically interpreted with the aid of the schemas of the metaphysical monadology. This severe verdict results in part, at the very least, from the strategies of analysis deployed by Gueroult. After having schematized the a priori path from statements of the theorems in the Dynamica de potentia, he bases himself essentially on the syllogistic codification of the assumptions, such as formulated in the correspondences with Johann Bernouilli and De Voider, and on the model which Wolff proposes to Leibniz and to which the latter reacts. In my opinion, the "rational a priori" character of the Leibnizian considerations deserves to be more adequately identified in the light of the Leibnizian epistemology: isn't there reason to take into account possible divergences between Leibniz and his Cartesian interlocutors concerning what can count as an a priori ground in physics? Putting to the side at least for the moment the idea of an artificial veneer of an a priori path, the analysis must take for its objective setting out the structure that Leibniz wants to give to physical theory. It is suitable, we think, to set out the "empirical" arguments which subtend the two types of Leibnizian demonstrations, then to show how Leibniz conceives theoretical construction with the aid of heuristic principles. It seems arbitrary purely and simply to relate these heuristic principles to metaphysics. Dynamics enters, indeed, into a category of knowledge which Leibniz presents as a mixed science;6 between the pure truths of reason and the pure truths of fact is inserted the analysis of necessary truths ex hypothesi. This representation may without doubt clarify the structure of the apparently divergent demonstrations among which dynamics is divided. We will try here to illustrate this approach by taking off from the Dynamica de potentia, a Leibnizian text illustrating

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57

illustrating the systematic articulation of a priori and a posteriori demonstrations. Perhaps we will thus contribute to reducing the tensions and difficulties that Gueroult revealed in the combination of paths followed by Leibniz. We will certainly not rule out that for polemical reasons Leibniz sometimes unduly emphasized the apodeictic aspect of certain arguments arising from the so-called a priori method. This is true in particular of those definitions with respect to which it is appropriate to establish to what extent they are real or simply nominal and to what extent one can establish a possible gradation from one to the other category. Such will be the stakes in a large part of subsequent controversies. But isn't it first necessary to reconstitute the modalities according to which Leibniz plans to construct physical theory, on the model of dynamics as a <;cience? The most important text of Leibniz on the fundamental laws of mechanics is also the least utilized by commentators. It is the long manuscript entitled Dynamica de potentia et legibus naturae corporeae which Leibniz wrote in 1689-90, following discussions which he had with the mathematician Auzout at the time of his trip to Italy. The text is thus directly contemporary with Leibniz's first reactions to the Newtonian physics of the Principia. Certainly, this major text did not see the light of day until 1860 in the edition of the Mathematische Schriften by Gerhardt, 8 but it is necessary to take into account its strategic insertion into the series of Leibnizian works on dynamics. If the unpublished writings suggest that from 1678 on Leibniz was in possession of his theorem of the conservation of living forces,9 the first official intervention against Cartesian mechanics was in 1686 with the Brevis demonstratio erroris memorabilis Cartesii, whose argumentation is reproduced in article 17 of the Discourse on Metaphysics. This is followed by correspondence relating to the quarrel about forces vives, the I/lustratio ulteriOr, and the Principium quoddam qenerale. In 1687, the

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Pbilosopbiae naturalis principia matbematica of Newton appeared. Leibniz, who was in Rome, learned of it through a compterendu which appeared in the Acta eruditorum of 1688. The Pboranomus, seu de Potentia et legibus naturae dates from the same time. It is then that Leibniz undertook the writing of a systematization of his principles, which the Dynamica de potentia gives us. Leibniz will, it seems, construct all of his subsequent analyses based on this synthesis. It is thus that the Specimen dynamicum (1695) constitutes, if not a detached part of it, then at least a condensed presentation, destined to bring out especially the metaphysical aspect of the Leibnizian theses. Other fallout: the Animadversiones in partem genera/em Principiorum Cartesianorum and a first Essay de dynamique (1692),10 clearly composed to sanction the superiority of Leibniz's positions over those of the Cartesians in the wake of the living forces controversy and the partial support of Malebranche. The composition of relational equations to form an absolute equation of quantity of action, without doubt the most remarkable feature of the more mature Essay de dynamique, (c.1700) results from the deductive chains constructed in the Dynamica. Contrary to other Leibnizian accounts, the Dynamica is founded on an apparently synthetic and more geometrico development of arguments. Moreover, the very tight conceptual tissue which the geometrization of fundamental mechanical phenomena generates, relegates the metaphysical presuppositions to another level of consideration, on the one hand, the arguments inferable from the mixed science on behalf of a metaphysical system, on the other hand. The Dynamica interests but little the interpreters of Leibniz's metaphysics, and even Gueroult refers to it only rather marginally. On the other hand, this scholarly and difficult text, in which Leibniz undertakes to express the systematic import of his analyses of the laws of mechanical phenomena, suggests Leibnizian solutions to the epistemological problems that interest us here. the

LEmNIZ'S DYNAMICS

S9

II

In the first part, devoted to the simple elements of dynamics, abstracted from concrete realities, a deductive procedure founded on definitions dominates. A single axiom is stated, which serves precisely to establish the propositions relative to the quantity of motive action from the formal effect of motion. This axiom says: "The fact that the same quantity of matter would be moved over the same length in a shorter time constitutes a greater action."12 We will come back to this point to establish the demonstrative status of the propositions which the axiom calls for. In a more immediate way, it is a question of determining what type of definition is at work in this first part of the Dynamica.. These definitions have, all of them, only a nominal status. Leibniz draws them from an abstractive reconstruction of what is phenomenally given in phoronomy, that is to say, in the relational analysis of displacements of material particles in space, displacements understood in terms of geometrical properties. However, at the very end of the first part, Leibniz adds a remarkable chapter devoted to the analytic calculus, "Pro phonometria dynamica." Since the instrumental concepts of the infinitesimal calculus exceed the possibilities of the geometrical representation of relations, it becomes evident that abstraction is no longer restricted to the faculty of projecting phenomenal relations according to the proportions between points, lines, and diverse figures. In a significant fashion, Leibniz thoroughly utilizes a serial theory of proportions allowing for the fixing of relations between elements or between quantities in virtue of term to term correspondences between series in simple phoronomy, that which bears on phenomena reducible to uniform motions. Certain proportions of an apparently paradoxical character are explained by

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the systematic intervention of these serial correspondences between factors. The justification of this procedure is the raison d'etre of the chapter "De ductibus seu de aestimationum compositione.» It is clear that by this mode of abstract representation, inherited from mathematical analysis, Leibniz intends to conceive these characteristic states (status) of empirically identifiable and phoronomically transposable processes: by way of the transposition, these states will appear as embryonic developments whose summation furnishes rational relations sui generis between mechanical "affections" of bodies. Thus he sees himself clarifying in a radical way the conception of gravitational effects, legacy of Galilean mechanics: [Galileo) reveals the acceleration of motions brought about by gravity. Encouraged by these beginnings, we discovered the true relation of estimation of corporeal power, from which certain general laws follow, laws which are surely remarkable and from which the heretofore little known nature of body and motion is more and more clarified ... 13 It is nonetheless necessary that the nominal status of these definitions be modified. In the context of a scientific investigation of the laws of nature, it is a question only of instrumental means allowing for sustaining the abstract representation of phenomena and the purely modal conception of elementary and general physical processes. This clarification must at the outside mold itself on a conceptual reform conforming to the demands of a phoronomy purged of all empirical residue that might be supposed to be not strictly representable. Not only will the theorem of conservation of motive action be thus called upon to furnish a basis for the phoronomic doctrine of force, but Leibniz envisages articulating there the Galilean lemmas on the fall of bodies and the effects of gravity. In the "Specimen praeliminare: de lege naturae circa corporum potentiam", Leibniz exposes the various

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possible demonstrations of the principle of conservation of living forces. The first three, corresponding to the a posteriori path identified by Gueroult, rest on a common lemma and on three distinct axioms, which one can suspect from the outset have the same status as the single axiom integrated into the first part on simple dynamics. These axioms are stated as follows: The same force is required to lift four pounds one foot and to lift one pound four feet. There is no perpetual mechanical motion. It cannot happen that the center of gravity of bodies rises by the force of gravity itself. 14 The lemma is the reprise of the law of Galileo under the form of a hypothetical construction whose concept is drawn from a doctrine of motion and of weight; the justification of this lemma rests in part on the possibility of geometrical representation, in part on empirical control: The perpendicular heights of the heavy bodies are as the square of the velocities which they can acquire in descending from these heights or by means of which they can ascend to them. It's a proposition of Galileo's, demonstrated from the nature of the uniformly accelerated motion of the heavy bodies, received by mathematicians and corroborated by multiple experiences. 15 The fourth demonstration is founded on the causes of the motions abstracted "a materia sensibili." It is a question of a deductive mise en scene of propositions on measurement and the conservation of motive action which one will find established with the aid of the subsequent axiom in the body of the first part. It is this deductive mise en scene which will be the object of discussions with Bernouilli and De Voider, and which Gueroult will signal as a degenerative procedure in Leibnizian construction. For the moment, the

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significance given by Leibniz to the demonstration arouses a particular methodological interest, as the attached scholium indicates: Although this last demonstration is probably not to the taste or reach of all, it must nonetheless please from the outset those who seek the clear perception of truths. Assuredly, just as it was discovered last, it seems to me the first in dignity, since it proceeds a priori and arises from the sole consideration of space and time without any presupposition about gravity or other a posteriori hypotheses. From which henceforth not only is a remarkable agreement between truths manifested, but a new path opens to demonstrate Galileo's propositions on the motion of heavy bodies without the hypothesis which he had to use, namely, that in the uniformly accelerated motion of heavy bodies, equal increases of velocity are acquired in equal times. This, as well as the lemma initially posed, can be concluded from our fourth demonstration in an independent fashion. This is worthy of notice and of the greatest importance for the perfection of the science of motions. 16 Manifestly, Leibniz undertakes to furnish an a priori derivation of the causes of motions which Galileo adopts by hypothesis and justifies by empirical corroboration. But the a priori character of the new demonstration keeps to its dependence on purely phoronomic considerations. It is not evident that one can carry out this derivation from totally abstract concepts of space and time. Would we not rather be concerned with "analogues," objects of abstract representation, and thus analyzable in terms of quantitative models using geometry and the infinitesimal calculus? And will these "analogues" not conserve from their initial empirical derivation certain characteristics which will serve implicitly to guarantee the

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coherence on the one hand and the objectivity on the other of the abstract representations? Section III, De actione et potentia, where the demonstration "a priori" is located, is preceded by a section "De motu et velocitate," where Leibniz composes in successive steps the abstract representation of an object of phoronomic analysis: at first, the motion is described by a contrast of homogeneous and heterogeneous relations between points at successive moments of timeP On the occasion of this first definition, judged nominal within the limits of the capacity to define which our understanding is allowed, Leibniz mentions that, thus constructed,the representation of motion can integrate that of rest, which is conceived as a motion of vanishing velocity; this representation can then be applied to aggregates of bodies, some in motion, others at rest. The definition proposed has the advantage that it permits treating conjointly that which moves and that which is moved, for an active substrate of homogeneous relations of displacement is implied in the notion: this substrate can be represented by global phoronomic relations for the set chosen. Leibniz insists on the functional role of the definition for every study regrouping correlative elements of motion and rest, including the interior of the point as a unity without parts but capable of homogeneous relations. Definition 2: "The time of the motion is the time of which no part can be assumed in which some point of a homogeneous element presumed in the moving body will not change place," 18 aims at relating the measurement of time to the relations of motion between homogeneous and heterogeneous elements up to the infinitesimal relation implied. From this fact, time is not a neutral element, but it accords with the intensio of the motion in its successive moments. The following definitions concern uniform motion, motion equally distributed, velocity and the length of the motion equally distributed, simply simple motion in so far as it is rectilinear, in so far

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as it is "consentiens", in so far as it is equidistributed, and in so far as it is equidirected. These definitions constitute a representation progressively constructed from formal elements of motions which can be measured in terms of analytic decomposition and recomposition. Note the particular character of the taking into account of the velocity in equally distributed motion: The velocity in an equally distributed motion is an affection of the moving body (formal affection existing there in virtue of the motion alone) which is proportional to the line which a point of the moving body would describe if the motion continued during a temporal interval of a given magnitude, this same affection of the moving body being conserved. The velocity remains the same, if the same point describes equal spaces in equal times. 19 Velocity (velocitas) in effect enjoys a double status: at the same time property (a!fectio) of the moving body and abstract measure of a parameter of displacement in space following the interval of time chosen, which can be as short as one wants, to the point of becoming unassignable: in which case this abstract measure rejoins the originating relation between homogeneous and heterogeneous elements allowing one to discern the moving body in its intensio. Because of this, the property is called formal or inherent in the moving body from the fact of its displacement (even virtual) alone. Certainly, the point concerns a first approximation of the conatus, fundamental architectonic element in Leibnizian dynamics. The conatus will itself be defined by an abstract reduction of the effects which result from it in impetus, since impetus is considered a summation of the conatus as a function of the mass to which they are applied. But at this stage, intensio as a contingent a!fectio, bound to the actualizable motion, is expressed by relation to the characteristics of uniform equidistributed motion. That is a significant recourse to

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the mode of expression in terms of proportional relations. The set of definitions relating to simply simple motion rests on an analogous methodological artifice, but this time with the intention of reducing by abstraction what the elements of motion can include of essential heterogeneity. The simple is thus the symbolic expression of a diversity infinitely reduced so as to form a continuity of geometrical type. The principal definition is stated thus: Motion is simply simple when the motions of the points of a moving body accord with themselves completely and between themselves in such a way that they cannot do it more. 20 And the explication furnished clarifies precisely the game of analogies on the basis of which motion can be represented as integrally homogeneous for a given concrete moving body, that is to say in so far as abstractly simple motion: This happens when neither a state of a point can be distinguished from another anterior or posterior, nor one point from another, so long as one considers their motions alone. The motions existing always similar and Similarly placed the ones in relation to the others, provide no distinguishing principle, even if one can probably distinguish the points with regard to the bodies by the situs which they have in the bodies, that which cannot be aVOided, and I omit here the qualities by which the parts of the body are differentiated, whose ground does not apply here. 21 It emerges by implication that the law of continuity serves to construct an abstract representation of the fundamental motion permitting one to surmount the aporia of pure and simple indiscernibility. Doesn't Leibniz suggest a state of essential difference between the physical elements entering into a simply simple motion? At the same time, doesn't the geometrical analogue

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constructed by the continued sequence and thus the serial homogenization of these elements suffice to express what there is of a constant relation in phenomenal variations, and as a result suffice to furnish the basis of a general phoronomy? The passage to action and force in the following section presents in my opinion the characteristic manner of inverting the course of the analogy. It is there, however, that the theme of an a priori demonstration of the fundamental theorem of dynamics is lodged, or rather of its equivalent, the principle of the conservation of the quantity of motive action. But the presuppositions of the demonstration scarcely allow considering that one has to deal with a pure form of a priori demonstration. At the outset there are again definitions of the nominal type, among which those of quantity of formal effect and quantity of formal action: Quantity of formal effect in motion is that of which the measurement is constituted by the fact that a matter of a cenain magnitude (the motion being equidistributed) is moved over a certain length. Quantity of formal action in motion is that of which the measurement is instituted by the fact that a certain quantity of matter is moved over a certain length (the motion being uniformly equidistributed) within a certain time. 22 These definitions contain a remarkable condition: they indicate material effects issuing from the actualized motion, abstraction made of other causes to be included in the nature of bodies and which would be outside of motion as a mode. But at the same time, these modal effects are analogically interpreted by relation to other modal effects expressing the intervention of forces behind the appearance of motions. It is in this way that we can without doubt interpret the significant remark accompanying the definitions:

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I have qualified as "formal" the effect as much as the action, because they are, following the definition which we have given of them here, essential to motion; they differ from other effects or other actions, arising from a particular obstacle, as from the force of gravity pushing bodies toward the center of the earth, or from the resistance due to the milieu or to a narrowing, or from the elasticity of a body to overcome, and from similar accidents of concrete matter. If using a metaphysical term in a mathematical subject is taken badly, then think that no other more satisfactory term offers itself and that once the definition has been given, all ambiguity has been removed from it. 23 The difference between the two types of modal effects stems from the fact that the former are revealed directly in unconstrained motion, while the latter will appear in an action caused by the inertia of matter and will be described as proportional to this inherent potential of resistance. Thus the first will be abstract, while the second will involve affections of corporel reality such as they are manifested in sensible experience alone. This same abstraction cannot result from a purely a priori conception, as would be the case with mathematical concepts; it corresponds only to a geometrical apprehension of concrete realities, which would not deny the inherence of the relations thus conceived with regard to the order underlying the phenomena. If there is formal effect and formal action as distinct and intrinsically distinguished notions, it is important to grasp the architectonic tie between these two notions and between the quantitative evaluations to which they give rise; it would be important even to postulate this relation if need be, to the extent that it did not emerge from the respective analysis of the notions. The strategy put to work consists in constructing from the definitions a convergence of proportions between the terms

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composing the motive effect and the terms composing the motive action. Let us take as an example proposition 3: " The formal effectS of the motions and, assuming an equal velocity, even the formal actions of the motions are a complex function of moving bodies and of lengths."Z4 It is then a matter of applying an axiom which ties this system of correspondence to the measurement of duration: "The fact that the same quantity of matter is moved over the same length in a lesser time constitutes a greater action."ZS In consequence, one is thus given a determination of the force of acting which is a complex function of the quantity of matter and of the spatial displacement, and inversely proportional to the time during which the action takes place. And this force of acting translates into a series of relations proportional to the quantity of motive action according to the speed of completion of the diverse sequences of formal effects. An additional definition then distinguishes the points of view of spatial displacement and of the force of acting: Diffusion of the action in the motion or the extensio of the action is the quantity of formal effect in the motion. The intensio of the same action is the quantity of velocity by which the effect is produced or by which the matter is transported over the same length. z6 Proposition 11: "Formal actions of motions are complex functions of diffusions and intentions,"27 poses the conjoint articulation of the two points of view under the form of a product of the factors of intensio and extensio. How is this integration justified? A simple, but without doubt unsatisfactory, response is given at the immediate level of the text. For one thing, aren't the formal motive actions complex functions of the effects and velOCities of translation? For another, doesn't the additional definition refasten the "extensions" or "diffusions" to the effects, the "intensiones" to the velocities (velocilas), in the line of application of the axiom postulating a basic relation, expressive of the

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force of acting? Certainly! but one cannot but proceed to compose an effect embryoed in an instant, the factor v, with an effect deployed in the space of translation, the factor s, assuming that the effect embryoed intervenes the whole extent of the translation. There is at this point an analogous transposition of the whole motive action as a sort of causal element enveloping at once, and in a homogeneous fashion, the ratio of a propensity to act and the ratio of a motive effect. Since all inertial resistance is suspended, it is presumed that the propensity is integrally conserved across the motive effect, in such a way that one would be able to add it to this effect as a permanent gain in terms of virtual translation. The equation of the formal motive effect poses a relation composed of moving bodies and of spaces of translation, or in an equivalent way, a relation composed of moving bodies, of velocities, and of times of execution of these velocities (prop. 16): it is then easy to state the propositions combining the motive effect with the virtual conserved effect to form the discursive analogue of the force of acting. That gives us proposition 17, for example: "Formal actions of motions are complex functions of the simple function of moving bodies and times and of the double function of velocities,,,28 and the propositions which draw from them diverse ingredients of proportionality, in particular, when one supposes the times equal, and that the actions are determined according to the relation combining the consideration of moving bodies and that of the square of the velocity (see the scholium to proposition 20).29 This formula itself depends on a prior formula where one assumed in addition equal quantities of matter (prop. 18).30 Once this mixed system of argumentation has been posed, Leibniz constructs the abstract proof a priori, which, moreover, he places at the beginning of the Dynamica, as the fourth demonstration of the "Spedmen praeliminare." This order is not haphazard, for the deduction depends in large measure on a prior analysis of the

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fundamental phenomena, and this analysis rests on some formalizing analogies. Going back to proposition 4, governed by the additional axiom, Leibniz suggests two possible parallel equations. 1) If the same length is traversed at four successive different velocities and among these BC is to EG as LM to NG, then in the same way the formal action corresponding to BC will be to the action EG as the action LM to the action NG. Whence, according to proposition 7, the fact that the length of the trip remaining the same, the actions are proportional to the simple or multiplied or divided velocity. Not having a sufficient reason to envisage a factor supporting multiplication or division, we retain the relation to simple velocity. Then, according to proposition 18 (itself depending on definition 4), the time being kept the same (- unitary time), the action is proportionally related to the square of the velocities (in my opinion by inscription of the velocity embryoed in the force of acting conserved across action, that is to say across the translation by unit of length). 2) Second possible way of putting the equation: it is a matter this time of proceeding from the hypothesis of successive velOCities occurring in a unit of time. Certainly, the same equation of actions

.ac - 1M EG

NG

is conserved and in this way the actions are called proportional to the simple velocities. It suffices from that point on to introduce the relation of the action thus characterized in terms of equal lengths in order to obtain anew an equation symbolizing the composition of the intensio and of the dijJusio of the action. In the reasoning called a priori which follows and which synthesizes the two approaches, it is necessary to note that everything is staked on a double consideration: on the one hand, the possibility of composing the extensive and intensive factors by the bias of the relations which velocity takes up respectively with the length of the distance covered and with time;

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on the other hand, the analogical tie between this symbolic composition of the velocity with itself, and a "physical" interpretation of the force (potentia) of acting, expressed by the verb "facere," which the additional axiom and definition 4 permit postulating nominally: To this point everything agrees; the only divergence consists in this: the length being assumed the same, the composition of functions does not take place; and thus the actions will be as the velocities, or reciprocally as the times; but the time being assumed the same, the composition of functions takes place, and thus it is not permitted to say that the times being assumed the same, the actions will be as the velocities. I thus show that there is place to compose. Let there be three actions: A does the double in a single time, B does the double in a double time, and finally C does the single in a single time; the relation A to C, which is that of actions of different velocities in the same time, is composed of the relation A to B, which is that of actions of different velocities over the same length, and of the relation B to C, which is that of a double action to a single action. Whence it emerges that A to C has a relation two times greater, which merited being shown separately by a particular proposition; and thus it was necessary to have precede what we have said of the relation B to c. 31 The problem would consist in holding on to the phoronomic relation from A to B, at the interior of which one cannot inscribe a simpler relation: in virtue of which one would be restrained to the proportionality of actions to velocities alone or in equal times to the continuous effects which result from these. At that point a phoronomy of the Canesian type, which would not be capable of subjecting geometrical symbolization to physical causes able to justify

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the combinatorial step, stops. Right away, the Leibnizian conception is constructed in a mixed way. To the argumentation which would be limited to the analysis of the phoronomic relation between A and B, Leibniz opposes a synthetic argumentation which he summarizes thus: The estimate of affections is composed of the estimate of effects, or of lengths and velOCities; but the lengths are a complex function of times and velOcities; actions are thus a simple function of times and doubled of velocities. If the times are equal, the actions are as a doubled function of the velocities. It would remain to account for the implications of this mixed step, "analogically" a priori, to examine the developments which Leibniz draws from it in chapter II, "De Potentia motrice absoluta demonstrata a priori." We will content ourselves with underlining in this respect that the a priori character of the demonstration stems here completely from a new definition connected with the axiom of the preceding chapter and with the demonstrations which are analogically a priori relative to the diffusio and intensio joined together in formal motive action. This definition is that of absolute force: Absolute force of what is moved is the affection of this thing, proportional to the quantity of action coming from the state of that which is moved, action continuing in a determined time, or proportional to the quantity of formal action which the moving body would exert if it uniformly followed its motion across a time of given magnitude. That is why the times of the action showing themselves equal and the formal actions assumed uniform, the absolute motive forces are as the formal actions. 33

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Leibniz here follows in a significant way the inversion of the abstract geometrical step proper to phoronomy. The dynamic representations tied to the composition of factors in motive action help him to reconstitute a sufficient reason for the force of acting in the moving body, and this sufficient reason is projected under the form of property (affeetus) of the model. Translated by elevation to the square of the relative factor y, the quantities of formal action are going to permit one to express, by the game of relations of proportion, this factor of absolute type which is the living force. It is evidently a question of living force embryoed and conserved in a body considered by itself, and not of living force translated into integral effects as in the case of free fall, indeed into respective effects as in the cases of percussion, of tension of springs, and of pendulum oscillation. It is in this conceptual context that Leibniz demonstrates proposition 4: "If moving bodies contain an equal quantity of matter, the motive forces are a doubled function of the velocities or as the square of the velocities." 34 This proposition is inferred from the definition of absolute force and from proposition 18 of the preceding chapter, account taken of a limitative condition: the equal quantity of matter of the moving bodies. The methodological artifice underlying the demonstration is clearly identified with the conservation of the force of action across the formal action which translates it. Whence the composition of the velocity with itself in order to assign to the mass an absolute force at the same time actualized in motion and maintained intact through motion in a unit of time. The first aspect constitutes the phenomenal and extensive expression of the second because of the parameter of velocity conjointly intensive and diffuse: "In the force is momentary that which is uniformly diffused in the action occurring in time. 35

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The whole demonstration is based on the consideration of bodies in motion without constraint or gravity factor. It is a matter of course that its application can extend to the actions and forces of motions diversified in such a way that one would be able to analyze them in terms of uniform and equidistributed combined motions. The interestiryg case emerges with the translations from accelerated or retarded motions, depending for instance on gravity. The junction of absolute force and of the effect of weight is established as a corollary of the principle of conservation of force represented by the quantity of formal motive action, but in such a manner that the a posteriori argumentation appears as an ineluctable analytic element in the context of a synthetic presentation. In the perpendicular descent to the horizon characteristic of the effect of weight, at equal quantities of matter the forces are proportional to the heights of fall, and it proves that these heights of fall are proportional to the squares of the velocities. But this last relation here is part of the concessa and thus conceived requires an a posteriori confirmation: "And since this is taken for agreed, it follows that our demonstrations are confirmed a posteriori.,,36 At this stage, Leibniz only uses it, moreover, to isolate such a dynamiC phenomenon from the regulative application of the Cartesian principle of conservation of the quantity of motion, properly without proof in the case conSidered, that of the fall of bodies by gravity. But this principle is equally subject to arouse paralogisms in all the cases where it is appropriate to grasp the implication of the action embryoed in the begetting of formal action. Well, precisely the concept of force, analogically expressed by the integration of the action embryoed in formal action, has as its function permitting the denouement of such paralogisms. This denouement is assured by an equation of absolute value, fixing the conservation of the total quantity of motive force in tempore quocunque. In order nonetheless to reveal the foundation of this principle in concreto, Leibniz will necessarily have to integrate an a

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posteriori derivation of the analogical equivalents of the potentia absoluta into the analysis. The case of uniform-difform motions, in the manner of weight effects on a moving body initially constrained, will permit him this radicalization of analysis. Lacking which, the significance of the potentia absoluta risks being maintained as an abstract subterfuge, like an algebraic unknown to which a semiarbitrary combinatory game on the factors intervening in the relations of formal actions will attribute the velocity value to the square. Certainly, this combinatorial is supported by the conceptual analysis representing in a progressively more adequate way the force (power) of acting in concrete moving bodies; but without junction with the a posteriori step, the a priori step could not suffice to constitute a coherent mixed science of fundamental dynamic phenomena. At this point let us suspend our analysis and list the elements of a provisional conclusion. We have taken account of only one part of the Dynamica de potentia, precisely that part which seemed to justify at first sight the thesis of an a priori demonstrative path which Leibniz would have followed in order to establish a theorem of dynamic conservation; this theorem would be that of the potentia absoluta, symbolized by the quantities of formal motive action in a given elementary time. Well, the strategy of geometrization of the phoronomic relations only meets by analogical approximations the conditions of a deductive progression. In fact, Leibniz progressively constructs, on the relations of proportions relative to the formal motive effect and to the formal motive action, a system of references to theoretical entities of another order; this system of representations is intended to express the force of acting in matter at the moment of a conjunction of intensive and extensive effects. By way of axioms and nominal definitions, the argumentation more geometrico integrates those concepts which escape the order of an analysis which proceeds entirely a priori. Not only do the principal

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phoronomic theorems translate an ascending progression in the game of analogies serving to interpret the geometrical and analytic relations in the phenomena of mass in motion; but at the moment of taking up the derivation of force, the order seems to invert itself in the argumentation: by the game of definitions and the axiom relative to the force of acting, the theoretical concepts representing physically sufficient reasons tend henceforth to regulate in a more deliberate fashion the construction of the systems of equations and of the theorems which allow the establishment of such systems. This tendency will become completely manifest when Leibniz at the beginning of the second part of the Dynamica will fix the definitions and the axioms allowing for the reconstruction of the a posteriori demonstration of the theorem of the conservation of force vive. In general, the epistemological problem one deals with in assessing the so-called a priori demonstrative path in Leibnizian dynamics, relates to what may count as adequate hypothetical constructions in science. The ideal demonstrative pattern for Leibniz is the logical one of a deductive chain of reasons or propositions stemming from real definitions; and real definitions express the logical possibility of their definienda. It so happens that definitions in physics cannot but remain ultimately nominal: they are viewed as provisionally real on the expectation that experience will in fact corroborate them. Dynamics, which Leibniz took to be the true ground of physics, can indeed be no exception. But Leibniz nevertheless believed that the finite understanding can frame up mathematical analogies to account for the more intelligible features of phenomenal reality. We are at liberty to develop the analytic implications of such sets of "geometrical" definitions as may combine in an architectonic fashion and represent the causal system of physical motion. If the arguments thus issued illustrate a true combinatorial procedure to express the transition from more simple to more complex features of physical motion, and if they result in a

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synthetic and comprehensive model, a kind of a priori consistency attaches to such a conceptual system. Because the fundamental laws of dynamics can be expressed through this type of combinatorial procedure, they would be supported by a priori as well as a posteriori proofs. But, in this sense, "a priori proof' means that a combinatorial construction of geometrical models can result in a distinct and comprehensive, though never fully adequate, expression for the theoretical concept of force, which is otherwise empirically warranted. Professor of Philosophy, University of Montreal

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Notes 1. Die philosophische Schriften von G. W. Leibniz~ ed. C. I . Gerhardt, (GP), Iv, p. 469: "Cujus rei ut aliquem gustum dem, dicam interim, notionem virium seu virtutis (quam Germani vocant Krafft, Galli b force) cui ego explicandae peculiarem Dynamices scientiam destinavi, plurimum lucis afferre ad veram notionem substantiae intelligendam." 2. On the significance of the controversy concerning "living forces" (forces vives), cf. Carolyn IItis, "Leibniz and the Vis viva Controversy," Isis, 62 (1971), pp. 21-35; George Gale, "Leibniz' Dynamical Metaphysics and the Origin of the Vis Viva Controversy," Systematics, 11 (1973), pp. 184-207; David Papineau, "The Vis Viva Controversy," in: R.S. Woolhouse (Ed.), Leibniz: Metaphysics and the Philosophy of Science, Oxford:Oxford Uniyersity Press, 1981, pp. 139-156. 3. Cf. Louis Couturat, La Logique de Leibniz, Olms: Hildesheim, 1%9, chap. VI, L' encyclopedie; chap. VI, La science generale; Francois Duchesneau, "Leibniz and the Philosophical Analysis of Science," Logic, Methodology and Philosophy of Science, Amsterdam: Elsevier Science Publishers, 1989, pp. 609-624.

4. Cf. De ipse natura sive de vi insita actionibusque Creaturarum, pro Dynamicis suis confirmandis iIIustrandisque (1698) GP, IV, pp. 505506: "Cujus inter alia indicium insigne praebetfundamentum naturae legum, non petendum ex eo, ut conservetur eadem motus quantitas, uti vulgo visum erat, sed potius ex eo, quod necesse est servari eandem quantitatem potentiae potentiae actricis, imo (quod pulcherrima ratione evenire deprehendi) etiam eandem guantitatem action is motricis ... "

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5. Martial Gueroult, Leibniz, Dynamique et metaphysique, Paris: Aubier Montaigne, 1967, pp. 110-153.

6. Cf. the declarations relative to the mixed propositions in the Nouveaux essais sur l'entendement humain, 4.11.14 G. W.Leibniz. Samtliche Schri/ten and ~rie/e, VI vi, 446-447, and those which concern the status of physics in 4.12.10:, ibid, 453-454 "I remain agreed that the whole of physics will never be a perfect science for us, but we will not lack the ability to have some sort of physical science; and indeed we already have some samples of it. For example, Magnetology can pass for such a science, for making little of suppositions founded in experience, we are able to demonstrate as a result of it a number of phenomena which are as reason leads us to believe." 7. Cf. Hide Ishiguro, "Leibniz on Hypothetical Truths," in: Michael Hooker (ed.), Leibniz: Critical and Interpretive Essays, Manchester: Manchester University Press, 1982, pp. 90-102; Fran~ois Duchesneau, "Leibniz et les hypotheses de physique," Ph ilosophiques, 9 (1982), pp. 223-238.

