What Mathematical Knowledge Could Be

What Mathematical Knowledge Could Be Jerrold J. Katz Mind, New Series, Vol. 104, No. 415. (Jul., 1995), pp. 491-522. Stable URL: http://links.jstor.org/sici?sici=0026-4423%28199507%292%3A104%3A415%3C491%3AWMKCB%3E2.0.CO%3B2-H Mind is currently published by Oxford University Press.

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What Mathematical Knowledge Could Be JERROLD J. KATZ One evening after sewices a Chelmite began searching the floor of the synagogue. "What are you looking for?" asked someone who was visiting Chelm from another town. "I lost a ruble on the road, so I'm looking for it." "You unfortunate Chelmite", the visitor taunted him, "why are you looking here when you lost it on the road? " "You're clever, are you" the Chelmite retorted, "The road is dark; here there b plenty of light. So, where is it better to look?"'

I . Introduction Many anti-realists think that the message of Benacerraf's "Mathematical T r u t h (1973) is that mathematical knowledge could not be knowledge of abstract objects because spatio-temporal creatures like ourselves can have no contact with what has no spatio-temporal location. Two such anti-realists are Gottlieb (1980, p. 11) and Field (1980, p. 98). Benacerraf himself is not prepared to go that far. Although, because of the impossibility of contact with abstract objects, he thinks that the realist's chances of coming up with a satisfactory epistemology are not very good, he is unwilling to put as much weight on the causal theory of knowledge as is necessary to conclude that mathematical knowledge could not be knowledge of abstract objects. He does not argue that realism, or any of the other philosophies that he criticizes, should be rejected. Rather, he argues that no philosophy of mathematics, as it now stands, can be accepted. None has both a satisfactory semantics for mathematical language and a satisfactory epistemology for mathematical knowledge. Benacerraf's message, I Chelm, according to Yiddish folklore, is where an angel distributing the souls of fools around the world accidentally spilled the entire bag.

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intended for realist and anti-realist alike, is "Back to the drawing boards! ". The present paper is a report from the drawing boards of one realist. It presents an account of how we can have mathematical and other formal knowledge when there is no possibility of making causal contact with the objects of such knowledge. Because there are too many issues for the space available, it is far from a full a report. But it is, I hope, full enough to meet Benacerraf's epistemological challenge to realists and to give the lie to "refutations" of realism based on the fact that the aspatiality and atemporality of abstract objects puts them beyond our causal reach.

2. Truth or knowledge? Benacerraf (1973) thinks that philosophies of mathematics have to provide both a plausible semantics for number-theoretic propositions and a plausible epistemology for mathematical knowledge, but that none of the available philosophies of mathematics can do this. Each satisfies one of the requirements at the expense of the other. "Separately", he says, "they are innocuous enough", but "jointly they seem to rule out almost every account of mathematical truth that has been proposed" (p. 410). The solution to the problem, as Benacerraf sees it, is to be found in a better account of mathematical truth than any of those in current philosophies of mathematics. Either not all the accounts we have of mathematical truth are properly formulated or we do not have all the accounts of mathematical truth. In so far as the problem is generated by an epistemic as well as a semantic requirement, its solution can lie in either epistemology or semantics. Hence, looking for the solution in the area of truth might be looking for it in the wrong place. No doubt there's plenty of light there, but the solution could lie in the darker area of knowledge. Nothing, as far as I can see, rules out an epistemic solution. Benacerraf rightly requires that "an account of mathematical truth ... must be consistent with the possibility of having mathematical knowledge" (p. 409), but this requirement, although weak enough to be generally acceptable, is not, as it stands, strong enough to rule out an epistemic solution. To do that, the requirement would have to be bolstered with a further clause restricting its notion of possibility to what the causal theory of knowledge allows as possible knowledge. Such a restriction is, of course, in line with Benacerraf's own sympathies. Referring to the "core intuition" of epistemologies such as Goldman's (1967, pp. 357-72), Ben-

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acerraf claims that "some such view must be correct" (1973, p. 413). Nowadays such a claim can no longer seem as reasonable as it did in the early seventies, but, even putting aside the difficulties with the casual theory of knowledge that have come to light over the years, imposing a causal condition on mathematical and other types of formal knowledge is too sttong. Such a strengthened requirement prohibits the realist from giving an epistemic solution to Benacerraf's problem, but the requirement is now no longer weak enough to be generally acceptable. Strengthening the requirement by generalizing the causal theory to all formal knowledge makes its acceptability depend on the acceptability of empiricism. After all, empiricism's claim that knowledge depends on experience (the justification condition cannot be met without experience of some kind) is just the claim that the causal theory in some form applies to all knowledge. Hence, there being no argument establishing empiricism, there is no argument that the strengthened requirement is one that everyone has to accept. Benacerraf wants an ontology which allows causal connection for epistemic reasons: if numbers are abstract objects, "then the connection between the truth conditions for the statements of number theory and any relevant events connected with people who are supposed to have mathematical knowledge cannot be made out", since our "four-dimensional space-time worm does not make the necessary (causal) contact with the grounds of the truth of [those statements]" (1973, pp. 413-4). The point is undeniable. But it does not show that we cannot come to know abstract objects, only that we cannot come to know them in the way we come to know concrete objects, that is, via a connection between ourselves and the objects of knowledge. For sure, the condition that has to be satisfied to know number-theoretic statements cannot be one whose satisfaction insures that we are causally related to the fact to which the statement corresponds. But, since empiricism hasn't been established, it is still an open possibility that the condition is one whose satisfaction insures correspondence to number-theoretic facts, not, as it were, courtesy of our senses, but purely a priori. Benacerraf's claim that the realist account of mathematical truth does not mesh with "our over-all account of knowledge" (1973, pp. 412-5) has no force against a realist account on which mathematical knowledge is purely a priori. Given what he means by "our over-all account of knowledge", his claim comes down to the assertion that a realist account of mathematical truth does not mesh with empiricism. But since there is no demonstration that empiricism is the best over-all conception of knowledge, the claim cannot be used to argue against a rationalist epistemology without begging the question.

