Truth in Context Chapter-03

Chapter 3 The Epistemic Challenge to Antirealism

3.1 The Challengeto Antirealism The challenge to realism was to explain how mathematical knowledge is possible if mathematical objects are abstract. The challenge to antirealism is to explain how mathematical knowledge is possible if mathematical objects are concrete. The challenge to antirealism derives from the traditional rationalist argument that knowledge of what contingently is cannot ground knowledge of what must be, but is a broader argument that applies to Quineans, who reject the notion that there is anything that must be. The challenge is to explain the special certainty of mathematical and other formal truths on the basis of a naturalist concrete ontology and an empiricist epistemology. This challenge is, in a sense, complementary to the epistemological . In both challenge to the realist to account for mathematical knowledge ' . s rationalism cases, ontology raises a problem for epistemology Realism seems the right sort of epistemology to explain a priori knowledge , but it is prima facie unclear how such an epistemology is supposed to work in the formal sciences when non naturalism says that what we know in those sciences are facts about abstract objects. Antirealism ' s empiricism seems the right sort of epistemology to explain a posteriori knowledge , but it is prima facie unclear how such an epistemology is supposed to work in the formal scienceswhen naturalism says that there is nothing to know except facts that are au fond about concrete objects. How can the special certainty of theorems about numbers , sets, and the like rest on a posteriori facts about contingent objects? That special certainty is an objective feature of those theorems. It is different from the subjective certainty we have about our beliefs in them . The former is something like an exclusion of all possibility of doubt based on the prevailing standards in the discipline . White (1972, 4- 5), for example, says that propositions are certain when the question of their truth is settled in virtue of the exclusion of all reasonable

64

Chapter3

1 is the mental state which possibility of falsehood. Subjective certainty " " " " we describe with such adjectives as " confident , assured, or certain " . The mathematical proposition that the circle cannot be squared is objectively certain, and it would remain so even if the general level of mathematical competence were to drop to the point at which everyone became a would -be circle squarer. The objective certainty of a proposition consists in the fact that different people who recognize its certainty are recognizing the same logical relations, whereas the subjective certainty of a belief consists in the fact that different people who are equally certain of the same proposition are in similar mental states but not in the same token mental state. Subjective certainty, as Rollins (1967, 67) observes, may be justified or not- and hence such certainty ' ' may be justified in one person s 'case but not in another s - while objective certainty obtains or doesn t. As with many significant notions in philosophy , there is no fully satisfactory explication of the notion of objective certainty. Questions of many kinds arise about the above description . How is possibility to be understood ? We apparently do not want to understand it as logical possibility becausewe want to allow some contingent statements to be certain. Again , this description of objective certainty takes the exclusion ' of all reasonable possibility of a proposition s falsehood to be a consequence of its relation to other propositions , but what relations are meant? Logical relations? Rational ones? A priori ones? Metaphysical ones? Inductive ones? Further, there is no demonstration that formal truths have a special certainty that common empirical truths lack. There are, however, clear signs that the certainty of the former is special. One is the relative imperviousness of knowledge in the formal sciencesto doubt based on counterfactual hypotheses, in contrast to the relative susceptibility of empirical knowledge . Contrary hypotheses in the case of empirical knowledge are far-fetched or off the wall , as is, for example, the hypothesis that the moon is made of swiss cheese, but contrary hypotheses in the case of formal knowledge are impossible , as is, for , example the hypothesis that four is not composite . Another sign of the difference can be seen in adverbial modifications of " certain." The certainty of truths in the formal sciencesis customar" " " ily described with adverbs such as completely" and absolutely, whereas such adverbs are typically eschewed in describing the certainty of even the most secure empirical truths . Instead, their certainty

The Epistemic Challenge to Antirealism

65

is described using adverbs such as " theoretically " and " empirically ." In the former case, we express an unconditional exclusion of doubt , while , in the latter, we qualify the exclusion of doubt , relativizing it to a theory or a body of empirical evidence. Fortunately, it is not necessary for our purposes to have either an explication of special certainty or a demonstration that formal truths have special certainty . On the one hand , the notion is clear enough, and, on the other hand , it is generally acknowledged that formal truths do have a special certainty. Even empiricists like Mill and Quine who deny the existence of necessary truth acknowledge that mathematical and logical truths have a special certainty, as is clear from the fact that they recognize the obligation to provide an empiricist explanation of it . The acknowledgment is also seen in Quine ' s dissatisfaction with Mill ' s explanation of the certainty of mathematics and logic in terms of the more extensive empirical confirmation of mathematical truths . Quine (1966, 100- 101) writes : [Mill ' s] doctrine may well have been felt to do less than justice to the palpable surface differences between the deductive sciences of logic and mathematics, on the one hand, and the empirical sciences ordinarily so- called on the other. Worse, the doctrine derogated from the certainty of logic and mathematics; but Mill may not have been one to be excessively disturbed by such a consequence. Perhaps classical mathematics did lie closer to experience then than now. . . . It is, of course, open to naturalists flatly to deny that mathematical and logical truths have such a special certainty, but such stonewalling is " implausible in the face of such wide recognition of the palpable surface differences between the deductive sciencesof logic and mathematics , on the one hand , and the empirical sciencesordinarily so- called on the other." Historically , empiricists who have kept faith with the doctrine that all truths are just a posteriori truths about contingent objects have had a hard time trying to explain how mathematical and logical truths can have the special certainty they are generally acknowledged to have. Mill ' s answer was an acute embarrassment to empiricists . Arithmetic truths are laws about natural objects, differing from what we commonly take as such laws only in being of a higher order. The arithmetic truth " l = I " is about natural objects weighing one pound , being a foot tall , and so on. Since they are a posteriori truths , they have to be falsifiable . Mill ' s (1874, vol . 2, ch. 6, sec. 3) explanation is that " 1 = I " could be false because one natural object weighing a pound need not have exactly the same weight as another weighing a pound . The