8. Mathemattsche Schriften von G. W. Leibniz, ed. par C.1. GerHardt (GM), VI, Pp. 281-514. I have translated the quoted passages into English. 9. Cf. Michel Fichant, "La 'reforme' leibnizienne de la dynamique d'apres des textes inedits," Akten des II. internationalen LeibnizKongresses. Bd 2. Wissenscha/tstheorie und Wissenschaftsgeschichte. Wiesbaden: F. Steiner, 1974, pp. 195-214; "Les concepts fondamentaux de la mecanique selon Leibniz en 1676," in A. Heinekamp (ed.), Leibniz a Paris (1672-1676), Wiesbaden: F. Steiner, 1978, I, pp. 219-232.

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10. Cf. on this point, Pierre Costabel, Leibniz et Ia dynamique. Les

textes de 1692. Paris: Hermann, 1960. 11. The same is true of the analyses of Gerd Buchdahl in his book

Metaphysics and the Philosophy of Science. Oxford: Blackwell, 1969, chap. VII. Leibniz: Science and Metaphysics, pp. 388-469. 12. GM,VI, p. 349.

13. GMVI, p. 283. 14. GM, VI , pp. 289-290.

15. GMVI, p. 282. 16. GM, VI, p. 292. 17. GM, IV, p. 320: "Movetur vel in motu est A, cui quidquid inest

homogenum seu comparabile B, aliquod punctum habet ut E, quod per unius ejusdemque temporis quamcunque partem in eodem loci puncto non est." 18, GMVI, p. 323. 19. GMVI, p. 330. 20. GMVI, p.

341.

2l. GMVI, pp. 341-342. 22. GMVI, pp.

345-346.

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23. GMVI, p. 346. 24. GMVI, p. 348. 25. GMVI, p. 349. 26. GMVI, p. 355. 27. GMVI, p. 355. 28. GMVI, p- 356. 29. GM VI, p. 359: "Actionum aestimatio compos ita est ex aestimatione effectuum seu longitudinum et velocitatem; sed longitudines sunt in ratione composita temporum et velocitatum; ergo actiones sunt in consideratione composita ex simp lice temporum et duplicata velocitatum. Ergo si tempora sint aequali a, actiones sunt in consideratione duplicata velocitatum; sed haec consideratio .....

30. GMVI, p. 357: "Si aequales sint materiae quantitates, et temp ora actionum aequalis, actiones motuum formales erunt in duplicata ratione velocitatum vel longitudinum motus." 31. GM, VI, p. 358. 32. GM, VI, p. 359.

33. GM, VI, p. 359. 34. GM, VI, p. 362.

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35. GlI-I, VI, p. 364.

36.

GM, VI, p. 367.

DUCHESNEAU

METHODE AXIOMATIQUE ET IDEE DE SYSTEME DANS L'OEUVRE DE JULES VUILLEMIN

Gilles Gaston Granger Le theme de l'axiomatisation, ainsi que celui de l'organisation systematique des theories; sont egalement presents dans l'oeuvre entiere de JULES VUILLEMIN. Soit qu'il s'attache a mettre en evidence ou a restaurer la structure axiomatique d'une theorie scientifique, comme it Ie fait des 1963 dans "Le principe du levier"; soit qu'il veuille mettre a l'epreuve une doctrine philosophique en en explicitant les presupposes, la conjonction des deux themes paralt etre l'une des idees directrices de sa philosophie de la connaissance et de sa conception de la pensee rationnelle. Je voudrais rassembler et commenter ici quelques aspects de cette conjonction, sans pretendre examiner toutes les formes et toutes les consequences qui s'en presentent dans une oeuvre aussi multiple, aussi technique et aussi riche, s'echelonnant sur plus de trente annees. Le probleme crucial et l'objet principal de la discussion m'apparalt alors comme etant celui du sens qu'il faut donner, et de la portee qu'il faut reconnaltre a une methode axiomatique appliquee a la philosophie meme. Cependant, comme il convient de preciser tout d'abord l'idee de methode axiomatique dans son usage ordinaire, j'adopterai successivement trois points de vue, considerant en premier lieu Ie role de l'axiomatisation dans les sciences, puis l'organisation interne des systemes philosophiques, enfin ce que j'appellerai la "metasystematique" teUe qu'eUe apparalt dans les oeuvres les plus recentes de notre auteur. Le role de I'axiomatique dans les sciences. Dans un ouvrage sur la Phllosophie de l' Algebre (962), J. VUILLEMIN se propose de montrer comment Ie renouvellement des idees et des methodes dans les mathematiques post-Iagrangiennes du xvm o siec1e-avec Gauss, Abel, GalOiS, Klein et Lie-se presente 83

G. G. Brittan (ed.), Causality, Method, and Modality © Kluwer Academic Publishers 1991

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comme Ie retentissement de I'avenement d'une Analyse abstraite (ibid .p.6S). 11 oppose alors cette methode a la methode geneJique, caracJerisJique, selon lui, de la geometrie cartesienne. Sans doute, Ie theme de I'axiomatisation n'est-il pas encore ici explicitement degage, mais il est bien deja sous-jacent a cette opposition. Le rapprochement constant qui est fait de cette mathematique nouvelle et de la reforme critique de la philosophie montre qu'il s'agit en effet d'une orientation de la recherche vers les conditions de possibilite des constructions et des proprietes de l'objet mathematique (et de sa connaissance), orientation qui naturellement devait conduire a l'explicitation axiomatique des structures. sur Ie paralU:lisme de la "philosophie theorique" et des "mathematiques pures", ( ibid pA), je me propose de bientot revenir; mais je voudrais souligner auparavant trois themes, dont je montrerai que, pour]. VUILLEMIN, ils tournent comme autour d'un pivot central autour de la these de ['importance essentielle d'une fonction axiomatisante de la pensee scientifique. 1. . "La science n'est pas seulement une langue bien faite, c' est a dire un systeme de concepts et d'axiomes propres a exprimer et prevoir la realite... On y distingue des parties qui possedent leurs propres axiomes et concepts; et entre ces parties la science rend possible l'etablissement d'une relation de traduction" ( 1982*, p.313)

Une theorie scientifique, une theorie assez complexe a tout Ie moins, si la formulation axiomatique en est suffisamment explicite, serait donc dissociable en plusieurs blocs, qui seraient en quelque maniere l'image l'un de l'autre. Les exemples donnes sont empruntes a la geometrie, a la dnematique-pour chacune desquelles la traduction * Les references aux articles et ouvrages de ].V. seront faites au moyen de la date de publication indiquee dans la Bibliographie.

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peut etre complete et reciproque, telle par exemple celie des geometrie euclidienne et non-euclidiennes et celie des differentes cinernatiques copernicienne et pre-coperniciennes pour Ie systeme solaire. Dans Ie cas de la physique, la traduction ne sera it jamais parfaite, soit qu'elle ne puisse s'effectuer que dans un sens, soit qu'elle ne vaille que dans des dornaines !imites, soit que Ie langage traduit n'ait qu'une portee approximative ( ibid, p.317; voir aussi 1983, p. 128, note 32). L'idee de J.VUILLEMIN est justement que Ie degre de perfection de cette traduction est la marque de l'ampleur des "engagements ontologiques" supposes par une theorie. Si la traduction est complete et reciproque, "nous la regardons comme un effet de la structure du langage" (ibid. p.319). dans les autres cas, c'est que la theorie comporte des conventions ou des Ii mites essentielles d'approximation, suggerees sinon imposees par l'empirie. Une telle interpretation est a mon avis tres convaincante. Elle est je crois justifiee par la nature meme de la representation axiomatique, qui, d'une part, appelle plusieurs realisations ou "modeles" specifique, d'autre part auto rise, dans les bornes de la coherence logique, des variations dans la specification des axiomes. Je rattacherais pour rna part cette possibilite de "traduction interne" a la dualue, plus ou moins parfaite selon les dornaines de la connaissance, entre un systeme d'objets et un syteme d'operations qui se codeterminent. La question demeure cependant posee de savoir si cette dualite se manifeste necessairement par la co-presence et la concurrence de plusieurs representations des phenomenes, interpretables l'une dans I'autre, comme dans les exemples analyses par J. VUILLEMIN. L'affirmer serait peuH?!tre trap prejuger de I'histoire et du futur de la science. Mais sous la forme plus generale que nous en praposons, il semble que l'on puisse en effet toujours reconnaitre, a chaque etape du developpement d'une science, un mouvement polarise vers la saisie, en un certain sens directe, d' objets - fussent-ils toto: coelo differents de ceux que nous

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manipulons et qui tombent sous nos sens -, et un mouvement qui privilegie un systeme d'operations de pensee et d'operations materielles. Mouvements qui se co-determinent . Leurs degres d'adequation, ou plutot d'inadequation, mutuelle caracteriserait ainsi la charqe en contenus ontologiques d'une theorie, allant de la vacuite des formes de la pure logique a la densite et a I' opacite des entites de la physique. 2. Un second theme aborde par J. VCILLEMIN et qui se situe dans la perspective du role de l'axiomatique est celui de la communicabilite des theories et de leur refutabilite par l'experience. Dans l'article sur "Les relations mutuelles entre philosophie des sciences et histoire des sciences" (1987), il examine et analyse avec objectivite et precision, pour les refuter par apres, les theses quelque peu provocatrices selon lesquelles il y aurait, au cours de l'histoire d'une science, succession de "paradigmes" incommunicables entre eux. Les theories qui se succedent sont au contraire comparables, une fois leurs presupposes, explicitement reconnus, et definie l'echelle ainsi que la zone d'approximation a l'interieur desquelles leurs deductions ont un sens empirique. Plus generalement, J. VUILLEMIN, en plusieurs occasions, developpe et precise l'idee de refutabilite des theories. (1987 et 19792). Contre l'indetermination des theories par l'experience-these radicale de Quine-il admet que, "des que l'echelle d'experimentation est suffisamment precise, l'experimentation decide de fa~on univoque" (1987, p.71). Mais une telle accessibilite a la refutation n'exige-t-elle pas que les theories aient recu une formulation rigoureuse, que seule une axiomatisation au moins partielle a pu garantir? Dans Ie cas des astronomies pre-coperniciennes et copemicienne qu'il prend souvent pour exemple, J. VUILLEMIN note que "c'est en apparence seulement que les astronomes antiques soustraient au doute les principes metaphysiques qu'ils proclament" (1987-2,p.65), et que l'hypothese de paradigmes isoles et rigides est insoutenable. Une assise axiomatique, ou en tout cas axiomatisable par

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l'analyse de l'epistemologue, non seulement n'exdut nullement la communicabilite entre theories, mais, loin d'etre la marque d'une impenetrabilite a l'experience, elle apparatt comme Ie gage d'une ouverture, parce qu'elle determine avec darte les conditions d'une prise en compte significative, non dogmatique, de ses donnees. Sur la nature exacte du rapport des axiomes a l'empirie en physique, il reste assurement beau coup a dire, et c'est a propos du difficile probleme pose par la physique quantique, et en particulier par l'interpretation d'une verification des inegalites de Bell, que l'on attendra les edaircissements de J. VCILLEMI:'oi 3. Cne autre these, liee aux precedentes, et qui depend me semble-t-il du role attribue a l'axiomatisation, serait la distinction entre verite, verite partielle et approximation de la verite. (983). "En disant que 1t - 3 ,1415 a 10-4 par defaut, je donne de fa~on exacte une approximation de 1t. Mon enonce lui - meme n'est ni partiel, ni approche. II est vrai." (ibid p.123) Appliquee aux sciences empiriques, la distinction prend toute sa portee, et permet d'edairer la notion de possibilite de traduction de deux theories l'une dans l'autre: on peut traduire une theorie approchee, par exemple la mecanique dassique, dans une theorie "plus riche", comme la mecanique quantique, l'operation inverse etant en general impossible (ibid. p. 128). Le progres de la connaissance ne consiste donc pas en une rupture radicale, entratnant l'incomparabilitei la verite d'une loi dans l'ancienne theorie "n'est alors qu'une verite partielle a la nouvelle precision requise, puisqu'elle n'a lieu que dans des cas particuliers et sous des conditions specifiques. Elle est en meme temps a l'ancienne precision pres une approximation de la verite propre a la theorie nouvelle" (ibid. p.131)

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Analyse qui rend parfaitement compte de la possibilite de ce progres et du sens d'un caractere cumulatif de la connaissance objective. Or l'assignation d'un degre de "richesse", tel que I'entend j. VUILLEMIN, a des theories suppose sans doute qu'il soit possible d'y distinguer precisement les presupposes et les regles determinant leurs objets. Mais il prend soin de faire remarquer que Ie progres de la science, "l'exigence de conservation des observations et des lois anterieures" n'entraine nullement que I'on imagine une "convergence de theories, vraies relativement a une echelle En vers une theorie vraie absolument." (1987, p.72). ]'ajouterai cependant que I'on doit pouvoir parler d'une convergence qui serait Ie signe meme de ce progres, mais non pas d'une convergence vers de la me me fac;:on que Ie geometre reconnait, dans Ie domaine des rationneis, des suites-de Cauchy- convergentes, dont il sait qu'aucun nombre rationnel n'est la limite. De preciser et decrire, dans chaque cas historique, Ie sens de cette "convergence" des theories serait alors I'une des tachesdifficiles- de l'epistemologue. Le moment de I'axiomatisation apparait donc bien a j.VUILLEMIN comme tout a fait essentiel a la constitution de la connaissance scientifique, qu'elle concerne les mathemata ou des objets empiriques. L'axiomatique, ecrira-t-il en 1986, "reveille l'esprit agite (restless) de son sommeil mythique" (1986, p103); sa fonction la plus generale serait de faire penetrer la raison dans Ie mythe. Fonction qui, pour notre auteur, ne se borne pas au domaine de la science, mais qui commanderait aussi la naissance et l'organisation d'une pensee philosophique. Sous quelles formes, jusqu'a quel point? C'est ce qu'il convient maintenant d'examiner. L'organisation axiomatique interne des philosophies ]'aborde ici un terrain beaucoup moins sur, et OU, faute d'un point d'appui comparable a celui que nous donnait et nous imposait la realite des sciences, les chemins de philosophes qui se sentent pourtant tres proches peuvent considerablement diverger. Dans Ie

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domaine de la connaissance scientifique, I'axiomatisation joue assez manifestement Ie role constitutij de determination de I'objet, c'est a dire que se trouve par elle explicitee la preparation du phenomene en vue de sa representation par une image abstaite. Peut-on deceler une fonction comparable dans Ie cas des philosophies? Les enonces philosophiques s'organisent-ils en systemes, a partir d'une base axiomatique, a la maniere de ceux de la science? 1. La reponse de J. VUILLEMIN s'est sans doute nuancee au cours du temps, mais elle est decidement affirmative. II la formulait de faeon abrupte en 1955, lorsqu'il enoncait l'un des postulats caracterisant sa methode en histoire de la philosophie sous la forme: "La philosophie est elle-meme une science ... " 0955, p.3). Et il definissait alors l'idealisme transcendantal comme "la science des actes intellectuels par lesquels I'homme pense la mecanique rationnelle". (ibid p,J) En 1962, dans La philosophie de l'Algebre, il se propose "d'utiliser les analogies de la connaissance mathematique pour critiquer, reformer et definir autant qu'il se pourra la methode propre a la philosophie theorique." 0962, p.5) Dans un texte de 1985, il souligne Ie rapport originaire des mathematiques a la philosophie, et affirme la communaute de leurs methodes: "... vraisemblablement chez Pythagore, les deux sciences se sont engagees dans une liaison plus etroite, et afin de delimiter leurs taches et de fixer leurs conditions de validite, se sont soumises a une methode commune: la methode axiomatique". 0985,p. 1) Cependant, lorsqu'il commente alors l'application du mot "science" it la philosophie, Ie seul trait decisif qu'il met en vedette est celui de coherence: "avant tout vaut pour la mathematique et la philosophie l'exigence de coherence" (ibid). Et deja dans l'etude sur Kant, il justifiait l'appellation de science en disant: "Lorsqu'on comprend quels problemes sollicitaient

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Kant, il est impossible de bonne foi d'attendre une solution differente de celie qu'il avance" ( 1 955,p.2) Ainsi Ie caractere scientifique de la philosophie se trouverait-il finalement reduit a un requisit de coherence deductive et de noncontradiction, ce que revelerait en particulier Ie role important joue par les raisonnements ab absordo dans cette discipline. On notera qu'il s'ecarte sur ce cernier point d'un texte sur La demonstration de l'irrationalite de 1t ou il semblait reconnaitre avec Kant, que, si les preuves en mathematiques peuvent etre apagogiques, elles doivent etre en philosophie ostensives 0961, p.430) . 2. II n'est certes pas douteux que la formulation axiomatique soit la presentation vers quoi I'on doit tendre, si I'on veut pouvoir mettre en evidence la non-contradiction d'une doctrine. Cependant, la fonction d'un systeme d'axiomes, dans la mathematique merne, peut-elle etre suffisamment decrite en disant qu' il expose Ie squelette d'articulations logiques qui doit gouverner toutes les propositions de la theorie? Outre cette fonction architectonique regulatrice, un systeme d'axiomes exerce une fonction proprement constitutive, selon I'expression employee plus haut. Alors me me qu'il parait dessiner une pure forme d'objets, comme en mathematiques, il introduit pourtant deja des contenus, soit qu'on en rattache I'origine, comme Kant, a la sensibilite, soit qu'on les considere comme inherents au fonctionnement d'une pensee symbo/ique et qu'on veuille, ainsi que j'ai tente ailleurs de Ie fa ire, les definir comme "contenus formels". La question est alors de savoir si la philosophie peut se contenter de tels contenus formels, ou si les conte nus propres a des axiomes philosophiques n'alterent pas de facon essentielle Ie jeu d'une axiomatique. Probleme qui n'est nullement etranger a la pensee de J.VUILLEMIN. Dans Ie texte cite de 1985, ou se trouvent etroitement associees la philosophie et les mathematiques antiques, il note en effet que

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"Les questions touchant la nature des concepts fondamentaux et l'evidence des principes sont l'affaire d'une autre discipline: la phitosophie" (ibid p. 1; c'est moi qui souligne). Si la reduction axiomatique est appliquee dans cette discipline, ce n'est done pas seulement pour mettre en lumiere la coherence de ses demarches, c'est aussi, et specifiquement, pour en preparer I' (interpretation) J. VUILLEMIN parle du reste d'axiomatique "materielle", et reconnait que: "Les premisses que manipule la philosophie sont si generales, si nombreuses et si complexes qu'elles sont rebelles a toute expression formelle particuliere, et meme defient peut-etre toute expression historique particuliere, si complete soit elle." 0986, p.113) A l'axiomatique formelle it assigne deux caracteres: elle procede par definitions implicites, et les demonstrations qu'elle fonde vont "des choses aux signes" selon la methode "metamathematique". Or Ie premier caractere peut aller, dit-il, sans Ie second, ce qui serait Ie cas en philosophie, de telle sorte qu'it ne s'agirait alors que d'axiomatique matmelle. (ibid p. 156, note 61). Si l'on en croit cette note, seul subsiste donc ici Ie trait de definitions implicites. On peut des lors se demander si une telle axiomatisation, ayant perdu la vertu qui lui venait de son possible deploiement sur Ie seul plan des symboles, continue veritablement de pouvoir exercer la fonction qui etait, dans son premier usage, essentiette. It me semble pour ma part qu'un systeme de definitions implicites ne peut meriter vraiment Ie nom d'axiomatique que dans la mesure ou, s'il vehicule des contenus, ceux-ci se trouvent convenablement representes dans les rapports des signes, et ou les deductions "metamathematiques" (au sens du texte cite) qui s'ensuivent sont susceptibles d'interpretations valides. Ou reste, j.VUILLEMIN ne manque pas d' avoir recours a une telle axiomatisation "formelle" lorsqu'it veut restaurer et mettre en forme un argument comme celui du Dominateur. Mais it s'agit alors

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d'un fragment de discours envisage dans sa demarche purement logique, dont il faut degager la tactique; son aspect proprement philosophique ne se manifeste, posterieurement, que dans sa fonction strategique, dans son insertion au sein d'un systeme de la contingence ou de la necessite. 3. Si l'on veut comprendre cependant pourquoi J.VUILLEMIN semble renoncer en principe aux pouvoirs des signes en philosoph ie, il faut se referer a sa juste mefiance a l'egard de la pensee "mythique". La philosophie, comme la science, sans doute, est bien nee du mythe, mais pour en transposer radicalement les orientations et les procedes. La philosophie, dit J. VUILLEMIN "resulte de la reorganisation des deux dimensions des signes mythiques. Le recit mythique fait place a la recherche des, vrais principes selon les canons de la methode axiomatique. Telle fut la premiere relation fondamentale de l'axiomatique a la philosophie. En meme temps, cependant, la philosophie vise a reformer et a restaurer l'ontologie mythique evincee par l'axiomatique. Une ontologie determinee prend la place d'une reference equivoque a la realite" 0986, p. 114) Nous voici donc revenus au probleme entrevu plus haut de la preservation des contenus. II est formule dans la perspective de J.VUILLEMIN comme rapport de l'axiomatique a l'ontologie. La philosophie lui parait repondre a deux defis, quasiment contradictoires. Elle do it, en effet, repondre aux exigences de la nouvelle methode axiomatique d'organisation d'un savoirj mais elle doit egalement repondre a "la question ontologique posee par l'avenement de l'axiomatique, lorsque celle-ci a jete par dessus bord Ie monde unifie du mythe, moule dans Ie langage ordinaire". (ibid p.vii) Dans cette fonction de substitut du mythe, la philosophie se presente comme une ontologie, c'est a dire, de l'aveu de j.VUILLEMIN Iui-meme, comme Ie contraire d'une axiomatique. Le contenu du mythe etait un codage du monde du sens commun (ibid.

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p. 116). L'ne philosophie devra d'abord donner une forme canonique aux modes de predication qui apparaissaient dans ce langage, afin de pouvoir exprimer par leur moyen des "premisses ontologiques" capables de fonctionner comme axiomes . C'est Ie choix d'un mode de presentation de i'etre par un langage et de sa distinction d'avec I'apparence qui va caracteriser les conte nus philosophiques. Une philosophie, formulee dans un discours, sera donc systematique en un double sens: premierement comme organisation logique de principes, et des theses qui s'en peuvent deduire; deuxiemement comme organisation pour ainsi dire semantique, qui projette sur notre experience du monde et de nous-memes un reseau de concepts, et lui applique un filtre propre a dissocier I'apparence de la realite. La systematisation ainsi entendue est assurement inherente au projet philosophique meme, et j.VUILLEMIN n'aura pas de peine a brievement rejeter les objections de ceux qui en nient I'interet, la pertinence, ou la possibilite. Pour rna part, j'en accepte la description. Je remarque toutefois que la methode proprement axiomatique n'y joue peut-etre pas un role aussi essentiel et determinant que celui, qu'en theorie, j.VUILLEMIN lui assigne. Au reste, dans sa pratique, on I'a vu, il ne la met guere efficacement en oeuvre que comrne auxiliaire, non pour la mise en forme d'un systeme dans sa totalite, mais pour I'analyse logique d'une argumentation particuliere. C'est alors seulement qu'elie joue pleinement son role, et fonctionne, rnais fragmentairement, comme axiornatiquejormeJIe. Or en prenant les mots avec leur plein sens, ne faut-il pas dire, en fin de compte, qu 'une axiomatique sera jormeJIe, ou qu 'elle ne sera pas? Quoi qu'il en soit de cette reserve, la conception de la systematicite philosophique proposee par notre auteur Ie conduit tout naturellement a considerer la pluralite des systemes, et a formuler une sone de metatheorie de la systematicite.

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La meta -Systematique de Jules Vuillemin 1. Une raison particuliere d'etablir une classification des systemes se

presente, du reste, dans l'ouvrage sur Necessite ou Contingence (984), ou JULES VUILLEMIN se propose d'abord de reconstituer la demonstration inconnue de l'argument "Dominateur". Interpretant comme rejets de l'une de ses premisses, les reactions au defi que cet argument proposait aux philosophes, il est amene a en expliquer les divers choix par reference a la coherence d'un systeme. II remonte ainsi de la premisse refusee a la conception d'un type de loi naturelle, puis au systeme philosophique global ou cette loi s'insere. De ce point de vue, c'est la notion de modalite qui apparait d'abord comme la clef d'une classification des systemes; mais s'elevant a des considerations plus generales, JULES VUILLEMIN esquisse dans cet ouvrage la meta-systematique qui sera developpee peu apres dans What are philosophical Systems? (1986), en la fondant sur la pluralite des types d'enonces predicatifs. II insiste alors sur Ie caractere synthetique de la methode, qui va "substituer I'ordre des principes a l'ordre des matieres, objet propre de la methode analytique" 0984, p.357). Et il reconnait qu'illui faut en ce cas justifier Ie groupement dans une meme classe de systemes qui prennent, a l'egard de la modalite, des positions "variees et me me contradictoires", montrer comment, en philosophie, I'unite d'un meme principe peut engendrer plusieurs systemes divergents. Montrer pourquoi il est permis, par exemple, de classer sous la meme rubrique de "conceptualisme" des philosophies dont les theses sur la necessite et la contingence sont aussi differentes que celles d'Aristote, de Leibniz, de Saint-Thomas et de Duns Scot. (ibid) 2. On ne se proposera pas ici d'exarniner dans sa realisation et dans ses applications la classification etablie par JULES VUILLEMIN, mais

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plutot d'en considerer Ie fondement et la portee. Le fit conducteur qu'it a choisi est, rappelons-le, la distinction entre types de predications deja presents aux formes du langage ordinaire, servant d'abord a decrire, schematiser et mettre en ordre l'experience originairement perceptive. Une prerni~re distinction est faite entre enonces de type "dogmatique" et jugements de type "subjectif". La prerni~re serie oppose les enonces "norninaux", pour ainsi dire neutres quant au monde per~u, aux enonces "de participation". Ces derniers distinguent alors predication "substantielle", "accidentelle" et "circonstancielle", la derni~re seule introduisant des indicateurs de subjectivite qui evoquent l'ultime singularite des experiences. La seconde serie se caracterise essentiellement par une sorte de reduplication de la pensee suggeree deja par Ie terme de "jugements": a la pensee d'objets se superpose un index d'evaluation . On en distingue deux especes: les jugements "de methode" et les jugements "d'apparence". On peut discuter, sans doute, d'un tel decoupage, s'interroger sur Ie degre de primitivite des categories qu'it expose. je me bornerai, Ie prenant tel qu'it se presente, a en souligner ce qui me parait en etre la motivation profonde. II s'agit, nous l'avons signale plus haut, de preciser les instruments symboliques qui permettent de donner un sens a une distinction de la realite d'avec l'apparence. Un syst~me philosophique, selon JULES VUILLEMIN, est precisement une certaine organisation, a la fois formelle et materielle, de cadres de pensee capables d'effectuer cette separation; et it s'oppose a d'autres syst~mes en ce qu'il assigne a l'apparence des fronti~res differemment situees. Tel serait, en fin de compte, Ie propre d'un syst~me philosophique; et l'on voit bien d~s lors que, selon cette conception, une philosophie ne peut qu'etre systematique, puisqu'elle doit nous proposer les regles et Ie fondement de cette distinction. Une reflexion qui se voudrait purement rapsodique faillirait necessairement a cette tache, et l'on pourrait montrer-je m'accorde ici pleinement avec JULES VUILLEMIN-que cette

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formulation rapsodique, lorsqu'elle est adoptee par un philosophe, dissimule toujours une organisation, un systeme. Bien entendu, il faut se garder, comme Ie remarque notre auteur, d'associer systematicite et dogmatismej c'est que Ia vaJidite d'un systeme n'est pas de J'ordre de Ia Vente. Bien que JULES VUILLEMIN n'exprime pas cette these de fa~on aussi abrupte, c'est bien la ce qui resulte de sa conception d'une pluralite des philosophies. II ne s'agit, en les class ant comme il fait, ni de pre parer un choix eclectique qui en combinerait illusoirement les benefices, ni d'en etablir une hierarchie qui decreterait la suprematie de I'une d'entre elles. II me semble donc que la classification des systemes, la metasystematique qu'il expose, peut etre entendue comme une sorte de substitut du critere absent de la verite. Grace:l elle, un cadre serait donne :l la refiexion qui preside au choix d'une classe de systeme. Choix en quelque sorte circulaire du point de vue de la stricte logique, puisqu'il decoulerait d'une "preference prealable a l'egard de [certains) interets de la raison", preference qui suppose elle-meme I'adoption d'une classe particuliere de systemes".(1986,p. 131). Cette circularite serait assurement inadmissible dans I'exercice d'une connaissance objective. En philosoph ie, elle est sans doute ineluctable, mais il suffit qu'elle soit clairement reconnue, et que Ie discours philosophique donne Ie moyen de la formuler et d'en assigner la portee. Or n'est-ce pas l:l justement ce que tend a assurer la systematicite et la methode "axiomatique"? C'est ainsi, pour rna part, que je comprendrai et accepterai, avec JULES VUILLEMIN, leur usage en philosophie, a condition de prendre ces deux termes, et particulierement Ie second, en des sens plus larges que ceux qu'ils ont lorsqu'on les applique ala science, dont c'est Ie propre,la vertu et la limite, d'en avoir durci les contours. Chaire d' epistemologie comparative, College de France

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Textes de J.Vuillemin cites.

1955 Physique et Metaphysique leantiennes, PU.F 1955 1961 "La demonstration de l'irrationalite de sophique, 1961, pp. 417-431.

1C

"Revue philo-

1962 La philosophie de 1'Algebre, PU.F 1962 "Le principe du levier", Revue internationale de philosophie, n064 1963.fasc.2. p. 1-23. 1977 Astronornie et metaphysique d'Eudoxe a Kepler (cours du College de France) 1979-1 "La raison au regard de l'instauration et du developpement scientifiques", La rationalite aujourd 'hui, Geraets ed. ,Presses de l'Un. de Montreal ,1979. 1972 "On Duhem and Quine's Thesis", Grazer Philosophische Studien, vol 9 (1979) p.69-96. 1982 "Internal translation and reality in science" Language and onotology 6th international Wittgenstein Symposium Holder-Pichler Wien 1982

1986 What are philosophical 1986.

System~,

Cambridge University Press,

1987 uLes relations mutuelles entre philo des sciences et histoire des sciences", Epistern%gia X (1987) p.57-82

ALGEBRA, CONSTRUCTIBILlTY, AND mE INDETERMINATE

Gordon G. Brittan, Jr. 1. There is little to be gained, I believe, in trying to puzzle

through the classic philosophical texts of the 17th and 18th centuries which have to do in important ways with on-going developments in mathematics and the various sciences without also having some knowledge of those developments. Lacking this sort of historical, although at times rather technical, knowledge, commentators often miss the point of the texts. Perhaps more important, they fail to bring out their full sophistication. The main figures of the 17th and 18th centuries simply knew a great deal more about mathematics and the various sciences than do the majority of their present commentators. The same cannot be said, however, about Jules Vuillemin. His historical studies, which in fact comprise no more than part of a vast work, are exemplary. They are based on a close reading of the texts, of course. But they are also marvelously well informed. Vuillemin has a detailed and encyclopedic knowledge of mathematics, the SCiences, and their respective histories. When he brings this knowledge to bear, there is a harvest of insight. Realizing this, I have tried, not always successfully, to pattern much of my own historical work after his. Three of Vuillemin's books have been particularly significant for me: Physique et metaphysique kantiennes (Presses Universitaires de France, 1955), Matbematiques et metaphysiques cartesiennes (P.U.F., 1960), and La Philosophie de l'algebre (P.U.F., 1962). I have tried to record my indebtedness in a book and several papers and will not say here in particular what I have learned from them. Rather, I want in this paper to develop a theme which is not much mentioned in the book on algebra and use it to throw new light 99

G. G. Brittan (ed.), Causality, Method, and Modality © Kluwer Academic Publishers 1991

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on the positions of Descartes and Kant. Adopting Vuillemin's approach, I hope to supplement what he has already said about each of them. I also hope, of course, to draw his critical reaction. 2. In the book on algebra, Vuillemin does not discuss the work of Fran~ois Vieta (I 540-1603). 1 Yet not only was Vieta the "father" of modern algebra, he was also the first to make clear the ways in which it differs from arithmetic and geometry and to suggest the specifically philosophical problems which it poses. A brief sketch of the innovations that he introduced should suffice to make these problems clear. 2 Among other things, Vieta was apparently the first to use a comprehensive and systematic symbolism in the development of algebra, using consonants for known quantities and vowels for unknown quantities. This system played an abbreviatory role, but more especially it allowed him to represent not just particular equations, 7x2 + 3x + 4 - 0, for example, but the form of these equations, ax + bx + c - 0 (in modern notation) in this case, and to study them as such. Algebra is for him the study of the forms of equations, the relations between objects and the combinatory operations performed on them, rather than a study of the objects which these equations are taken to describe. In a word, Vieta was apparently the first to conceive of algebra as the study of abstract structures. . Thus he made a very sharp distinction between algebra, which he called /ogisttca spectosa... a method of operating on "species" or forms of things, and arithmetic, /ogis/tca numerosa. . which is specifically a calculation with numbers. Algebra has to do not simply with numbers, or line segments, or anything else in particular, but with magnitudes or quantities as such, anything in general which it is coherent to add, subtract, multiply, or divide. In this way it has to do with the (quantitative) forms of things, and not with the things themselves.