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Benacerraf recognized this danger. In the unpublished manuscript from which much of the material in "Mathematical Truth" came, he notes that the focus in that manuscript might also have been on the concept of knowledge (1968, p. 53). He indicates that the assumption that the source of the problem is our understanding of mathematical truth rather than our understanding of human knowledge reflects his personal confidence in a causal theory. And he explicitly recognizes that arguing from that assumption might strike people as "stacking the deck". Benacerraf says that his

... claim is that with the concepts of knowledge and truth, extricated as I have suggested, we do not seem to have adequate accounts of mathematical truth and mathematical knowledge. I am open to suggestion on how the analysis of either concept might be improved to remedy this defect. (1968, p. 53) Hence, realists have the option of locating the defect in the empiricist concept of knowledge rather than the Tarskian concept of mathematical truth. 3. Towards an epistemological solution Although Benacerraf leaves room for an attempt to formulate an "improved" concept of mathematical knowledge, he, as we have seen, takes the position that "some such view [as the empiricist's concept of knowledge] must be correct". This strikes me a significantly stronger position than is warranted by the prima facie facts about the current situation in the philosophy of mathematics. On the one hand, Benacerraf's own semantic argument for realism over anti-realism-the positions Benacerraf (1973, pp. 4 0 6 7 and p. 416) calls "combinatoria1"-is much stronger than the argument seems on his presentation, and, on the other, there are doubts about the generality of an empiricist concept of knowledge that have not been taken into consideration. In this section, I want to show that the grounds for seeking an improved concept of mathematical knowledge may well be stronger than the grounds for seeking an improved concept of truth. Benacerraf pointed out that realism has the advantage of allowing a uniform semantic treatment of mathematical and non-mathematical sentences (1973, pp. 405-12). On realism, a sentence like (1) and a sentence like (2) are both straightforward instances of the logical form (3). (1) There are at least three perfect numbers greater than seventeen. (2) There are at least three large cities older than New York.

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(3) There are at least three FGs that bear R to a. On the anti-realist's combinatorial approach, (2) is not an instance of (3). Benacerraf (1973, pp. 410-2) bases his preference for a semantics that treats the logical form of mathematical sentences and corresponding nonmathematical sentences in the same way to one that treats them in a different way on the success of Tarskian semantics generally, coupled with the absence of an appropriate semantics for combinatorical approaches and with the difficulty of coming up with one.2 These considerations are certainly a reason for skepticism about the prospects for a satisfactory semantics within an anti-realist framework, but there is another reason which, when added to these, makes the prospects seem dim indeed: arguably, a multiform semantics would be undesirable even if we could come up with one, precisely because it treats the logical form of mathematical sentences and corresponding non-mathematical sentences in the same way. This argument against anti-realism must also contain an argument against a non-combinatorial nominalist view like Field's (1980 and 1989) that takes reference to numbers to be reference to fictional entities and mathematical truth to be truth in a certain type of fiction. As Field says, "The sense in which '2 + 12 = 14' is true is pretty much the same as the sense in which 'Oliver Twist lived in London' is true" (1989, pp. 2-3). For Field, the former statement is true according to the well-known arithmetic story, while the latter statement is true according to the well-known Dickens story. Some such view is needed to distinguish truths like "2 + 2 = 4" from falsehoods like "2 + 2 = 17". But truth in mathematics is not the same as truth in fiction. The striking difference is that consistency is a necessary condition for truth in mathematics but not a necessary condition for truth in fiction. A fictional character's having incompatible properties (in the fictional work) does not rule the character out of fictional existence, but a mathematical object's having incompatible properties does rule that object out of mathematical existence. If I remember correctly, Dr. Watson has incompatible properties: his one army wound, caused by that famous "Jezail bullet", is both in his shoulder and it is in his leg. It would be wrong to take this to show that Dr. Watson does not exist (i.e., does not fictionally exist in the sense in which Hamlet's wife does not fictionally exist)--and also wrong to say that the "inconsistent" adventures of Sherlock Holmes entail everything, e.g., that Holmes is Lestrade. In contrast, it is right to say that there is no largest number. The criticism does not depend on any actual example. It is pointless to quibble about the above case, for example, by arguing that the locations are mentioned in different Conan Doyle's stories (so one of them must be false in the Sherlock Holmes corpus). Hypothetical cases do just as well. Further, it does not help to try to distinguish mathematical fiction from, as it were, fiction fiction, for example, by saying that it is part of our logical pretence about mathematics (but not about fiction) that nothing can have incompatible properties. If arithmetic and literature are both fiction, why couldn't it be the other way around, logical pretencewise? We can't have inconsistency in mathematics-that's a logical impossibility-but we can in fiction. The best explanation of why consistency is a constraint in mathematics but not in fiction is that fiction is fiction and mathematics is fact.

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Burgess (1990) makes a start toward developing such a reason. He observes that the choice between a uniform and a multiform semantics for natural language belongs to "the pertinent specialist professionals" in linguistics (1990, p. 7), and expresses strong skepticism that a multiform semantics would pass muster with them. I think Burgess is certainly right that the choice has to be made on the basis of which semantics best fits into the linguist's account of the grammar of natural language. The argument against multiform semantics emerges when we see why its treatment of sentences like (1) and (2) would not be acceptable in linguistics. Benacerraf observed that (1) is as much an instance of (3) as (2). This suggests that what is wrong with treating (2) but not (1) as an instance of (3) is that sentences which are essentially the same in grammatical structure are being treated as essentially different in grammatical structure. But there are cases in linguistics where sentences which appear to have the same grammatical structure are treated as different in grammatical structure. However, such cases differs from the combinatorialist's treatment of (2) but not (1) as an instance of (3). The difference is that the linguistic cases are ones where the appearance of sameness of grammatical structure does not go below the surface. So, the linguist's treatment is sacrificing no more than a surface grammatical similarity and the sacrifice is for the higher good of preserving a deeper, more pervasive grammatical similarity. A classic example of this treatment in the history of linguistics is the postulation of different underlying structures for the sentences (4) and (5). (4) John is easy to please. (5) John is eager to please. The critical point about this example is that the postulation is grammatically driven. For, despite the surface similarity of (4) and (5), speakers have a strong linguistic intuition that "John" is the direct object of the infinitive in (4) and the subject of the infinitive in (5). The linguist's desire to account for those intuitions-in accord with the prevailing theory of grammatical relations (see Chomsky 1965hmotivated a transformational analysis of (4) and (5) on which their surface similarity masks deep grammatical differences. Sentences like (4) and (5) are the "exceptions" that prove the linguist's rule that grammatically similar sentences are to be similarly described in the grammar. The transformational analysis of (4) and (5) is both appropriate to linguistics and properly implemented. The linguist's aim is to obtain a more encompassing analysis of grammatical structure than is possible on a description which preserves surface similarity. The analysis is properly implemented because the assignment of different underlying syntactic