66

Chapter3

" " arithmetic equation 3 = 2 + I is about natural objects because it expresses the physical possibility of grouping three ungrouped" natural objects into two groups - to which Frege (1953, 9- 17) replied , " What a mercy, then, that not everything in the world is nailed down . Mill explained the special certainty of mathematical truths as a matter of their receiving far more extensive empirical confirmation than other a posteriori truths . We know the truth of a mathematical equation like " 3 = 2 + I " either on the basis of direct induction from a sample of three natural objects or on the basis of inference from more general ' " " inductive truths such as The sums of equals are equals. Frege s (1953, 9- 17) criticism of this account is generally regarded as about as close ' to a total refutation as philosophical criticisms get. Moreover, Mill s explanation of the certainty of mathematical truths transparently fails to establish a genuine difference between their certainty and that of " " a posteriori truths . On Mill ' s view of I = I , it can at best have a probability of I , but even that does not capture the certainty of that equation . If anything is required to drive the point home, note that an objective probability of the occurrence of one event with respect to another can be 1 even though there are indefinitely many instances in which the one event occurs without the other, whereas the certainty of an arithmetic equation is lost with one counterinstance. Further, it is hard to see why esoteric mathematical truths should be better confirmed than commonplace empirical ones. Unable to accept Mill ' s inductivism , empiricists flirted for a time with a conventionalist account of logical and mathematical truth , but , as Quine (1936) showed , it , too , fails as an explanation . It was only with ' turned around . Quine s empiricism that the empiricist ' s situation was ' It made full -blooded empiricism respectable. Quine s (1961c, 46) empiricism treats posits of mathematical objects and hypotheses about their relations on a par with posits of physical objects and hypotheses " about theirs, simply as a means to expedite our dealings with sense " ' experiences. Quine s holism solves both the problem of accounting for the certainty of logical and mathematical truths and the problem of for them as truths about the natural world . In a nutshell , accounting 's Quine (1961c) explanation of the certainty of mathematical and logical truths is that " total science" is a single system in which those truths , though ' revisable in principle , are nonetheless so far removed from the system s experiential boundary and so intertwined with everything else in the system, that revision of them is not really a live option in normal science. This doctrine saved the uncompromising empiricist ' s account of mathematics as a posteriori knowledge . Since Quine, empiricists have stopped worrying about the certainty of mathematical and logical truths .

The Epistemic Challenge to Antirealism

67

In this chapter, I will argue that Quine ' s epistemology is, in fact, no more successful than inductivism and conventionalism . Since the failure of Quine ' s epistemology is tantamount to the failure of contemporary uncompromising empiricism , if my argument goes through , contemporary empiricism will be shown to have no explanation of the special certainty of mathematical and logical truths and no explanation of how they can be about natural objects. Without those explanations , it will have no account of mathematical knowledge . 3.2 Antirealist Responses Since the antirealist claim is that everything that exists is a concrete object in the sense of something that can have spatial or temporal location , antirealists may be monists or dualists with respect to the mind / body problem . Antirealists can subscribe to the existence of both Cartesian egos and material objects like sticks and stones, and hence be mind / body dualists . Or they can be mind / body monists, subscribing to idealism , materialism , or neither, i.e., they might straddle the issue because, taking their cue from Russell (1927), they are neutral on the basic nature of reality . Our argument in this book is directed at antirealists of every kind , but we are most concerned with materialist antirealists becausethe most vigorous contemporary forms of antirealism ' were inspired by naturalism in Oanto s (1967, 448- 50) sense of a monism that says that everything that exists is a material or physical part of a single causally interrelated spatiotemporal realm. American naturalism in the early part of this century was an ontoand epistemological naturalism . It was an across the board logical " " revolt against dualism , opposed as much to the dualism of the abstract and the concrete as to the dualism of the mind and the body or the dualism of the natural and the supernatural . Such global natu ralists had a rationale for their comprehensive stand against dualism . They saw dualism per se as the common cause of a number of philo sophical problems . In bifurcating the world , dualism creates the problem of reconciling onto logically incommensurable entities. Cartesian dualism creates the problem of explaining causal interaction between the mind and the body, supernatural dualism creates the problem of explaining causal interaction between the spiritual and the natural , and Platonic dualism creates the problem of explaining causal interaction between the abstract and the concrete. Since all dualisms pose essentially the same problem , there can be a single solution to them all : reject dualism per se. In spite of the attraction of eliminating the source of all of these philosophical problems in one stroke, some philosophers adopt a