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Significantly, Vieta characterized his algebra as an "analytic art." He thought thereby to connect his work to the long tradition of Greek analysis, of which he pictured himself as the restorer. But despite certain resemblances, there are at least three crucial differences between the Greek tradition and Vieta's new conception. In the first place, on the Greek tradition analysis is directionally symmetrical with synthesis. On that tradition, one proceeds analytically when one assumes the theorem that is sought (the unknown) and "works backwards," so to speak, to the axioms of geometry (the known) from which it can be derived. To proceed synthetically, on the other hand, is to begin with the axioms and proceed to the demonstration of individual theorems, to advance from the known to the unknown. But these two procedures are logically symmetrical only if all the steps involved are convertible. If they are not, then analytic procedures are not necessarily valid. This fact led Greek mathematicians, in principle, to require that all of the steps in an analytic "demonstration" be convertible and that in every case an analysis be followed by a synthesis which was necessarily valid. For them, genuine proof is synthetic, a deduction of theorems from axioms. But Vieta gives up both of these requirements, and with them the claim that synthetic proofs are uniquely genuine. Even in cases where synthetic proofs cannot be given, the analytic result is allowed to stand. In the second place, the Greeks thought of analysis as an auxiliary method of demonstration, a heuristic way of finding results which could then be proved synthetically, because they thought that mathematical objects were completely determinate, in the way that a particular number or a given geometrical figure is. From their point of view, apparently indeterminate magnitudes are simply abstractions from mathematical objects which in themselves are completely determinate and the "equations" (series of proportions) concerning them are similarly abstractions. But the analytic method often leads to indeterminate magnitudes, as for example when we have one

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equation in two unknowns. Hence for the Greeks the method is incomplete until ways of computing determinate numerical solutions have been found. For Vieta, on the other hand, indeterminate solutions are as admissible from a strictly mathematical point of view as are determinate ones, and to this extent he does not subordinate analysis to synthesis. In the third place, algebra, i.e. "analysis," is for Vieta completely general, whereas for the tradition of Greek mathematics it had a definite subject matter, numbers (arlthmoi) and ultimately geometrical figures. This is to say that Greek mathematics is in an important sense intuitive; it has no concept of number apart from a definite number of things, geometrical space is at most threedimensional, and the "unknowns" appearing in equations are limited to only certain kinds of quantity. As a result, admissible operations are bounded by characteristics of the objects operated on; at most cubic equations, for example, are considered} Vieta, however, considers equations of any arbitrary power and any magnitude, numerical or geometrical or whatever, can serve as the value of the unknown variable. At the same time, as is perhaps true of most revolutionaries, Vieta was unable or unwilling to carry out completely the implications of his symbolic and purely combinatorial program. Thus, although they could be "analytically" characterized as the solutions to certain types of equations, Vieta refused to accept either negative or complex numbers as real, this despite the fact that rules giving correct results for negative numbers in particular had been known for many hundreds of years. That is to say, despite the fact that it was coherent to add, multiply, subtract, and divide them, and certainly compatible with the generality his algebra sought, Vieta rejected negative numbers, as values for coefficients as well as for variables, because they did not have the intuitive significance that the positive numbers

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did. If irrational numbers were admitted, it was only because they could be represented geometrically.4 3. Descartes simply took over Vieta's algebraic program,S modifying and extending it in a number of ways, the most important of which involved the introduction of reference lines or axes with respect to which points could be plotted and loci problems solved. The distinction between arithmetic and algebra, the emphasis on structures as against objects, the admission of indeterminate equations and solutions - all of this is part of Descartes' program as well. His algebra, like Vieta's, is a theory of magnitudes, or "proportions," in genera1. 6 At the same time, however, Descartes' view of the relation between analysis and synthesis, and between algebra and geometry in particular, is more subtle than Vieta's and turns on distinctions which his great predecessor did not make. Like Vieta, he was unwilling to carry out completely the implications of a purely combinatorial program, and thus admitted a role for "intuition" to play, but he drew the line at a different place and construed "intuition" more precisely in terms of a concept of constructibiJity. These claims can be spelled out as follows. One of the many remarkable results Descartes gives (without proof) in his Geometry is that Every equation can have as many distinct roots (values of the unknown quantity) as the number of dimensions of the unknown quantity in question. 7 But he adds immediately: It often happens, however, that some of the roots are false or less than nothing. He obviously has negative roots in mind. In saying that they are "false," he suggests that they do not exist. Not only are they less than nothing (in numerical terms), they cannot be represented geometrically. To this point he is with Vieta, and most 17th century mathematicians. But then he goes on to make another distinction that takes him beyond Vieta. The result that an equation can have as many

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roots as its degree holds only when complex, or as Descartes termed them "imaginary" numbers are also included. Moreover, neither the true nor the false roots are always real but are sometimes wholly imaginary; that is, we can, of course, always imagine as many roots of an equation as I have named. But sometimes there is no quantity which corresponds to those we imagine. 9 In this passage, Descartes grants that at least some false roots are real, even though they do not exist, while no wholly imaginary roots are real. How does he understand this distinction? Since imaginary roots are extracted using otherwise legitimate algebraic operations, the claim that they are not real can only rest on the fact that they cannot be given a geometrical construction. On the other hand, the claim that at least some false roots are real must similarly depend on their constructibility. And in fact Descartes does provide such a construction: negative roots can be made "real" by transforming the equation in which they occur into another equation whose roots are positive. Equations containing imaginary or complex roots cannot be transformed in the same way. But although they are real, as against Vieta, negative or false roots do not exist. Constructibility is the hallmark of "reality," or as I prefer to say "real possibility," but it is not, as it seems to have been among the Greeks, the hallmark of existence. 10 If one assumes, as Descartes appears to, that the mathematician is charged with the responsibility of demonstrating the real possibility of his or her objects (and not of their existence), then it follows that every admissible object and operation must be constructed. In fact, this is just Descartes' practice. Every "analytic" characterization of a curve in his geometry is followed by its "synthetic" construction (or an indication of the route to same). This is to say that Descartes rejects the sufficiency of a merely analytic or symbolic method. Two aspects of his practice are noteworthy.

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One is that Descartes broadened the concept of constructibility. The Greeks seem to have been committed to the claim that an object was constructible only if it could be constructed using straightedge and compass. 11 In Book I of the Geometry.. Descartes works within the Greek constraints, taking up problems "the construction of which requires only straight lines and circles." His method, as we might expect, is "analytic" or algebraic, utilizing (again without proof) the theorem that any algebraiC expression of degree S; 2 is constructible using straightedge and compass, and conversely. Then in Book II he admits other operations and posits a new set of constraints. Book II is entitled "On the Nature of Curved Lines." Two aspects of his discussion in it are important for us. On the one hand, he rejects the idea that the demonstration of "reality" depends on the accuracy of particular constructions, since in mathematics it is only the exactness of the reasoning that counts. On the other hand, he relaxes the constraints on constructibility. There is no longer, as with the Greeks, any restriction on dimension or degree. Descartes suggests two new (presumably equivalent) criteria. According to one, curves are admisSible, "constructible" in the sense required, if "they can be conceived of as described by a continuous motion or by several successive motions, each motion determined by those which precede; for in this wayan exact knowledge of the magnitude of each is always obtainable.,,12 According to the other criterion, "constructible" curves are those that can be expressed by a unique algebraic equation (of finite degree) in lS: and ~. Still, it would be a mistake simply to identify the constructible with the algebraic, as Vuillemin seems to do in his book on Descartes, for as we have seen complex numbers can be characterized algebraically, yet they cannot (by Descartes' lights) be constructed. These criteria admit such curves as the conchoid and the cissoid which, Descartes asserts, the Greeks were wrong to banish from geometry, and they require for

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their construction instruments other than straightedge and compass, certain types of linkages for example. The other noteworthy aspect of Descartes' practice is its presumed philosophical foundation. Why are merely analytic or symbolic methods not sufficient, even when the curves in question are algebraic and not, for example, transcendental? I suggest that it is because the rules used in algebraic calculation are not, in an appropriate sense, "intuitive;" there is no way in which they can be justified on the basis of an appeal to more fundamental evidence. They give the "correct" results, but this correctness can only be measured by way of a synthetic proof which provides the means for a geometric construction. Only a diagram, we might put it, is intuitive, directly and immediately grasped, only a diagram can be seen to be true. A merely symbolic representation is, as such, neither clear nor distinct, and hence does not suffice to ground the "reality" of the object represented. 13 4. These summary remarks on Vieta and Descartes are, I hope, relatively uncontroversial. They are intended largely to provide a context for a discussion of Kant's philosophy of mathematics, almost every aspect of which remains contested. Very briefly, on my reading Kant's position represents in important ways a still further modification and extension of the algebraic program, most important in his willingness to grant a relatively large degree of autonomy to "analytic" methods and to enlarge the class of "real" numbers and with it the concept of constructibility. Although to my knowledge he was not directly acquainted with the work of Vieta, Kant had a copy of Descartes' Geometry in his library and was thoroughly familiar with the methods and results of many 18th century "analysts," the direction of whose thought paralleled his own. 14 Indeed, I suggest that reflection on the character of algebraic methods, and in particular on the "indeterminate" character of algebraic variables, was

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a source of his central philosophical problem, how to secure the "objectivity" of at least some of our judgments about experience. I begin with a much commented passage at A716/B744 of the Critique of Pure Reason: But mathematics does not only construct magnitudes (quanta) as in geometry; it also constructs magnitude as such (quantitas), as in algebra. In this it abstracts completely from the properties of the object that is to be thought in terms of such a concept of magnitude. It then chooses a certain notation for all constructions of magnitude as such (numbers), that is, for addition, subtraction, extraction of roots, etc. Once it has adopted a notation for the general concept of magnitudes so far as their different relations are concerned, it exhibits in intuition, in accordance with certain universal rules, all the various operations through which the magnitudes are produced and modified. When, for instance, one magnitude is to be divided by another, their symbols are placed together in accord with the sign for division, and Similarly in the other processes; and thus in algebra by means of a symbolic construction, just as in geometry by means of an ostensive construction (the geometrical construction of the objects themselves), we succeed in arriving at results which discursive knowledge could never reach by means of mere concepts. 15 This passage is problematic in several respects. But it is a key to understanding Kant's position. As is invariably the case with Kant, it turns on distinctions, here between magnitudes (quanta) and magnitude as such (quantitas) and between ostensive and symbolic constructions. The distinctions are clearly and quite closely related. The paradigm of an ostensive construction is provided by geometry,

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where the axioms and definitions afford the "synthetic" means to carry out the construction of particular figures which as such are "ostended" and the eventual objects of (visual) perception. But it is important to note that for Kant arithmetic as well is "ostensive.,,16 As much is indicated by the passages at B15-16 and elsewhere which indicate how numbers can be "ostended" by fingers, strokes on a page, and so on. In both cases, arithmetic as well as geometrical, the ostension is taken to provide us with a kind of evidence on the basis of which the corresponding claim (that 7 + 5 - 12, that the sum of the interior angles of a triangle is equal to two right angles) may be verified. The ostension, we might say, controls the claim. At this point it is necessary to be more precise about arithmetiC, for one thing in order to locate some of the differences between my interpretation of Kant's view and that of other commentators. Kant distinguishes between arithmetic and general arithmetic and couples the latter with algebra. 17 The evident parallel between the two types of arithmetic is that they involve the same combinatorial operations. The difference between them, as Kant puts it in #2 of the First Reflection of the Enquiry Concerning the Principles of Natural Theology and Ethics is that general arithmetic has to do with indeterminate magnitudes and arithmetic proper with numbers ("where the ratio of the magnitude to unity is determinate"). Michael Friedman suggests that this means "that the arithmetic of numbers is concerned only with rational magnitudes, whereas general arithmetic or algebra is also concerned with irrational or incommensurable magnitudes." 18 This is fine as far as it goes, but it does not go far enough. First, the reason why arithmetic is concerned only with rational magnitudes is that only rational magnitudes can, in the appropriate arithmetic sense, be "ostended," that is, represented as determinate ratios of whole numbers. There are no determinate magnitudes, no homogeneous combinations of ostensive representations of numbers, corresponding to the irrationals.

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Second, a point to be developed later, general arithmetic has to do not only with rationals and irrationals, but with negative and complex numbers as well. Kant, as we shall see, has a much more general concept of the "indeterminate than Friedman allows him. For him as for Vieta, algebra is logistica speciosa, and applies as much to geometry as to arithmetic. But the most important and immediate point is simply this: arithmetic proper has an obviously "intuitive" foundation, ostensive constructions on the basis of which sample arithmetic claims may be verified, algebra or general arithmetic does not. 19 The subject matter, in the sense already introduced, "controls" the adequacy of its descriptions. In the case of algebra, on the other hand, the "subject matter" does not exist unless and until various combinatorial operations have been performed, i.e., the quantities involved are reached by way of an "analytical" characterization. Given that they cannot be "ostended,"zo what other sorts of controls might there be on their admission? Or, in the absence of appropriate ostensions, do we simply dismiss them as do Vieta (without qualifications) and Descartes (with qualifications)? This is the problem that is central for Kant. We might also put it in this way: given that algebra is, from one point of view, a system of rules licensing various combinatorial operations, what in turn licenses or legitimizes the rules? These same points can be reinforced from a different direction and in an alternate vocabulary. Ostensive constructions can be identified with the use of synthetic methods, symbolic constructions can be identified with the use of analytic methods. The question, then, is this: does every analytic construction have to be backed, as in the case of Descartes, with a synthetic proof, and, if not, what guarantees the adequacy of analytic constructions? Kant responds directly to the first of these questions in the controversy with Eberhard. The situation is straightforward with respect to the tradition of Greek geometry, he indicates, for on that tradition every

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proof given is synthetic and hence in a now familiar sense constructive. Thus, one of Kant's favorite examples, Apollonius first constructs the concept of a cone, i.e., he exhibits it a priori in intuition (this is the first operation by means of which the geometry presents in advance the objective reality of the concept). He cuts it according to a certain rule, e.g., parallel with a side of the triangle with cuts the base of the cone at right angles by its summit, and establishes a priori in intuition the attributes of the curved line produced by this cut on the surface of the cone. Thus, he extracts a concept of the relations in which its ordinates stand to the parameter, which concept, in this case, the parabola, is thereby given a priori in intuition. Consequently the objective reality of this concept, i.e., the possibility of the thing with these properties, can be proven in no other way than by providing the corresponding intuition. 21 But Kant is also very much aware that since the time of Vieta, and particularly in the work of Newton, analytic methods had been generally adopted. 22 So with this fact in mind, he continues: One could ... address to the modern geometers a reproach of the following nature: not that they derive the properties of a curved line without first being assured of the possibility of its object (for they are fully conscious of this together with the pure, merely schematic construction, and they also bring in mechanical construction afterwards if it's necessary), but that they arbitrarily think for themselves such a line (e.g., the parabola through the formula ax - y2), and do not, according to the example of the ancient geometers, first bring it forth in a conic section. This would be more in accordance with the elegance of geometry, an elegance in

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the name of which we are often advised not to completely forsake the synthetic method of the ancients for the analytic method which is so rich in inventions. 23 With some trepidation, I read this passage as follows. Contemporary geometers, despite their use of analytic methods, are always in a position to demonstrate the objective reality (real possibility) of the curves with which they work. But this does not require that every analytic construction be backed by a traditional synthetic proof. That is, it would be more "elegant" to bring it forth in a geometrical construction and we should not "completely forsake" the synthetic method of the ancients, but these are not required by the "possibility" of the curve analytically characterized. Undoubtedly Kant was aware of the fact that many of the curves thus characterized could not be brought forth by the conic sections or any of the other constructive methods available to the Greeks, or to Descartes for that matter. How, then, is the possibility of such curves to be established? The passage originally quoted at A716/B744 of the first Critique is, I think, even more explicit that geometrical diagrams do not have to be supplied for algebraic results in its sharp separation between ostensive and symbolic constructions. For the requirement that every analytic demonstration be backed by a traditional synthetic proof just is the requirement that all constructions be ostensive. At the same time, Kant wants to provide some sort of "foundation" for these algebraic results. He does it by introducing the notion of a symbolic construction which is in some sense "intuitive" and therefore "synthetic," in the process going well beyond what Descartes, and a jortiori the Greek tradition, had meant by "construction," "intuition," and "synthetic.,,24 What we "intuit" are the operations performed on certain otherwise unspecified "objects" by way of their symbolic representation. That is to say, the symbols are manipulated according to certain "universal" rules and, I take it, the application of these rules is guided, or controlled, by our intuition

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of the manipulation of the symbols. 25 It is never made clear what the status of these "rules" - for addition, subtraction, multiplication, and the extraction of roots - is. He denies that arithmetic or algebra have axioms in several different places,26 nor would it help to solve the question of status if the universal rules were axioms for Kant, for he never really says why the undoubted axioms of Euclidean geometry have a special status. The point, rather, seems to be that intuition is needed to verify the correct application of these rules (we can see whether the symbols are being manipulated correctly), that these rules are used to construct magnitudes, and that therefore algebra, like arithmetic and geometry, is synthetic. 27 For the rest, whatever else may be said about them, the rules and the combinatorial operations they define are, at least for Kant, a priori. A great deal more, of course, needs to be said about the notion of a symbolic construction. I want here only to draw attention to two very general and fundamental implications of it. The first is that whereas in geometry and arithmetic the objects (numbers and figures) ostended "control" the resulting claims about them, in algebra the claims, in the form of the various combinatorial operations we perform, "control" the objects. An object is admissible on this latter tack if it is constructed in the right sort of way. We have no independent access to the object, no ostension to which we can make appeal; this is just the difference, Kant rightly saw, between a variable and a constant. The other, related implication is that Kant is working his way towards the concept of a relational structure. What a symbolic equation "pictures," so to speak, is a relation between magnitudes (as such, and otherwise unspeCified). The "objects" are simply whatever satisfies the relational structure. Thus "algebra abstracts completely from the properties of the object that is to be thought in terms of such a concept of magnitude." But this said, the problem for him becomes one of making the case for the "objectivity" of particular relational structures. We cannot appeal

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to objects ostensively given, as we can in geometry and arithmetic, to insure such "objectivity." Or to put it in a different way, the "objects" which satisfy a relational structure, algebraic objects, are importantly "indeterminate." What "determines" them to the point that the structures which they satisfy are "objective"? This problem, generalized, is at the core of Kant's whole enterprise in the critical philosophy. After these sweeping and possibly cryptic claims, we can return to the subject at hand, the notion of an algebraic or symbolic construction. One litmus test for philosophies of mathematics in the 17th and 18th centuries is to see how they handle irrational, negative, and complex numbers. On my interpretation, "objects" are to be admitted so long as they result from a correct application of the combinatorial rules. In this respect, as I have mentioned, Kant moves well beyond Descartes, and in fact many of his 18th century contemporaries, in asserting the relative independence of algebra from arithmetic and especially geometry. Irrational, negative, and complex numbers result from a correct application of the combinatorial rules. So it should follow on my interpretation that he admits them. Is this so? Let me begin by restricting attention to irrational and complex numbers. Though they were more troublesome for mathematicians and philosophers in the 18th century than either irrational or complex numbers, negative numbers raise rather special problems and Kant's most extended discussion of them28 is intertwined with a great deal of extraneous material. It happens, in fact, that Kant discusses both irrational and complex numbers in the aforementioned letter to A. W. Rehberg. This is not all that Kant has to say on the subject, but there is enough in the Rehberg letter to show what kind of difficulties my interpretation faces and how they might be overcome. 29

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The question that Kant tries to answer in his letter is this. "Since the understanding has the power to create numbers at will, why is it incapable of thinking ...J2 in numbers?" 30 He first says that one can "regard every number as the product of two factors, even if these factors are not immediately given to me or even if they could not be given in numbers." This follows from the combinatorial rules used to multiply and divide numbers, and not from a survey of the numbers themselves. "If neither of the factors is given but only a relationship between them" - as in algebra - "for example, it is given that they are equal - so that we have a and the factor sought is x, where l:x x:a ... then, since a - x 2 , x must - Va." Once again, this follows from the universal rules embodied in our combinatorial activities. It is precisely the generality of these rules that leads to the introduction of irrational numbers. For since the number 2 is an admissible value of a, the value of X must be V2 , which is irrational. But the "possibility" of v2 , which has just been demonstrated by way of a symbolic construction, leads to two paradoxes, the second more serious than the first. The first paradox is that we cannot find a (rational) number for this quantity. But this paradox is quickly dispelled when the two non-sequiturs on which it rests are indicated. The first non-sequitur is that it follows from the fact that every number can be represented as the square of some other number that the square root must be rational. But as Kant points out, the rule enabling us to extract roots is completely general; it requires only an "inter-relationship" between quantities and not a "determinate relation to unity." The second nonsequitur is that it follows from the fact that a quantity can be "named (as in algebra)" that it can be "calculated (as in arithmetic), ..31 any more, we might add, that it follows from the fact that we can describe a curve analytically that we can draw it. Calculation is subject to the conditions of production in time, which entail that although we can approximate an irrational number to an arbitrarily close degree, 32 we can never carry out its complete calculation.

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The other, more difficult paradox, to put it in our earlier terms, is that although a symbolic construction of ..J2 is possible, and an ostensive geometrical construction is possible (as the diagonal on a unit square), an ostensive arithmetic construction (necessarily in terms of rational numbers) is not. Why this should be so, Kant concludes, is something we do not understand although, again we might add, it is not a particular difficulty for his position that we do not. There is no particular difficulty either in all of this for my interpretation. Indeed, what I think is the most natural reading of the letter is not simply compatible with, but provides additional evidence for it. More troubling is the comment Kant makes in passing that "Geometry shows us, by the example of the diagonal of the square, that the mean proportional quantity between the quantities 1 and 2 can be found, and that ..J2 is consequently not an empty, objectless concept. " The last part of this sentence would seem to be incompatible both with my interpretation and also, I think, with what is said in the Critique about symbolic construction. It would be tempting to read the sentence to mean that geometry in this instance indicates something more than the real possibility of the concept in question, to wit that there is a quantity corresponding to it (and that it is in this sense not "empty and objectless"), were it not also the case that at the end of the letter Kant says "the understanding is not even in a position to assume the possibility of an object ..J2, since it cannot adequately present the concept of such a quantity in an intuition of number. " This does suggest that a merely symbolic construction cannot demonstrate (real) "possibility." But surely this sentence cannot be taken at face value, for otherwise it involves a non-sequitur. It doesn't follow from the fact that..J2 cannot be adequately represented in an intuition of number that the understanding is not in a position to assume its "possibility;" Kant has just told us that geometry demonstrates that "possibility." What Kant must mean is something like this: the understanding is not on arithmetic grounds

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in a position to assume the possibility of an object V2 (since such an object cannot be represented ostensively as a ratio of whole numbers), but that does not preclude either a geometric or a symbolic (algebraic) demonstration of "possibility" being given. Kant says the following about complex numbers: .... one can see the impossibility of the square root of a negative quantity, V-a (where the same relation would have to hold between the positive quantity, unity, and another quantity, ~, as holds between ~ and a negative quantity) .. .33 This is to say that he was not in a position to carry his views about symbolic construction to what might seem to be their logical conclusion. In light of this comment in the Rehberg letter, we apparently have to say that symbolic construction shows the "real possibility" of the "objects" so long as the description of that object is not self-contradictory.34 5. The introduction (or as he preferred to think, re-introduction) of analytic and algebraic methods by Vieta led to a changed concept of mathematics and its subject matter. The simple burden of this paper has been to sketch the roles played in that development by Descartes and Kant, and to suggest how it bears on Kant's philosophy and its subsequent generalization. Whatever the merit of my conclUSions, the history of mathematics can be separated from the history of philosophy only at the price of misunderstanding both. This is one of the many lessons that Jules Vuillemin teaches. Regents Professor ofPhilosophy. Montana State University

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NOTES 1My

own interest in vieta was first aroused by Jacob Klein's neglected classic, Greek Mathematical 7bought and the Rise of Algebra, first published in German in 1934-36, then reissued in translation by the MIT Press in 1%3. 2In addition to Klein, which contains a translation of Vieta's Introduction to tho Analytical Art, there are helpful textbook accounts in Carl Boyer, History of Analytic Geometry (Scripta Mathematica, 1956) and Morris Kline, Mathematical Thought from Ancient to Modern Times (Oxford University Press, 1972). Michael Mahoney, "The Beginnings of Algebraical Thought in the Seventeenth Century," in S. Gaukroger, ed., Descartes: Philosophy, Mathematics, and Physics (Barnes & Noble, 1980), indicates Vieta's central role and underlines the philosophical implications of his work. Yrhe Greek classification of equations as linear, plane, and cubic (as well as the numbers used in their solution) reveals its geometrical and intuitive origins. 4It is important to note here, and in the sequel, that for the Greeks mathematics was the science of whole numbers and their ratios only. Incommensurable ratios could only be construed geometrically.

5-r'he extent to which Descartes was "influenced" by Vieta has always been a matter of controversy. Two things are clear: Descartes was well acquainted with Vieta's work, however much he may have arrived at the same conclusions independently, and he went beyond it.

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6For more on his program and its application to the physical world, see Stephen Gaukroger, "Descartes' Project for a Mathematical Physics," in Gaukroger, ed., Descartes: Philosophy, Mathematics, and Physics. 7 Geometry, Smith and Lapham translation (Dover Books, 1954), p. 159.

&rhe qualification marks the fact that for Descartes not every root of an equation is invariably positive, hence acceptable.

9Geometry, Smith and Lapham translation, p. 175. lOMargaret Wilson points out in Descartes (Routledge & Kegan Paul, 1978) that the distinction between real possibility and existence is crucial for Descartes in a number of contexts, including the evil demon argument in the Second meditation. The existence per se of mathematical objects does not seem to have been central. lITo my knowledge, there is no explicit discussion of this commitment, or of its sources, in the Greek texts. See Ian Mueller, Philosophy of Mathematics and Deductive Structure in Euclid's Elements (The MTT Press, 1981), p. 27: "... book I of the Elements is unique in Greek mathematics in containing an explicit list of permissible constructions. In connection with other texts one has to say simply that the constructions permitted are the constructions actually used." 12 Geometry,

Smith and Lapham translation, p. 43.

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131 agree with Emily Grosholz, "The Cnification of Algebra and Geometry," in Gaukroger, op. cir., p. 160, who after a much more careful study of the text reaches a similar conclusion: "For Descartes, proofs had to be carried out in terms of the geometrical diagram, and curves introduced by mechanical constructions analogous to construction by compass and ruler. He made use of algebraic notation in proofs, but only if each equation could be geometrically constructed; and he never introduced a curve by means of an algebraic equation alone." 14An adequate description of the algebraic background of Kant's thought would have to include Leibniz, whose work on algebra is extensive, and Euler, whose Anleitung zur Algebra (published in 1770) is widely regarded as the best algebra text of the century. 15n,.is and all subsequent quotations of the Critique of Pure Reason are from the translation by Norman Kemp Smith (Macmillan and Co., Ltd., 1933).

16Michael Friedman, "Kant on Concepts and Intuitions in the Mathematical Sciences," Synthese, forthcoming, argues for including both arithmetic and algebra under the heading of symbolic construction. In so doing he seems to me to miss the point of Kant's distinctions. J. Michael Young, "Kant on the Construction of Arithmetical Concepts," Kant-Studien. 1982, thinks that it would be a proper extension of Kant's view to regard calculations as symbolic constructions, but he notes correctly that in the basic case arithmetical constructions are ostensive. Indeed, he finds it more plausible to regard our knowledge of arithmetical truths as resting on ostensive constructions than geometrical truths, for a collection of n things, in his view, can represent the number n in a way that allows us

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to use the collection to gain knowledge about the number, whereas the parallel claim about geometrical constructions has little support. 17As

in the letter to Johann Schultz of November 25, 1788. The letter is included in Arnulf Zweig, Kant: Philosophical Correspondence 1759-99 (University of Chicaqo Press, 1%7). 18op.

cit.

19And it is simply a fact, according to Kant, that these constructions show the "reality" of whole and rational numbers and their various rule-governed combinations. All "determinate" mathematical objects are in certain respects qualitative, ".. .for instance, the difference between lines and surfaces, as spaces of different quality, and with the continuity of extension as one of it qualities ... " (CPR, A715/B743). But algebra considers these objects in their merely quantitative guise, as magnitudes as such. See the letter to K.L. Reinhold of May 19, 1789: "The mathematician cannot make the least claim in regard to any object whatsoever without exhibiting it in intuition (or, if we are dealing merely with quantities without qualities, as in algebra, exhibiting the quantitative relationships thought under the chosen symbols) .. ," in Allison, The Kant-Eberhard Controversy, p. 167. The intuition to which the arithmetician appeals is not in any way "symbolic" (it might, for example, be a series of stroke-marks on a piece of paper), although it is, of course, "representative" in that one

intuition of a number does duty for all the others. Moreover, the qualitative/quantitative contrast which is used here to characterize algebra would seem to have little to do with the rational/irrational contrast that Friedman wants to take as primary. is a mistake, I think, to focus so exclusively as Friedman does on the problematic character of the irrational numbers since for Kant 20It

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(and virtually the entire 17th-18th century mathematical tradition), (a) irrational numbers could be "os tended" geometrically and (b) negative and complex numbers were more difficult to deal with. I will return to this question, and Kant's letter to A.w. Rehberg (before September 25, 1790) in which irrationals are taken up, shortly. 21From The Kant-Eberhard Controversy, translated by Henry E. Allison (Johns Hopkins Press, 1973), p. 110. Note how close to Descartes' view the last quoted sentence is. In an important gloss on this passage in his letter to Reinhold of May 19, 1789, Kant reinforces the point: if Eberhard understood the example, he would see "that the definition which Apollonius gives, e.g., of a parabola, is itself the exhibition of a concept in intuition, namely, the intersection of a cone under certain conditions, and in establishing the objective reality of the concept, that the definition here, as always in geometry, is at the same time the construction of the concept." In Allison, The KantEberhard Controversy, p. 168. He adds that the actual drawing of the parabola, given the parameter, follows as a practical corollary and has nothing to do with the theoretical point at issue. 22The case of Newton is complex and illustrates the lack of clarity concerning the relationship between geometry and algebra, synthetic and analytic methods, and the foundation for each, which in fact extended well into the 19th century. On the one hand, he says in his Universal Arithmetic (1707) with apparent approval that "the Moderns advancing yet much further than the plane, solid, and linear loci of the Greeks have received into Geometry all Lines that can be expressed by Equations," and we know that in his own work he made rather free-wheeling use of analytic methods. Yet on the other hand, in a letter to David Gregory (1661-1708), Newton remarks that "Algebra is the analysis of bunglers in arithmetic," and he continued to take the Greek mathematical tradition as his standard of rigor.

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Perhaps for reasons of the "elegance" which Kant mentions in the Eberhard controversy (see below), ~ewton composed the Principia according to the synthetic geometrical method, apparently no more than assuming that the synthetic proofs he offered were adequate to reach the results that he had in fact arrived at analytically. 23From The Kant-Eberhard Controversy. p. 111. 24According to Morris Kline, Mathematics: The Loss of Certainty (Oxford University Press, 1980), p. 125, Newton tried to ground algebra by arguing that "the letters in algebraic expressions stand for numbers and no one can doubt the certainty of arithmetic." What this means, apparently, is that algebraic results are simply generalized versions of arithmetic truths and in some sense reducing to them. It is noteworthy that Kant does not avail himself of this strategy, nor could he, on my interpretation, since arithmetic proper affords ostensive constructions on the basis of which tts truths are immediately verifiable. 25See the Critique of Pure Reason. A734/B762. 26Notably in the Axioms of Intuition, A163/B204. 27More than anyone else, Gottfried Martin in Arithmetic and Combinatorics: Kant and His Contemporaries (Southern Illinois University Press, 1985) has stressed the role played by algebra in Kant's philosophy of mathematics and the ways in which it is "creative" and hence "synthetic." Martin goes astray in insisting that Kant was the first to formulate certain axioms of arithmetic (such as the rule of commutation), but I now think his emphasis on the "rules" used in combinatorial operations was fundamentally correct.