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structures is based on grammatical facts about the sentences. In contrast, the multiform analysis of (1) and (2) is neither appropriate to linguistics nor properly implemented. The combinatorialist's aim is the linguistically irrelevant desire to avoid Platonism in the foundations of mathematics. The implementation of the analysis is improper because not only is there no strong grammatical intuition reflecting a difference in their grammatical structure, there is not even a hint of the kind of underlying grammatical differences we find in connection with (4) and (5). From a grammatical perspective, ( I ) and (2) have equally good claims to be instances of (3). (1) is not, moreover, an isolated case. There are infinitely many sentences of English in which number terms occur in referential position. Hence, if the concerns of a partisan viewpoint in the philosophy of mathematics are allowed to decide questions of grammatical structure, distinctions reflecting no grammatical differences will be made over a wide segment of the language. Since such distinctions are only philosophically motivated, a multiform semantics would compromise the autonomy of linguistics, making linguistic argumentation degenerate into philosophical debate. (Why should the linguist let the concerns of anti-realism decide; why not those of realism?) To presenfe the autonomy of linguistics and the integrity of its argumentation, grammatically irrelevant considerations cannot be allowed to determine the description of sentencese3 We now turn to doubts about generalizing the causal condition from cases of empirical knowledge, such as Hermione's knowledge that she is holding a truffle, to cases of formal knowledge, such as (6H8). (6) Seventeen is a prime number. (7) No proposition is both true and false. (8) An anagram is an expression made by transposing the letters of another expression. Given that empiricism has not been established, there is no reason to think that the causal condition generalizes to all knowledge, including cases that are not clearly empirical like (6H8). In the case of numbers, propositions, and expressions (in the type sense), prima facie there doesn't seem Someone might reply that it would be a good thing if the present disciplinary boundaries were to disappear so that questions within disciplines could be decided by arguments from other disciplines in a hlly interdisciplinary way. It is not clear to me what would be so good about this, either for scientific disciplines or for philosophy. If decisions within disciplines became so radically interdisciplinary, it would wreak havoc with argumentation about the nature of phenomena in a scientific discipline, since its practitioners could then substitute philosophical arguments for scientific description. We would forfeit the constraint to save the phenomena. Moreover, it wouldn't be so good for philosophy to be without an independent scientific characterization of knowledge by which to judge philosophical accounts of knowledge.

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to be anything in the natural world to which our knowledge can be causally connected. Not only is there, as it were, no truffle, but there doesn't seem to be anything to which such knowledge can be causally related in the way that theoretical truths in physics can be causally related to events in cloud chambers or pictures from radio telescopes. There seem to be neither natural objects to serve as the referents of the terms in (6H8)nor natural facts to which such truths may be taken to correspond. Another doubt about whether the causal condition generalizes is that (6)-(8) are necessary truths. Quine and other naturalists have denied that there are necessary truths but, in the fullness of time and modality, his arguments have not proven to have much force.4 Given that (6H8) are necessary truths, they are prima facie counter-examples to a causal condition on all formal knowledge. There is nothing in the natural world to explain their necessity. Experience, as Arnauld, Leibniz, and Kant observed, can account for the truth of such statements but not for the fact that they couldn't be otherwise. Benacerraf quite correctly claims that a realist account of mathematical truth does not mesh with an empiricist account of knowledge. But this, as we have seen, need mean no more than that we cannot know truths about mathematical objects in the same way we know truths about'natural objects. Assuming realism, then, the existence of mathematical knowledge shows that there is another way of knowing mathematical truths. Since it is unlikely that realism meshes with anything but an account of knowledge in the Cartesian and Leibnizian traditions, that other way of knowing mathematical truths should be what traditional rationalists called "the light of reason".

4. Mysticism and mystery Anti-realists like Gottlieb, Field, Chihara, and Durnrnett would dismiss this approach on the grounds that a rationalist epistemology is mysticism Given that the argument for a causal condition in the case of mathematical knowledge is so weak, it seems that the prevalence of the naturalistic outlook is the only thing that instills confidence in the prospects of empiricism as a general theory of knowledge. Naturalism supplies the premiss required to generalize from a causal condition on justification in the uncontentious case of empirical knowledge to all cases of knowledge. Having an ontology which says that all objects of knowledge are uniformly natural objects and also having accepted a causal condition on knowledge of natural objects, there is overwhelming pressure to generalize the causal condition to knowledge of numbers, sets, propositions, sentences, meanings. Katz (1 990a) argues that the principal arguments that twentieth century philosophers have given for naturalism are too deeply flawed to make it acceptable.