68

Chapter3

naturalist stance on one issue or another but not on all . For example, Fodor (1990, 132 n. 6) is opposed to mind / body dualism , but not to abstract/ concrete dualism . Those who forgo the simple and effective solution of rejecting dualism per se do so because they think there are overriding considerations, usually tied in with their needs in the case of the dualism that most matters to them , to allow certain non-natural objects full onto logical citizenship . Accordingly, a more descriptive for contemporary philosophers calling themselves " natu terminology " ralists would not categorize them as naturalists tout court, but rather as naturalists on the abstract/concreteissueor naturalists on the mind/ body issue or naturalists on the natural/supernatural issue. Thus, on the abstract / concrete issue, global naturalists and local naturalists see the source of the problem of reconciling the incommensurable entities somewhat differently , but they have the same view about its cause: dualism . Although methodological naturalists can maintain an abstract/ concrete dualism , as Quine does, they take the scientist' s and the " epistemologist ' s object "of study to be a natural phenomena, viz ., a physical human subject (Quine 1970, 83), and they take an uncompromisingly empiricist stand on the a posteriori status of all knowledge . Hence, unlike our dualism , the dualism that methodological naturalism tolerates is kept within the bounds of ontology . It is not allowed to spill over into epistemology. The two most influential forms of naturalism in contemporary phi ' ' losophy are Quine s and Wittgenstein s. Whereas the former is dualist , empiricist , and scientistic, the latter is monist , critical , and ascientistic. These two forms of naturalism have given rise to two distinct natural istic explanations of the special certainty of mathematics and logic . I will discuss Quine ' s paradigm in this and the next section, and Wittgen stein' s in section 3.4. Uncompromising empiricism explains all our knowledge as based, to some extent or other, on information about the natural world obtained through our senses. Such information can undergo certain ratiocinative processing once inside us, but our reasoning, in and of itself , never provides the sole grounds for a piece of knowledge .2 Since only natural objects can affect our senses, all scientific knowledge , the formal and the natural alike , must ultimately , if very indirectly , depend on 2. Pure naturalismrequiresan epistemologythat seesall scientificknowledgeas about the natural world. The general requirementgoes the other way too. An empiricist epistemologyrequiresa naturalistontology. A rationalistepistemologydoesnot mesh with a naturalistontologyin which thereareonly objectswith contingentpropertiesthat i , and an empiricistepistemologydoesnot meshwith a realist can be known a posterior ontology in which there are abstractobjectsthat have necessarypropertiesand can be

The Epistemic Challenge to Antirealism

69

sensory information about such objects . But if so, how can logical and mathematical knowledge be so much more certain than natural knowledge when it rests on the same sensory information as the latter and when that information itself would seem to be of a distinctly lower order of certainty than the knowledge that rests on it ? ' The palpable inadequacy of Mill s view led logical empiricists to deny that mathematical and logical knowledge is a posteriori . Carnap ( 1963, 64 ) wrote : Since empiricism had always asserted that all knowledge is based on experience , this assertion had to include mathematics . On the other hand , we believed that with respect to this problem the rationalists had been right in rejecting the old empiricist view that " " the truth of 2 + 2 = 4 is contingent upon the observation of facts , a view that would lead to the unacceptable consequence that an arithmetical statement might possibly be refuted by new experiences . Our solution . . . consisted in asserting empiricism only for factual truth . By contrast , the truths of logic and mathematics are not in need of confirmation by observations . . . . " " " logical truths , which do not state Carnap ( 1963, 64) distinguished anything about the world of facts [but ] hold for any possible combination " " " of facts , from factual truths , which make empirical statements and do not hold for any possible combination of empirical facts . This ' reconstruction of Hume s distinction between relations of ideas and ' matters of fact gave rise to a new empiricism . Thanks to Frege s logical semantics , particularly his conception of analyticity , arithmetic truths " " like 7 + 5 = 12 are no longer synthetic a priori , but analytic a priori ' " " truths in Carnap s extended sense of analytic . Hence , logical empiri cists are not subject to Kant ' s criticism of Hume for throwing out mathematics along with metaphysics . This left logical empiricists with the question of what kind of truth Carnapian analytic truth is if it is neither factual nor observationally confirmed . There were basically two answers . One was that it is truth ' " by convention . Hempel s ([ 1945] 1983) The Nature of Mathematical " Truth is a classic formulation of this answer . The other answer abandons the idea that stipulation can deliver truth in favor of a noncognitivism about logical and mathematical truth . As Carnap ( 1963, 64 ) puts it , theories " for the construction of a metalanguage for the analysis of " the language of science are meant " not as assertions , but rather as known, if at all, only a priori. This generalrequirementis part of the pressureto construct . In its fullest form, the controversybetween rationalism and philosophicalsystems empiricismis a controversybetweenentire systemsof philosophy.

70

Chapter3

" proposals. Driven by his perennial positivism about the nature of philosophical disagreement, Carnap claimed that issues about logical or mathematical truth that arise outside linguistic systems with explicitly formulated rules for theoremhood are meaningless. Moreover, since there is no cognitive basis for external debate about the correctness of such systems, philosophers can do no more than propose linguistic systems. Anything (everything ?) is okay as long as it is explicitly formulated . I shall discuss this position in chapter 6. Quine ([ 1936] 1983) famously refuted the first answer, showing that stipulation cannot even explain the truth of logical laws (much less their certainty ). Logical truths , being infinite in number, must be captured as instances of general principles , but , as logic is required for this offers no explanation of logical truth . (See enterprise, conventionalism also Benacerraf' s ([ 1973] 1983, 419- 20) for an important addendum to Quine : convention does not guarantee truth ; it gives us no distinction between " those casesin which it provides for it [and ] . . . those in which it does not " .) Quine (1961c) also mounted a powerful attack on the second answer. ' Carnap s noncognitivism has two components : a positivistic motivation for the claim that linguistic systems are essentially unconstrained proposals and a conception of analyticity as a basis for the understanding of logic and mathematics. Since positivism was widely seen as a failed program , the issue came down to whether Carnap ' s apparatus of meaning postulates can explain analyticity . Quine (1961c, 32" " 37) argued that such a recursive specification of the analytic sentences of a language tells us which of its sentences are to be assigned this label, but it does not tell us what property is attributed to the sentences with this label. As Quine (1961c, 33) put it , the label " might better be as ' K ' so as not to seem to throw light on the untendentiously written " ' the recursively interesting word analytic' . There is nothing to distinguish " " specified class of sentences with the label analytic from any other class of sentences, except for the fact that someone chose to label them such. than other Carnap (1963, 918) replied that analyticity is no worse off ' notions of formal logic , but this reply is grist for Quine s mill . Construed lines, " 5 is a logical truth in L," " 5 implies 5' along Carnapian in L," and so on are, of course, no worse off than analyticity , but , as Quine seesit , they are no better off either. Quine (1961c, 36) wrote : Appeal to hypothetical languages of an artificially simple kind could conceivably be useful in clarifying analyticity, if the mental or behavioral or cultural factors relevant to analyticity - whatever