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28In an early work of 1763, Versuch, den Begriff der negaliven Grossen in die Weltweisheit einzufahren. 29Friedman, op. cit., takes the letter as an important document in support of his own very different view of Kant and offers an interesting and detailed reading of it. 3
ON WHETHER AN ANSWER TO A WHY-QUESTION IS AN EXPLANATION IF AND ONLY IF IT YIELDS SCIENTIFIC UNDERSTANDING-

by Karel Lambert I. Introduction

Probably the reason the topic of scientific understanding has received relatively little independent attention from philosophers of science is because most regard the expression merely as an honorific synonym for the expression "scientific explanation." And this surely often is the case. There is no loss of meaning, for instance, when the expression "scientific explanation" is substituted for the expression "scientific understanding" in the remark, "We simply have no adequate scientific understanding of the way T-receptors function in acquired immune deficiency." But there are occasions when this substitution would be preposterous--for instance, in the statement, "The ultimate goal of scientific explanation is to obtain scientific understanding." Indeed, yielding scientific understanding is used by many philosophers of science as a measure of scientific explanation. • Jules Vuillemin, France's foremost philosopher, has written with power and persuasion for decades on a range of topics extending from Kant to the history and philosophy of science. Though he cannot be blamed for the contents of this essay, his own work is in large measure responsible for stimulating its author to produce them. This essay also profited from conversations with Gerhard Schurz in Salzburg during the fall of 1986, and with Wolfgang Spohn in the spring of 1987, on the topic of scientific understanding. 125

G. G. Brittan (ed.), Causality, Method, and Modality © Kluwer Academic Publishers 1991

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Witness, for example, Wesley Salmon. In his 1978 Presidential Address to the Pacific Division of the American Philosophical Association entitled "Why ask 'Why?'?",l he uses scientific understanding as a means of determining when something is or is not a scientific explanation. Apparently his view is that an answer to an explanation seeking why-question is a scientific explanation if and only if it yields scientific understanding. For instance, suppose an ancient Greek mariner were asked the question, "Why was the tide level at Alexandria two meters high at 4 o'clock PM on December 12th?" And suppose he answers, "Because of the current position of the moon and its phase; whenever those circumstances occur, the tide level at Alexandria will be as you described it." For Salmon, since this answer yields no scientific understanding of the stateof-affairs in question, it is not a scientific explanation of that state-of-affairs. 2 On the other hand, suppose in response to the question, "Why is Olivia's hair blond?," one receives the answer, "Because she is the offspring of parents in a population about 3/4 of whom have dark hair, and about 1/4 of whom have blond hair. " For Salmon, this answer does yield scientific understanding of what the why-question is about and thus would constitute a scientific explanation of that state-of-affairs. 3 But in the absence of any independent account of scientific understanding, and of the relation of yielding, triviality threatens any attempt to so isolate the class of scientific explanations. The initial purpose of this essay is to supply such an account; the remote purpose is to assess the stated relationship between scientific explanation and scientific understanding.

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II. Scientific Understandinq and its Relation to Scientific Explanation

To begin it is helpful to winnow the concept of scientific understanding relevant to scientific explanation out of the many irrelevant conceptions. Here are some examples of the latter. There is a conception of sCientific understanding which says that scientific understanding is the reduction of the novel or the unusual to the familiar. This essentially subjective conception emphasizes a sense of relief from uncertainty or puzzlement. Whatever its appeal, this conception of scientific understanding simply is not the sense of scientific understanding relevant to scientific explanation. No one has been more eloquent on this matter than Carl Hempel. He writes: The view that an adequate scientific explanation must, in a more or less precise sense, effect a reduction to the familiar, does not stand up under close examination. To begin with, this view would seem to imply the idea that phenomena with which we are already familiar are not in need of, or perhaps incapable of, scientific explanation; whereas in fact, science does seek to explain such "familiar" phenomena as the regular sequence of day and night and of the seasons, the phases of the moon, lightning and thunder, the color patterns of rainbows and of oil slicks, and the observation that coffee and milk, or white and black sand, when stirred or shaken, will mix, but never unmix again. Scientific explanation is not aimed at creating a sense of at-homeness or of familiarity with the phenomena of nature. That kind of feeling may well be evoked even by metaphorical accounts that have no explanatory value at all, such as the "natural affinity" construal of gravitation or the

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conception of biological processes as being directed by vital forces. What scientific explanation, especially theoretical explanation, aims at is not this intuitive and highly subjective kind of understanding, but an objective kind of insight that is achieved by a systematic unification, by exhibiting the phenomena as manifestations of common underlying structures and processes that conform to specific, testable, basic principles. If such an account can be given in terms that show certain analogies with familiar phenomena, then very well. Otherwise, science will not hesitate to explain even the familiar by reduction to the unfamiliar, by means of concepts and principles of novel kinds that may at first be repugnant to our intuition. This has happened, for example, in the theory of relativity with its startling implications concerning the relativity of length, mass, temporal duration, andsimultaneityj and in quantum mechanics with its principle of uncertainty and its renunciation of a strictly causal conception of the processes involving individual elementary particles. 4 A second conception holds that to have scientific understanding is to be in a position to explain scientifically.5 Clearly this cannot be the sense in which scientific explanations are said to ~ scientific understanding. Rather, it constitutes a kind of knowledge and/or expertise which one must have to give scientific explanations. A third conception of scientific understanding envisages it to be the unification of diverse phenomena under a few generalizations or laws. 6 As such, it is a global property of theories. But this cannot be the sense of scientific understanding appropriate to single

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scientifice explanations -- or at least to most of them. It might be the product of a large class of scientific explanations, but most single explanations of the sort one encounters in a science simply do not yield the global characteristic described above. Finally, there is a fairly common sense of scientific understanding in which one is said to have sCientific understanding to the extent one knows the facts about the universe and/or the laws governing it. This sense is reflected, for example, in remarks by scientists who assert that our scientific understanding of Saturn's rings has been greatly increased by the Voyager probe. But the acquisition of new information in the sense of uncovering new facts or new laws is not the sense of scientific understanding relevant to scientific explanation. An example will establish the point. Consider the fact of superconductivity. One can easily imagine even Laplace's famous demon to be in a quandry--as is current physics--about superconductivity; in other words, to be in a quandry about how it fits into the network of facts and laws helping to make up electrical theory. The situation is not really different from the quandry about how to fit a specific piece into a picture puzzle despite having all the pieces before one as well as knowing the "laws" governing construction of the picture puzzle. The moral is that though achievement of scientific understanding involves the acquisition of new information, not all new information constitutes scientific understanding. The metaphor of "fitting into," and its stylistic variants such as "incorporated into" or "integrated into," seem especially germane vis a vis scientific understanding as it relates to scientific explanation. This metaphor, or something Similar, crops up repeatedly in discussions of scientific understanding. Thus, Braithwaite, speaking of "intellectual satisfaction," his own rubric for scientific understanding, writes:

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Whether we take incorporation [of a fact] as being an answer to a "How" or a "Why" question is immaterial: what matters is that we know more than we did before of the connectedness of the fact .. with fundamental laws .. ? Again, Hempel writes: A class of phenomena has been scientifically understood to the extent that they can be fitted into a testable, and adequately confirmed, theory or system of laws. S Finally, Salmon asserts that: The aim of scientific explanation, according to the ontic conception, is to fit the event-to-be-explained into a discernible patternY The idea behind the "fitting into" metaphor is that scientific understanding consists in placing the phenomenon to be understood in a theoretical network thereby rendering it intelligible. IO An historical example will help convey the sense of scientific understanding involved in scientific explanation. The English chemist Joseph Priestley actually isolated oxygen before Lavoisier, but its significance eluded him. It remained for Lavoisier, who fitted oxygen into a scheme of quantitative chemical changes, to uncover the significance of oxygen--to render it intelligible. I I The sense of scientific understanding relevant to scientific explanation may be characterized as an answer to the question, "How does state-of-affairs S fit into theory P" Here are two illustrations. First, suppose I is the current physiological theory of appetite, and suppose .s. is the state-of-affairs that Eddie Rickenbacker--the famous American World War II aviator--has no appetite,12 Scientific understanding of the state-of-affairs which is the topic of concern of the question, "How does the state-of-affairs that Rickenbacker has no appetite (under the conditions mentioned in

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footnote 12) fit into the physiological theory of appetite?", might be constituted by the answer, "It is inferable, in that theory, from the fact that he was near starvation and extremely fatigued and the law that rhythmic contractions in the duodenum tripping off blood chemistry changes initiating appetite are blocked by extreme fatigue." Or it might be constituted by the answer, "It is the product of the (statistically relevant) factors that Rickenbacker was in a state of near starvation and was extremely fatigued, factors which, according to that theory, block rhythmic contractions in the duodenum that in turn cause the blood chemistry changes initiating appetite." Or it might even be constituted by the answer, "According to that theory, it serves to decrease the rate of depletion of bodily tissues, under the physical history outlined." Second, suppose T is the theory of inherited characteristics, and 5 is the state-of-affairs that Olivia's hair is blond. Scientific understanding of the topic of concern of the question, "How does Olivia's hair being blond fit into the current theory of inherited traits?", might be constituted by the answer, "It is the chance result, according to that theory, of the fact that her parents were members of a genetic population about 3/4 of whom had dark hair, and about 114 of whom had blond hair." These answers constitute different ways in which a state-ofaffairs can be fitted into a theory, and it is their variety no doubt that makes it difficult to provide anything more than a metaphorical statement of the sort of scientific understanding relevant to scientific explanation. Moreover, other answers are in all likelihood possible and hence other ways in which a state-of-affairs can be fitted into a given theory. On the other hand, if any of the above ways of integrating phenomena into a theory are deemed not to be cases of "genuine" scientific understanding, it seems clear that the ground for such a belief must be outside the logical theory of how-questions and answers.

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Consider the attitudes of the three major contemporary theorists on scientific explanation, Hempel (circa 1966), Salmon and van Fraassen vis a vis the four alleged examples of scientific understanding specified above. Hempel would reject the second answer above--the causal answer, the third answer--the functional answer--and apparently, at one time, the stochastic answer,13 to the question about Olivia's blond hair because for him only deductive or inductive consequence is an acceptable way of fitting states-of-affairs into a theory. Salmon apparently would exclude the first answer--the deductive answer--and the functional answer as constituting scientifically acceptable ways of integrating a state-of-affairs into a theory because for him the only scientifically acceptable answers to an understanding seeking how-question must indicate in some degree the underlying causal network. Van Fraassen, however, may count all the answers above as cases of genuine scientific understanding since apparently one way of fitting the phenomena into a theory is as good as any other, the appropriateness of the manner of incorporation varying with the inquirer's goals. 14 It is useful to distinguish between global and local scientific understanding. In the case of local understanding, the more common case in applied science, the theory is known or taken for granted but not how a given state-of-affairs fits into that theory. In the case of global understanding a batch of diverse states-of-affairs (phenomena) are in need of a theory, or another theory is discovered, to accommodate them. Friedman's analysis of scientific understanding concerned the global notion whereas this essay concerns local understanding. The question arises how scientific understanding-conceived as an answer to a question of the form, "How does S fit into theory 17"-is related to scientific explanation. The examples above suggest the following response: to say that an answer to an explanation seeking why-question--a scientific explanation--yields (provides, produces)

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understanding is to say that it shows or exhibits what is described by an answer to a how question of the form described above. To illustrate, consider the why-question, "why did Rickenbacker have no appetite?", and the answer, "Because he was nearly starved and exhausted and whenever an organism is extremely fatigued, rhythmic contractions in the duodenum are blocked and whenever that is so, the blood chemiStry changes initiating appetite are not tripped off'. This answer shows, but does not mention, that Rickenbacker's lack of appetite is inferable in the theory of appetite from other relevant states-of-affairs with the help of a pair of laws. It also shows, but does not mention, how Rickenbacker's lack of appetite is the causal product of certain statistically relevant factors in the theory of appetite. Another illustration. Consider the same question about Rickenbacker's lack of appetite and the answer, "Because it reduced the rate of tissue depletion." This answer, again, shows but does not mention, that Rickenbacker's lack of appetite served or functioned to reduce the rate of tissue depeletion and hence to sustain life. Talk of scientific understanding as something shown or exhibited by a scientific explanation is also common among those who speak of scientific explanations yielding understanding. The initial contribution of this essay is to put together explicitly two common intuitions about scientific understanding and scientific explanation. It outlines what is yielded by a scientific explanation, that is, what the piece of new information constituting scientific understanding is, and it specifies explicitly the nature of the relation of that new information to scientific explanation reflected in words like "yield" or "provides." This analysis of the yielding relation between scientific explanation and scientlfic understanding does not imply that the only purpose in seeking a scientific explanation is to achieve scientific understanding. On the other hand, the existence of other purposes

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for seeking a scientific explanation does not mean that scientific explanations do not always yield scientific scientific understanding; it may simply be a by product of scientific explanations in such cases. For instance, a scientist may seek an explanation of a novel state-ofaffairs solely in the interest of sustaining an already well deserved reputation as the best explainer in his field. But, though it may not have been his purpose to do so, the subsequent explanation he uncovers may nevertheless exhibit what is described by an answer to the understanding seeking how-question. Or the scientist may seek an explanation of some state-of-affairs simply to illustrate to his students what successful explanation seeking activity in his science is like. But this goal surely does not preclude the possibility that the explanation in fact yields scientific understanding. The foregoing analysis of yielding seems to remove any threat of triviality that might be feared to be lurking in the dictum: an answer to a why-question about a state-of-affairs S is a scientific explanation if and only if that answer yields scientific understanding of S. The question now arises whether the dictum, so explicated, is true. III. On Yielding Scientific Understandinq: Is it a Measure of Scientific Explanation?

It is convenient to divide the question into two parts: (a) Is yielding scientific understanding a sufficient condition for being a scientific explanation? and (b) Is it a necessary condition? Let us deal with these questions in order. First, there are various defective arguments against a positive answer to question (a)--one is due to the writer of this essay. In a recent essay, Bas van Fraassen considers the case of a mother who asks why her syphillitic son got general paresis (in contrast to other syphillitics) and receives the answer that there is no explanation. 15 Van Fraassen believes that this answer nevertheless provided the mother with increased understanding. So, at first glance,

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we have here a case of an answer to a why-question yielding scientific understanding but which is not a scient'ific explanation. But is the declaration, "There is no explanation," really an answer to the mother's why-question? No, because it rejects the question. As far as I know, no theory of why-questions holds that a declaration which rejects a why-question is an answer to that why-question; "There is no explanation" is a response to the mother'S why-question about syphillis, but it is not an answer to it. So though one might agree with van Fraassen that the response to the mother'S why-question increases her understanding, such agreement is not evidence against a positive response to question (a). A similar objection applies to a negative answer to (a) based on inference to the best explanation. For example, consider the state-ofaffairs that 65 million years ago plankton temporarily disappeared from the surface of the Earth's oceans (S'). The geologist, Michael Rampino of New York University, abductively inferred this state-ofaffairs recently from the fact that the dimethyl sulfide necessary for condensation of water droplets in clouds was drastically reduced 65 million years ago (S) and the supposition that if plankton disappeared 65 million years ago, then no dimethyl sulfide would have been produced. Now suppose a person to enter a room just as a distinguished colleague mentions S'. Suppose further that he asks his distinguished colleague, "Why did plankton temporarily disappear from the surface of the Earth's oceans 65 million years ago?", and suppose, fmally, his distinguished colleague replies, "Because if one assumes that state-ofaffairs, then one can deduce that for some period of time no dimethyl sulfide was produced 65 million years ago" .16 Now the distinguished colleague's reply surely describes the way in which S' fits into the theory about how the dinosaurs became extinct. But it does not constitute good grounds for a negative answer to (a). For though a response to the question "why S'?" it is not an

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explanatory answer. An appropriate answer would instead look something like this: "Because the dust in Earth's atmosphere caused by a meteorite bombardment temporarily blocked the process of photosynthesis necessary for plankton to live". In general, if a reply to a why-question about a state-of affairs uses words like "deduce" it is not an answer to that why-question given the analysis above that explanations exhibit but do not mention the way in which states-ofaffairs fit into a theory when they provide scientific understanding; deduction is a way of fitting a state-of-affairs in a theory. In an earlier essayl7, I suggested that even though a Newtonian might have an adequate answer to the question, "How does the fact that a body moves when acted upon by forces fit into the theory of motion?", this does not mean that one has an explanation why a body moves when acted upon by forces because the why-question does not arise for the Newtonian. Even so, this does not constitute evidence against a positive response to (a). For if the why-question does not arise, then it has n2 answers. But (a) is exactly the question whether an answer to a Why-question which yields scientific understanding is a Scientific explanation. Second, there are, nevertheless, other substantial sources of concern over a positive answer to (a). Let us return to the question about Olivia's hair color, "Why is Olivia's hair blond?". The answer, "Because she is the offspring of parents in a population about 3/4 of whom have dark hair and about 1/4 of whom have blond hair," yields scientific understanding for van Fraassen, Hempel (circa 1966), and Salmon, but only Salmon counts that answer a scientific explanation. Why? Because van Fraassen and the earlier Hempel think there is more to a scientific explanation than yielding scientific understanding; in common language it must also provide a "reason" for believing in the state-of-affairs being explained. According to Hempel, for an answer to a why-question to qualify as a scientific explanation, it must be rationally acceptable.18

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Moreover, any such answer will also be a potentially adequate answer to a reason-seeking (or epistemic) why-question of the form "What are the grounds for believing that 5? ~ 19 Despite the otherwise vast differences between the structuralist theory of scientific explanation of the earlier Hempel and the nonstructuralist theory of van Fraassen, they agree on this matter, though their answers to the question, "What constitutes a rationally acceptable answer to an explanation-seeking why-question?" are quite different. For Hempel (circa 1966) such an answer must offer information making clear that the state-of-affairs was to be expected;20 for van Fraassen, such an answer must favor the state-of-affairs being explained. 21 It is easy to see on either account of sCientific explanation why the earlier answer to the question, "Why is Olivia's hair blond?" does not constitute a scientific explanation while yet yielding SCientific understanding in the sense explained in this essay. For the state-of-affairs constituting the topic of concern of that question is neither to be expected on the basis of, nor favored over its alternatives by, the answer to that same question. Turning now to (b) the question whether it is to be answered negatively is the question whether there is sometimes less to scientific explanation than yielding scientific understanding. For example, suppose that there are pre (or non) theoretical explanations of a given states-of-affairs. Then, since there is no theory for the explained state-of-affairs to fit into, there can be no scientific understanding--as elucidated in this essay--of that state-of-affairs. Indeed this verdict is exactly the verdict required if the answer to the question about the tide level at Alexandria cited earlier in this essay is judged to be an explanation even if only a superficial one and one supposes, as Salmon seems to suppose, that there is no theory--or at least not the right kind of theory--underlying that answer. It was for precisely this latter reason that Salmon rejected that answer as a scientific explanation.

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Clearly, assessment of this example and others, e.g., explanations via Kepler's laws prior to Newton, depend on a precise statement of what counts as a theory (or as the right kind of theory). Recent discussion in the foundations of microphysics makes the issue an especially poignant one. I refer specifically to the Einstein Podolsky-Rosen thought experiment, Einstein's ruminations about the implications of that experiment for quantum mechanics, and ].S. Bell's torpedoing of any causal hidden variable account of the results of EPR. Einstein thought the EPR set-up showed the incompleteness of quantum theoretical explanations and suggested that a fuller description of the EPR situation would turn up a compelling causal explanation. But Bell's celebrated proof has cast considerable doubt on the idea that such an explanation is possible especially where causes are taken to be key segments of some kind of continuous process. But then it would seem that if scientific understanding is interpreted a la Salmon, either quantum theoretical explanations are not genuine scientific explanations, or they do not yield scientific understanding as he has characterized it, and hence the answer to (b) is no; that is, sometimes scientific explanations do not yield scientific understanding. It should be pointed out, however, that this is a localized problem affecting only Salmon's views and not necessarily the view of others who sponsor a yes-answer to (b). For if "fit into" means "is inferable", then quantum theoretical explanations do yield scientific understanding because the EPR results are inferable in quantum mechanics. But there is an argument that yielding scientific understanding, as characterized in this essay, is not a necessary condition of scientific explanation. The argument turns on an important caveat in the current explication of scientific understanding. The caveat is this: the theory T into which the explained state-of-affairs S is being fit (or incorporated) can't be just any old scientific theory for there to be

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scientific understanding of S; rather T must be the correct theory in a given area of interest. To illustrate, an explanation of why a certain substance bums might very well exhibit what is described by an appropriate answer to the question, "How does the state-of-affairs that substance A burns fit into phlogiston theory?" but that doesn't mean, the objection goes, that the explanation provides scientific understanding. It would only do so if it exhibited what is described by an appropriate answer to the question, "How does the state-of-affairs that substance A burns fit into oxygen theory?" To be sure there was a time when the two theories were in competition for acceptance by the community of chemiSts, but this only means that chemists were then in conflict about what the correct theory of combustion was and hence what constituted scientific understanding in this special case. If the intuition expressed in these past several sentences is correct, then (b), which reads: (b) If an answer to a why-question is a scientific explanation of a state-of-affairs S, then it yields SCientific understanding of S, is false. The example above about the explanation of a burning substance A in the phlogiston theory is the evidence. Supporters of the view that yielding scientific undersanding is a measure of scientific explanation-and hence of (b)-might very well reply that the argument against (b) fails because it ignores the distinction between real understanding and understanding, and hence fails to see that while correct scientific explanations yield real scientific understanding, incorrect explanations do not. But it does not thereby follow that incorrect scientific explanations do not yield scientific understanding. Scientific, as opposed to nonscientific, understanding of the state-of-affairs that substance A bums certainly is a product of the (incorrect) phlogiston explanation in the sense

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that it shows how that state-of-affairs fits into the scientific theory of phlogiston. Whether a distinction between real understanding and understanding is a real distinction is an important and troublesome question. Nevertheless, it is not prima facie implausible, and to that extent those who might object to (b) and, hence, to scientific understanding as a measure of SCientific explanation, have got more work to do. To sum up, in this essay I have tried to explicate what yielding scientific understanding in the sense relevant to scientific explanation amounts to and to assess the proposition that yielding scientific understanding is a measure of scientific explanation. I do not claim to have provided an up or down answer to this proposition, but I have sought to focus some important points of contention. Finally, it is clear that further progress on this issue awaits precise explication of the metaphorical notion of "fits into". Professor of Philosophy. University of California. Irvine. and University of Salzburg

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Notes 1. Proceedings and Addresses of the American Philosophical

Association, 51 (978), pp. 683-705. 2. Ibid, p. 684.

3. See, for example, Wesley Salmon's SCientific Explanation and the Causal Structure of the World, Princeton Univ. Press, Princeton, New Jersey, (984), p. 109. 4. Carl Hempel, Philosophy of Natural Science, Prentice Hall, Englewood Cliffs, New Jersey (966), pp. 83-84. 5. Bas van Fraassen, The Scientific Image, Clarendon Press, Oxford (980), p. 154. 6. Michael Friedman, "Explanation and Scientific Understanding," Journal of Philosophy, 71 (974), pp. 5-19. 7. See his Scientific Explanation, Harper, New York (954), p. 349. 8. See his Aspects of Scientific Explanation, The Free Press, New York 0%5), p. 329. 9. See his Scientific Explanation and the Causal Structure of the World, Princeton University Press, Princeton (984), p. 121.

10. Op. cit., C. Hempel, Aspects of Scientific Explanation, p. 488. 11. See Henry Guerlac's, Lavoisier: The Critical Years, Cornell University Press, Ithaca 0%1).

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12. Rickenbacker crashed into the Pacific Ocean in World War II and was found three weeks later on a raft in a state of extreme fatigue and near starvation.

13. Op. cit., C. Hempel, Aspects of Scientific Explanation, p. 337. 14. This is a very speculative remark given the sketchiness of van Fraassen's remarks on scientific understanding. It seems to accord, however, with the spirit of his general attitude toward scientific explanation.

15. "Salmon on Explanation," The Journal of Philosophy, 11 (1985),

pp. 639-651.

16. Hence no condensation in the clouds would be produced allowing the Sun's rays to be reflected back into space, and, therefore, temperatures on the Earth would rise dramatically leading to extinction of the dinosaurs. 17. "Scientific Understanding; Preliminary Considerations" in Erklilrung und Verstehen (Ed. Gerhard Schurz), forthcoming. 18. Op. cit., Aspects of scientific Explanation, pp. 367-378. 19. Ibid., pp. 367-368.

20. Hempel no longer accepts this standard of a rationally acceptable answer to a why-question.

SOME REVISIONARY PROPOSALS ABOUT BEUEF AND BEUEVING

Ruth Barcan Marcus There is consensus about some general conditions on a theory of believing and belief such as (l) believing is a relation between a subject, the believer, and an object or set of objects as given in the grammatical form of the sentence, IX believes that S'; (2) beliefs, whatever they are, can be acquired, replaced, or abandoned; 0) they enter along with desires, needs, wants and other particular circumstances into an explanation of action; and (4) for some circumstances and for some beliefs it is appropriate to describe a subject'S beliefs as justified or unjustified, rational or irrational, and the like. But such general features are in contrast to a tangle of unreconciled views when one tries to flesh out a theory or give the concepts more content. There is disagreement about the nature of the belief state of the subject, the nature of the object of the believing relation, the efficacy or causal role of belief in shaping actions, and the role of language in an account of belief. There is disagreement about whether there can be unconscious beliefs, about where one draws the line between believing and acting, and about whether nonlanguage users can have beliefs. The inventory is large. This tangle has led some philosophers l to claim that discourse about belief is folk psychology to be replaced by proper science. The language of belief they say will fall into disuse, just as the theory of humors as an account of emotions fell into disuse. • Corrected version of a lecture presented in May of 1986 at the Coll~e @

de France.

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I should like in this paper to sketch an account which may resolve some of the disagreements. In so doing I depart from some received views 2 and hence the account may seem revisionary as an explication of belief. But it is not so revisionary as to be wholly without precedent or flagrantly disparate with features of our ordinary understanding. What is central in the account here presented is the departure from the dominant, language- oriented accounts of belief, which take it that the objects of believing are always linguistic or quasi-linguistic entities such as Frege's propositions or "thoughts" or Davidson's interpreted sentences. Since Frege, the preoccupation with belief claims, belief reports, which are linguistic, and the efforts of formal semanticists to provide a semantics for sentences with epistemological verbs have sometimes obscured our understanding. We are concerned to give an account of " x believes that 5." I should like to begin with a critical examination of language oriented views. LANGUAGE CENTERED lliEORIES OF BEliEF AND SOME DIFFICUL71ES WIlli SUCH lliEORIES

In "Thought and Talk," Davidson argues that 'x believes that 5' is equivalent to 'x holds a certain sentence true' in a shared interpreted language. That sentence is '5' or some translation of '5' in the shared interpreted language. Believing is a conscious relation of subjects to their utterances. He goes further and claims that even desires relate a subject to utterances. No language, then no desires or beliefs. And fmally, a non-language user, he says, cannot even have thoughts. There is a further stated baffling claim; that we cannot have thought, beliefs, and even desires, without the concepts of thought, of belief, and of desire. With respect to belief he says, "Can a creature have a belief if it does not have the concept of a belief? It seems to me it cannot and for this reason. Someone cannot have a

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belief unless he understands the possibility of being mistaken and this requires grasping the contrast between truth and error--true belief and false belief. But this contrast I have argued can emerge only in the context of interpretation [of a language). ,,3 The view on reflection is implausible. Consider the following example: A subject, call him 'Jean,' and his dog, call him 'Fido,' are stranded in a desert. Both are behaving as one does when one needs and desires a drink. What appears to be water emerges into view. It is a mirage for which there is a physical explanation; such a mirage occurs when lower air strata are at a very different temperature from higher strata so that the sky is seen as if by reflection, creating the optical illusion of a body of water in the distance. Both hurry toward it. On what possible ground can we deny Fido a desire to drink, a belief that there is something potable there? Jean and Fido an? both mistaken, but only a language user, Jean for example, has the concept of a mistake and can report it as a mistake. That does not require on the dog's part a concept of belief, a concept of desire, a concept of truth and error. The pre-verbal child hears familiar footsteps and believes a person known to her is approaching, a person who perhaps elicits behavior anticipatory of pleasure. It may not be the anticipated person, and when the child sees this, her behavior will mark the mistake. But must there be some linguistic obligato in the child to attribute to her a mistaken belief, or a disappointment? Must an agent have the concept of a mistake to be mistaken? The important kernel of truth in such a linguistic view is that arriving at a precise verbal description of another'S beliefs and desires is difficult, and especially so when the attribution cannot be verbally confirmed by the subject. We will not go far wrong in attributing thirst to the dog Fido, or the belief that there is the appearance of something potable. Whether we can attribute to the dog the recognition of water would depend in part on whether dogs

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can select water from other liquids to roughly the extent that we can. That is an empirical question. Of course, to attribute a belief, a desire, a thought to oneself or others, or to assert that someone has a desire or a belief or a thought requires language. But in the example given, Fido and Jean need be making verbal claims, vocalized or nonvocalized, about what they desire, what their thoughts and beliefs are. What this language-based view entails is that without a verbal obligato or without an identifiable linguistic representation there are no thoughts, desires or beliefs. Nor does the problem of correct belief attribution disappear among language users, particularly if linguistic confirmation from the believer is unavailable. It is, of course, conSiderably reduced. To decline to attribute desires and beliefs to non-language users is reminiscent of Descartes' declining to attribute pain to higher non-human animals despite the similarity with the causes of pain and pain behavior in non-human animals and language users. The case of belief is analogous. Descartes argues that in the absence of conscious introspective thoughts about our states, such as pain thoughts or belief thoughts, we do not have those pains and beliefs. In the revamped current version it is the absence of language, rather than mind, which deprives a subject of thoughts and beliefs. Not all who have language oriented theories of belief take so strong a stand. Some like Ramsey saw the distinction, but attributed it to an ambiguity in the notion of belief. It is of interest to note that Ramsey4 allows a sense of 'belief in which we may, using his curious example, attribute a belief to a chicken who has acquired an aversion to eating a species of caterpillar on account of prior unpleasant experiences. Here again an overly rich attribution is difficult to avoid. We surely cannot attribute to the chicken the belief that the caterpillar is pOisonous, but surely we will not go too far afield if we attribute the belief that the caterpillar is not for eating. And indeed if presented with a caterpillar which had the appearance of the

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despised kind but was in fact of an edible kind the chicken would be mistaken about its inedibility. Still, although acknowledging such a use of 'belief,' Ramsey finally concludes that believing as it applies to language users is so disparate with that of non-language users that the term 'belief' is ambiguous. He goes on to say, "without wishing to depreciate the importance of this kind of belief, ... I prefer to deal with those beliefs which are expressed in words ... consciously asserted or denied .... The mental factors of such a belief I take to be words spoken aloud or to oneself or merely imagined, connected together and accompanied by a feeling or feelings of belief.... " For Ramsey then, 'assenting to a sentence S', 'asserting that S' and 'believing that S' are equivalent alternative usages. Ramsey takes those utterances, spoken aloud or to oneself, as mental factors and hence the objects of belief, the S in 'x believed that S', are events of a linguistic character, sentences spoken or thought. Those are the sentences toward which we have an assenting attitude, a feeling for Ramsey. This attitude or feeling performs in Ramsey the role of 'holding true' in Davidson. The identification of the objects of believing with sentencelike objects has some familiar consequences. The believer, the subject, has those beliefs. But how does the subject have them? Ramsey singles out those "mental factors," sentences spoken aloud or to oneself or imagined. To say instead that they are the quasi-linguistic entities, the propositional contents of sentences as in Frege does not alter the picture in a helpful way. Such propositions mimic the structure of sentences. They have properties which interpreted sentences have like truth and validity. Sets of them can be consistent or inconsistent. They can be contradictory. They can enter into the consequence relation and so on. Frege5 had the view that propositions were the abstract nonmental contents of sentences toward which we had mental attitudes. But the mind-centered locus of the objects of belief is not wholly

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evaded. We "have them in mind" when we entertain them, believe them, disbelieve them. He did after all call them 'thoughts'. A recent account which claims to demystify Frege's propositions and is more explicitly language~entered may be described as the computational mode1. 6 Highly simplified, the subject is seen to have an internal register of basic concepts and basic sentences which are mental representations of actual sentences in his language. Syntactical rules for mental representation sentences generate complex sentences; deductive rules generate consequences of sets of sentences. Mental representation sentences are associated with mental analogues of yes or no responses on appropriate cues which will generate mental yes or no responses to more complex sentences on appropriate cues. The subject is said to believe that S just in case his correlated mental representation sentence elicits a mental yes response. Jerry Fodor7 presents such a view which I have much simplified. He says straight out that attitudes toward propositions are in fact attitudes towards formulae in "mentalese," the language of thought, formulae which are internally codified and are correlated with the external sentences of a given language. Propositions have given way to sentences in mentalese. The objects of belief are linguistic entities placed squarely in the mind. There are failings in such language-centered, wholly mindcentered accounts of belief. They exclude belief attributions to nonlanguage users or alternatively, insist that if one can make such attributions and if the manifestations of a public language are absent, the language of thought sententially organized must still be there in the non-verbal child or dumb animal's mind or brain. They also create difficulties for making sense of unconscious beliefs. The exclusion of unconscious beliefs is explicit in Ramsey and it would seem in Davidson. For Fodor, there remains the question of what would count evidentially in the attribution of an unconscious belief to an agent. Is it unconscious assent to a sentence in mentalese?