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as much as the ontology it serves. Gottlieb says, "Abstract entities are mysterious and must be avoided at all costs" (1980, p. 11). Field says that the realist "is going to have to postulate some aphysical connection, some mysterious grasping" (1982, p. 59). Chihara says that the realist's appeal to Godelian intuition is "like appealing to experiences vaguely described as 'mystical experiences' to justify belief in the existence of God" (1 982, p. 2 15). Durnmett says that Godelian intuition "has the ring of philosophical superstition" (1967, p. 202). Such philosophers think that realism is mysticism because they think that, in the absence of objects acting directly or indirectly on us, the realm of abstract objects might be any way whatever for all we, in our windowless state, would know. Given that realists claim that we can know how things are in the realm of abstract objects in spite of our having no natural connection to them, realists can only believe in some form of supernatural connection. No doubt there is a mystery about how we can have knowledge of abstract objects. But philosophy is full of such mysteries. Every philosophical problem is one. Anti-realists like Gottlieb, Field, Chihara, and Durnmett thus make much too much of that particular philosophical mystery. Further, obscurity about the workings of a cognitive mechanism cannot be grounds for distrusting it. The goings on behind the phenomenological scenes in mathematical intuition and reason are certainly obscure, but not more so than the goings on behind the phenomenological scenes in sense perception. The mystery of how spatio-temporal creatures like ourselves obtain their knowledge of objects that are causally inaccessible is precisely what sober realists have set out to solve. It is hard to see why their project deserves more disparagement than other partisan philosophical projects such as, for example, Field's or Durnmett's. Some realists do stray off into mysticism, but to criticize all for the sins of some is just the politics of guilt by association. A philosophical mystery is no grounds for crying "mysticism". Mysticism involves a claim to a means of attaining knowledge beyond our natural cognitive faculties. Those who cry "mysticism", "superstition", and the like need to be reminded of the fact that our sensory faculties do not exhaust our cognitive faculties. Sophisticated empiricists recognize an autonomous rational faculty. Benacerraf himself seems to recognize that the operations of reason can't be reduced to the operations of our other faculties when he says that "knowledge of general laws and theories, and, through them, knowledge of the future and much of the past" is "based on inferences based on [perceptual knowledge of medium-sized objects]" (1973, p. 413). The possibility of an epistemology based on one of our natural cognitive faculties ought to go quite far in dispelling irresponsible charges of mysticism.

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5. Toward a rationalist epistemology In blaming empiricism for a defective concept of knowledge and trying to exonerate realism of the charge of mysticism, I do not mean to suggest that realism is blameless. Realism has one major drawback: it has no plausible epistemology available to meet Benacerraf's challenge. Too many realists have followed Plato's example, basing their epistemology on some form of perception, and, hence, on what is in reality an empiricist notion of acquaintance. The result is a combined ontological and epistemological position which incoherently claims that we (or our souls) are causally acted on by causally inert objects. From this perspective, Frege's silence on the topic of grasping is golden. And Godel, who has been so badly maligned on this score, is, if anything, even more admirable in making clear that the "kind of perception, i.e., in mathematical intuition" on which mathematical knowledge rests "cannot be associated with actions of certain things upon our sense organs" or with "something purely subjective, as Kant asserted" (1947, p. 484). Godel also pointed realists in the right direction in going on to say that the presence of such intuitions in us "may be due to another kind of relationship between us and reality" (1947, p. 484). I think these remarks set the task for realist epistemology, which is to provide a coherent account of this other kind of relationship that explains how, in spite of having no window onto the realm of abstract objects, we can nonetheless know what things are like there.

5.1. Epistemic conditions Like Benacerraf (1973, p. 414), I will assume that knowledge is justified true belief. The notions of justification, truth, and belief are not understood in any special sense, but just in their familiar philosophical sense. Someone has a belief about something when he or she takes a proposition about it to be true. A proposition is true when the facts are as it says they are. (Certain disquotational notions of truth are appropriate, but, as Benacerraf's challenge to realism assumes a correspondence notion, our response can assume it, too.) We have a justified belief when we have adequate grounds for thinking it true. Since all of this is reasonably uncontroversial, our explanation of how we can have knowledge of abstract objects should be compatible with a wide range of views in the theory of knowledge. 5.2. The belief condition The constraint that mathematical knowledge is based on reason alone applies as much to the belief and truth conditions as it does to the justifi-

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cation condition. No aspect of the realist's epistemology can presuppose contact with abstract objects. In the case of the belief condition, this means that no aspect of the content of our mathematical beliefs can depend on contact with abstract objects. This is a principal reason that nativism has always been an essential component of the rationalist conception of our faculty of reason. The basic rationalist claim is that the concepts required to form beliefs about inter alia abstract objects are either themselves constituents of the faculty of reason or else derivable from concepts which are constituents of that f a c u l t y 4 n the basis of combinatorial principles that also belong to the faculty. (See Katz 1979 and 1981, pp. 192-220.) Of course, realists are not committed to any particular version of nativism. Plato's version, as just mentioned, is unacceptable because it depends on the soul's acquaintance with abstract objects. Other versions of nativism may not be rich enough to account for the full content of our formal beliefs, or have other defects that make them unsuitable. The realist is only committed to there being some version of nativism which does the job.' One candidate might be Chomsky's nativism (1965, pp. 47-59). This theory hypothesizes that the child has innate knowledge of the grammatical structure of natural language, and also innate principles for putting this knowledge to use in acquiring (tacit) knowledge of a natural language (competence). Given that the child's innate knowledge represents the full range of possible competence systems for natural languages, the child's task is to discover, on the basis of a sample of its utterances, which competence system from its innate space of such systems underlies the sample. The specific way in which the child performs this task, whether on the basis of a parameter setting device, a hypothesis testing device, or some as yet unformulated device, is of no concern here. What is of concern is that the child's innate knowledge of semantic structure be rich enough to make it unnecessary for experience to provide any of the concepts out of which we form mathematical beliefs, that is, to make Nativists have presented arguments for a rich system of innate concepts (see, for example, Fodor 1980, pp. 143-9). It is not clear how rich our innate semantic system must be in order to provide the account of our beliefs about abstract objects that realism requires, but it must be rich enough to provide an account of the semantic competence of speakers. Given that there are infinitely many non-synonymous expressions in a natural language, our full stock of concepts will be infinite. Since this infinite stock must be representable in the speaker's finite m i n d brain, a nativism acceptable to realism must posit a recursive mechanism for expanding a finite set of primitive concepts into the full infinite stock of concepts. As I see it, the simplest, and hence most desirable, nativism is one which characterizes the finite set of primitive concepts and a combinatorical mechanism in such a way that the latter does the maximum work. The simplicity of a nativism