character is unlikely to throw light on the problem of explicating analyticity . This criticism of Carnap ' s analytic / synthetic distinction was the first battle in Quine ' s campaign to replace logical empiricism with an uncompromising ' empiricism in the Millian spirit . Quine s (1961c, 37- 42) criticism of the verificationist account of synonymy was the second. The story is familiar : the relation of statements to experiences, on which the notions of confirmation and disconfirmation depend , presupposes the confirmability and disconfirmability of statements in isolation , and this , in turn , presupposes " a cleavage between the analytic and the " " " synthetic . When , as a consequence, the unit of empirical significance " " is taken to be the whole of science rather than the term or the statement, we get a new uncompromising empiricism free of the defects of Mill ' s empiricism . This is Quine ' s (1961c, 42- 46) famous conception of knowledge as a single fabric of empirically revisable beliefs with maximally revisable beliefs at the observational periphery and minimally revisable ones at the center. ' Quine s epistemology does a far better job of explaining the special ' certainty of logical and mathematical knowledge than Mill s. More central beliefs are more certain, not just because they have been more frequently confirmed in experience, but because of their larger role in knitting together the entire fabric . Central beliefs, being involved in more interconnections, receive more empirical support from the total predictive success of the system, but there is another, perhaps more , source of their certainty. Quine (1961c, 44) points out that significant " our natural tendency [is] to disturb the total system as little as possible ." This orders beliefs in the system so that the more centrally located statements are sacrificed only after other, less drastic , measures have failed to accommodate " recalcitrant experience." Hence, maintaining the equilibrium of the system means sacrificing mathematical and logical beliefs only as a last resort. Further, mathematical and logical statements enjoy a special certainty because the fate of the total system is bound up with the central mathematical and logical beliefs. Abandoning them removes the strands that hold the web of belief together, so that the system becomes a tissue of lies. This is worse than the opening up of an embarrassing hole here or there as the result of abandoning less central beliefs , because the unit of significance is the whole system. Loss of the main connecting strands maximally disrupts the entire network of

72

Chapter3

interconnections in the system, making the whole of our experience 3 literally incomprehensible . 3.3 The Revisability Paradox Full appreciation of the power of Quine ' s empiricist account of logic and mathematics must compel admiration from even the staunchest rationalist . It is quite surprising that an account based on so extreme a form of empiricism can steer contemporary empiricism past the Scylia of Millian inductivism and the Charybdis of logical empiricism and come so close to capturing the special certainty of mathematics and ' logic . Nonetheless, Quine s holistic conception of knowledge does not in the final analysis enable contemporary empiricists to provide a satisfactory account of the special certainty of logical and mathematical truth , because the conception is inconsistent. ' Quine s epistemology is an account of the way we adapt our system of beliefs to changing experience. It is an epistemology of reevaluation : A conflict with experience at the periphery occasions readjustments in the interior of the field . Truth values have to beredis tributed over some of the statements. Reevaluation of some statements entails reevaluation of others, because of their logical interconnections . . . . Having reevaluated one statement we must [sic] reevaluate some others, which may be logically connected with the first or may be the statements of logical connection themselves. (1961c, 42) There are three principles that are constitutive of this epistemology. One tells us when we " must " reevaluate, one tells us where we can reevaluate, and one tells us how we should reevaluate. The when -principle is, in effect, the principle of noncontradiction . It mandates revision when there is inconsistency. The system as a whole must be restored to consistency within itself and with its observational ' of universal revisability periphery . The where- principle in Quine s principle " " . It says that no statement of the system is immune to revision . As Quine (1961c, 43) puts it : 3. This is why it is child' s play for Quine to meetGriceand Strawson's (1956) objection that the distinctionbetween" not believingsomethingand not understandingsomething " can be used to draw the analytic/ syntheticdistinction. Quine canagreewith them that " it would be absurdto maintainthat the former distinction doesnot exist" but he can [ ] , saythat thelatterdistinctionis nothingmorethanthedistinctionbetweenthe very central and the not very central.

The EpistemicChallengeto Antirealism

73

Revision even of the logical law of excluded middle has been proposed as a means of simplifying quantum mechanics; and what difference is there in principle between such a shift and the shift whereby Kepler superseded Ptolemy, or Einstein Newton , or Darwin Aristole ? In permit ting no exceptions to the rule that " the tribunal of sense " experience is the supreme court in 'which all knowledge is judged , this principle is the heart of Quine s uncompromising empiricism . The how -principle is the principle of simplicity . It provides guidance about which statements are the best to revise in conflicts with Quine (1960, 20- 21) on conservatism.) experience. (See On Quine ' s epistemology, noncontradiction , universal revisability, and simplicity are different from other principles in our system of beliefs in this respect: they are constitutive of the epistemology of the system. The epistemology is a belief revision epistemology and those principles comprise the basic mechanism of belief revision . They thus serve as essential premises in every argument for reevaluating a belief. Every such argument has to assume the principle of noncontradiction as a rationale for departing from an assignment of truth to logically conflicting statements. The principle is required to initiate the process of revising presently accepted statements, otherwise we have to tolerate a radically laissez-faire epistemology on which anything goes. Further, every argument has to assume that the class of revisable statements is the class of all statements of the system, otherwise the epistemology will no longer be the uncompromising empiricism it was intended to be. Finally , every argument has to assume simplicity or something like it to narrow down the class of potentially revisable statements. Here is the paradox of revisability . Since the constitutive principles are premises of every argument for belief revision , it is impossible for an argument for belief revision to revise any of them because revising anyone of them saws off the limb on which the argument rests. Any argument for changing the truth value of one of the constitutive principles must have a conclusion that contradicts a premise of the argument , and hence must be an unsound argument for revising the constitutive principle . Consider a special case of the paradox . Given universal revisability , the principle of noncontradiction is revisable in principle . If it is revisable in principle , there is a possible belief -revision argument for its reevaluation . But , as we have seen, since the principle of noncontradiction is a constitutive principle , it must appear as a premise of the argument . But if it is right to revise a belief in the system, that belief was wrong all along , and if it was wrong all along , it cannot be a part