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Language-centered views also tend to define rationality in terms of attitudes to sentences which are consistent, contradictory, logically true, or related by deducibility and the like. Since it is agreed that rational, language-using agents as ordinarily viewed are not omniscient or perfect logicians, they may still be rational to a point yet fail to believe ClOthe consequences of their beliefs and hence may even come to hold true or assent to a sentence which is equivalent to a blatant contradiction. Where to draw the line and preserve the attribution of normal rationality is difficult to mark. But this is not an insurmountable failure. Nor am I suggesting that considerations of consistency and validity of inference are irrelevant to an account of rationality. It is rather that there is a broader notion of rationality and irrationality which language-centered theories are incapable of accommodating. There is, for example, the irrationality of the subject who sincerely avows that S, or holds'S' true, yet his non-verbal actions belie it. Such cases need not be centered, although they often are, around questions of akrasia such as that of the subject who sincerely avows that smoking is harmful yet continues to smoke. There are plausible psychological claims that our explicit avowals of belief, our sincerely stated claims about our own states such as our desires and fears, or the objects of our affections and disaffections often do not serve us as beliefs are supposed to in the explanation of action. Actions may belie our most sincerely reported "beliefs". These are not cases of deliberate deception or insincerity and may have some explanation in theories of self deception or false consciousness. But those latter theories are often grounded in the absence of an agent's conscious formulations in language of the contrary impliCit beliefs which explain the dissonant actions. Consider the subject who assents to all the true sentences of arithmetic with which he is presented and rejects the false ones; who can perform the symbolic operations which take him from true sentences of arithmetic to true sentences of arithmetic, and who also

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has toward them the belief feeling. Yet if you ask him to bring you [wo oranges and three apples, he brings you three oranges and five apples. He never makes correct change. Are his assents and assertions sufficient for ascribing to him correct arithmetic beliefs? Shouldn't non-verbal behavior also count as an indicator, or a counter-indicator of belief? And then there is the obvious fact alluded to in the example of the thirsty desert wanderers that we often, very likely more often than not, do not consciously entertain propositions or sentences held true when acting, although they are actions, explicable as consequences of beliefs and desires. Language users may assert such "propositions" if asked why they are acting as they are. Indeed being asked why we are acting as we are may lead us to discover or describe a belief which had never been verbalized. I usually walk a route to my office which is not the shortest and am asked why. It requires some thought. It isn't out of habit, I decide. I finally realize that I believe it to be the most scenic route. Verbalization as a necessary condition of believing precludes our discovering and then reporting what we believe. AN OBJECT-CENTERED ACCOUNT

We are concerned with beliefs purported to be about the actual world, not about fiction or myth or the like. This is not to deny the use of "belief' locutions in discourse about fiction. Its role will be understood from the context. If I am asked what it was that was converted into a chariot by Cinderella's godmother and do not have a clear recollection of the story, 1 might respond that "I believe it was a pumpkin that was converted into a chariot," but contextual cues make it clear that I am not making an historical claim about the actual world.

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What follows is a sketch of a world-centered, object-centered account which may be better fitted to our understanding of some epistemological attitudes. 8 Believing is understood to be a relation between a subject or agent and a state of affairs (not necessarily actual) but which has actual objects as constituents. We may think of states of affairs as ordered structures of actual objects which includes individuals as well as properties and relations. The structure into which those objects enter need not be an actual world structure. The state of affairs described by the sentence 'Socrates is human' is a structure containing Socrates and the property human. In taking believing as a relation to a state of affairs not necessarily actual, the believing subject may also be related to the constituents of the structure. 9 As an analogue, my ancestors may be structured as a set. I as a descendent am related to that structure and also to each of its constituents. Believing has often been called a propositional attitude. On the present account, if we wish to retain the locution 'proposition' for an object of believing, it is an atypical use. Since a proposition is more commonly viewed as a linguistic or quasilinguistic entity, it is best to deploy another vocabulary such as 'states of affairs' or 'structures.' One recognizes here a Russellian thrust. IO In one of Russell's early accounts of epistemological attitudes, constituents of propositions are actual objects which include abstract objects such as properties and relations. One of the departures from Russell herein is that no reductionism for constituents need be supposed. Ordinary indiViduals, properties, and relations may be constituents of states of affairs. On the subject or agent side of the relation we give a dispositional account. II 0: x believes that S just in case under certain agentcentered circumstances including x's desires and needs as well as external Circumstances, x is disposed

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to act as if S, that (actual or non-actual) state of affairs, obtains. Note the absence of a truth predicate in D. Actual or nonactual states of affairs are not truth bearers. If we employed the truth predicate as in "disposed to act as if'S' were true," S would have to be a linguistic or quasi-linguistic entity. It was such continued use of truth and falSity as properties of his propositions which made Mt. Blanc's intrusion into one of Russell's "propositions" so baffling to Frege. 12 WA YS IN WHICH SUCH AN ACCOUNT ACCOMMODATES SOME NATURAL VIEWS OF BElJEF (1) On the proposed view speech acts public or private are

only a part of the range of behavior which manifests belief; they are not as in language- centered views, a necessary condition. This allows belief attributions to non-language users. Naturally nonlanguage users will fail to have beliefs only possible to language users; beliefs about language for example. Linguistic items, whether type or token, are objects and can be constituents of states of affairs, but they are inaccessible to non-language users as linguistic items. Nonlanguage users will therefore not have beliefs about describing or referring; about truth or falSity, validity or logical consequence; about grammar and the like. Inference as a psycbological phenomenon will be severely limited in non-language users since complicated inference would seem to require stating, describing or reporting what we believe, setting it out in language. Prediction, deception, counterfactual speculation and long-range planning at a certain level of complexity would also seem to require language, as would second order beliefs, although there are recent empirical studies which claim that non-language users can and do plan and make long-range

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"decisions. But that is not to deny beliefs to non-language users altogether. (2) There is ample evidence that when acting out of belief we need not precede or accompany such actions with verbal, sentential accompaniments. We need not be entertaining before the mind, sentences or meanings. Jean, the desert wanderer, is racing to the water he believes to be out there. Both Jean and Fido have a belief in that they are related to a non-actual state of affairs and given their circumstances, act accordingly. (3) Such an account of belief views believing as a relation between a subject and a state of affairs not necessarily actual, where the subject, given psychological states such as wants, needs and in the presence of other circumstances will act as if that state of affairs obtained. Speech acts are among the acts which may and often do manifest a belief and one such speech act when x believes S is that x may sincerely assent to a sentence which is descriptive of the state of affairs S. It is not supposed that the act of sincere assent even where evoked must be an overriding indicator of belief. A range of circumstances will evoke such a speech act of assent. x may want to report his beliefs, to communicate them in language to others, to carefully examine for himself what follows from them, to testify, and so on. But note that if circumstances and desires are such that x wants to deceive he may not assent to a sentence which describes his beliefs even though that would be counted as insincere. But deliberately denying a sentence which describes what one believes is not the only way a speech act may mislead others about one's beliefs. Given that speech acts are important in reportina beliefs, the agent's particular way of reporting is a function of local circumstances as well as other beliefs which may not be shared by others. It is also a function of a subject'S mastery of the language. A language user of minimal competence may not even be able to perform the speech act which describes the state of affairs to

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which he is in the believing relation, but he is in that relation nevertheless. Indeed, despite the widespread assumption of privileged access to one's own beliefs, it could and does happen that someone other than the agent may better be able to report an agent's beliefs than the believer. Also, we can often assist a believer to describe more accurately' the state of affairs to which he is in the believing relation; that is a common phenomenon of language acquisition. (4) The proposed view accommodates the possibility of unconscious beliefs. They may be the very ones I have and do not or (on some psychological theories) cannot report in a speech act. Reporting brings them into consciousness. (5) It is a feature of the proposed non-language-centered view of belief that it permits a more adequate and natural account of rationality. Language-centered theories tend to define rationality in terms of sentences or sets of sentences or their quasi-linguistic "contents." On a language-centered account a rational agent is one who, for example, will not assent to surface contradictions. For a perfectly logical agent, belief is closed under logical consequence (pace Dretske and Nozick). 13 Given the empirical fact that we are not faultless logicians, belief for a rational agent is closed under logical consequence to some acceptable level of complexity of proof. A !lQIID of consistency for sentences we assent to is preserved. We abandon assent to sentences known to be inconsistent or necessarily false. Extended to inductive reasoning, such an account of rationality still focuses on a relation between accepted sentences and probable conclusions. But this language centered account is an impoverished view of rationality. It lacks explanatory force. Why should we dissent from known contradictions or inconsistent sets of sentences? A computer would pay no price for that, nor presumably would a brain in a vat. 14

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There is a wider notion of rationality than those of strongly language centered views where we say of a rational agent that such an agent also aims at making all the behavioral indicators of belief "coherent" with one another. For example, the agent's assentings and avowals should be coherent with his chOices, his bets, and the whole range of additional behavioral indicators of belief. We may say of someone who avows that he loves another yet often harms the one he claims to love that his behavior is "dissonant." He is in a wider sense, irrational. He is not logically irrational in the narrow, languagecentered sense. The set of sentences he assents to are consistent. He assents sincerely to sentences 'I love A'; he does not assent to sentences 'I don't love A'. He would deny the latter if asked. Agents who become aware of their incoherent behavior may try to "rationalize" that behavior, make it coherent, get a better fit. On such awareness the ambivalent lover may no longer assent sincerely to 'I love A'. He notes that other actions are incoherent with his speech acts or he may alter his cruel behavior to fit his avowals. He may argue that the concept of love is confused. That does not exhaust the possibilities. Such considerations are usually viewed as central to questions of akrasia. An agent might sincerely avow that smoking is harmful and that he wants to preserve health but continues to smoke. Recurrent in the literature on akrasia are explanations of such seemingly irrational behavior. One explanation is that the akratic has conflicted beliefs, one conscious and reportable in a speech act, one unconscious and unreponed which is the action guiding belief which overrides if he continues to smoke. A second15 is that the akratic agent has conflicting reportable beliefs and although the acknowledged grounds which justify one are stronger than the grounds which justify the other, he acts in accordance with the less justified belief. A third is where akratic action is seen as action not grounded in belief at all, but as "compulsive." Still, in all such

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explanations of irrationality it is our non-verbal acts which believe our words. Nor are we suggesting that adding coherence to strict logicality is exhaustive of a wider account of rationality. There are psychological syndromes, paranoia for example, in which an agent's actions may be described as remarkably coherent albeit irrational. A still wider account of rationality must also include acting in accordance with norms of evidence, norms of justification and the like that warrant an agent's believing as he does. Our proposal is only that coherence is an important feature of a more general explication of rationality. Assentings and avowals do play an important role in a wider notion of rationality, for an agent is often the best describer of what he believes and a judgment of incoherence is more determinate. But more important, a wider notion which includes coherence is explanatory of why a norm of logical consistency is preserved. Why, for example, is it claimed that a rational agent does not assent to a known contradiction? If a sentence '5' describes a specific state of affairs and if sincere assenting to '5' is taken as an act, a speech act, which marks our believing that 5 obtains, then if without relinquishing our original assent we also assented to 'not-S' under the same circumstances we would be believing that 5 both obtained and did not obtain in those circumstances; an objective impossibility which might render many of our actions incoherent and self-defeating. 16 Rationality in the wider sense is not preserved. If the assents to contradictions are not ferreted out, beliefs could lose their crucial role as guiding and explaining actions. BEliEF, ASSENT AND 11IE DISQUOTA110N PRINCIPLE

The object-centered position sketched here, and it is just a sketch, does not preclude special relationships between some

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speech acts,such as acts of assent, and belief, under some conditions. What has been rejected is the idea that an agent's believing S entails that the agent performs or can perform an appropriate speech act of assent even among language-using agents. There may be other markers of belief. We have also suggested that there are cases where assent to a sentence even on the part of a sincere, competent, reflective language user need not be a sufficient condition, an overriding guarantee of belieVing, since it denies that a person's nonverbal actions which seem to run counter to his avowals can be evidence against his having that belief. Briefly stated, we have questioned a principle generally accepted as non-controversial; Q: x sincerely assents to a sentence'S' entails that x

believes that S where the conditions sincerity,competence and reflectiveness obtain.

of

The widespread acceptance of this weak disquotation principle Q is not wholly without ground. My suggestion is that on a broader view other actions might belie the agent's words and sincere assenting might not be the privileged marker of believing. Let us suppose, however, for the discussion which follows, that our agent's other actions are coherent with his sincere assentings and he makes no logical mistakes. If that is the case, then on principle Q his sincere assenting to'S' does go over into a belief, for it is assumed that a linguistically competent, reflective speaker can reliably report what he believes. Where conditions including coherence hold, there remain on the language centered view some puzzles which can be resolved where Q is viewed as falling under D; i.e. where in assenting to a sentence under the appropriate conditions an agent is acting as if a state of affairs described by that sentence obtains.

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Q AND A PUZZLE ABOUT BELIEFS IN NON-EXISTENCE

In "Speaking about Nothing" Donnellan 17 recounts the actual case of 1be Horn Papers which were purported to be the published diaries of one Jacob Hom, a colonial American, and were so viewed until historians disclosed that it was a hoax. Consider someone, call her 'Sally', who having read 1be Horn Papers and unaware that they were part of a fabrication, sincerely assents to 'Jacob Hom lived in Washington County, Pennsylvania'. The syntactical proper name 'Jacob Hom' is not a genuine name but an invention of the hoaxster. Then, elaborating on Donnellan's views, the assenting to what seems a perfectly formed sentence 'Jacob Hom lived in Washington County, Pennsylvania' should not carry over into a belief. Backtracking her acquisition of the name does not terminate in a person named 'Jacob Hom'. But Sally need not know that. It is simply that the purported state of affairs described is not a complete state of affairs. It is as if she had assented to 'z lived in Washington County, Pennsylvania where 'z' is a variable. What she assented to does not describe a closed structure. Of course, Washington County, Pennsylvania ~ a constituent, and lived in ~ a relation; the structure does not wholly lack constituents. It lacks a constituent to make it count as a state of affairs. Yet Sally's was a sincere assent to 'Jacob Hom lived in Washington County, Pennsylvania'. She is competent, not conceptually confused, and reflective. We have also assumed that she makes no logical errors and is broadly rational. She just lacks relevant knowledge that the name 'Jacob Hom' is without a referent. In such a case on disclosure that the syntactical name 'Jacob Hom' does not refer, Sally should say on the proposed analysis, that she only claimed to believe that Jacob Hom was a resident of Washington County, Pennsylvania, in the first instance. Of course the disquotation principle simply has as antecedent 'x assents to'S" and it may be an implicit assumption of the disquotation principle Q that linguistiC

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competence and absence of conceptual confusion will rule out assent to a sentence which is not fully interpreted. But, as in the present example, that is unjustified. A rational competent agent can, on the psychological side of the believing relation, appear to "hold" a sentence "true" which lacks a truth value altogether. Nor will interjecting Jacob Horn as a possible person work, for reasons discussed elsewhere. 18 Suffice it to say that Sally took 'Jacob Horn lived in Washington County, Pennsylvania' as making an historical claim, as the author of The Horn Papers intended. Of course in the context of the example, Sally may assent to related sentences which are fully interpreted such as There was a person named 'Jacob Horn' born in Washington, Pennsylvania who kept a diary' which will carry over into a belief. Her non-verbal behavior may also be affected in predictable ways. She may engage in what she believes to be historical research such as searching out 'facts' not narrated in The Horn Papers, about the purported person as she might have been had there been such a person. On the present object-centered view, her behavior is rational and explicable although she could not have believed that Jacob Horn was born in Washington, Pennsylvania. Principle Q requires an additional condition, that '5' be a fully interpreted sentence. In the case of failed reference, as in the above example, what goes wrong is that the troublesome sentences appear to have a interpretation which they do not fully have. In discourse which purports to be about our actual world it is presupposed that proper names refer. 19 Sentences with such reference failures mislead. In such cases we should disclaim having had a belief despite sincere assenting. It should be noted that this case of reference failure does not present a problem for a Davidsonian language centered view of Q since he requires that '5' be fully interpreted and in the absence of a referent for 'Jacob Horn'; the sentence 'jacob Horn was born in

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Washington, Pennsylvania' lacks a complete interpretation and is not a truth bearer.

Q AND A PUZZLE ABOUT IDENTITY AND CONTRADIC110N Although for the purposes of discussion of puzzles we have assumed that the epistemological agent is rational and a faultless logician it appears to follow from Q on a language-centered view that such agents can come to "believe" a contradiction. What I want to argue is that under 0 and the object-centered view, that does not follow. At worst, what may occur is that a rational, logical agent may be seen to be in a believing relation to an impossible state of affairs. Consider the following example. 20 Someone, call her Sally, is rational in the wide sense and an impeccable logician. She also succeeds in maintaining in her actions, including her speech acts, a norm of coherence. In such a case we can normally take her sincere assentings as privileged markers of believing. In the 1930's Sally became acquainted with Alexis Leger of the French Foreign Office. On the baSis of information available to her she assents to the sentence 'Alexis Leger is not a poet'. Some years later Sally meets a poet, St. J. Perse, at the United States Library of Congress. Time has not been kind. St. J. Perse is not recognizably Alexis Leger, and she assents to the sentence 'St. J. Perse is a poet'. Since unbeknownst to her 'Alexis Leger' and 'St. J. Perse' name the same person, he is the constituent in the states of affairs encoded into or described by the sentences 'Alexis Leger is not a poet', 'St. J. Perse is a poet'. Each of those sentences taken separately describes a possible state of affairs. Leger might not have written poetry. Circumstances could have prevented that, as well as his serving in the foreign office. Given that Sally is logical, she will also assent to the sentence 'Alexis Leger is not a poet and St. J. Perse is a poet' which unbeknownst to her describes an impossible state of affairs.

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Given that she knows about identicals having to have all properties in common, including uniqueness, she assents to 'Alexis Leger is not St. J. Perse', which on the necessity of identity also describes an impossible state of affairs. She would of course not assent to 'Alexis Leger is not Alexis Leger' which, unbeknownst to her, describes the same impossible state of affairs as 'Alexis Leger is not St. J. Perse'. She would also assent to 'Alexis Leger' and 'St. J. Perse' name different persons', but that sentence is not descriptive of an impossible state of affairs. Our background theory only requires that names, once given, retain a fixed value. This puzzle is irksome to those who have that thought or language- centered view of belief where the objects of believing are either sentences or are those propositions which mimic sentences in having properties like true, false, contradictory, valid and the like. On their account, given the disquotation principle, Sally's assentings to 'Alexis Leger is not identical to St. J. Perse' and 'Alexis Leger is not a poet and St. J. Perse is a poet', go over into beliefs. But on the language or thought-centered views, the objects of believing are sentences or propositions. Since, semantically speaking, and unbeknownst to her, the same person is assigned to both names, she is assenting to contradictory sentences or propositions, so she seems to believe contradictions. Yet she has justification for arriving at her beliefs and has made no logical mistakes. She lacks other information. Why should the mere lack of information lead one to believe contradictions having begun only with seemingly justifiable albeit some false premises? The language-centered theorist is baffled. What does Sally believe he asks? If she believed that Alexis Leger is not a poet, then that is the same proposition as St. J. Perse is not a poet. Does she or does she not believe St. J. Perse is a poet? But note how differently our presently proposed view accommodates the language-centered theorist'S puzzle. Sally was

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introduced to Alexis Leger in the French Foreign Office. The individual in the state of affairs described by the sentence she assents to, Le., 'Alexis Leger is not a poet', is that person: not the essence of Alexis Leger, not the concept Alexis Leger, not the sense of the name 'Alexis Leger'. When later she meets St. Jean Perse, he, that person, is a constituent in the state of affairs described by the sentence 'St. Jean Perse is a poet'. What Sally lacks is information which would permit her to reidentify Leger. The sentences she assents to are not surface contradictions. The syntactical form of some of the sentences she assents to which describe impossibilities do not have the surface form of logical falsehoods. It is surely possible for different syntactical names to name the same thing. Sally acquired the names 'Alexis Leger' and 'St. J. Perse' on two different occasions and under different circumstances as very likely did Leger. The chain of communication in the public language will carry the second name into the first and finally to the object named. The value of those names, if they refer, is fixed but not by some known set of identifying descriptions which permitted a determination by the agent. As a practical matter even a chain of communication may be empirically and practically irretrievable. Unlike the Jacob Horn case or the Leger-Perse case, historical records may not be available sufficient to make a determination of reference or absence of reference. It is such situations which prompt research (often frustrated) into, for example, the historical Homer or the historical Robin Hood. On our proposed view of believing, x believes that S when x has a disposition to act as if a certain state of affairs obtained. A sincere assenting to'S' by a rational, logically impeccable but nonomniscient agent can serve as a privileged marker of believing that S. But unlike linguistic propositions, states of affairs obtain, or do not obtainj must obtain or cannot obtain. They are not true or false,

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contradictory, valid or invalid. Some of the sentences Sally unwittingly assents to would seem to lead her to be in a believing relation to an impossible state of affairs but though she assents to a logically false sentence, she doesn't believe a contradiction as demanded by the language-centered account. It is sentences which are contradictory. Note that on the dispositional account given by principle D, agent-centered circumstances as well as external circumstances are conditions on x's being disposed to act as if S, that state of affairs obtained. There are (1) circumstances under which Sally is disposed to act as if Leger (Le. St. J. Perse) is not a poet and (2) circumstances under which she is disposed to act as if Leger (Le. St. J. Perse) is a poet. Indeed, among those circumstances in (1) are those in which she also assents to 'Alex Leger is not a poet' and does not assent to 'St. J. Perse is not a poet'. Similarly for the circumstances in (2) under which she assents to 'St. J. Perse is a poet' and does not assent to 'St. J. Perse is not a poet'. Note that assenting is a speech act which occurs at a time and place and under circumstances. Under those variant circumstances her assents in accordance with Q, each go over into a belief. Indeed on Q, the above assents each carry into a believing relation to a possible state of affairs. But she will also assent in some circumstances to 'St. J. Perse is a poet and Alexis Leger is not a poet' and on Q that would put her into a believing relation to an impossible state of affairs. In assenting to a sentence which, given still unknown reidentifications might come, on logical grounds but unbeknownst to her, to have on substitution the form of a surface contradiction, she is on D disposed to act as if an impossible state of affairs obtained. The answer to the question on such a theory of what Sally believes about Alexis Leger and St. J. Perse is not baffling. If, as we have proposed the objects of believing are not linguistic entities, then on the disquotation principle, we can say that she believes what she

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says she believes; that Leger is not a poet, that St. J. Perse is a poet and given that she will assent to logical consequences of sentences assented to she may also act as if some impossible states of affairs obtained to the extent that such actions are describable. Her assents, her logical reasoning and her evidential grounds are not incompatible with having such dispositions. She is not omniscient. So, seeing no incompatibility between rationality in the narrow sense and believing impossibilities, we should let the matter rest. A puzzle has been solved. Furthermore, there seems to be ample evidence that impossibilities are believed. A CONTROVERSIAL PROPOSAL

Given the present account we are not led to the puzzling conclusion that a logically rational agent believes a contradiction but only that under certain circumstances she is in the believing relation to an impossible state of affairs describable by a sentence to which she assents. A puzzle has been solved. Nevertheless I should like to propose, on other conSiderations, a modification of Q, which disallows believing impossibilities. In the suggested proposal an agent can claim to have such beliefs but will be mistaken in so claiming. Just as norms of truth lead to retroactively revisable knowledge claims; norms of rationality should lead to retroactively revisable belief claims. What is being proposed is that whatever the psychological dimension of the belief state, on disclosure of impossibility the belief claim should be viewed as mistaken. Despite prevailing views to the contrary, the thesis that one cannot believe impossibilities has its advocates. Berkeley 21 argues that to believe propositions which entail contradictions is illusory. He says of such propositions that "they are an instance wherein men impose upon themselves by imagining they believe those propositions. "

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There is, of course, agreement that a rational agent does not assent to simple formal surface contradictions such as '5 and not 5', but in such a case of assenting to an overtly and formaliy contradictory sentence one can claim that a condition on the disquotation principle had not been met. The agent is conceptually confused. He hasn't grasped the m~aning, and doesn't comprehend the semantics of words like 'not', 'and' and so on. Cases of deductive failure which lead to assenting to sentences which are equivalent on substitution to contradictions can sometimes be attributed to mistakes in calculation. But among those like Berkeley, Wittgenstein and those positivists who rejected belief in impossibilities which are represented by logically or formally false sentences, there was an underlying argument for rejecting such beliefs which would seem to apply to all cases of impossibility, including those which have their origin, as in Sally's case, in deficient information. Wittgenstein22 is concerned in the Tractatus with those necessities and impossibilities which are given by tautologies and contradictory propositions of whatever complexity. He argues that a significant proposition has to describe a definite situation which allows that the situation mayor may not obtain. A proposition must admit an alternative truth value. Where a proposition does not admit of alternative values, i.e. is not contingent, it lacks sense, although he adds that it isn't exactly nonsense either. "Tautologies and contradictions ... do not represent possible situations." "The truth of a tautology is certain, of contradiction impossible" and therefore they lack Significance. He says that 'x knows that 5' is senseless even where '5' is a tautology. On the same ground he would say that 'x believes 5' where '5' is a contradiction is also senseless. But if what informs his argument is that it is the impossibility of a proposition 5 being false or the impossibility of a proposition S being true which makes it an improper object of a propositional attitude, then the

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2Iiiin of those attitudes should not matter. A false identity claim, for example, is necessarily false, never mind how it was arrived at. A later Wittgenstein 23 softened somewhat toward the range of significant propositions. Necessary truths and falsehoods are no longer denied sense. But he does hint at a distinction between believing and claiming to believe impossibilities. He says, "I feel a temptation to say one can't believe 13 x 13 - 196 ... But at any rate I can say 'I believe it' and act accordingly." (Italics mine) That hint, if it is a hint, can be elaborated into a proposal. Present received views insist that if we ascribe to an agent that he knows that Sand S turns out not to be the case, we alter our ascription. This can be done retroactively. Disclosure of the falSity of'S' would falSify the knowledge claims of all who ever made such claims. But it would seem that we do not retroactively disclaim belief in impossibilities unless one takes something like Wittgenstein's early radical view that impossible states of affairs are not states of affairs at all and hence not proper objects of epistemological attitudes. There were after all those who believed that angles could be trisected with a compass and a ruler, or that the consistency of arithmetic was provable on some canonical criterion of proof, or that Shakespeare was the Earl of Oxford. Those claims are necessarily false, yet what is claimed seems to have been believed. But I do see an advantage to a revision of the disquotation prindple; not Wittgenstein's early radical proposal (for tautologies and contradictions are surely meaningful) but a proposal which allows a distinction between believing and claiming to believe. We propose a modification of Q as follows. In addition to linguistic competence, sincerity and reflectiveness of the agent, we add the condition that her actions, including her speech acts, are coherent and preserve a norm of rationality in the wide sense. This is required if assenting is to count as an overriding marker of belief.

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assents to'S' (2) 'S' is a fully interpreted sentence in x's language and (3) possible S, entails that x believes that S. If conditions (2) and (3) are not met, x's assenting does not carry over into a belief. The two puzzling cases are accommodated. In The Horn Papers example, (2), the condition on complete interpretation of'S', is not met. Nevertheless a rational account can be given of why Sally claimed to have a belief that Jacob Horn lived in Washington County, Pennsylvania despite the absence of a fully structured state of affairs which is a proper object of believing. In the Alexis Leger-St. J. Perse example the state of affairs is properly constituted. Here the disclosure of the truth of an identity sentence would reveal the logical falsehood of some sentences to which Sally assents. In such a case Sally might say that she only claimed to believe that Alexis Leger was not the same as St. J. Perse, for such a belief comes to believing of a thing that it is not the same as itself and that does not meet logical norms of rationality. Just as the falsehood of 'P' excludes knowing that P, the necessary falsehood of 'P' excludes believing that P whatever the agent's knowledge claims or belief claims, respectively. The revisionary proposal places conditions on when a speech act of assent goes over into a belief. It does not place a possibility condition on the beliefs of non-language-using agents. The condition of possibility is grounded in norms of rationality just as in the case of knowledge the condition of actuality is grounded in norms of truth. Such norms would seem to require second order conceptualization and reflection open only to languages users.24 If believing S is normally a disposition to act as if a certain state of affairs obtained, and if such a state of affairs could not possibly obtain, a rational, language-using epistemological agent is in a position to ask what would count as acting as if it did obtain? Many actions would be rendered incoherent, many ends frustrated. If the Q': (1)

X

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speech act of assenting is one of those actions which marks our believing, then we would be acting as if'S' were true. But what sense can we make of acting as if'S' were true when either'S' has no truth value or'S' is necessarily false? Of course on the revisionary proposal we could not ascribe belief to those who claimed to believe that the Fountain of Youth is in Florida (where the Fountain of Youth is taken as a directly referring name) or that an angle can be trisected with compass and straight edge or that Shakespeare is identical to the Earl of Oxford. We could only say that they claimed to have such beliefs, they are not irrational, and we can explain why they made such claims, but that does not do too much violence to plausible usage. For they did have other proper beliefs which explained their assent to sentences which describe impossible states of affairs. SOME CONCLUDING REMARKS: It must be emphasized that the dispositional account of belief (D)

may be detached from the proposed controversional constraint on the disquotation principle. However, the dispositional account and principle Q' do have consequences for the semantics of belief sentences and belief reports of a radical kind such as whether the belief "operator" can be factored out of or distributed over a conjunction and the like. Also and more radically it suggests that the "sentential" or "quasi- sentential" account of epistemological attitudes and mental representations is inadequate and must be replaced by some more radically expanded or different model of mental states and information processing than the simple language centered model, which takes sentences or mental or metaphysical analogues of sentences as the objects of believing. Halleck Professor of Philosophy, Yale University

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Notes

1 See S. P. Stitch, From Folk Psychology to Cognitive Science: The Case Against Belief(Cambridge, Mass.: M.I.T., 1983). Also, D. Dennet, "Beyond Belief' in A. Woodfield, ed., Thought and Object (Oxford: Oxford Press, 1982). 2 The present paper in later sections amplifies and revises my "A Proposed Solution to a Puzzle about Belief," Midwest Studies in Philosophy: Foundation of Analytic Philosophy, vol. VI, ed. P. French et al. (1981): 501-510 and "Rationality and Believing the Impossible," The Journal of Philosophy" LXXXI (1983): 321-339. Those papers focused on S. Kripke's "A Puzzle about Belief" in Meaning and Use, ed. A. Margalit (Dordrecht, Reidel: 1979). The disquotation principle discussed below was set out by Kripke. 3 See S. Guttenplan, ed., Mind and Language (Oxford, Oxford Press, 1975): 7-23. 4 The Foundations of Mathematics (New York: Humanities Press, 1950): 44. Ramsey also claimed that such introspective feelings are an insufficient guide when it comes to judging the difference between believing more or less firmly. 5 See "On Sense and Nomination" and other essays in G. Frege. TranslatiOns from the Philosophical Writings tr. P. Geach and M. Black (Oxford: Blackwell, 1952). The present discussion of propositions as linguistic entities mapped by sentences which "express" them does not apply to those more recent accounts of propositions as functions from worlds to truth values.

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6 Computer scientists concerned with such "artificial intelligence" models actually use the language of belief in discussing their programs. 7 See The Language of Thought (New York: Crowell, 1975) and Representations (Cambridge, Mass. M.I.T., 1981). 8 Russell maintained throughout his work an object-oriented view of epistemological attitudes which is sometimes obscured by the use of "proposition" which has a linguistic connotation. "Propositions" for Russell contain non-linguistic constituents. 9 For Russell, believing relates the agent to the constituents of the proposition, and not the proposition. This suggests that one precludes the other, but it need not. 10 See the papers mentioned in Footnote 2. Also R. Chisholm, "Events and PropOSitions," Nous4, (1970): 15-24. The recent work of J. Perry and J. Barwise on "Situation Semantics." See SituatiOns and Attitudes (Cambridge, Mass. M.I.T., 1983), N. Salmon, Frege's Puzzle (Cambridge, Mass., MIT, 1986). 11 R. B. Braithwaite in "The Nature of Believing," Proceedings of the Aristotelian Society, 33 0932-3): 129-146 has a dispositional account but it is also a language-bound account. For Braithwaite "x believes S" is analyzed as follows: S (a proposition) must be entertained, and under relevant internal and external circumstances, x is disposed to act as if S were true. Such a language-oriented dispositional account also excludes unconscious beliefs, excludes beliefs of non-language users, and supposes that believing always entails entertaining linguistic or quasi-linguistic objects. The dispositional account I am proposing

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is rather unlike Quine's in that he does not take states of affairs as objects of believing. 12 G. Frege, Philosophical and Mathematical Correspondence (University of Chicago Press, 1980):169.