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it unnecessary for experience to do more than provide the occasion for correlating expressions with concept^.^ Some adherents of the causal theory of reference might claim that mathematical realism precludes our having mathematical beliefs, since, even if we have innate numerical concepts, our words can only come to refer to numbers causally. If we are cannot refer to numbers, we cannot have beliefs about them. There are two responses to this criticism. First, the causal theory is not etched in stone. I have argued that there is a new version of the description theory which avoids the problems that have been taken to refute it (1990b, 1994, and in preparation). Second, not every version of the causal theory of reference supports this criticism of realism. Only a completely general version which says of every term what Putnam says of proper names (1975, p. 2 0 3 F t h a t some member of the community must have or have had causal contact with the bearer of the name-would deny the possibility ofreference to abstract objects. A completely general version of the causal theory of reference, like the comcan be measured by the size of the set of primitive concepts relative to the output of the recursive mechanism. Generally speaking, the smaller the list of primitive concepts, i.e., the fewer the case-by-case specifications, the simpler the nativism (given a factually warranted combinatorical mechanism). This implies that, other things equal, the most desirable nativism is one whose semantics is decompositional because, in such a nativism, the senses of lexical items are built up from a small stock of primitive senses on the basis of the same combinatorical relations found in compositionally formed senses above the lexical level (see Katz 1972 and 1977). Such a nativism contrasts with Fodor's (1981, pp. 257-3 16) on which the stock of primitive concepts for a language contains a distinct concept for every non-synonymous, syntactically simple word in the language. When we recognize, as Chomsky frequently urges, that the infant is an all-purpose language learner, the stock of primitive concepts, even assuming it is finite, is, as many have noted, too incredibly vast for Fodor's nativism to be plausible. At the very least, it is an Occamesque nightmare-perhaps even too large for the storage capacity of the brain. When we recognize that, theoretically, the possible non-synonymous lexical items can be infinite, Fodor's nativism is a logical impossibility. Fodor bites this bullet, presumably because he sees the decompositional semantics as worse yet. Apart from the fact that it is hard to imagine how anything could be worse, the arguments Fodor gives against decompositional semantics are inadequate, mistakenly assuming that the notion of sense in such semantics has to be the standard Fregean one (see Katz, 1992 and in preparation, and Pitt 1994). From a realist standpoint, the knowledge acquired in acquiring competence and the language that the competence is knowledge of are different things-the former being a concrete state of the mindibrain of the speaker and the latter an abstract object. Thus, the process of acquisition has to be understood as one in which the child acquires the linguistic information necessary for it to stand in the relation "having knowledge of" to an abstract natural language. This is more fully discussed in Katz (1981 and forthcoming).

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pletely general causal theory of knowledge criticized above, presupposes a successful argument for empiricism. Modest versions of the causal theory--or "pictures" like Kripke's (1972, p. 88, fn. 38 and pp. 96-7), do not go so far as to entail causal contact with numbers, sets, and the like. Such versions sacrifice none of the virtues of causal theories of reference generally, since those virtues are only demonstrated in connection with natural objects. Finally, since abstracts objects have their defining properties necessarily, none of Kripke's modal arguments provide support the ambitious claim that the causal theory of reference should take a completely general form (e.g., 1972, pp. 40-9; note the remark about mathematics, p. 43). Thus, whatever theory we adopt for terms referring to concrete objects, we can entertain a classical account of the semantics of terms referring to abstract objects. On such an account, constituents of sentences have senses, their senses are bundles of properties or relations, and a sense (or, derivatively, a sentential constituent) refers to something in case it has the properties or relations which comprise the sense. Accordingly, causal contact is unnecessary for beliefs about abstract objects.'

5.3. The truth condition The correspondence of a mathematical proposition and the mathematical fact in virtue of which the proposition is true involves no contact between an abstract and a concrete object. The reason is that, on the one hand, correspondence does not involve causal contact and, on the other, both the Fregean and Russellian conceptions of propositions enable us to construe number-theoretic and other formal propositions as abstract objects. On the Fregean conception, propositions are senses of sentences, and it is quite natural for the mathematical realist to say, in accord with linguistic realism, that senses of sentences are abstract objects. On a Russellian conception of propositions, a proposition is a pair consisting of the object(s) the proposition is about and the property or relation the proposition ascribes to it (them). Here, too, mathematical propositions can be taken as abstract objects because, for realists, all of the components of such propositionsthe mathematical objects in the sequence as well as the properties and relations-are abstract objects. Since mathematical propositions can be regarded as abstract objects, and since the facts which they are about are Since a belief about something is a state of taking a proposition about it to be true, and since mathematical propositions, for realists, are abstract objects, realists will understand such states as involving a mental representation of the proposition together with the thought that the facts are as the proposition says they are. Such thoughts relate us to abstract objects-propositions-but, since the relations are extrinsic and acausal, there is nothing wrong with saying that we bear such relations to them.

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facts about abstract mathematical objects, mathematical truth is simply an abstract relation between abstract objects.

5.4. The justijication condition On our rationalist epistemology, what counts as adequate grounds for the truth of a proposition depends on its nature, and that, in turn, depends on the nature of the objects the proposition is about. Propositions about natural objects are empirical, and we can understand adequate grounds for them in terms of something like Benacerraf's account of empirical knowledge (1973, p. 413). Propositions about abstract objects are non-empirical. Knowledge of abstract objects is a priori knowledge, and must be grounded in reason alone. In an earlier work, I presented a suggestion about how we can have knowledge of abstract objects (198 1, pp. 200-1 6). The suggestion involved two thoughts. The first is that the entire idea of basing our knowledge of abstract objects on perceptual contact is misguided because, even if contact were possible, it could not properly ground beliefs about such objects. The epistemological function of perceptual contact is to provide information about which possibilities are actualities. Perceptual contact thus has apoint in the case of empirical propositions because natural objects can be otherwise than they actually are (non obstante their essential properties), and, hence, contact is necessary to discover how natural objects actually are. In some possible worlds, gorillas like bananas while in others they don't. Hence, an information channel to actual gorillas is needed in order to discover their taste in fruit. Not so with abstract objects. They couldn't be otherwise than they are. They have all their intrinsic properties and relations Purely mathematical properties and relations of mathematical nece~sarily.~ objects cannot differ from one world to another. Unlike gorillas, numbers have their intrinsic properties necessarily, and, hence, there is no question of which possibilities are actualities. The way abstract objects actually are with respect to their intrinsic properties is the way they must be. Six must be a perfect number and two the only even prime. Since the epistemic role My claim is that all intrinsic or formal properties and relations of abstract objects are necessary. Some properties and relations of abstract objects are contingent, e.g., the relation I bear to the number seventeen when I am thinking of it. On my claim, this follows from the fact that such a relation is not part of pure mathematics. A somewhat trickier case is that a word has the property of being coined at a particular time (but might have been coined at another time). Since I want to treat the words of a language as types in Peirce's sense, and, hence, as abstract objects, coining a word is not creating a word of the language. As I argue elsewhere (198 1 and forthcoming), what happens when a word is coined is that speakers of a language begin to use tokens of the word as tokens of that type under conditions which lead to a change in their competence. Clearly, whether a word type has a representation in the competence of speakers is not one of its intrinsic or formal properties and relations.