74

Chapter3

of a sound argument . The argument for revising the belief would be unsound and provide no grounds for the revision . Hence, there can be no sound argument for revising the principle of noncontradiction , and it is not open to revision . Nonetheless, since all beliefs are revisable, the principle of noncontradiction must be revisable, and hence it is both revisable and not revisable. The paradox was set out on the basis of Quine ' s own notion of reevaluation , namely, redistribution of truth values. But reevaluation might mean just dropping a principle from the web of belief. This ' option construes revision to consist in a statement s changing from being marked true to disappearing from the system, rather than construing revision to consist in a statement' s changing from being marked true to being marked false. This option does not avoid the basic paradox . All that happens with this change is that the argument for revising the principle of noncontradiction goes from being unsound to being invalid . For, instead of the argument having a false premise, it now lacks a premise essential to drawing the conclusion that the principle of noncontradiction ought to be revised (i.e., dropped from the web of belief) . The argument for revision is now invalid . Since all beliefs are revisable, noncontradiction must be, but since, again, noncontradiction is not revisable, it is, again, both revisable and not revisable. Looked at from the right angle, universal revisability already flashes the signal Paradox! Paradox! Paradox! Unrestricted universality sanctions the dangerous move of self-application , which is a familiar feature of paradox . From the application of the belief -revision epistemology to itself , it follows that a revisable principle is unrevisable. Hence, just as the barber paradox proves that there is no actual barber who shaves all and only those who do not shave themselves, so the revisability paradox proves that there is no actual epistemology that says that everything including itself is revisable. It might make senseto contemplate ' replacing any and every plank in Neurath s boat, but it makes no sense to contemplate replacing the basic principle of ship construction that says that there have to be planks between us and the water. The revisability paradox shows that no form of uncompromising empiricism , at least none that we are presently aware of , can meet the epistemic challenge to antirealism . The paradox also undercuts the Quinean explanation of how truths of mathematics and truths of logic can be taken to be about natural objects in the Quinean (1961c, 44) sense of being part of " a device for working a manageable structure into the flux of experience." Thus, both of the prima facie advantages of Quine ' s uncompromising empiricism over Mill ' s empiricism are lost.

The Epistemic Challenge to Antirealism

3.4 Wittgenstein's Naturalism Wittgensteinian naturalism provides an alternative way of taking an implacable stand against the rationalist position on a priori knowledge , the only way left . The price is, of course, abandoning empiricism as a philosophical position , since, for Wittgenstein , empiricism is another " " philosophical theory, as much a house of cards as any other. Indeed , the alternative entails the radical reconceptualization of philosophy of ' Wittgenstein s (1953) late philosophy, which rejects philosophical theories and the metaphysical issues that define them. Wittgenstein sought to eliminate what he took to be the linguistic confusions that underlie metaphysical issues. For Wittgenstein , traditional philosophy is not bad philosophizing as it was for Kant , nor bad science as it is for Quine , but simply nonsense resulting from metaphysical pictures that get us tangled up in the rules of language and prevent us from having a clear view of our linguistic practices. Good philosophy is linguistic therapy. It tries to give us a clear view of those practices by providing reminders of what our actual use of language is, accurate descriptions of particular , and so on. linguistic techniques ' s account of Wittgenstein certainty seems to be the only option for onto and remaining logical epistemological naturalists to account for the special certainty of logical and mathematical knowledge . " Wittgenstein (1956, pt . 2, sec. 39) says: To accept a proposition as to use it as a grammatical unshakably certain I want to say- means rule : this removes uncertainty from it ." For Wittgenstein , characterizations of mathematical results as absolutely necessary are only a " somewhat " way of putting things (1956, pt . 5, sec. 46). The " must " that hysterical mathematicians and philosophers typically use to express mathematical or logical compulsion is no more than the expression of an attitude towards the technique of calculation . . . . The emphasis of the must corresponds only to the inexorableness of this attitude both to the technique of calculating and to a host of related techniques. The mathematical Must is only another expression of the fact that mathematics forms concepts. (1956, pt . 5, sec. 46) Thus, Wittgenstein (1969, sec. 47) says, " Forget transcendent certainty, which is connected with your concept of spirit ." Wittgenstein (1969, sec. 39) thinks that absence of doubt in the case of mathematical calculation is a feature of our linguistic techniques and " practices: This is how calculation is done, in such circumstances a calculation is treated as absolutely reliable , as certainly correct."