13 F. Dretske, "Epistemic Operators," j. of Phil. 67 (970): 1007-1023 and later R. Nozick, Philosophical Explanations (Harvard 1981) question the claim that belief is closed under logical consequence but those purported counter examples are not critical in our present account. 14 Andrew Hodges in Alan Turing: The Enigma (Simon,& Schuster, NY, 1983): 154, reports a conversation between Turing and Wittgenstein on contradiction which includes the following exchange. Wittgenstein (citing the paradox of the liar) :.. .it doesn't matter...it is just a useless language game .... Turing. What puzzles one is that one usually uses a contradiction as a criterion for having done something wrong .... Wittgenstein: Yes - and more: nothing has been done wrong ... where will the harm come? Turing: The real harm will not come unless there is an application in which a bridge may fall down or something of that son. Wittgenstein: ... But nothing need go wrong, and if something does

go wrong - if the bridge breaks down - then your mistake was of the kind of using a wrong natural law. Turing: Although you do now know that the bridge will fall down if there are no contradictions, yet it is almost cenain that if there are contradictions it will go wrong somewhere. 15 D. DaVidson, "How is Weakness of the Will Possible?" in Moral Concepts, ed. J. Feinberg (Oxford, Oxford Press, 1970).

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16 It is difficult to make a case for rejecting contradictions unless we

see the connections between rationality, coherent action and plausible outcomes. A computer programmed with proper deductive rules and a contradiction will allow any sentence in its register of affirmations. In the absence of further action to be guided by those outputs, there is no problem of coherence as here described. Similar considerations apply to examples of brains in vats. See footnote 14. 17 Donnellan, "Speaking of Nothing," Philosophical Review 83(974):

3-30. The Horn Papers were launched as history hence the characterization "hoax" rather than "fiction." 18 "Dispensing with Possibilia," Proceedings of the American

Philosophical Association, Newark (977). Also "Possibilia and Possible Worlds", Grazer Philosophische Studien, Vol. 25, 26, 1985/86 The latter is a revision of the first of two lectures presented at the College de France May 1986. The present paper is a revision of the second lecture. 19 In my "Modalities and Intentional Languages," Synthese. 13 0%1): 303-321 such a directly referential view of proper names was

proposed. There, p. 310, I say "To give a thing a proper name is different from giving a unique description. This [identifying] tag, a proper name, has no meaning (as contrasted with having reference). It simply tags. It is not strongly equatable with any of the singular descriptions of the thing." It should be noted that on this view, proper names are llQ.t assimilated to descriptions, even "rigid" descriptions. Kripke, in "Naming and Necessity," Semantics of Natural Language. ed. G. Harman and D. Davidson (Reidel, 1972) classifies proper names as "rigid designators" along with rigid descriptions, thereby obscuring the different sernantical relationship

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between a proper name and the object named as compared with the relationship between a rigid description and the object described. Kripke, in commenting on my 1961 paper when it was presented at a symposium in February 1962 interpreted my views as taking the position that "the tags are the essential denoting phrases for individuals." That was not part of my account but we can see in those 1961 remarks Kripke's move toward his 1971 theory of "rigid designators. " His comments appear in Synthese XIV (962): 132-143 in "Discussion on the paper of Ruth B. Marcus." See especially p. 142. 20 See Kripke, op cit, footnote 2. My example is an analogue of Kripke's case of Pierre cOming to believe a "contradiction."

21 Principles of Human Knowledge, ed. C. M. Turbayne (New York, Bobbs Merrill, 1970): 273. 22 Tractatus logico-Philosophicus (London, Kegan Paul, 1922). See especially 4.461-4.466,5.1362,5.142, 5.43a, 6.11. 23 Remarks on the Foundations of Mathematics (Oxford, Blackwell, 1956): 1-106. 24 This is what seems to be at the center of Davidson's view discussed at the outset of this paper.

QUANTIFICATION, MODAUTY, AND SEMANTIC ASCENT

Brian Skyrms

Introduction: Since Professor Vuillemin has studied several aspects of the thought of Bertrand Russell, it is perhaps appropriate here to juxtapose a program for metalinguistic interpretation of modality considered by Quine with a much more sweeping metalinguistic program earlier advocated by Russell. Some philosophers, including Quine himselfl , have expressed skepticism that Quine's program could be fully carried out for quantified modal logic. However, a general result by Cresswell for any intensional propositional logic taken together with a reinterpretation of the semantics of quantification suggested by Mates, make Quine's project almost trivial. Quine and Russell· Quine has argued that the only theory of modality which makes sense is one which construes the modalities as metalinguistic predicates rather than object linguistic operators. Earlier, in An Inquiry into Meaning and Trutb, Russell articulated a more ambitious program of semantic ascent. In chapter IV he argues for a hierarchy of metalanguages: Tarski, in his important book Der Wabrbeitsbegriff in der !ormalisierten Spracben, has shown that the words "true" and "false n as applied to sentences of a given language, always require another language, of higher order, for their adequate definition. The conception of a hierarchy of languages is involved in the theory of types, which, in some form, is necessary for the solution of the paradoxes; it plays an important part in Carnap's work as well as in Tarski's . I suggested it in my introduction to Wittgenstein's Tractatus, as an escape from his theory that syntax can only be "shown", not expressed in words. The arguments for a hierarchy 175

G. G. Brittan (ed.), Causality, Method, and Modality © Kluwer Academic Publishers 1991

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of languages are overwhelming, and I shall henceforth assume their validity." In chapter V Russell indicates the scope of the program: In the present chapter I wish to consider certain words which occur in the secondary language and in all higher languages, but not in the object language. I shall especially consider "true", "false", "not", "or", "some" and "all". Here Russell regards the quantifiers and even the truth functions as metalinguistic. We are faced with the prospect of a metalinguistic theory of almost everything, with only simple predications left in the object language . Is the idea mad or sensible? The interesting question is not whether we must look at things this way - to which the answer is most surely "no"- but rather whether there is something interesting to be learned by trying to adopt this point of view. Let us grant Russell some working assumptions. Language is bivalent, i.e. every sentence is true or false. And the metalanguages may refer by something like logically proper names, whose meanings are their denotations. Given some such assumptions it is possible to see how one could paraphrase "a is not F" as "'a is F' is false". More generally, one might have metalinguistic relations such as NANDY( , ) which holds between two sentences just in case not both are true, etc. There are the truth functions. As for the quantifiers, one could paraphrase "( xXFx)" as "Fx is satisfied". Here, however, I would like to think about an alternative reductive strategy which brings out some relationships between Quine's program and that of Russell. The leading idea is that quantification can be thought of as a kind of modal operator, and if it is thought of in this way then by Quine's lights as well as Russell's, it belongs in the metalanguage. Mates Quantification: In Elementary Logic Mates introduces a novel approach to quantification whereby truth-in-a-model is defined for

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quantified sentences without passing through a concept of satisfaction. Mates takes (xX~x) to be true in a model just in case ~ is true in all ~ variants of that model, where ~ is the first individual constant not occurring in ~ and a ~-variant of a model is a model just like that model except that ~ may be assigned a different denotation in the domain of that model. Formally, this is not so different from the classical treatment of the semantics of quantification by Tarski, and it is equivalent to it when quantifying into referentially transparent contexts. However, Mates' formulation of the semantics of quantification is conceptually interesting for several reasons. In the first place, it is a kind of hybrid between substitutional and referential quantification. 2 Like substitutional quantification, it is perfectly well-defined for quantification into referentially opaque contexts. But the objections that Quine raised to substitutional quantification about cardinality of the domain in his debates with Marcus do not apply. In the second place, the Mates semantics for quantification looks very much like the Kripke semantics for modal operators. We could carry over the semantical idea into the syntax and dispense with variables in the language. Suppose that we take a standard first order language purged of individual variables and quantifiers, but with a denumerable number of individual constants and which has added for each such constant, a, an associated modal operator, N a. Na (J) will be held to be true in a model just in case r is true in every model just like that model except perhaps for the denotation of a. For every sentence in an ordinary first order quantificational language, one can find an equivalent sentence in this language. One must only be careful, in the translation procedure, to keep on picking new names to replace variables, to avoid illicit quantifier capture. The business of illicit quantifier capture makes for an instructive peculiarity of this reformulation of quantification theory. The contexts in question now become referentially opaque, and

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quantification theory in this guise becomes a branch of intensional logic. For we cannot validly infer: Na (a-b) (Everything is b) from: (a is b) a-b and: Na (a-a) (Everything is itself) This sinister aspect of quantification suggests the desirability of a metalinguistic paraphrase. 3 Validity de dicto and de re: In medieval philosophy, modality was always thought of as a predicate rather than an operator. For example, in the Summa Logicalis of Peter of Spain, we have: There are two kinds of composition. The first kind arises from the fact that some dictum can suppose for itself or a part of itself, e.g. That he who is sitting walks is possible'. For if the dictum 'that he who is sitting walks' is wholly subjected to the predicate 'possible', then the proposition is false and composite, for then opposed activities, sitting and walking, are included in the subject, and the sense is: 'he who is sitting is walking'. But if the dictum supposes for a part of the dictum, then the proposition is true and divided, and the sense is 'that he who is sitting has the power of walking.' To be distinguished in the same fashion is: 'that he who is not writing is writing is impossible'. For this dictum 'that he who is not writing is writing' is subjected to the predicate 'impossible', but sometimes as a whole and sometimes in respect to a part of itself. And similarly: 'that a white thing is black is possible'. And it is to be known that expressions of this kind are commonly said to be de re or de die to. 4

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In the spirit of Peter, we distinguish between validity de dicta and de reo Validity de dicto is properly predicated of the whole sentence (or the dictum which it expresses). A sentence is valid de dicta just in case it is true in all models. Validity de re is more restricted; the supposition of possible situations is only implemented with respect to part of the dictum, with the denotations of the designators in the rest of the dictum being held constant in the process. There are various kinds of validity de re according to which designators are allowed to vary their denotation and which have their denotations frozen. A sentence is valid de re just in case it is true in all models in which the "frozen" designators have their actual denotation. 50 r is valid de re with respect to set 5 in model M just in case r is true in all models in which the designators in set 5 designate what they designate in M. 5 is the set of designators for which we do not suppose. Validity de re with respect to the null set of designators is validity de dicta and validity de re with respect to the set of all designators is truth. In between we have, among other things, the metalinguistic counterparts of our Mates-quantificational modalities. Corresponding to N(D we have r is valid de re with respect to the complement of (n). Maral: A general metalinguistic analysis of sentential modal operators already includes within itself the resources for quantification. Metalinguistic Modality; Are such meta linguistic accounts possible? Yes, and they can be done in a number of ways- not all of which have been fully explored. As long as their are no restrictions put on the hierarchy of metalanguages, the project is straightforward and almost trivial. In "An Immaculate Conception of Modality" I pursued this analysis with validity and provability being taken as the metalinguistic necessity predicates. Fairly natural ways of doing this yield 5-5 for validity and 5-4 for provability. These are not, however the only ways of doing this. In particular, in order to get 5-4 for provability one

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must assume a hierarchy of stronger and stronger meta theories, each of which is able to prove the consistency of the theory below it. Different assumptions would get different results. 5 In this treatment, the structure of the metalinguistic semantics for 5-4 and 5-5 is closely related to that of the Kripke possible world semantics for the modal operators. In fact, Cresswell has shown that any intensional operator with a standard possible world semantics has an equivalent metalinguistic semantics. Thus, given a suitable hierarchy of metalanguages, the metalinguistic approach must have at least as much power and scope as the Kripke semantics, and the Kripke semantics must inherit whatever philosophical legitimacy its metalinguistic counterpart possesses. Concluding Duestion: There are things to be learned in pursuing Quine's metalinguistic program. One is that if there are no special restrictions on how one is allowed to proceed, a metalinguistic account of quantified intensional logic is not that hard to get. Another is that some seemingly trivial reformulations can put questions of referential opacity in an entirely different philosophical light. What more is to be learned in pursuing the bold program put forward by Russell? Professor ofPhilosophy. University of California. Irvine

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Notes 1. in "Three Grades of Modal Involvement". 2. I discuss this in some detail in "Mates Quantification and Intensional Logic". The reader may also want to compare the treatments of quantification in Bencivenga, Lambert and van Fraassen (1986); Lambert and van Fraassen (1972); and Scott (1970). 3. Compare Montague (1960) on this point. 4 . Summa Logicales 7. 26.

5. Of particular interest is the modal logic of the canonical proof predicate of arithmetic. See Boolos (979); and Smorynski (1985).

References: Bencivenga, E. , Lambert, K. and van Fraassen, B. (1986) Logic Bivalence and Denotation (Ridgeview: Atascadero, California). Boolos, G. (1979) The Unprovability of Consistency: A n Essay in Modal logic (Cambridge University Press: Cambridge). Cresswell, M.J. (1985) "We are all children of God" in Studies in Analytical Pbilosopby: A contemporary Perspective ed. Shaw, J,L and Matilal, B.K. (Reidel: Dordrecht). Kripke, S. (1959) "A Completeness Theorem in Modal Logic" journal of Symbolic Logic 24, 1-14.

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Kripke, S. (1963) "Semantical Considerations on Modal Logic" Acta Philosophica Fennica 16,83-94. Lambert, K. and van Fraassen, B.(1972) Counterexample (Dickenson: Encino, California).

Derivation and

Marcus, R. (1962) "Modalities and Intensional Languages" Synthese 27, 303-322. Mates, B. (1965) Elementary Logic (Oxford University Press: Fairlawn, N.J.)

Montague, R. (1960) "Logical Necessity, Physical Necessity, Ethics and Quantifiers" Inquiry 3, 259-269. Montague, R. (1963) "Syntactical Treatments of Modality" in Acta Philosophica Fennica: Modal and Many Valued Logics 153-166. Quine, W. VO.(1943) "Notes on Existence and Necessity" Journal of Philosophy 40, 113-127. Quine, W. YO. (1953) "Three Grades of Modal Involvement" Proceedings of the XIth International Conqress of Philosophy 14, 6581.

Quine, W. YO. (1962) "Reply to Professor Marcus" Synthese 27, 323330.

Russell, B. (1940) An Inquiry into Meaning and Truth (New York).

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Schweizer, P. (987) "Necessity Viewed as a Semantical Predicate"

Philosophical Studies 52, 33-47. Scott, D. (970) "Advice on Modal Logic" in Philosophical Problems in Logic ed. K. Lambert (Reidel: Dordrecht) Skyrms, B. (978) "An Immaculate Conception of Modality" The journal of Philosophy LXXV 368-387. Skyrms, B. (981) "Mates Quantification and Intensional Logic"

Australasianjournal of Philosophy 59, 177-188. Smorynski, C. (985) Self-Reference and Modal Logic (Springer: New York). Tarski, A. (1956) Logic, Semantics and Metamathematics (Oxford University Press: Oxford).

Temporal Necessity, Time and Ability: a philosophical commentary on Diodorus Cronus' Master Argument as given in the interpretation of Jules VuiUemin David Wiggins "You couldn't have it if you didwant it," the Queen said. "The rule is, jam tomorrow and jam yesterday-but never jam today." "It must come sometimes to 'jam today'", Alice objected. Lewis Carroll Alice through the Looking Glass.

According to the report of Epictetus (Dissertationes II, 19), Diodorus' Master Argument comprised three propositions:

nav napO.T)AUeoS' clAT)eES' clvaYKaLOV Eon (8) 8uvaT41 cl8uvaTov OUK clKOAOueEt (C) 8uvaTOv Eonv (3 OUT' Eonv clAT)eES' oiJT' EOTal (A)

(A) Every past true thing is necessary [once it is past). (B) The impossible does not follow from the possible. (C) There is some possible thing that neither is true nor will be. I shall follow Jules Vuillemin and others in seeing the modal adjective "necessary" in (A) as essentially tensed and in giving "follows" in (8) a logical (non-temporal) interpretation. In this interpretation, (B) is a general principle that applies to all genuine modalities including the modality introduced by "it is necessary that p, given that at t it is already the case that p". 185

G. G. Brittan (ed.), Causality, Method, and Modality © Kluwer Academic Publishers 1991

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Vuillemin sees the contradiction that Epictetus says the ancients found between (A) (B) (C) as depending interalia upon certain extra premises. Vuillemin's first extra premiss 2 is one that he adduces from his analysis of the disagreement that Aristotle expresses with Plato at De Caelo 285 b6-17. The net effect of the extra premise (D) is to extend (A) to the case of things that are present. If we want to give this premiss expression in Greek, it may seem that we can hardly do better than use Aristotle's words at De Interpretatione 19a23,

(D) To ~€v

OTav ~~

ouv dvm. TO OV
D, clvciYKT).

(D) The being of what is when it is, and the not being of what is not when it is not, is necessary. Just as the past cannot now be helped, one might reason, so what is already happening has already begun to happen and, at least to that extent, cannot now be helped. The "necessary" here is similar to the "necessarily" in (A) in being essentially tensed. There is one other premise that Vuillemin invokes in his derivation of the inconsistency between (C) and (A) and (B). Vuillemin calls it the 'Principle of Synchronic Contraction' (the contraposition of that of 'Diachronic Expansion'): (E) If there is an instant to such that it is possible at to that p at t, then there is on the interval to-t an instant t 1, where to~tl~t, such that it is possible at tl that pat t 1. Vuillemin remarks (page 39) 'Le principe contracte synchroniquement sur un instant de l'intervalle un possible pose diachroniquement sur cet intervalle.' We shall come back to (E). In the meantime, we only need to find a thought that could have given rise to something like (E) and that

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was so obvious for Diodorus or for some reporter intermediate between Diodorus and Epictetus that it didn't merit explicit recognition. Perhaps the idea was something like this. If you can 0, then there must be a time t such that you can at t 0 at t, where this is the very same time t. We shall encounter problems with the formulation of this thought ,(VII below), but there is a way of seeing this kind of claim (if only one could state it correctly) that would make it seem almost as natural and unavoidable as Alice's presumption that, if she were to accept the White Queen's employment in consideration of two pence a week and jam every other day, then there would have to be some days on which she could say "It's jam today" and be right about that. II Faced with (A) (B) (C) (D) (E), our first or chief need is to understand better the meanings of these claims and to see their import brought out in an example. But what sort of an example? Here, as so often with Greek texts where we encounter neuter participles and adjectives and/or any form of the verb to be, we have to contend with the fact that the familiar words that Diadorus is reported to have used admit alternative interpretations. They can be interpreted as concerned with either events (unrepeatable dated things), or actions (taken as a subset of events), or acts (things we do at different tlmes)3, or facts, or states of affairs or propositions. Experiment with these possibilities suggests that we shall do best, neither losing generality nor trading upon ambiguity, and keep closest to the interpretation Vuillemin intended for his abstract symbolism ('p at t' etc.), if we choose as our specimen Diodorean thing an act an act being something done, which can have various instances at various times (particular actions) or no instances at all (as in the case of the acts of squaring the circle and jumping over the moon). If we make this choice, then it will be natural to say that Diadorus' 'true' means 'can

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be truly reported as having been done', 'past'means 'was done or has been done' and 'be' as in principle (D) means 'be done'. And an act such as catch the 27 bus or scale Everest or find the fountain of youth will be the kind of thing that it is possible or not possible [for person xl [at time til [at such and such costl to do [at time til .... 4 For some important philosophical purposes, it will be necessary not to prescind from consideration of what is to be supplied to any of these slots. But what matters for the Master Argument is that two time slots are provided, and that, as things stand in advance of putative linguistic reform, there are many true statements that represent an act as an act that someone can at some time ti do or perform [or that it is possible for them at some time ti to do or performl at a later time ti. That all such abilities to 0 have their foundation or vindication or cashable consummation in some ability at a time t to do some act (identical or not identical with the act of 0-ing) at the very same time t is a further claim That is what the principle (E) is concerned with. III

Suppose the kingdom is lost because the battie is lost. The possibilist or upholder of (C) says that on the day of the battle the kingdom could have been saved because the defenders could have won the battie. The actualist Oiodorus, who rejects (C) and is armed with (A) (B) (D) (E), might reply at this point: Do you mean that it was possible at that point for the defenders to prevail at that point? Or do you mean that, if all sorts of unfulfilled conditions had been satisfied, then it would have been possible at that point for the defenders to prevail? If you mean the first (poterant in Latin), I disagree. If you mean the second (potuissent in Latin), of course I

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agree. But the second is consistent with the denial of (C). What I deny and believe I can refute is the claim (C). The possibilist may be expected to reply: What I mean does not simply reduce to the second (potutssent as you interpret that). I mean that to win that battle on that day was at some point a real but unactualized possibility for the defenders. The actualist might reply: I don't know when it was a real possibility. Do you think that this unactualized possibility once existed because you think that it was possible on the day for the defenders to try harder on the day? Or do you think it existed because you think that earlier the defenders could have prepared better for the battle? Let us suppose that the possibilist replies that depending on further details he might think either or both of these things and that the disputants agree to examine each in rum. 5 The actualist may say: As for the defenders trying harder in the battle, there are two points. There is the ordinary point that nobody actually says they were faint-hearted or inconstant. And then there is an altogether different sort of pOint, which trumps the first one. Surely at each moment as it arrived they were doing what they were doing at that moment and it was too late then to do anything else then. See (D) and see (A). So at no time could they have acted differently from the way in which they acted at that time. Possibilist: I am not sure that either of your points really shows they could not have tried harder. Do you mean to say that they could not have tried harder even when they were trying hardly at all? But the most important thing I have to say is that, if what already holds at t must hold at t, then the real (actualized or unactualized) possibility for someone to do act A cannot be boiled down without residue to the simple possibility for him at the instant t to do act A at t. Actualist:

Let us concentrate on your second plea. Your

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second plea is that, as things were, the defenders could have won if they had prepared better; by which you mean that it was possible (not merely that it would have been possible) for them to win by preparing better. What that plea depends on, I suppose, is the idea that, things being what they were, it was possible for them earlier, by preparing better at an intervening time, to bring it about that on the day of the battle itself their later efforts would succeed. 6 Let us suppose that the possibilist agrees to this interpretation of the second plea. Then the actualist may be expected to continue as follows: If their winning the battle is conditional upon (or entails) their having prepared better, then the real possibility (pater-ant) of their winning depends on the real possibility (poter-ant) of their earlier preparing better. Recall principle (B). But if we accept that, then we must ask: was there any real possibility of their preparing better? Certainly that real possibility, if it ever existed, had disappeared by the time of the battle (by principle (A)). By the time of the battle they could not (non poterant) win the battle by virtue of having prepared better earlier. They had not prepared better. So now consider the time of their preparations. They were doing then that which they were doing then, viz. not preparing quite well enough. Recall principles (A) and (D). By the time of any given piece of preparation that they undertook, they were doing it as they were doing it, right down to putting in the last horse-shoe nail. So if you accept (B) and if "they could at the time of their preparations have brought it about that their later efforts would succeed" is to mean any more than "if their preparations had been better, then they would have been able (potuissent) to win", which is perfectly consistent with the denial of (C), then either you must deny (D) -- but can you deny this when you have accepted (A)? -- or, doubting (E), you must find the unactualized possibility of the defenders winning the battle in an irreducibly temporally stepped possibility, the possibility at an earlier time, t2 say, before the time of their actual preparations (for

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then it was too late) of their making the better preparations at t3 required to ensure their victory in the eventual battle. But I say that is like jam tomorrow. What they need and cannot have is the possibility at some time tj of making better preparations at ~. IV

There is of course no way of telling whether this imagined exchange between a possibilist and an actualist rehearses any of the actual arguments of Diodorus Cronus. We have to guess. But let us begin by noticing that, even though this reconstruction of the Master Argument redeploys materials that are familiar to us from Aristotle's sea battle discussion, it brings new and distinctive ideas into play. Diodorus' line of argument does not depend on bivalence or on any other questionable point about truth that Aristotle could have exercised the option to reject. What is more, the derivation never invokes a premise that depends on physical determinism. These are positive points. Yet the theoretical situation is not yet completely satisfactory. Surely (A) (B) (D) (E) are too many principles. In Vuillemin's own presentation, Diodorus' (A) seems to be upstaged by a principle Epictetus does not even mention, namely the extra premise (D), while Epictetus' (B) only figures as a principle for the assessment of the modal status of the Diodorean refutation of (C) (see op.cit., pp. 136-7). Admittedly, one can give (B) a more active role in the actualist's derivation or argument. I attempted this in my illustration. But what one must suspect - and Jonathan Barnes has already emphasized this 7 -- is that, in the presence of (D) and (E), (A) and (B) are simply superfluous. Yet (A) and (B) are all we know of Diodorus' argument. Whatever else we attempt by way of reconstruction, (A) and (B) must be preserved. The suspicion that (A) and (B) are in danger of becoming redundant may be confirmed. Suppose we formalize (C) (D) (E), letting capital letters from the beginning of the alphabet symbolize

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verbs that stand for acts and combine with subject terms like "John" (omitted here), with or without the auxiliary "do", and a time specification 't' to make a complete sentence, which can itself fall within the scope of temporally qualified modal operators such as 0t or 0t. (Where they do so fall, as in "Ot John bicycles at t''', we may make trial with a putative transformation and read the result as "It is possible at t for John to bicycle at t' ".) Then, given these conventions it appears to be demonstrable that (D) and (E) render (A) and (B) redundant for the disproof of (C). For what we then have is: (C)

(EA)(Et)(Et')(Ot At ,)&(t

~

t')&(t")(t

~

til -+ not A at t')]

Suppose (C)is true, and let A be an act that will never be done but which the possibilist claims is still possible. Then there must be times t, t' such that it is possible at t to do A at t'. Then either t - t' or t < t'. Suppose t - t'. But, given the stipulation about A, not A at t'. So, by t - t' and (D), t (not A at t'), in which case not 0t(At ,). Suppose then t < t'. But by (E), if (Ot(At then there must be a t} such that t ~ t} ~ t' and On (An). Meanwhile, though, we are already assured that A is never done at any time after t. But in that case, however t} is chosen, 0u (not A at t}). So (not 0n(A u»' So, however t and t' are chosen, it is not possible for it to hold that t ~ t' and Ot(At ,), where A is never done. So if (D) and (E) are true, (C) is not true.

°

,»,

V

Faced with this proof that, in the presence of (D) and (E), (A) and (B) are superfluous, our first remedy is surely to try to dispense with

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Vuillemin's (D) as an extra premise. Perhaps we can re-engage Epictetus' reports of (A) and (B) by using premises (A) and (B) to derive (D) - perhaps that was the first part of Diodorus' stunt - and then we can revert to Vuillemin's construal of Diodorus disproof of (C), using (E) as one extra assumption. Such a line of argument would both improve the fit with Epictetus, by showing how crucial (B) was to Diodorus' chain of reasoning, and also preserve the deductive route marked out by Vuillemin to the conclusion not-(C). Consider everything I am doing now at this instant. And consider also something I am not actually doing that the possibilist might say I can now do now. Read the newspaper, say. I am not reading a newspaper. If it be possible now for me to read a newspaper now, then by principle (B) whatever is implied by my now reading a newspaper is possible now. But then it is possible now for me to have done something different from what I did moments ago. But it is not possible now for me to have done something different then. 8 By principle (A), what happened then is necessary now. Therefore, generalizing from this case, I cannot now do anything different from what I am doing now. So (A)(B) 1-(0). That is the first leg of the reconstructed argument. The next goes as follows. If there are any unrealized possibilities, they cannot have the form (not At & not Cl tAt). The only other form for them to have is (not At & to S; t & OteAt). But, if anything of that form held, then, if (E) is true, there would have to be a time t} such that to S; t} S; t and Ot}A t}. But, given the possibilist stipulation about A, that would have to be a time such that not At}. And by (0), not At} ~ Clt} (not A t1). So for no t is the unrealized act of doing A at t possible either at t or at some to before t. That completes the Oiodorean argument. But more needs to be said about the derivation of Vuillemin's (0) at the first stage. Later CS VII) we can attend to the second stage and the difficulties of (E).

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Both (A)and (B) were at work in the deduction of (D). Of course they did not work unaided. The demonstration turned on the understanding of what reading a newspaper is, the thing I am not doing, and the understanding of what it is to do whatever I am actually doing now. One who knows what any of these acts is knows, for instance, that they do not occur at an instant or discontinuously with what precedes them. No doubt someone will ask where that last claim comes from. The answer is that the point might be put down as resulting from the (a posteriori but semantical) meaning rule for a particular verb, and a consequential generalization about how any act stands in relations of both synchronic and diachronic entailment and exclusion with certain other acts. It is important to add that the deduction of (D) does not exclude a measure of indeterminism in the world in which we speak of actions. For the point about the verbs in question to hold good, there simply has to be enough regularity in the world for us to be able to speak as we now do of the things that we do. We speak that way in the actual world without assurance that determinism (proper or full determinism) applies to it. VI

With this amendment and redeployment, Vuillemin's reconstruction of Diodorus invokes, apart from the meaning rules for verbs, just one extra premise, namely (E). And the Master Argument promotes actualism by using this principle to refute the natural seeming supposition that there are possible acts that will never happen or be done. Should we simply accept that? Is there really nothing wrong? What I think we can be reasonably sure of is that there is no incoherence in the temporally indexed modality that Diodorus' (A) depends upon. a tP is true if there is nothing to be done at t about whether or not p. It is fIxed by time t that p.