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of contact is to provide us with the information needed to select among hypotheses about the different ways something might be, and since perceptual contact cannot provide information about how something must be, such contact has no point in relation to abstract objects. Lewis says something similar (1986, pp. 111-2): the necessity of a mathematical proposition exempts it from the requirement on empirical propositions to show that they counterfactually depend on the facts to which they correspond. According to Lewis, counterfactual dependency is not required for mathematical propositions because "nothing can depend counterfactually on non-contingent matters. For instance, nothing can depend counterfactually on what mathematical objects there are .. . . Nothing sensible can be said about how our opinions would be different ifthere were no number seventeen" (1986, p. 111, my italics). Field rightly objects that we can sensibly say how things would be different if the axiom of choice were false (1989, p. 237). Furthermore, if what Lewis says were so, there would be no reductio proofs, since such proofs begin with the supposition that a necessarily false statement is true, which, on Lewis's view, is "nothing sensible". For example, in a reductio proof that the square root of two is irrational, we suppose counterfactually that (9) is true (9) There is a rational number equal to the square root of two and then, reasoning from this supposition, we go on to spell out the difference its truth would make. We say such things as that there are numbers which are both even and odd. The absurdum of a reductio is precisely what it is sensible to say about how things would be different on the supposition of the truth of the necessarily false proposition. At one time, Lewis held the opposite view (1973, pp. 24-6). The change came about, I think, because of the difficulty of reconciling his earlier view that something sensible can be said about how our opinions would be different if a necessarily false proposition were true with his possible worlds conception of propositions. In so far as we have no commitment to the possible worlds conception of propositions, we have no reason to stick to that conception at the cost of not being able to make sense of reductio proof.9 On possible worlds semantics, propositions are sets of possible worlds: the proposition P is the set of possible worlds at which P is true. Either contradictions are true in the null set of possible worlds or they express no proposition. But, if we say that they are true in the null set of possible worlds, it does not seem possible to maintain the distinction in Lewis' earlier position between it being sensible to say "p! + 1 is prime" and "p! + 1 is composite" on the supposition (i) "there is a largest prime p", but not "there are six regular solids" or "pigs have wings". Since the former are taken to be sensible things to say, it would also have to be sensible to say "pigs have wings" on the supposition (ii) "pigs have wings and are wingless". But since, ex hypothesi, a contradiction is true in exactly the null set of

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To unproblematically express the otioseness of contact in the case of formal knowledge, we require an alternative conception of propositions, one upon which necessarily false sentences express a proposition. One such conception is that propositions are senses of sentences (see Katz 1972 and 1977, and Smith and Katz, in preparation). With this conception, we can say that the condition for supposability is the meaningfulness of the clausal complement of the verb "suppose". A sentence expresses no supposition when and only when, like (lo), (10) Suppose that seventeen loves its mother the clausal complement of "suppose" has no sense, and hence provides no object for the propositional attitude. When the clausal complement has a sense, even one which is necessarily false as in (11): (1 1) Suppose some propositions are both true and false there is an object for the propositional attitude, and the whole sentence expresses a supposition. Thus, on our alternative conception of propositions, there is something to be supposed in the case of necessarily false sentences, and hence we can make perfect sense of reductio proof. Given this condition for supposability, reductio proofs are tests of necessary truth based on an exploration of the logical consequences of supposing a necessary truth to be false. In such proofs, we first suppose that what a necessary truth asserts is not the case, then, by deriving an explicit inconsistency, we expose the fact that the proposition supposed is a contradiction, from which fact we infer that the denial of the proposition we are testing is a necessary truth. Hence, something sensible can be said about how things are on a necessarily false supposition, namely, things are every way they can supposably be.Io I have now explained the first of the two thoughts about how we can have knowledge of abstract objects. The second thought also depends on the fact that the only way an abstract object can actually be is the way it must be. The thought is that it doesn't matter that the function of contact does not carry over to the acquisition of knowledge about abstract objects. If reason is an instrument for finding out how things must be, as traditional possible worlds, all contradictions express the same proposition. Since (i) and (ii) express the same proposition, if it is sensible on (i) to say "p! +1 is prime" and "p! + 1 is composite", it must also be sensible on (ii) to say "pigs have wings". So, it seems that, to avoid having to concede that the latter is sensible, Lewis decided to say that neither is. l o Moreover since, on that notion, necessarily false sentences with different sense content express different propositions, separate reductio proofs are to be kept separate. We have to keep them separate to be able to say that things are the same on the supposition that there is no number seventeen as they are on the different suppositions that there is no number two or that there is a largest prime.

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rationalists claim, then it can tell us how numbers and other abstract objects actually are, and contact with abstract objects was never needed in the first place. If the light of reason enables the mind's eye to see that seventeen must be a prime number, of what relevance can it be that the body's eye cannot see numbers? It is a commonplace of mathematical life as well as a philosophical claim that reason is an instrument for determining that mathematical objects could not be otherwise than as mathematical truth presents them. When mathematicians claim that there could not be a largest integer or more than one even prime, they are using the notions of possibility and necessity in the absolute Leibnizian sense. Thus, on the basis of what has been said above, to assert that reason can provide adequate grounds for the claim that two is the only even prime is to assert that it can show that the truth conditions of the proposition are satisfied no matter what one supposes about the numbers. Reductio proofs provide the most straightforward way of showing that a mathematical proposition is a necessary truth. This is because such proofs (explicitly) begin with the supposition that the proposition is false and go on to exclude every possibility of the supposition being true. They show that there is only one possibility of how the mathematical objects in question might be because all other "possibilities" are impossibilities. The argument (A) that two is the only even prime is an example. (A) We see that two is an even prime. Supposing that another number is even and prime, that number must be either less than two or greater than two. If the number is less than two, it has to be one, but then it is not even. If the number is greater than two, then, since it is even, it is divisible by two, and hence not prime. Since the law of trichotomy cannot fail here, there is no even prime other than the number two. A proof provides us with adequate grounds for knowledge of a proposition P about abstract objects 0 , , O,, ... by showing that it is impossible for O,, O,, . .. to be other than as P says they are. But the question arises of how proofs establish the necessity of their conclusion when, typically, their conclusion is a statement that 0,,O,, ... have or lack a certain property, but contains no further statement expressing a modal predication about that statement. For example, the conclusion of (A) simply states that the numbers greater than two lack the property of being even and prime. It isn't itself a modal statement. How can such an argument establish the necessity of its conclusion? The answer is that an argument for a conclusion P-not itself a modal statement-establishes the modal statement "Necessarily, P" in virtue of the fact that the argument is a proof of P, The essence of proof consists in reasoning so close textured, so tight, that it excludes every possibility of