76

Chapter3

Furthermore , the possibility of doubt is itself built into those techniques and practices, but in such a way as to make some expressions of doubt and others not. As Wittgenstein (1969, sec. 154) puts it : intelligible " There are casessuch that if , someone gives signs of doubt where we do not doubt , we cannot confidently understand his signs as signs of doubt ." While Quine ' s account of the certainty of mathematical and logical truth is unacceptable logically, Wittgenstein ' s is unacceptable linguistically . He claims that an expression of philosophical doubt in connection with a truth like " 3 = 2 + I " or with a properly carried out , fully conclusive mathematical demonstration is literally unintelligible. Wittgenstein (1953, sec. 119) would say that the words of a philosopher who expresses a skepticism about the certainty of a piece of logic or mathematics are " plain nonsense" , " bumps that the understanding has " got by running its head up against the limits of language. (See also Wittgenstein (1969, secs. 446 and 370).) To be sure, we would be confused if someone were to express doubt about a simple arithmetic truth . If Hermione says, quite sincerely, " ' 3 = 2 + l ' might be false," we would be at a loss to know what ' s up with her. But it is just because the English sentence she used, " ' 3 = 2 + l ' " might be false, is meaningful that sheis unintelligible to us. We are at a loss to know what she was up to in saying what she said. We know she knows the language and we also know she knows more than than to doubt the addition . It is, enough mathematics to know "better therefore, becausethe sentence ' 3 = 2 + l ' might be false" is meaningful in English , because the context seems to leave us no choice but to understand her utterance as a token of this sentence type and hence to have the meaning of the type , and because we know enough mathematics ourselves, that we can' t figure out what her motive might be for asserting it . Thus, Wittgenstein ' s account runs counter to clear intuitions about what is and is not meaningful . Consider this scenario. We learn that Hermione has been professor of mathematics at the College de France for many years. Has she gone nuts? No , she acts perfectly sane and passes every psychological test for normality with flying colors. What' s the answer? Finally , we are told : she' s read Descarteson what it means for God to be all -powerful , and that has shaken her faith in the certainty of arithmetic . Here, too, it is clear that it would be quite wrong to say that Hermione ' s assertion is " plain nonsense." Here Hermione ' s utterance is even intelligible to her audience. If Wittgenstein were right , the passages expressing skepticism in ' Descartes's Meditations, Hume ' s Treatise , Goodman s Fact, Fiction, and ' Forecast , Kripke s (1982) Wittgensteinon Rulesand Private Language , and

The Epistemic Challenge to Antirealism

77

' even Wittgenstein s (e.g ., 1953, sec. 185) Philosophical Investigations would be literally unintelligible , plain nonsense, literally signs without sense. But , whatever complaints there have been about such works , none has been about the unintelligibility of the language in them expressing the skeptical arguments of those philosophers . When we want genuine examples of meaningless language, we have to turn to 's Jabberwockyor to coinages from literary works like Lewis Carroll " " s Colorless . e. ( green ideas sleep furiously ). g , Chomsky' linguists There is real plain nonsense. The contrast between such cases and expositions of skepticism in and out of philosophical texts show that Wittgenstein was simply wrong about the linguistic distinctions between meaningfulness and meaninglessness, and hence wrong about certainty . ' As we saw in chapter 2 in connection with Burgess s (1990, 7) point about mathematical sentences, the question of what to say about the and meaninglessnessof sentencesof natural language meaningfulness " " the to pertinent specialist professionals in linguistics . Linguistic belongs intuition seems quite clearly to say that the philosophical texts in question , whatever else their defects, are not meaningless. In phi losophy, we are puzzled about how to answer the skeptical questions in those texts, but in linguistics , which abstracts away from philosophi cal content, there is not even a hint of meaninglessness about the sentenceswhich express the questions. (In chapter 6, I shall present a ' similar argument against Camap s positivism .)4 3.5 The Necessityand Certainty of Formal Truths If the arguments in the previous sections are correct, antirealism faces an epistemological challenge that it has no resources to meet. I now want to show that realism, which of course faces the same challenge, has the resources to meet it . If realism can account for the special certainty of mathematical and logical truths , then, combining such an ' s accountof 4. Wittgenstein certaintyrestson argumentsin the sectionsof the Philosophi . Those rule about the to and cal Investigations following including paradox leading up argumentspurport to show that there are no objectivefacts about meaning in the traditional senseof the term. But, asI (l990b) argued, the assumptionof thosearguments is that the basicfactsfor decidingquestionsaboutmeaningin natural languageare facts about use. This assumptionis not defended, and is open to questionbecausethere is another conceptionof the basic facts about meaning in natural language. On this , the basicfactsarefactsaboutthe sensepropertiesand relationsof expression conception " " and sentencetypes (e.g., that the word " bank is ambiguous, the expressions sister" " " " " , and " femalesibling" aresynonymous , the words blind and sighted areantonymous and the expression" freegift" is redundant). Factsaboutuseare derivative. They derive ' semantic from the speakers knowledgeof the sensesof expressionand sentencetypes,