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As for Diodorus' (B), it is hard to follow Chrysippus, as Epictetus reports him, and find any fault with this principle or doubt its applicability to the temporal modalities. If there were a possible world like the actual world in respect of moments up to but not including t in which I read a newspaper at t, and if every world (a fortiori every world like the actual world in respect of moments up to but not necessarily including t) in which I read a newspaper at t is a world in which I have a newspaper in my hands at ta, appreciably before t, then surely there would have to be a possible world like the actual world in respect of all moments up to t and in which I have a newspaper in my hands at to' appreciably before t. So far, so good. Yet (D), which we derived from (A) and (B), is not so straightforward as it seems. First, there may be exceptions to the generalization of the argument in paragraph 3 of section V. (There seems to be the possibility of exceptions wherever a situation can lead indeterministically but without discontinuity into different or alternative sequel situations9). More importantly, there is a question of interpretation. Where t is an instant, it is hard to doubt that At ..... CI tA t. Indeed, so long as we confine our attention to instants of time and so long as the proper interpretation of "I cannot now do a different act from the one I am doing now" makes "now" the designation of an instant, (D) can appear irresistible. Where t is an instant, Aristotle's very statement of (D) comes close to convincing one of the principle. But that in itself should not instantly persuade us that whatever I do today I cannot help today doing today. That seems incredible (unless full physical determinism is being appealed to). For today isn't over yet. And understanding (D) in the natural way in terms of times that are not instants, we have not been given any argument to believe that we cannot help doing what we are actually doing today. What we must note is that, unlike 7:30 p.m. on November 14th, times such as today, tomorrow, next Thursday afternoon and even time such as

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now ... are not instants but stretches of time (or intervals, as some say). They are times in the ordinary sense, which is what we use to chronicle activities such as speaking or writing or reading a newspaper. Once this point appears, we may be tempted to cut matters short without even looking again at (E), and simply declare that what has lowered our resistance against Diodorus' picture of the world -the picture of a world shifting as if stroboscopically from a state of being frozen stiff in one way to a state of being frozen stiff in another way-and what will always tend to bewitch us in the face of arguments like Diodorus', is the suspect idea that today is nothing over and above a collection of instants 10; the suspect idea that any ability I have that is dated for today must repose upon a foundation of abilities each of which is dated to an instant; or the suspect idea that any process that is going on today must, in the last analysis, break down without residue into a multitude of events or actions each of which can be dated to some instant of today. We may then be tempted to urge that the sanitized language of instants and of capacities/states/acts instantaneously qualified is an abstraction from the more familiar language in which we are more truly at home, the language in which we have imperfective as well as perfective verbs, 'travel', 'run', 'sleep', 'work', as well as 'arrive', 'win', 'fall asleep', 'finish building the house'.l1 In this language, what imperfective verbs require (in the first instance, at least) is not punctual but stretch-like time indicators; and there is nothing to suggest that what we use the language of events and instants to describe is foundational of everything else. Unless (D) operates under a restriction to instants, we lack any reason to affirm it in its generality. VII

There is something important here, and we shall come back to it. But it is scarcely enough to make Diodorus' problem go away. For we are

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still left with the point that, if (D) is true in the restricted interpretation involving only instants, then whatever I start to do at t, I necessarily at t start to do at t. At instant t, it's too late to start to do anything other than whatever it is I do start to do at t. So we still face the conclusion that, if there are to be any unrealized possibilities, and if there are to be things I ,do but could have helped doing (in the sense of "poteram"), then these must depend on the existence of the temporally stepped possibility at t to do F later than t conjoined with temporally stepped possibility at t to do G later than t, where F-ing excludes G-ing. And this was the kind of possibility the actualist was poking fun at in section III, when we were discussing the idea that the defenders could have saved the kingdom because it was possible for them at various times before the battle began to prepare better for the battle and bring it about that their later efforts would succeed. Returning a second time to this point in the dialectic, one now finds it natural to speculate that Diodorus would not have found it difficult to make this kind of possibility seem completely vacuous (never jam today implies no jam, etc) and to do so without advance articulation of any general principle. Hence the absence of such a general principle from Epictetus' report. But hence also the great philosophical (albeit not necessarily strictly historical or philological interest) of trying to articulate some not incredible assumption that Diodorus was making and was able to count on his audience making. So I applaud what Vuillemin is trying to do with his (E). It is harder, however, to applaud his formulation of that prindple. 12 Not only is his (E) too awkward in expression to be anything we can imagine Diodorus resorting to. (E) is too simply and undeceptively false. Suppose it is possible at 4:10 p.m. to catch the 4:45 pm train from Paddington to Stroud. Then Vuillemin's principle (E) asserts that, if it is possible at 4:10 pm. for Tom to leave for Stroud at 4:45 on that day, then there will be an instant tl such that it is possible at t for Tom to leave Stroud on the train at t. (Note that here t has to be the same as

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4:45 p.m.). Is that necessarily true? Surely not. Of course, if Tom gets his suitcase shut, talks his mother in law off the telephone line and then finds a taxicab to take him to the station etc., then it will indeed be possible at 4:45 for him to leave by the Stroud train at 4:45 p.m. But what if Tom will fail in fact to do some one of these vital things? That need not alter the fact that it was possible at 4:10 p.m. for him to leave for Stroud at 4:45 p.m. If we want to find something that is both true and a bit like (E), perhaps it will have to be this: (EI). If it is possible at to for x to A at t, then there are things that can at some time tn happen or be done at tn' where to ~ tn ~ t, such that, if they happen or are done, then it will be possible at t for x to do A at t. But it is unclear that (EI) will serve Diodorus' purpose of deducing (not-C) from (D), where (D) has been got from (A) and (B). To try to get closer to Diodorus, we have to reconsider the task of discrediting the temporally stepped possibility that was recorded in the claim that before t3 (when this or that preparation was to take place) it was possible -- at tz, say -- for the defenders to prepare better at t3 for the battle that was due to take place later at tlO. What does this possibility itself amount to? Presumably it involves inter alia a modest measure of indeterminism and the thought that at tl and tz it is not fixed that the preparations at t3 will be inadequate nor yet is it flXed at tl and t2 that they will be adequate. That gets fixed later than t2, before or during the preparations themselves. What we are concerned with here are necessary not sufficient conditions. But the important thing is that, on the strength of the satisfaction of this indeterministic condition and any other condition that the possibilist wants to impose, we shall not say that the defenders could not help the badness of their preparation. Indeed, we shall say they could help it (poterant melius) because at some

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time before or during, the time of the preparations it was open to them (or so we shall say) to mount adequate preparations at t3, including no doubt the proper shoeing of every single horse. 13 The possibilist can say that. But it is easy to imagine that Diodorus will have found it easy to reply that the mere possibility earlier, at t 1, of making better preparations later, at t3, is insufficient ground for saying that the defenders could have helped the inadequacy of their preparations. One speculates that Diodorus will have pointed out that that is insufficient ground for saying that the defenders could really have helped the inadequacy of their preparations. And one speculates that Diodorus will have pointed out how hard it is to see at any particular moment any actual exercise of the freedom the defenders supposedly enjoyed to make better or worse preparations. As each moment comes up it is too late (he says) to do differently at that moment. What comfort is it to anyone if earlier it was not deterministically excluded that the later preparations should have been adequate? If that was the line of argument, then I suspect that the best one is likely to do in formulating the extra premise Diodorus needed is likely to be (E") If one can truly say something in the form 'x can help A-ing at t', then either x can at t help A-ing at t or there is a B such that XiS B-ing earlier than t would have enabled x to avoid A-ing and there is a time t', t' ~ t, such that x can at t do B and can at t' help B-ing at t'. Now I am not sure whether we should in the end accept (E"). But simply letting (E") stand uncriticized for the time being, I now revert to the comment that I made first about (D), after we had tried to deduce it from (A) and (B). This is the comment about perfective and imperfective verbs. If we note that "can" and "can help" are imperfective, and require not punctual but stretch-like temporal qualification, and if we take seriously times that are not instants, then it

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is surely much easier to satisfy (E") than it appears to be when our historical narrative is gratuitously chopped up into instants and then run stroboscopically in the language of states at an instant and achievements and non-achievements at an instant. Suppose we go back to tl and t2, before the preparations at t3 and the battle at tlO. Surely, despite anything Diodorus could really have shown, there is a time, namely this week (say), such that the defenders can this week help making defective preparations this week. In which case, one may persist in saying, it is still a real possibility for the battle to be won and the kingdom to be saved. Or so we can say if we will use the imperfective aspect to understand better what Diodorus as we reconstruct him would have wanted to belittle as a temporally stepped possibility. The dialectical situation is then this. Maybe whatever the defenders do or achieve or fail to achieve by way of preparation at instant t 3, it is necessary at t3 for them to achieve or fail to achieve at t3. But this does not count against the fact that they could have done differently and better at t3. And there is nothing comical or irrelevant in stressing this ability. For we can also say truly enough that, if that's how it was, then in that week including t1, t 2, t3, t4 it was possible for them to prepare fully adequately in that week for the battle that was to come later at tlO. If a relevant kind of ability must be temporally smooth (the foundational one), well, this one is smooth. What is more the imperfective is the right aspect in which to describe such an ability. 14 Diodorus cannot see the wood for the trees. He is too close up. And that is where he is determined that everyone else shall look from. The only aspect he will take seriously or permit anyone else to take seriously is the perfective. It is surely no aCcident, one now sees, that the author of the master argument was a thorough-going atomist who carried this position to the length of arguing, apparently seriously, that one can say "it has moved" but at the same time deny

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that "it is moving" was ever true, offering as a specious but desperate parallel the fact that Helen was married to three husbands is true while "Helen is married to three husbands" was never true!15 VIII

A summary of what has happened may be useful. I replaced Vuillemin's disproof of (C) on the basis of (A) (B) (D) (E) by a longer chain of argument intended to make use of Epictetus' (A) (B) to derive (D). I offered that as the first leg of Diodorus' argument, but later, in criticism, I only allowed (D) to stand so long as its interpretation was restricted to instants. I reconstructed the second leg of Diodorus' argument as an intuitive demonstration of the absurdity of counting the possibility at t2 of shoeing every single horse properly at t 3, as a real possibility of shoeing every single horse properly at t3' Vuillemin's (E) is intended to bring out that absurdity. Then, in criticism I commented that there was a problem about this Diodorean demonstration of absurdity. For (E) is false, its natural replacement (E') seems insufficient for Diodorus' purpose, and its other replacement (E") calls clamantly for an interpretation in terms of states and intervals or stretches. Once that other interpretation is supplied, what the reconstructed Diodorus calls a temporally stepped (or jam-tomorrow) possibility admits of description as a temporally smooth possibility--a possibility that is both real and unproblematic, once you get your nose further away from the picture frame and look from the right distance to try to find what Diodorus himself claimed to disprove the existence of. X

Diodorus Cronus, in the fresh reconstruction that we owe to the philosophical enterprise of Jules Vuillemin, deserves our thanks not (I still believe) for proving by logic the truth of fatalism or determinism or discrediting our ordinary ideas about freedom, but for a striking

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and signal demonstration of how indispensable certain descriptions, the descriptions from which science and exact measurement abstract, must always be to any view of the world that shall find in it genuine process, time, change, causality or action. To seek to understand the world as actually containing these things, we must take continuous and imperfective verbs for what they are, as irreducible to punctual or perfective vocabulary; and we must descry within our empirical experience not only events but also continuous and irreducible states and processes, not only instants but also intervals -or (as I find it so natural to say) times. The necessity I have just claimed is of course only a conditional necessity. I have not proved that there is any categorical metaphysical duty to take imperfective verbs non-punctually qualified as irreducible. At best I have shown how we have to take them as irreducible if we want to find change, process etc .. This can hardly be the moment for me to speculate what it would take properly to license the detachment of the 'have to' from the 'if. I only reiterate how strongly it appears that the alternative to taking the imperfective or progressive verb seriously is to embrace the staccato, homme machine necessity of the Diodorean argument. Professor of Philosophy. Birkbeck College-University of London

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Notes 1. In writing and rewriting this essay I have had the benefit of comment and encouragement from various people who have heard or read it in some form, most notably Jules Vuillemin, Jonathan Barnes (see also notes 7 and 12), John Ackrill, Gerald Cohen and Myles Burnyeat.

2. See Jules Vuillemin, Necessite ou Contingence: l'Aporie de Diodore et les Systemes Philosophiques (Editions de Minuit , Paris 1984), pp.32-39ff., 50ff. 3. Cpo Jennifer Hornsby, 'Which Physical Events are Mental Events?, Proceedings of the Aristotelian Society 1980, ad init; Richard L.Cartwright, Philosophical Essays (MIT Bradford Cambridge Mass. 1987) p.36. A related possibility would be lekta as the Stoics understood them. But to appeal to these would import a strain of anachronism, perhaps. 4. Cpo Vuillemin, p.34; David Wiggins, 'Towards a Reasonable Libertarianism', Essays on Freedom and Action. ed. Ted Honderich (London: Routledge & Kegan Paul 1973), § III, to be reprinted (with the correction of an error in the first reprinting) in Needs, Values, Truth (2nd edition, Oxford: Blackwell 1990-91). See § 4 there. Note that if the second time specification is absorbed into the act deSCription, what results is still the denomination of an act (something general), but one whose instances are then limited to the specified time. The idea of cost is important for many purposes. But more traditional concerns are restored if we fix this parameter so that the incompatibilist's "cannot" means "cannot at any cost".

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5. Lest there be difficulty in finding any agreed background for the dispute or we run out of historical knowledge, let it stan out from this:

For the want of a nail, the shoe was lost; For the want of a shoe, the horse was lost; For the want of a horse, the rider was lost; For the want of a rider, the battle was lost; For the want of a battle, the kingdom was lost; And all for want of a horseshoe-nail. (Benjamin Franklin: Poor Richard) (It was amiably pointed out to me by Barnes that I am not the first to refer to this rhyme. See for instance A.N.Prior, Time & Tense(Oxford 1968), pp.54-55.) 6. Cpo Rod Bertolet and William L Rowe, 'The Fatalism of "Diodorus Cronus"', Analysis 39, 1970, p.130, who write, referring to [Diodorus Cronusl 'Time, Truth and Ability', Analysis 25, 1965, that "[Diodorus Cronusl mistakenly infers from 'Stilpo is unable to do something at t1' that is logically sufficient for the falsity of 'Stilpo walks through the Diomean Gate at t2' that Stilpo is unable at tl to do something logically sufficient for the falsity of 'Stilpo walks through the Diomean Gate at t 2"·

7. Classical ReviewXXXVI no. 1 (1986), pp.70-79.

8. For a challenging attempt to domesticate this kind of claim, see Martin Davies, "Boethius and others on Divine Foreknowledge", Pacific Philosophical Quarterly 64 (983), especially pages 319,322. I hope that those who understand Chrysippus better than I think I do will study the possibility of interpreting him in the light of Davies' idea.

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9. Suppose there is an indeterministic process which, for all that is fixed by its nature or its content, can at t issue in different outcomes after t. At t, the newspaper is right by my hand, say. It mayor may not end up in my hands. And suppose now is arbitrarily close to t, so that the principle (A) does not suffice to necessitate that which happens at t. (There are subtle questions here, on which see also the next paragraph.) 10. Cpo the position apparently taken by Chrysippus in Stobaeus 1 142, 2-6, translated as follows in A A.Long and D.N. Sedley, The Hellenistic Philosophers I (Cambridge 1987), page 297. "... bodies are divided to infinity, and likewise things comparable to bodies such as surface, line, place, void and time. But although these are divided to infinity, a body does not consist of infinitely many bodies, and the same applies to surface, line, and place .... " (My italics.) 11. The terms "perfective" and "imperfective" begin life in grammars of explicitly aspectual languages like Russian. There is a rough and ready equivalence between "imperfective" and "progressive" in the usage of theoretical linguistics. For the usage of Russian grammarians, see B.O.Unbegaun, Russian Grammar(Oxford 1957), chapter XII. 12 . Here again I record a debt to Barnes. 13. Cpo again Bertolet and Rowe tited note 6. 14. Never content with anything short of everything, some will now ask what it would really be like to be able now to do otherwise now. My own answer is an incompatibilist one. Let now be a stretch, not an instant. Then my ability now to do A now when I am actually doing B consists not in my capacity at this instant to do something different at this instant but in the fact (if it is a fact and I really can now do now

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each of A and B) that within the stretch of time designated as now there is a juncture at which the event consisting of my shortly starting to do B is possible, and at which the event consisting of my shortly starting to do A is possible. That requires the falsity of universal determinism. But the falsity of that general thesis is consistent both with "every event has a cause" and with the simple exclusion principle between acts that was used in V for the derivation of (D) from (A) and (B). And surely this gives what the possibilist wants. For if the situation is as described and there is the said juncture, then in virtue of all this obtaining I can count this afternoon as able this afternoon to do A this afternoon and able this afternoon to do B this afternoon. I suppose I have here to reiterate the view (op.cit. note 4; Needs, Values, Truth. p. 290ft) that it will not follow from such a description of the situation, unless by an argument that simply begs the question (e.g. the assumption that "not physically determined" means or implies "random"), that my doing A, when that is what happens, is random. It is consistent with the denial of universal determinism that, when I do A, I can have had my reasons: and that my having those reasons can explain why I did A and, on occasion, rationalize my doing A. 15. Cpo Long and Sedley, op.dt., page 51.

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Jules Vuillemin The recipient of this volume d 'hom mage warmly thanks the authors who have done him honor. He has read their contributions with pleasure, interest, and profit. In vain has he tried to respond to each one. The diversity of subjects and methods, and the precision which is required these days in philosophical analysis, have reduced his response to a dry and adventurous summary which would not at all help the reader, already sufficiently sustained by the intrinsic interest of these essays. I will therefore limit myself to several reflections having to do with questions posed by Gordon Brittan, Gilles Gaston Granger, and David Wiggins. Although Patrick Suppes' study does not refer to my work, these reflections will be seen to lead me to hunt briefly on his own ground. 1. Considered by themselves and without regard to the requirements of the orders of discovery and exposition, the methods of analysis and synthesis are entirely symmetrical. It will be said that a theological proof a posteriori begins with the fact of motion and explains this fact by a Prime Mover, although the existence of motion cannot be deduced from the Prime Mover. The objection is not pertinent. Analysis poses the fact of motion and the principle of causality; synthesis refinds the final production of the effect under the assumption, always necessary, of the same fact. The illusion of an asymmetry between these two methods arises from the circumstance that the synthesis of the ancients is compared with the analysis of the moderns. The analysis of the moderns contains more than the synthesis of the ancients; it suffices, however, to develop this analysis synthetically, by deducing the consequences from the basic principles instead of ascending to the prinCiples from the consequences in order to restore the symmetry. 207

G. G. Brittan (ed.), Causality, Method, and Modality © Kluwer Academic Publishers 1991

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One might nevertheless subordinate these two methods to supplementary conditions. Such subordination, as Gordon Brittan rightly says, is explicit in Descartes and in Kant. In the case of Descartes, analysis and synthesis are valid only in going from clear and distinct ideas to clear and distinct ideas. In mathematics this condition means that the theory of proportions is the whole of geometry, from which it is necessary to exclude mechanical constructions involving the infinite. This explains the disparity between Descartes' Geometrie and the solutions to problems given only in his correspondence. Kant goes further than Descartes, by reinforcing the conditions imposed on the two methods. The "I think" can, according to him, produce objective synthesis only by constructing clear and distinct ideas in the sensible (and not purely intellectual) intuitions of space and time. This explains the restrictions which Kant imposes on geometry and arithmetic. The algebra of his time obliged him, although it is a very accessory part of his system, to make room for symbolic constructions; Brittan judiciously analyses what renders these constructions possible. A risk of confusing Kant and Leibniz would arise if one thought that the demonstration of the consistency of an uninterpreted symbolism is itself enough; in effect, what legitimizes reductio proofs in mathematics is the constructibility of concepts in sensible intuition, according to the transcendental doctrine of method. Without the irreducible given of these intuitions, the symbolic constructions of algebra would remain chimerical. Descartes and Kant are both opposed to Leibniz. The only limitation which Leibniz imposes on mathematical and philosophical concepts is consistency. For Descartes and Kant, this limitation is notoriously inadequate. To analysis and synthesis they join constructibility, characteristic of mathematical as well as philosophical intuitionism, Kant restricting what is to count as intuition more rigorously than Descartes.

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2. We have just associated mathematics and philosophy and both in tum with analysis and synthesis. Synthesis is nothing other than the axiomatic method. Let us recall for the benefit of those whom these words upset that Descartes and Spinolza set their metaphysics out synthetically and that the Critique of Practical Reason begins with definitions, theorems, and problems. In mathematics three types of axiomatic development are distinguished: material axiomatics (Euclid's geometry, Newton's Principia), formal axiomatics (Aristotle's syllogistic, Book V of Euclid's Elements), and formalized axiomatics (Stoic syllogistic? A fourth type, which does not use Hilbert's Geometry). conventionally defined symbols and restricts itself to ordinary language, preceded them. Numerous Aristotelian texts belong to this type, for example the exposition and refutation of the arguments of Zeno of Elea in the Physics. it is the same for the Master Argument related by Epictetus. The questions which Gilles Gaston Granger poses a propos of these distinctions are fundamental. The status which must be ascribed to philosophy depends on the answer given to them: does the validity of the systems in terms of which philosophy is expressed have to do, at least partially, with truth? The history of "classical" mathematics presents a development which passes inexorably from the first to the second and even to the third type of axiomatic; it is in this sense that one says that mathematicians don't know what they are talking about, a claim already implied by the speech of Socrates in the 1beaetetus. Intuitionists, however, have and always will oppose this inexorable development.

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In the natural sciences, there are numerous formal developments. Even the most unlike phenomena come to be placed under a single equation. Nevertheless, if this commonality of form allows for the transposing of solutions, it is also necessary, when all is said and done, to come back to the specific character of the phenomena studied and it is not permitted in physics to pass beyond the stage of material axiomatics. This subjection to content is still more rigorous in philosophy, which generally does not go beyond the limits of ordinary language. When it does go beyond them, it is either because a discipline detaches itself to become an autonomous science or, as Granger says, in order to make vague notions precise, as for example the formulation of the Master Argument by Epictetus requires. The incompatibility between axioms shows itself, in effect, only when their sense is fixed unambiguously by symbolic conventions, that is to say, when they are translated into formulas. Once the translation is made, the sense of a notion - here of the possible - not only is made precise, but it depends on the set of axioms which determine its use. Depending on the choice made between incompatible axioms for the Master Argument, a different meaning will be given to the word "pOSSible. "It is in this sense, in the note cited by Granger, that I have spoken of implicit definition, not in the sense where, in formalized set theory, one defines what a set is by the operations which the system of axioms permits. (The material variations authorized by the notion of the possible, diachronic as well as synchronic, are moreover a priori limited by the intuitive content intended by the symbol t of the temporal parameter, with the possible exception of the system of eternal recurrence). Such a symbolic translation will be criticized for its excessive precision. But such is the price to pay when one analyses a thought by stripping it of the ambiguities in ordinary language. On the other hand, I do not see why a material axiornatics susceptible of expressing a philosophy - whether this be in an imprecise natural language or,

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locally, in a symbolic translation - would not involve considerations of truth. Not only is it natural and inevitable to ask if a particular translation of the Master Argument is true or false - the first version I gave of it was false since it did not prove the desired incompatibility of the premises - but as the claim made here by David Wiggins itself testifies, an incompatibility between the axioms compels the incrimination of at least one of them and shows its falsity. I understand that this claim can bear on principles so general that it undermines the truths recognized as scientific only after a long detour. But without it what would be the point ofaxiomatics? It is the validity of certain mathematical or logical principles in terms of truth which the intuitionist doubts of a Descartes or a Kant, of a Kronecker or a Poincare, point to. It is true, and it is thus that I interpret the reservations of Granger, that to this degree of generality, when one appeals to the whole of mathematics or to the totality of experience in order to verify or falsify principles, the words "true" and "false" lose the precise meaning which they have in mathematical demonstration and in experimental and even historical confirmation. Thus its essential rapport with truth does not suffice to transform a philosophical system into a scientific discipline. 3. Let us test these assertions with the example of the Master Argument.In order to interpret the three premises of Epictetus cited by David Wiggins, I had appealed to the text of Aristotle's De Caelo. The proposed reconstruction, however, adding to Aristotle a premise (E) and superposing on the premises with a double temporal index (A,C,D,E) a premise of pure modal logic (B), did not demonstrate what it should have demonstrated, to wit, logical incompatibility between the premises. I have since, without violating the letter of De Caelo, and holding to the premises with a double temporal index (E, slightly modified, taking the place of B), re-established a demonstration of incompatibility. These premises are as follows: 2

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First, three remarks a propos of these axioms. 10/ Epictetus' text text simply asserts their incompatibility. It is Diodorus (not I) who infers the falsity of (C). Others argue against (A) (Cleanthes), (B) (Chrysippus). Plato contests (D). 2°/ Wiggins tries to derive (D) from the conjunction of (A) with the axiom (1) L(p::)

cD ::) (Lp ::) Lq)

(which does the duty of (B) in my first formulation). I don't see that this derivation is formally possible: (A) stipulates that t < N. 3°/ Wiggins, following M. Barnes, thinks that (B) is so clearly false that it loses all plausibility. It could be, in effect, that that which we now think possible in t is revealed to be no longer possible in 1. In this case there is no contraction of the two indices of the modality, contrary to what (B) affirms. But let us go into detail. I proposed explicating a text of Aristotle (De Cae/o, I, 283 b 617). What does this text say? That it is excluded for a certain possibility to produce an impossibility by its actualization. The possibility in question is a possibility with double temporal index that for Simplicity one might note "MNPt", where N indicates the actual instant,Nunc, and t an indeterminate instant. What Aristotle excludes

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is that in realizing itself, and it will be able to realize itself only in contracting to a given moment, Mtpt, it produces an impossibility, that is to say before Mtpt. Aristotle utilizes, without distinguishing them formally, two principles of modal logic corresponding to the formula of Epictetus: "From the possible the impossible does not follow (logically). "The first has nothing to do with time. It is written in the form (l) or in the equivalent forms: L(p :::> q) :::> (Mp :::> Mq) or again (2) L(p:::> q) :::> - (Mp . - Mq). One will take p, for example, as: "d is the diagonal of the square with side.a - 1," and, for q. "there do not exist whole numbers!. and.§. such that

.d. - r. ." a

s

(De Cae/o, 281 b15; Anal. pr., I, 13, 32 aI8).

It is remarkable, however, that Aristotle illustrates this modal principle by referring equally to propositions which are clearly dated (De Cae/o, 281 bIO-20). Thus, for 12.: "a given person is standing" and

for g:"this same person is seated. "It is clearly possible for the person, assumed to be in good health, to sit (later) when he is standing and to stand (later) when he is seated. There is then, here, reason to flank the modal propositions with two indices, of which the first chosen will be N. Since it is not possible to be simultaneously standing and Sitting, one can put - Pt in place of qt for the same instant t. It is then seen that -MN (Pt· - Pt), while MNPt· MN - Pt indicates a compatible state of affairs, the variable t referring to eventually different instants in the two constitutive propositions of the conjunction.

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The question is thus to know how to express the Aristotelian exclusion of the state which will lead from a temporal possible with double index to an impossible consequence. In his critique of the Megarians, who confuse potency with act, Aristotle writes

(Metaphysics,

e, 3, 1047 a23-24):

a being "may be capable of walking

and yet not walk, or capable of not walking and yet walk, ,,3 thus, MNPt· -PN' MN - Pt· PN' He adds (24-25): "A thing is capable of doing something if there will be nothing impossible in its having the actuality of that of which it is said to have the capacity. "There would be impossibility in the first case if MNPt was actualized in No for one would then have MNPN' PN and (by D) PN - PN' in the second case if MN - Pt was actualized in N, for one would then have MN - PN· PN and (by D) - PN • PN • The same impossibilities would be produced from the premises MNPt • - Pt ' MN - Pt • Pt (which correspond exactly to what is given in the text of De Caelo), if one actualized in the first case MNP t in Mtpt in the instant where - Pt and in the second case MN - Pt in Mt - Pt in the instant where Pt' More generally, and without having to invoke (D) further than to explicate the state assumed to prevail at the instant of contraction, being given a possible MNPt> this possible rules out that in contracting (Mtp t) it produces an impossible ( - Mtp t). In the general case, the instant of contraction remains indeterminate or is submitted only to a clause fixing the lapse of time in which the possible must contract. What is excluded is thus the impossibility, at any intermediary moment, of contraction. And this is what (B) says. Let us follow Wiggins and deny (B). The direct consequence of the falsity of (B), (- B), is as follows: (3t) - LN - Pt • (tl) (N :5; t1 :5; t ::) - Mt1Ptl).

There is thus an instant 1 such that it is now possible that LL in 1, although at all instants from now until 1 it would be impossible to

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contract this possibility. There are two limiting cases. If N = t, a contradiction is obtained. The other limiting case is that where the diachronically possible stipulates that the contraction is expected at a particular instant fixed in advance. But, in order for the event in question to be an authentic possible and not a word devoid of sense, it is further necessary, in the eyes of Diodorus and Aristotle, that the impossibility of the contraction at the instant fixed is excluded and that the various hindrances which might intervene are ruled out in advance, something we sometimes express by explicitly associating the assertion of conditions of non-hindrance with the assertion of the diachronically possible. Aristotle mentions these conditions: "To add the qualification 'if nothing external prevents it' is not further necessary, for it has the potency on the terms on which this is a potency of acting, and it is this not in all circumstances but on certain conditions, among which will be the exclusion of external hindrances; for these are barred by some of the positive qualifications."

(Metaphysics.

e

5, l048 a 17-21).

In order for a possible to be

authenticated as such, it is necessary that the exterior hindrances to its realization are put aside. It is because he accepts them rather than putting them aside that Wiggins falsifies (B) so easily. Here we have Aristotle condemned for the first time in the name of some suggestions of natural language. It happens a second time, on the same grounds, when Wiggins contests the axiom of conditional necessity. This axiom, he says,eliminates the consideration of that alone on the basis of which we act, the always slightly extended temporal present, in favor of such abstractions as spatial and temporal points, and the instantaneous velocities and momenta of kinematics and dynamics. Such chimeras stand in the way of the experience of freedom. Henri Bergson and William James, in the name of metaphysics and psychology, and often taking themselves as critics of everyday logic and language, had defended a similar idea, which Gaston Bachelard attacked, while sharing their point of view.

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Is the testimony of natural language in favor of this idea more solid? It has the merit of providing the philosopher with a simple decision procedure and of keeping morality independent of scientific conflicts, But how can we believe that this testimony on the modalities is coherent or even univocal, when it allows philosophers to express such contrary opinions on the subject? Aristotle, to whom we owe the distinction between limits and parts of time, did not expect to draw out of it, as has Wiggins, the solution to the problem of freedom. He intended it rather to extract us from the labyrinth of the continuum. To safeguard freedom, he attacked nothing less than the principle of bivalence. But enough of Aristotle. What separates Wiggins and me is not so much the acceptance or rejection of particular axioms as the method on the basis of which we do it. The history of philosophy has persuaded me that there are irreducible conflicts between human beings concerning first principles: questions about continuity and freedom are not grammatical questions, they are real, in such a way that our conception of physics very naturally invites us to accept a particular response or to doubt it. Let us show this in connection with Diodorus' aporia. As Wiggins remarks, this aporia presupposes nothing about determinism or causality. It should thus apply, in complete generality, to actions and events, and, among the events, to those which are certain as well as to those we take as contingent and to which we can assign no more than a probability. In this sense, it expresses the most abstract conditions with respect to which we can think of motion or rather change. Axiom (A) is a retrodiction, Axiom (C) is a prediction. Together they fix the initial and final conditions of change, such as common sense perceives it. The two remaining axioms are intended to assign the state of change to intermediary instants. The contraction axiom (B)evokes in certain respects the principle of superposition

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which allows for the description of the state of a physical system, since the possibles, in themselves not observable, must be capable of being superposed, their synchronic contraction not menacing directly their superposition (which is prohibited only by the effect of (D)). Finally, the axiom of hypothetical necessity (D), effectuates the "reduction" of the modal into the real and puts us in the presence of the "proper" value confirmed by experience. We have seen linguistic analysis take the extremal axioms (A) and (C) as evident, and raise doubts about the conjunction of (B) and (D) taken to entail determinism. Physical analysis will overthrow these appearances. In order to simplify the problem to the maximum, let us make use of Young's two-slit set-up, which quantum mechanics uses as an appropriate thought-experiment to illustrate the inadequacy of classical mechanics. This set-up includes (1) a source of projectiles or of waves (too imprecise in the case of projectiles to control angular dispersion), (2) a screen pierced with two slits Oland 0 1 conveniently placed, (3) a detector of the number of projectiles (we suppose that those which pass through the slits arrive whole) or of the intensity of the wave (the square of its height for a water wave).4 This is an elementary model with which to test the Master Argument: there are two possibles only (to pass by 0 1 or to pass by O2); by slowing down the firing of the projectiles, one could follow a unique event and avoid the misleading suggestions which are present when contingency is generically defined by a type of event which sometimes takes place and sometimes does not. Instead of an infinite time, we reason on the basis of a lapse of time limited by the emission in t1, the detection in t2 , and the passage through a slit in an intermediary instant. Even if it presents the remarkable and paradoxical phenomenon of interference, since in being added the peaks and valleys are annulled, a phenomenon probably guessed at by Posidonious, the propagation of a wave does not pose any particular problem for physics. The propagation is continuous and not local.