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the conclusion being false. Mathematicians sometimes say that there are no "holes" or "gaps". This is not just a matter of logical structure. The tightness of a proof derives not only from the absence of a counter-example to one of the steps from premisses to conclusion, but also from the absence of a counter-example to one of the premisses. The argument (A) is a proof that two is the only even prime not only because none of its inferential steps can be faulted, but also because its premisses cannot be either, e.g., there is no possibility of two not being an even prime. Thus, (A) shows that two is necessarily the only even prime, not in virtue of a modal conclusion, but in virtue of reasoning so tight that every possibility of another even prime is ruled out. Tightness is different from the properties of informativeness and depth. A completely tight proof may be less informative about what is going on mathematically-give less insight into the mathematical structurethan a proof with a hole. Further, tightness and informativeness are both different from depth, what the proof tells us about the bigger mathematical picture. Truth table proofs of tautologies are completely tight and completely informative, but, though not as shallow as solutions to chess problems, they are not particularly deep. Thus, a proof with a hole in it can be mathematically significant if it is informative or deep, but, of course, for the purpose of having mathematical beliefs on which we can rely absolutely, an uninformative or shallow proof is as good as an informative or deep one."

6. The order of knowledge The foregoing account of the epistemic conditions for formal knowledge falls short of meeting Benacerraf's challenge to realism in two respects: we need to add an explication of further methods for acquiring formal knowledge to obtain a full picture of a rationalist epistemology; and we need to add a philosophical explanation of why a rationalist epistemology meets Benacerraf's challenge. The present section completes our sketch " It is widely held that knowledge is reliable in the sense of being theoretically and practically dependable. In empirical knowledge, reliability is typically explained in terms of its resting on evidence from either direct or indirect causal contact with the natural objects it is about. Since the process of arriving at empirical beliefs monitors those objects, empirical investigation provides grounds for confidence in the evidence, and hence the beliefs it supports. In the case of mathematical knowledge, we can explain reliability in terms of tightness of proof. The tightness of mathematical proofs underwrites our confidence that their conclusions represent the numbers as they are. Every possibility of the numbers being otherwise has been excluded because there are no gaps.

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of the epistemology. The next section provides the philosophical explanation.

To complete the sketch, we have to explain why the step from knowledge of simple mathematical facts to knowledge of mathematical laws and theories does not depend on contact with abstract objects. Of course, such dependency seems on the face of it unlikely, since causal contact was not invoked to explain knowledge of the simple mathematical facts which underlie knowledge of mathematical laws and theories. Nevertheless, to be sure there is no such dependency, we need to look at how a rationalist epistemology handles the ascent to knowledge of mathematical laws and theories in order to make sure that every step is a priori. The ascent from basic facts to laws and theories is a feature of the development of both a priori and a posteriori knowledge. Not all cases of empirical knowledge concern simple natural facts like the case of Hermione's knowledge that what she is holding is a truffle. Recognizing that there are more complex cases of empirical knowledge involving laws and theories, Benacerraf qualifies his causal condition as "an account of our knowledge about medium-sized objects, in the present" (1973, p. 413). He goes on to say Other cases of knowledge can be explained as being based on inference based on cases such as these ... This is meant to include our knowledge of general laws and theories, and, through them, our knowledge of the future and much of the past. (1973, p. 413) This is, in effect, to introduce an order of knowledge into empiricist epistemology: there is basic knowledge of "medium-sized objects, in the present", and transcendent knowledge of "general laws and theories, and, through them, . . . knowledge of the future and much of the past". Arationalist epistemology also posits an order of knowledge to account for knowledge of general laws and theories in the formal sciences. Corresponding to Benacerraf's basic, observational knowledge of properties of medium-sized objects, our rationalist epistemology posits basic ratiocinative knowledge of evident properties of abstract objects, for example, the knowledge that four is composite or the knowledge that two is the only even prime. Corresponding to Benacerraf's transcendent knowledge of empirical laws and theories, our epistemology posits transcendent knowledge of formal laws and theories. The basicltranscendent distinction is no easier to draw in the formal sciences than in the natural sciences.12Nonetheless, a rough distinction can l2

The observationitheory distinction in natural sciences has proven notori-

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be drawn. In the case of the natural sciences, the more basic the knowledge, the less observation depends on such artificial devices as electron microscopes, radio telescopes, etc., to boost the power of our natural faculties. Since the reliability of the evidence from such devices depends on laws and theories, the more basic the knowledge is, the less the extent to which theoretical considerations play a role in connecting what we directly perceive with what we count as observation. Similarly, in the case of formal knowledge, the more basic it is, the smaller the extent to which reasoning relies on laws and theories about the general structure of objects in the domain. In the most basic formal knowledge, we can determine that there is a necessary connection between an object and a property thanks to knowing only the definition of that object and related ones in the system. The traditional rationalist's most prominent example of the source of basic mathematical knowledge is intuition, an immediate, i.e., non-inferential, grasping of structure and the limits of possibility imposed by structure. Although rationalists over-emphasize the role of intuition in the acquisition of formal knowledge to the neglect of other sources of basic knowledge, they are right that it is the source of much of our basic knowledge.13 For, in many cases, there is no other explanation of why the premisses and inferential steps of a proof have no counterexample. Some anti-realists cry "mysticism" at the mention of mathematical intuition. The earlier comments on the use of this epithet apply here but, in the present case, it can be rejoined that anti-realists owe us an alternative explanation of the numerous examples of the immediate grasp of formal truths such as the pigeon-hole principle, the indiscernibility of identicals, and the ambiguity of sentences like "Visiting relatives can be annoying" and "I saw the uncle of John and Mary". Prior to learning the proof of the pigeon-hole principle, even mathematically naive people can ously difficult to draw. Many philosophers of science have given up trying to make a sharp distinction between what can known through observation and what requires theory. A sharp distinction between basic and transcendent cases in the formal sciences, though, as far as I know, investigated little if at all, is unlikely to prove more tractable. But, in the present connection, it is no more necessary for us to draw a sharp distinction for the formal sciences than it was for Benacerraf to draw one for the natural sciences. l 3 Over-emphasis on intuition has obscured the fact that reasoning and computation are also sources of basic mathematical knowledge. Two cases in point are the theoretically unmediated inferences which underlie our basic knowledge that two is the only even prime, and the theoretically unmediated computation that underlies our basic knowledge that a cube has twelve edges. Basic knowledge in such cases does not seem to depend on rational operations encompassed within a single grasp of structure-though, in certain instances, it might be thought of as concatenated intuitions.