78

Chapter3

account with the results of the last chapter , we have a strong case for the epistemological superiority of realism over antirealism . Realism explains the special certainty of formal truths in terms of their necessity, and their necessity in terms of the abstractness of the objects they are aboutis Necessary formal truths are necessary because they describe unchangeable properties and relations of unchangeable objects . Given that abstract objects can have neither spatial nor tempo rallocation , no property of their intrinsic structure and no relation they bear to one another solely in virtue of their intrinsic structure is subject to change .6 If no intrinsic aspect of an abstract object is subject to change , then no aspect of an abstract object can be otherwise than it is , their pragmaticknowledgeabout the useof tokens, and their knowledgeof featuresof the contextof use. Hence, the basicfacts for deciding questionsabout meaningdo not have to be, as Wittgensteinassumed , factsabout use. I (1990b , 21- 133) arguedfurther that sucha conceptionof the basicfactsaboutmeaning leadsstraightforwardlyto a theory of meaningfor sentencetypesthat is not subjectto ' Wittgensteins criticisms, and that the factsabout use that Wittgensteindeploysagainst theoriesof meaningcanbederivedfrom factsaboutsentencetypesand thecircumstances of their tokens. For example, the factsabout family resemblance that Wittgensteinuses againstthe classicalintensionalistclaim that meaningsare universalsare shown to be derivablefrom the meaningof generaltermsand featuresof the circumstances of their ' application. To take anotherexample, Wittgensteins rule- following paradox is shown not to arisebecause , on this theoryof meaning , we canspecifythe semanticfact in virtue of which a speaker's applicationof a word is corrector incorrect. (In the next chapter, I ' presenta similar treatmentof Kripke s relatedrule- following argument.) 5. If Fregewere right about the analyticity of mathematicaland logical truths, another ' explanationof their necessitywould be at hand. But the failure of Freges logicismblocks such an explanation. I would also arguethat it is blockedby the failure of his attempt to provide a sufficientlycomprehensive referentialsemantics . Frege's semanticspurports to bring the entire rangeof definitional truth under his broadenednotion of analyticity. This enabledFregeto explicateanalyticity in a way conducivenot only to arguing for logicism, but alsoto arguingfor the analyticityof logic. Thebasisof his broadenednotion of analyticity was his definition of senseas " mode of referentialdetermination," which enabledhim to understandcontainmentin termsof principlesof the theoryof reference . I (1992) argued that Frege's definition of sensemakesreferencethe centralconceptin the theory of sense , thereby ineluctably blurring the boundariesbetweensenseand referenceand therebybringing purely linguistic truths within the referentiallyspecified classof analytictruths. To draw the proper boundaries , sensehasto be definedwithout . It hasto be defined, not as the determinerof reference referenceto reference , but as the determinerof sensepropertiesand relations, suchas having a sense , having more than one sense , having the samesense , having oppositesenses , having a redundantsense , and so forth. (SeeKatz 1992.) Oncethis is done, the boundariesof linguistic truth are drawn sufficientlynarrowly so as properly to circumscribethe areaof the necessitythat is a matter of language. (For a systematicdevelopmentof theseideas, see Katz [in preparation].) 6. Only extrinsicaspects , suchasbeing thoughtaboutby someoneor beinginstantiated in this or that concreteobject, are subjectto change. Seethe further discussionof the point in chapter5.

The Epistemic Challenge to Antirealism

79

and connections that depend on only such aspects are necessary. Since the predication expressed by a formal truth holds in virtue of the fact that the property or relation that it attributes to the abstract objects it is about is one of their intrinsic (structural ) properties and relations , there is no possibility that the objects do not have the property or relation . Hence, such a formal truth is a necessary truth .7 This account is uniform . Necessity is the same thing in connection with mathematical , logical , and linguistic truths . To be sure, linguistic , logical , and mathematical objects differ in ways that reflect differences among linguistic , logical , and mathematical structure . The necessary connection between being the number two and being the only even prime is an aspect of the abstract arithmetic structure of numbers. The necessary connection between being a consequence of true premises and being true is an aspect of the abstract logical structure of propositions . And the necessaryconnection between a sentence's being analytic and its being nonsynonymous with every synthetic sentence is an aspect of the abstract semantic structure of senses. But, even though necessaryconnections can be aspectsof different structures in different kinds of abstract objects, such differences do not affect the nature of the necessity. The unchangeableness of the connections within and among the structures of abstract objects turns on their common abstractness . In note 1 of chapter 1, I indicated that I would be using the term " formal truth " as an abbreviation for " mathematical , logical , or simply " linguistic truth . I will continue to use it this way outside the context of realistic rationalism , but when speaking about matters inside this " " , formal truth will be used with the more descriptive meaning position " " proposition that is true of abstract objects in virtue of their form . 3.6 Eliminativism and Supervenience The above account of the necessity and certainty of formal knowledge makes no effort to explain them in nonmodal terms. I have not tried to eliminate the modal idiom because I think there isn' t a prayer of ' doing it . Furthermore , I don t think that modal properties and relations stand in need of an account in terms of nonmodal properties and relations to avoid circularity , as eliminativist naturalists about modality 7. A natural question about the above explanation is whether realism says that abstract objects have necessary existence. If some possible worlds do not contain some abstract ' objects, then the realist can t say that formal truths about abstract objects are necessary truths , since formal truths about such abstract objects are not true in all possible worlds . The realism I have developed in this book takes abstract objects to have necessary existence. Chapter 5 explains why realists should take this position on abstract objects.

80

Chapter3

suppose. Rather, their circularity criticisms reflect a misconception on their part of the nature of explanation in the area. The situation here is parallel to Quine ' s (1961c, 27- 32) e1irninativist criticism of the theory of meaning . As I (1990b, 175- 202) have argued, the requirement of a noncircular explanation to certify synonymy or any other concept in the theory of meaning is just a reductionist fiction . Quine promoted this requirement in the philosophy of language because he (1961a, 11) himself had so strong a prejudice against meanings as entities that he dismissed them without so much as a hearing . Quine (1961b, 48) urges us to: . . . resolve to treat " alike in meaning " in the spirit of a single word " " synonymous , thus not being tempted to seek" meanings as entities. . . . [and to ] treat the context having meaning intermediary " in the " " spirit of a single word , significant , and continue to turn our backs on the suppositious entities called meanings. The only reason Quine doesn' t advocate the same strategy with sentences - treating " is a well -formed sentence" in the spirit of a single " word grammatical " and turning our backs on sentences- is that he does not have it in for them. Somehow one kind of abstract object, a sentence type , is kosher, but another kind , a meaning, isn' t (see Quine 1987, 216- 19) . This intolerance led Quine (1961b) to embrace the similarly intolerant Bloomfieldian linguistics . Quine ' s (1961c, 27- 32) argument against synonymy in " Two Dogmas of Empiricism " is an application of Bloomfieldian operationalism . Quine himself makes this clear when he (1961b, 56- 57) adopts Bloomfieldian operationalist substitution criteria as the proper approach to the problem of synonymy , and points out that his argument against synonymy in " Two Dogmas of Empiricism " is an application of this operationalism to semantics. Quine ' s (1961b, 57) claim that the attempt to make objective senseof synonymy results in " something like a vicious circle" is an artifact of the operationalist demand that the attempt be carried out without the use of intensional concepts, and his overall argument against intensional concepts is a non sequitur because it leaves the possibility of theory construction as a way of explaining them open. The requirement of a noncircular explanation for modal properties and relations is similarly just a reductionist fiction , motivated by a prejudice against unreduced modal notions . It is certainly not the case that reduction is required for semantic or modal concepts to do their work . I (1972, 1987) have tried to show that explanations of sense properties and relations in terms of the structure of sensesareseman tically illuminating . Explanations of modal properties in terms of the