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The wave passes through the two slits and that is why there is interference. The analogue of a choice between two possibles which exclude one another does not obtain. The resulting intensity, 112, measured at the detector, does not reduce to the sum of the intensities I} and 12 which would obtain respectively if one closed first the slit 12, then the slit I}. It contains an interference term which depends on the phase difference between the two waves diffracted in and 02' Let us now use classical corpuscles. Suppose that one of them is detected at its arrival at the point ~ of the detector. Thanks to experiments made earlier on projectiles, the probability p} for a projectile to arrive at ~ in passing by 01 and the probability P 2 to arrive at the same point in passing by 02 have been determined, the total probability for a projectile to arrive at ~ in passing either through 01 or 02 being the sum of p} and P 2 . Actually, this probability, having become a certitude, is equal to 1. Moreover, even if we have not observed through which slit the projectile has passed, we can say, once the event has taken place, that the probability that it has passed through 1 is 1 and in this case the probability that it has passed through 02 is zero, or vice versa. This assumption corresponds to the axiom (A). Let us suppose that the projectile has passed through 01. Whatever might have been its state of motion before leaving 01 ' we can be sure that it is at the latest at the instant of this passage that its trajectory has been determined. It is at this instant 1 at the latest that the diachronically possible: "It is possible in t} that the object in motion will pass in t through 1" is transformed into the synchronically possible: "It is possible in 1 that the object in motion passes in t through 1'" in conformity with axiom (B). The contraction of the modality is besides immediately accompanied by the realization; the object in motion passes in t through 01 in conformity with (D). But, given this, is the possible which is not realized: "It is possible in t} that the projectile. passes in t through 2" an authentic possible? Let us call "authentic possible" a possible 01

°

°

°

°

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219

which is such in virtue of the nature of things, and not only of our ignorance. We have attributed to the projectile, at the instant t1, a non-zero probability of passing through 02' But we have no reason to think that between t1 and t any change has taken place in the state of the object in motion; our set-up has been made carefully enough to avoid any influence which might alter the direction of the trajectory between the source and the screen. Except by attributing to the projectile an unintelligible clinamen (the Epicurean "swerve"), we have no reason for attributing to it at the instant t1 a nonzero probability to pass through 02 or a probability different from 1 to pass through 01' other than our ignorance of the initial conditions. Thus a possible which is not realized is not an authentic possible. It is necessary, with Diodorus, to abandon axiom (C). Classical mechanics adopts the solution of Diodorus. Everything changes with quantum mechanics. With electrons as projectiles, experience gives the following results: If a measurement is made at the moment when the electron passes through one of the slits and thus permits determining the slit through which the electron passes, one is led to the case of classical projectiles: it is true to affirm that the electron passes through one of the slits to the exclusion of the other, there is no interference, and the total probability of the electron arriving in x on the detector is equal to the sum of the probabilities that it arrived there in passing through 1 and that it arrived there in passing through 02' We add the probabilities and have the situation of statistical mechanics (of statistical mixture); probability is due only to our ignorance. But let us give up measurements designed to inform us about the passage slit. Then interference is re-established. The probability of the electron being detected in x is no longer the sum of its probability of arriving there by way of 01 and of its probability of arriving there by way of 02' A "crossed" term is added to the sum, which comes from the fact that one no longer sums the probabilities, but the amplitudes of probability. By the same token it is

°

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impossible, once the electron has been detected in x, to say that it has passed through 01 or that it has passed through 02 The interference term falsifies axiom (A) and the disjunction at time tz: the electron has passed through 01 with a probability of 1, in which case it has passed through 02 with a probability of zero, or the electron has passed through 1 with a probability of zero, in which case it has passed through 02 with a probability of 1. This interference term equally falsifies axiom (- C): the two possible paths cease to be exclusive, there is no longer room to reject possibles which will never be realized. The Master Argument renders the compossibility of two simultaneous contrary events impossible. In effect, in virtue of (B), there is an instant t of contraction where one would have at the same time Mtpt and Mt - Pt and, in virtue of (D), the realization of these synchronic possibles would produce the contradiction Pt and - Pt. That is why everything is decided at the latest at the instant of passage through one of the slits. In classical mechaniCS, everything happens as if at each instant of time one measured or could measure where the projectile passed. In quantum mechanics, to the contrary, in the absence of measurement the general state of motion is a state of superposition and this superposition in no way implies diachrony. At the moment of passage through the slits of the screen, in t, one may then say that there is an amplitude of positive probability for the electron to be in the process of crossing 01 and equally for it to be in the process of crossing 02' even though these possibles are contrary. All that quantum mechanics requires, then, is that one does not make a measurement which would transform a state of superposition into a proper state. Axiom (B) is thus truistically confirmed, on the supposition that no measurement is made. We can even reinforce and extend it to all of the intermediary instants of the two diachronic boundaries. What binds the axioms (B) and (D) in the case of Diodorus and in classical mechanics, as well as in the case of Wiggins and linguistic

°

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analysis, and what hinders the superposition of two synchronic and contrary possibles, is the implicit principle of the observability at all moments of every physical magnitude. Quantum mechanics, to the contrary, in associating measurement with the selection of a proper value of the observable, subordinates to the effective act of measurement the reduction of the wave packet corresponding to the collapse of the modalities. If the measurement takes place only in ~ (detection) and not at the moment of the passage of the slits, this measurement will register the presence of the interference effect. Axiom (D) is only verified by the act of measurement. So long as this explicit act is lacking, axiom (B) is verified without the superposed contrary possibles risking collapse and producing a contradiction, to be avoided only at the cost of (B). The following table will help the reader to compare the three pOSitions that have just been described. D10DORUS

A

WIGGINS yes

B

no

(classical mechanics) yes yes

C D

yes

no

AXIOMS

no

yes (B and D) :::> determinism

antum mechanics

no yes yes yes, if measured - [(B and D) :::> determinism)

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4. By brilliantly dissociating predictability and determinism, Patrick Suppes shows - with the possible exception of quantum mechanics how mechanical instability grounds the practical and theoretical indifference of the choice between determinism and indeterminism. As regards quantum mechanics, the experimental evidence in terms of probabilities is too weak to legitimate a choice. Science, in these two cases, imposes skepticism. Probabilistic experience legitimates indifference with respect to two conditions. In the first place, when a deterministic system of equations and the experimental set-up which illustrates it lead to chaos by an evolution depending eventually on critical values assigned to the parameters, is it possible to describe what order remains in the chaos in making abstraction from the determinist hypotheses at the outset? In the second place, can one simulate and express in scientifically acceptable determinist terms5 all of the probabilistic consequences which orthodox quantum mechanics impliesl> Whatever response must be given to these questions in physics, skepticism, understood in the sense of Suppes' probabilism, appears to offer a third possible solution to the Master Argument, given the poverty or rather the imprecision of the psychological facts concerning freedom, if there exist such facts. Confronted with a decision which he must make, an agent might reasonably shrug his shoulders and say: "Diodorus or Heisenberg, what's the importance?" Except if the belief that there are possibles which are not realized was found to furnish our will, if not with a determining motive, then at least with a hope practically grounded and suitable to support our efforts.

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Notes I add these words in order to delimit the part of philosophy which has to do with epistemology, insofar as it is distinguished, for example, from ethics. 1

2For the old, faulty version, see J. Vuillemin, Necessite et Contingence(Les Editions de Minuit, 1984), chapter 2, pp. 27-57. For the new version, see J. Vuillemin, "Zur Rekonstruktion des Meisterschlusses, Antwort an Helmut Angstl. "Allgemeine Zeitschrift jur Phi/osophie, Frommann Holzboog, 113, 1986. The premise (A) has another form in De Cae/o, whose logical equivalence to the present expression is easily shown. On the other hand, in the text cited from De Cae/o, the premise (D) is expressed by an incidental clause contained in the premise (B). 3rhis and subsequent quotations of the Metaphysics are from the translation by W D. Ross. 4See, for example, R. Feynman, R. Leighton, M. Sands, The Feynman Lectures on Physics (Addison-Wesley, 1965), v. II, p. 1; Cl. Cohen-Tannoudji, B. Diu, F. Laloe, Mecanique Quantique (Hermann, 1973), v. I, pp. 11-15,51-53,255-259 51 understand by this: "responding intuitively to that which one expects of determinism"; as does, for example, Einsteinian locality. 6c. Vidal concludes the excellent book of P. Berge, Y. Posneau, and

L'ordre dans /e chaos, Vers une approche deterministe de la turbulence (Hermann, 1984) by assigning respectively to

C. Vidal,

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determinist chaos and quantum indeterminacy the role of limiting principles of physics in the macroscopic and microscopic orders. • This third section responds to a version of Wiggin's article which has been slightly altered. But since the causes of our disagreements have not changed, I did not think it necessary to change my response. The first of these disagreements concerns the economy of Diodorus' axioms; it results in part from a misunderstanding, my critique not having taken account of my retractio (see note 2). The second disagreement concerns the original axiom E, for which Wiggins substitutes the forms E' and E", without having reflected on the clauses of external hindrance assumed by the Ancients in the notion of the possible. The third, fundamental disagreement is of a philosophical kind: in order to found the analysis, one of us reverts to the suggestions of natural language, the other of us criticizes them in the name of scientific language.

PUBLICATIONS OF JULES VUILLEMIN

Avec L. Guillerrnit : Ie sens du destin, Neuchatel, Ed. La Baconiere, 1948.

Essai sur fa signyu:ation de

fa

mort, Paris, P.U.F., 1948.

L'etre et Ie travail, les conditions dialectiques de la psycho\ogie et de la sociologie, Paris, P.u.F., 1949. Traduction en espagnol, EI ser y el trabajo, 1961. L'beritage kantien et fa revolution copernicienne, Fichte, Cohen, Heidegger, Paris, P.U.F., 1954. Physique et metaphysique kantiennes, Paris, P.U.F., 1955; 1987. Mathbnatiques et metaphysiques chez Descartes, Paris, P.U.F., 1960 ; 1987. Introduction a fa philosophie de I'algebre, Paris, P.U.F., Tome I, 1962, "Recherches sur quelques concepts et methode de l'Algebre Modeme" Tome 2, 1963, "Structure, infini, ordre". Ie miroir de Venise, Paris, Julliard, 1965. De fa log;que ala theologie, cinq etudes sur Aristote, Paris, Flammarion, 1967. Iefon sur fa premiere philosophie de Russell, Paris, A. Colin, 1968. I, Paris, Fayard, 1968. Reblltir /'Universite, Paris, Fayard, 1968. Ie Dieu d'Anselme et les apparences de la raison, Paris, Aubier, 1971. La logique et Ie monde sensible, etude sur les theories

contemporaines de l'Abstraction, Paris, Flammarion, 1971.

Necessite ou contingence, I'aporie de Diodore et les systemes philosophiques, Paris, Les Editions de Minuit, 1984. What are Philosophical Systems?, Cambridge University Press, 1986. 225

226

LIST OF THE PUBLICATIONS OF JULES VVILLEMIN, 1947-1989

- "La mort dans la philosophie de Hegel", in Revue Philosophique, Paris, P.u.F., pp 194-202, 1947. - "L'imitation dans I'interpsychologie de Tarde et ses prolongements" in Journal de psychologie normak et pathologique, pp. 420-449, Oct.Nov. 1949. - "La dialectique negative dans la connaissance de I'existence, note sur i'epistemologie et la metaphysique de Nikolai Hartmann et de JeanPaul Sartre", in Diakctica, pp. 21-42, 1950. - "Le monde de I'esprit selon Dilthey", in Revue Philosophique, Paris, P.U.F., pp. 508-519, 1950. - "Les classes sociales chez Schumpeter et dans la realite", in Revue d'Economie Appliquee, Paris, P.U.F., pp. 571-614, 1950. - "Note sur l'evidence cartesienne et Ie prejuge qu'elle impJique", in Revue des Sciences Humaines, pp. 42-49, Janvier-Mars 1951. - "La mort, Ie travail et la Revolution copernicienne", in I'lnformation philosophique~ pp. 14-17, nO I, Janvier-Fevrier 1951. - "Traduction de deux textes de Max Weber sur la notion de classe sociale", in l'Information philosophique, nO 5, Novembre-Decembre 1951. - "Les syndicats ouvriers et les salaires", in Revue d'Economie Appliquee, Paris, P.u.F., pp. 261-336, 1952. - "La signification de l'humanisme athee chez Feuerbach et I'idee de nature", in Doucalion, Ie diurne et Ie nocturne, Neuchatel, pp. 17-46, avril-sept. 1952. - "Lavoratore nell'universo, Saggio di una filosofia del lavore de Mario Rossi", in Revue philosophique, pp. 593-596, 1952. Collaboration

a la Revue Les

Temps Modernes ;

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227

- "Les statues et les hommes", I, pp. 1922-1955, nO 55, 1950. - "Le souffle dans I'argile", II, pp. 1225-1257, nO 63, 1951. - "Experiences de verite : sur I'autobiographie de Gandhi", nO 56, pp. 2259-2268, I 951 - "Nouvelles traductions de Marx.A propos de la correspondance Marx-Engels-Sorgo", nO 59, pp. 548-563, 1950. - "Vivre et peindre : sur les lettres de Van Gogh et Van Rappart", nO 60, pp. 746-750, 1950. - "Nietzsche aujourd'hui", nO 67, pp. 1921-1954, 1951. - "Tiepolo a Venise, c.R. de I'exposition de la Biennale de Venise", nO 74, pp. 1133-1138,1951. - "La personnalite esthetique du Tintoret", Essai sur les relations entre la peinture venitienne et la civilisation de la Contre-Reforme, nO 102, pp. 1%5-2006, 1954. - "Economie europeenne et economie mondiale", a propos du livre de Francois Perroux l'Europe sans rivages, nO 106, pp. 398-441, 1954. - "Edgar )ene,-peintre de I'oubli", nO 109, pp. 1149-1151, 1955. - Collaboration a I'ouvrage collectif publie sous la direction de M. Merleau-Ponty : Les philosophes celebres, Paris, L. Mazenod, 1956. Articles sur Kant, Fichte, Schelling, Lessing, Herder, Maimon, Renouvier, H. Cohen, Feuerbach et Engels. - "Sur la generalisation de I'estimation de la force chez Laplace", TbaJes, pp. 61-75, 1958. - "Le probleme phenomenologique : intentionalite et reflexion", in Revue Philosophique de la France et de l'Etranger, tome 149, pp. 463470, Paris P.U.F., 1959. - "La philosophie de l'algebre de Lagrange (Reflexions sur Ie memoire de 1770-1771)" Conference du Palais de la Decouverte, nO 71, pp. 310335,1%0.

228

LIST OF THE PUBLICATIONS OF JULES VUILLEMIN, 1947-1989

- "Sur la difference et l'identite des methodes de la meta physique et des rnathernatiques chez Descartes et Leibniz et sur la conception classiques des principes de causalite et de correspondance", in Archiv fur Geschichte der Philosophie, bd 43, Heft 3, pp. 267-303, 1961. - "Reflexionen iiber Kant's logik", in Kant studien philosophische zeitschrijt, bd 52, Heft 3, 1960/61. - "La demonstration de l'irrationalite de 1t chez Leibniz, Lambert et Kant", in Revue Philosophique, pp. 417-431, 1961. College de France, Chaire de Philosophie de la Connaissance : Lecon inaugurale fatte Ie mercredi 5 decembre 1962. Nogent Ie Rotrou, imp. Daupeley-Gouverneur, 1963. - "Le principe du levier", in Revue Internationale de Philosoph ie, Bruxelles, nO 64, fasc. 2, pp. 1-23,1963. - "Preface de l'Ecole de Marbourg", in Henri Dussort, L'Ecole de Marbourg, Paris, PUF, 1963. - "L'origine et Ie mecanisme des antinomies dans la premiere philosophie de Russell (1903)", in Bulletin trimestrlel du Centre National de Logique et Anaryse, nO 25-26, pp. 59-95, avril 1964. - "Sur Ie jugement de recognition (wiedererkenn ungsurteil) chez Frege", in A rch iv fur Geschichte der Philosophie, bd 46, heft 3, pp. 310-325, 1964. - "La methode indirecte de M. Merleau-Ponty", in Critique, Paqris, Gallirnard, pp. 1007-1016, 1964. - "Sur les proprietes formelles et rnaterielles de l'ordre cartesien des raisons", in Homrnage :l Martial GuerouIt, l'histoire de la Philosophie, ses problemes, ses methodes, Paris, Fisbacher, pp. 43-58, 1964. - "Probleme de validation (Bregriindung) dans les axiornatiques d'Euclide et de ZermeIo", in The Foundations of Statement and decisions, Actes du Congres International de Logique et de Methodologie des Sciences de Varsovie, pp. 179-203, 1965. - "L'elirnination des definitions par abstraction chez Frege", in Revue

LIST OF THE PUBLICATIONS OF JULES VUILLEMIN, 1947-1989

229

Philosophique de Ia France et de l'Etranger, nO 1, pp. 19-40; 1966. - "Sur les conditions qui permettent d'utiliser les matrices russelliennes des antinomies (905) pour exprimer des theoremes de limitations internes des formalismes", in Notre Dame Journal of Formal Logic, Notre Dame, Indiana, Vol VII, nO 1, pp. 1-19, 1%6. - "Faut-il fermer les Grandes Ecoles?", in Janus 12, pp. 103-111, septnov. 1966.

- "c.R. Hockett CF. "Language, Mathematics and Linguistics, in current trends", in Linguistics, Vol. 3, Theoretical Foundations, The Hague, A. Ferguson, Paris, Mouton, pp. 155-304, 1966.

- "La theorie kantienne de l'espace ala lumiere de la theorie des groupes de transformations", in The Monist, Vol. 51, nO 3, pp. 332351, 1967. - "Mesure, verification et langage", in Entretiens de I'Institut International de Philosophie, Liege, pp. 183-195, 1967. - "Sull'influenza innovatrice della logica matematica in filosofia", trad. Pia Bozzi, in Annuario Internationa/e della scienza e della tecnica, Milan, Mondadori, 1967. - "Preface pour la reedition de Poincare, in La Science et I'Hypothese, Paris, Flarnmarion, pp. 1-20, 1968. - "Les indicateurs de subjectivite (Egocentric Particulars) la derniere philosophie de Russell" I et II in l'Age de Ia Science, nO 1, pp. 41-64, nO 2, pp. 99-135, 1968. Fondation en collaboration avec G. Granger de la revue l'Age de Ia Science, Paris, Dunod, Janvier-Mars 1968. - "L'influenza innovatrice della logica matematica in filosofia" in Enciclopedia della scienza e della tecnica, Mondadori annuario della Est, Verona, pp. 420-426, 1968. Avec G. Granger: - "Tendances de la Philosophie des sciences en France depuis 1950", in Contemporary Philosophy A survey, Firenze, La Nuova ltalia, pp. 161-163, 1968.

230

LIST OF THE PUBLICATIONS OF JULES VUILLEMIN. 1947-1989

- "Preface pour la reedition de Poincare", in La Valeur de Ia Science, Paris, Flammarion, pp. 1-18, 1969. - "Remarques sur 4.442 du Tractatus", in Revue Intemationale de Philosophie, 23eme annee, nO 88-89, fasc. 2-3, pp. 299-318, 1969. - "La Constitution selon Camap : la construction logique du Monde", in l'Age de la Science, nO 4, pp. 303-333, oct-dec. 1969. - "Expressive statements", in Philosophy and Phenomenological research, Univ. of Buffalo, vol. XXIX, nO 4 pp. 485-497- June 1969. - "La theorie kantienne de l'espace a la lumiere des groupes de transformations" (The Monist, Vol. 51, nO 3, Juillet 19(7) traduction anglaise dans Ie livre collectif sur Kant publie par L.W Beck, Open. Court publications, La Salle, III., pp. 141-159, 1969.

c.R. de Nelson Goodman: "Language of art",

London, Univ. Press., in l'Age de la Science, Vol. 3, nO 1, pp. 73-88, janvier-mars 1970.

c.R. de c.P. Hockett: "Language, Mathematics and Linguistic, in

current trends" in Linquistics, Vol. 3, Theoretical Foundations, Ed. A. Ferguson, extrait in l'Age de Ja Science, Vol. 3, p. 157, 1970.

- "Remarques philosophiques sur l'aspect createur du langage" in Echanges et communications, Melanges offerts a Claude LeviStrauss. The Hague, Paris, Mouton, pp. 981-997, 1970. "La philosophie des mathematiques", in Revue Philosophique de la France et de l'Etranger, Paris, PUF, 1971, pp. 333-334.

- "Sur la tolerance" in Revue Internationale de Philosophie, nO 95-96, fasc. 1-2, pp. 198212, 1971. - "Uber die innere Moglichkeit eines rationalen Gottesbegriffs Id quo nihil maius cogitari potest"', in A rch iv fur Geschichte der Philosophie, bd 53, heft 3, pp. 279-299, 1970. - "Platonism in Russell's early philosophy and the principle of abstraction" ,

LIST OF THE PUBLICATIONS OF JULES VUILLEMIN. 1947-1989

231

in Bertrand Russell, a collection of critical essays, pp. 305-324, Edited by D.E. Pears, Anchor Books, Doubleday, New York, 1972. - "Logical flaws or philosophical problems: on Russell's Principia Mathematica" in Revue Internationale de Philosophie, 26eme an nee, nO 104, pp. 334-336, 1972. - "Das problem der identitat in der Beweistheorie und die Kantische forstellung", in Kant Studien, pp. 289-304, 1972. - "Poincare's philosophy of space", in Synthese, an international journal for epistemology, methodology and philosophy of science, Dordrecht, Boston, Reidel, pp. 161-179, 1972. - "Sport and conflit", in Sport in the modern world, chances and problems, Scientific congress Munich, Ed. O. Grupe, Springer, Berlin, Heidelberg, New York, pp. 77-78, 1973. - "Reponse a un compte-rendu de Me Von Savigny" in Archiv fur Geschichte der Philosophie, bd 57, heft 2, pp. 210-211, 1975. - "Quine's concept of stimulus-meaning" in Philosophic exchange, Brockport, New York, pp. 4-13, 1975. - "1st systernatisch Philosophie moglich ?", in Stuttgarter Hegel Kongress, pp. 327-339, 1975. - "Le Platonisme dans la premiere philosophie de Russell et Ie

principe d'abstraction",

in Dialogue, pp. 222-240; 1976.

- "Le concept de signification empirique (stimulus meaning) chez

Quine",

in Revue Internationale de Philosophie, Epistemologie et langage, pp.

350-375, 1976.

- "Conventionalisme geometrique", in Archives de l'lnstitut International des Sciences Ibeoriques, Bruxelles, OFFILIB, nO 20, pp. 65-105, 1976.

232

LIST OF THE PUBLICATIONS OF JULES VUILLEMIN, 1947-1989

- "Kant aujourd'hui", Actes du Congres d'Ottawa sur Kant dans les traditions angloamericaine et continentale, 10-14 septembre 1974. Ottawa, Ed. de l'Universite, pp. 17-35, 1976. - "Definition und ratio : das paradigma der griechischen mathematik", in Objecktivitat in den natur und geisteswissenchajten, pp. 125-131, 1976. - "Martial Gueroult : Fichte", in Arcbiv fur Geschichte der Philosophie, bd 59, pp. 289-293, 1977. - "Notice necrologique sur Martial Gueroult", Association amicale des anciens eleves de l'Ecole Normale Superieure, pp. 59-63, 1977. En collaboration avec G. Dreyfus, L. Guillermit et V. Goldschmidt: "Martial Gueroult", in Arcbiv fur Gescbicbte der Pbilosophie, bd 59, heft 3, pp. 289-312, 1977. - "Definition et raison : Ie paradigme des mathematiques grecques", in Actes du congres d'Athenes-Pelion d'octobre 1975. Athenes, HelleniC Society for Humanistic Studies, pp. 273-282, 1977. - "Caracteres et fonctions des signes", in Die Aktualitat der transzendental-pbilosophie, Hommage a H. Wagner, Bonn, Bouvier, Grundmann, pp. 93-119,-1977. - "De la biologie a la culture", c.R. du livre de J. Ruffle in Informations sur les Sciences Sociales, XVI 5, pp. 621-633, 1977.

- "La puissance selon Aristote et Ie possible selon Diodore" in Manuscrito, Campinas, Vol. 1, pp. 23-69, 1977.

LIST OF THE PUBLICATIONS OF JULES VUILLEMIN, 1947-1989

233

- "De la philosophie analytique a l'idee d'un systeme critique", 1st systematische Philosophie Moglich ? Stuttgarter, Hegel Kongress, 1975, Bonn, Bouvier, pp. 327-339, 1977. - "L'analogie astronomique de la philosophie critique", in Manuscrito, Sao Paulo, Vol. II, nO 1, pp. 79-88, 1978. - "Concetto", in Enciclopedia, tome III, citta-cosmologie, pp. 710-756, Torino, Einaudi, 1978. - "Kant hoje", in Ciencia ejilosoffa, nO 1, Sao Paulo, pp. 141-159, 1979. - "L'argument dominateur", in Revue de Metaphysique et de Morale, n° 2, pp. 225-257, 1979. - "La raison au regard de l'instauration et du developpement scientifique" , in La Rationalite d'aujourd'hui, Ed. Th. Geraets, Univ. Ottawa, pp. 6784,1979.

- "Fenomeno", in Enciclopedia, tome VI, Familia-Ideologia, pp. 53-73, Torino, Einaudi, 1979. - "Forma", in Enciclopedia, tome VI, Familia-Ideologia, pp. 314-323, Torino, Einaudi, 1979. - "Idea", in Enciclopedia, tome VI, Familia-Ideologia, pp. 1067-1088, Torino, Einaudi, 1979. - "Ideologia", in Enciclopedia, tome VI, Familia-Ideologia, pp. 1144-1164, Torino, Einaudi, 1979. - "On Duhem's and Quine's theses", in Grazer Philosophische studien, Vol. 9, pp. 69-96, 1979. - "Is homo cumt identical with homo est currens ?", in Language, Logic and Philosophy, Proceedings of the 4th International Wittgenstein symposium, pp. 61-68, 28.8.79-2.9.79 .

234

LIST OF THE PUBLICATIONS OF JULES VUILLEMIN, 1947-1989

- "The influence of reason on the origin of science", in Rationality in Science, pp. 203-208, 1980. - "Qu'est-ce qu'un nom propre ?", in Fundamenta Scientiae, Vol. 1, pp. 261-273, 1980. - "Proposizione e giudizio", in Enciclopedia, tome XI, Prodotti-Ricchezza, Torino, Einaudi, 1980 - "Les lois de la Raison pure et la supposition de leur determination complete" in J. Kopper und W. Marx, 200 jahre Kritik der reinen vernun/t, Gerstenberg, Hildesheim, 1980, pp. 363-384. "Trois philosophes intuitionnistes : Epicure, Descartes, Kant", in Dialectica, vol. 35, nO 1-2, pp. 21-41, 1981. "Internal Translations and Reality in Science", in the Proceedings of the 6th International Wittgenstein Symposium, Language and Ontology, 23th to 30th August 1981, Kirchberg/Wechsel, Austria. "Le paralogisme du bain (Aristote, Poetique, 1460 18-26)", in Revue des Etudes Grecques, tome XCIV, n- 447-449, JuilletDecembre 1981-1982, pp. 287 294. "Comparative Philosophy as Applied to the Concept of Natural Law", in the Monist, vol. 65, nO 1, january 1982, pp. 3-12. "Note sur Ie Laocoon de Lessing", in Archiv jur Geschichte der Phi[osophie, 64, Band 1982, Heft 1, pp. 39-55. "Eternel retour et temps cyclique: queIIe solution Cleanthe a-t-il donnee de I' "Argument Dominateur" ?", in Archives de Philosophie, tome 45, cahier 3, Juillet-Septembre 1982, pp. 287-294. "La theorie kantienne des modalites", in Akten des 5. Internationalen Kant-Kongresses Mainz 4.-8. APril, 1981 Bouvier-Verlag, Grundmann, Bonn, 1982, pp. 149-167. "Concetto", in EnciclopediaEinaudi, vol. 15,1982, pp. 128-133, Torino.

LIST OF THE PUBLICATIONS OF JULES VUILLEMIN. 1947-1989

235

"On lying, Kant and Benjamin Constant", in Kant Studien, 73, Jahrgang, Heft 4, 1982, pp. 413-424. "Portrait d'un historien de la philosophie", in Victor Goldschmidt 0914-1981),Journees d'hornmage du 17 janvier 1982, Universite de Picardie. "Physicalism Relativity", in Grazer Phtlosophischen Studien, vol. 16/17, 1982, pp. 313-326. " Verite partielle ou approximation de la verite", in La Gazette des Sciences Mathematiques du Quebec, vol. VII, nO 2, Janvier 1983, pp. 7-30. avec Kurt Hubner (herausgegeben von), Wissenschajtliche und nichtwissenschajtliche Rationalittit, Ein deutsch franzosisches Kolloquium, Frornman Hoisboog, Stuttgart, 1983. "L'empirisme logique, science et metaphysique", Revue lnternationaie de Philosophie, nO 144-145, 1983, fasc 1-2, 37°annee, pp. 214-216.

c.R., J. Hopkins, Nicholas of Cusa's Metaphysic of Contraction, Minneapolis, The Arthur]. Banning Press, 1983, 196 p. in Revue lnternationale de Philosoph ie, 37' annee, fasc. 1-2, 144-145, 1983, pp. 214-216. "Le carre Chrysippeen des modalites", in Dia/ectica, vol. 37, fasc. 4, 1983, pp. 235-247. "Bemerkungen uber Lessings Laokoort', in Das Laokoon-Projekt, Hrsg. G. Gebauer, Stuttgart, ].B. Metzler, 1984, pp. 167-182. "Les formes fondamentales de la predication : un essai de classification", Recherches sur /a philosophie et Ie /angage, Langage et philosophie des sCiences, cahier nO 4, DREUG, Grenoble, Vrin, Paris, 1984, pp. 9-30. "La Reconnaissance dans l'Epopee et dans la Tragedie" (Aristote, Poetique, Chap. XVI), in Archiv fur Geschtchte der Philosophie, alter de Gruyter, Berlin, New York, Bd 66, heft 3, 1984, pp. 243-280.

236

LIST OF THE PUBLICATIONS OF JULES VUILLEMIN, 1947-1989

"Un systeme de fatalisme logique : Diodore Kronos", in Revue de Philosophie Ancienne, nO 1, 1984, pp. 39-72 (Ousia, Bruxelles). "Intuitionnisme et 'criteres' epicuriens", in Histoire et structure, a la memoire de Victor Goldschmidt, etudes reunies par J. Brunschwig, Cl. Imbert et A. Roger, Paris, J. Vrin, 1985, pp. 201-211. "Physique quantique et philosophie", in Ie Monde Quantique, ouvrage collectif, ed. du Seuil, 1985, pp. 201224. "Die Aporie des Meisterschlusses", in Allgemeine zeitschrljt fur Philosophie, Frommann-Holzboog, 10.2., 1985, pp. 1-19. "Une morale est-elle compatible avec Ie scepticisme ?", in Philosophie.. ed. de Minuit, nO 7, 1985, pp. 21-51. "Verite partielle ou approximation de la verite", in IA nature de Ia Verite Scientijique, Archives de l'Institut international des sciences theoriques, CIACO, Louvain-Ia-Neuve, 1985, pp. 123-139. "Sur deux cas d'application de l'axiomatique a la philosophie : l'analyse du mouvement par Zenon d'Elee et l'analyse de la liberte par Diodore Cronos", Fundamenta SCientae, vol. 6, nO 3, 1985, pp. 209-219. - "Guillermit et Kant", Louis Guiliermit, historien de la philosophie In MemOriam, Publications de l'universite de Provence, 1986, pp. 55-70. - "Rapport sur la communication de D. Follesdal", in Merites et limites des methodes logiques en philosophie, Fondation Singer Polignac, Paris, Vrin, 1986. "On Lying - Kant and Benj amin Constant", in G . Geisrnann und H . Oberer, Kant und das Recht der Luge, KonigshausenNeumann, Wilrzburg, 1986. "E compativel uma moral com 0 cepticismo ?" in Analise, nO 4, 1986, Lisbonne, pp.5-36.

LIST OF THE PUBLICATIONS OF JULES VUILLEMIN. 1947-1989

237

"Die Wlichterin der Vernunft. 'Der Menschenfeind' als Mythos der KomOdie" in Zur Krltik der wtssenschaftlichen Rationalitat, Alber, Munchen, 1986, pp. 515-530. "On Duhem' s and Quine's Theses", in The Philosophy 0/ w. V. Quine, L.E Hahn, P.A. SchUpp, La Salle, Open court, vol. XVIII, 1986. "Zur Rekonstruktion des Meisterschlusses, Antwort an Eelmut Angstl", in Allgemeine Zeitschrljt fur Philosophie, Frommann Holzboog, 11.3, 1986. "Les relations mutuelles entre philosophie des sciences et histoire des sciences, examinees a propos du developpement de 1 ' astronomie de position jusqu' a Kepler", in Epistemologia X (1987), pp. 57-72. "La justice par convention; signification philosophique de la doctrine de Rawls", in Dialectica, vol. 41, fasc. 1-2, 1987, pp. 155-166.

"Le bonheur selon Descartes", in TiJdschrlft voor Filoso/ie, 4ge Jaargang, nummer 2, Juni 1987, pp. 230-240. "Remarques critiques sur la doctine kantienne de la causalite", Kant, Analysen, Probleme, Krltik, herausgegeben von. Oberer und G. Seel, KOnigshausen und Neumann, 1988, pp. 91-101. ceL' intuitionnisme moral de Descartes et Ie 'Traite des passions de

l'ame''', Kant-Studien, 79 Jahrgang, Heft I, 1988, pp. 17-32.

"Remarques sur la convention de justice selon Rawls", L'Age de la Science, I, ed. o. Jacob, Paris, 1988, pp. 55-71. "Le chapitre IX du De Interpretatione et la connaissance probable," Aristote Aujourd'bui, Etudes reunies sous la direction de M.A. Sinaceur, Eres, Paris, 1988, pp. 77-93. "Les preuves cartesiennes et la preuve du Proslogion", Anselm Studies, II (Proceedings of the fifth International Saint Anselm Conference), ed. by Schaubelt, Losoncy, Vlan Fleteren, Frederich, Kraus, New York, 1988, pp. 185-199.

238

LIST OF THE PUBLICATIONS OF JULES VUILLEMIN, 1947-1989

"Methode transcendantale, morale et metaphysique", Metapbysik nancb Kant? Stuttgarter Hegel-Kongress, 1987, herausgegeben von Henrich und Horstmann, Klett-Cotta, Stuttgart, 1988, pp. 137-143. "Note sur l' Ecole allemande de mCcanique quantique", L' Age de Ia science, II, L' epistemologie, editions Odile Jacob, Paris, 1988, pp. 177-179. En collaboration avec Fran~oise Letoublon, compte rendu de Ivor Eke1and, Le calcul de I' imprevu, les figures du temps de Kepler a Thorn, editions du Seuil, Paris, 1984, L' Age de Ia Science, II, L' epistemologie, editions Odile Jacob, Paris, 1988, pp. 247-251. "Le Misanthrope, mythe de la comedie", Dialectica, vol. 42, fasc. 2, 1988, pp. 117-127. "Comparabilite et incomparabilite des theories physiques", Recbercbes sur la pbilosopbie et Ie langage, Semantique formelle et philosophie du langage, actes du colloque du 9-12 septembre 1987, Cahier du groupe de recherches sur la philosophie et Ie langage, Universite des Sciences sociales de Grenoble, 1989, pp. 73-98. "La methodologie de Kepler", Traditionen und Perspektiven der Analytiscben Pbilosopbie, Vedag Holder Pichler Tempsky, 1989, pp. 24-33. "Kant's 'Dynamics' : comments of Tuschling and Forster", Kant's transcendental deductions, Stanford Series in Philosophy, Stanford, 1989, pp. 239-267. "Sobra a Tolerancia" (traduction Michel Lahud) Conbecimento, Linguagem, Ideologia, Marcelo Dascal org., Editora perspectiva, Sao PAULO, 1989, pp. 241-257. "Le comique comme idee. Ella toquade a l' extravagance et au rire dechalne : comparaison entre Racine et Aristophane" in Hommage a Henri Joly, Cabiers Pbilosopbie et Langage, nO 12, 1990, pp. 443-450.