The Epistemic Challenge to Antirealism

81

structure of abstract objects are metaphysically illuminating . In connection with the explanation of the necessity of mathematical and logical truth , illumination comes from their necessity being grounded ontologically, rather than simply being left as a consequenceof the definition of necessary truth as truth in all possible worlds . Our realist account explains why mathematical and logical truths are true in all possible worlds on the basis of the onto logical status of the objects that such truths are about and the nature of the mathematical or logical structure of those objects. These considerations bear on Kim ' s (1993) attempt to develop a naturalistically based normative epistemology. Kim eschews areduction of the normative to the natural , but he requires that the normative supervene on the natural in a way that makes it possible to provide criteria for justified belief in terms of subvenient naturalistic phenomena . This requirement has teeth because Kim (1993, 235- 36) makes the significance of an evaluative or normative concept depend on its being governed by criteria expressible naturalistically. Kim (1993, 216- 19) adopts the quasi-eliminativist requirement that the criteria of justified belief must be formulated on the basis of descriptiveor naturalistic termsalone, without the useof any evaluative or normative ones, whether epistemicor of another kind. (Italics in original ) I applaud Kim ' s rejection of reductionism , but I can' t go along with this requirement . He (1993, 218) says that the difficulty with the use of terms is just that they " are themselves essentially normative epistemic " . I find this explanation unhelpful because Kim ' s requirement is " already "written to preclude the use of any evaluative or normative [terms], presumably on grounds of circularity . But there is no argument to show that there is vicious circularity, and hence to preclude an ' explanation of justification in normative terms. Why isn t it legitimate to use normative concepts to explain the concept of justification , just as it is legitimate to use the concept of sense to explain the concepts of analyticity and synonymy, to use logical and mathematical concepts to explain other logical and mathematical concepts, or to use the onto logical concept of an abstract object to explain a modal concept? How would Kim handle justification in mathematics and the other formal sciences? Global supervenience obtains trivially : since mathematical and logical truths hold in all possible worlds , worlds that coincide with respect to truths involving natural properties and relations coincide with respect to truths involving mathematical and logical properties and relations. But the supervenience relation required for

3 82 Chapter Kim ' s naturalistically based epistemology must provide criteria for particular justified mathematical and logical beliefs in terms of particular subvenient naturalistic phenomena. This requires a local supervenience relation between particular mathematical and logical truths and particular natural truths , but this supervenience relation , I submit , cannot be generally supposed to obtain in the case of theorems about numbers , sets, spaces, and even sentences. There is nothing in the natural world that can serve as an appropriate supervenience base for the explanatory task in the case of proofs of theorems about such entities. Moreover , if realism is right , there is no alternative to normative ' concepts serving as the basis for other normative concepts. If , as I argued in the last chapter, we can have knowledge of abstract objects a priori and the formal sciencesare essentially different from the natural sciences, and if , as I will argue in the next chapter, abstract objects in the domain of a formal science are the norms governing correctness in the science, then there is no need for a requirement of supervenience on natural aspects of reality . Normative judgments in the formal sciences (or, as I shall argue in the final chapter, in their philosophical foundations ) are true just in case they reflect the structure of the relevant abstract objects. For example, since the norms forsentencehood in English are the sentence types of English , the claim that an utterance or inscription is well -formed in English is true just in case it is a token of a sentence type of English . Kim (1993, 236) claims that " rejecting the supervenience of the normative on the natural would sever the essential connection between value and fact on which . . . the whole point of our valuational activities depend " . This claim has no force within a realist framework where valuational activities derive their normativity from abstract aspects of reality .8 3.7 Conclusion Naturalists do not have a satisfactory account of the certainty of formal truths , and hence they fail to meet the epistemological challenge to antirealism . I do not pretend to have ruled out all possibility of anti realists' coming up with another plausible naturalistic explanation of the special certainty of logical and mathematical truths to replace ' ' Quine s (or Wittgenstein s). But , whatever hopes for the future natural 8. Though one cannotbut havesomesympathyfor Moore's (1942 , 588) position, it has neverbeenclearto me what natural fact the good might superveneon. Mooredoesnot " " " " " " satisfactorilyexplainmoralusesof good asin goodperson and goodhumanbeing. Kim (1993, 235) doesnot cometo his aid.

The EpistemicChallengeto Antirealism

83

ists may entertain , at this point , they have no explanation of the special one. If this certainty of such truths and no prospects for obtaining conclusion is taken together with the rationalist epistemology set out in the preceding chapter, the true picture of the realist / antirealist antirealists have controversy emerges. In contrast to the picture that been promoting over the years, the true picture is that it is antirealism , rather than realism , that faces an epistemological challenge it has not met and , as far as one can see, lacks the resources to meet.