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see immediately that, if m things are put into n pigeon-holes, then, when m is greater than n, some hole must contain more than one thing. What other explanation is there than intuition? Transcendent formal knowledge, like transcendent empirical knowledge, is based on inferences based on cases of basic knowledge. Such inferences generalize basic knowledge and bring items of basic and transcendent knowledge into an integrated, coherent, total system in which the relations of the various components of the whole are derivable from the theoretical principles of the system. The inference principles and the ideal of systematization are best thought of as part of the general concept of knowledge, rather than as special features of the concepts of empirical and formal knowledge. Thus, many aspects of knowledge which are methodologically imposed, e.g., the simplicity of theories, are consequences of reason's general concept of knowledge. Despite our best efforts to sharpen intuitive discrimination and improve reasoning, the status of some cases will be left undecided or erroneously described. In such cases, theoretical considerations can be brought to bear to resolve intuitively unclear cases and overturn intuitive and theoretical judgments mistakenly accepted at earlier stages. We compensate for the fallibility of the processes of acquiring basic formal knowledge and of framing systems of transcendent formal knowledge by inferring the appropriate revisions from laws and theories when systematization has been carried far enough. The conclusions of reason are a priori and revisable. But their revisability is revisability in the light of further a priori, rational deliberations, not revisability in the light of new a posteriori, empirical discoveries. This sharp separation of rational and empirical revisability is a consequence of the fundamental difference between the epistemology of formal and empirical knowledge that was developed in the preceding sections. It explains why it is not possible to argue against a priori knowledge of mathematics as, for example, Kitcher (1984) does, on the grounds of the revisability of our mathematical beliefs. Such an argument would have to show that beliefs of pure mathematics are subject to revision on the grounds of experience. In criticizing Kitcher's argument, Hale rightly observes that "if revisability is to conflict with apriority, it must be revisability for empirical reasons" (1987, p. 148). A well-known argument against a priori knowledge claims that mathematics is revisable for empirical reasons because it would be irrational to continue to maintain a putative a priori truth if a better overall empirical theory could be obtained by giving it up. Examples have proved hard to come by. There is a good reason for this. Without empiricism to insure that

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the disputed a priori truth is part of the empirical theory or connected to it in "the web of belief', we will be able to distinguish the a priori truth from the proposition(s) of the empirical theory which must be given up to obtain a better overall theory. As an illustration, consider the claim that Putnam once made that the abandonment of Euclidean geometry in twentieth century physics is a counter-example to apriority: "[slomething literally inconceivable had turned out to be true". He writes:

I was driven to the conclusion that there was such a thing as the overthrow of a proposition that was once a priori (or that once had the status of what we call "a priori" truth). If it could be rational to give up claims as self-evident as the geometrical proposition just mentioned, then, it seemed to me that there was no basis for maintaining that there are any absolutely apriori truths, any truths that a rational man isforbidden to even doubt. (Putnam 1975, pp. xv-xvi) Putnam's case does not support his conclusion. Putnam is wrong to say that the proposition that was overthrown "was once a priori" or "once had the status of what we call 'a priori' truth''. To be sure, people once believed that it is a priori that Euclidean geometry is a true theory of physical space, but it was never a priori that it was a true theory of physical space. From the truth of "it is believed that p is a priori", it does not follow that p is a priori. Moreover, had the proposition that Euclidean geometry is a true theory of physical space actually been a priori, rather than merely believed to be, it could never have been refuted on the basis of the empirical considerations. "A priori (p) -+a priori (a priori (@))". Putnam's case concerns the revision of a physical theory. What everyone believed-and what Einstein showed to be false-was that the geometric structure of physical space is Euclidean and that it is inconceivable that it is not Euclidean. Thus, Putnam's case, which concerns a proposition in empirical science, shows at most that there are no absolutely indubitable propositions in empirical science. The case shows something about an empirical application of Euclidean geometry, but nothing about it per se. Thus, the case of the Einsteinian revolution provides no basis for the claim that pure mathematics is in some broad (perhaps Quinean) sense empirical. The case is one in which an a posteriori applied geometry was falsified on empirical grounds, not one in which an a priori pure geometry was.I4 l 4 On a realist view, a pure geometry, Euclidean or otherwise, is a theory of a class of abstract spatial structures. In a complete theory, its principles express the possibilities of figures and relations among them within a space. Anything that

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The ideal of systematization gives a certain holistic character to justification in the formal and empirical sciences, but that holism has nothing to do with Quine's semantic holism. It is not a semantic doctrine like Quine's claim that "the whole of science" is the smallest independently meaningh l linguistic unit (Quine 1961, p. 43). Rather, our holism is a methodological one. concerning the ways in which propositions in a particular formal or empirical theory obtain support from one another and from the basic knowledge on which the theory rests. As we saw above, rationalist epistemology is not a "bottom up" affair in which all basic knowledge is established prior to and independent of transcendental knowledge and systematization. Exactly how much foundationalism there is is something I will leave open, just as Benacerraf left the parallel question open in his account of empiricist epistemology. The extent to which transcendent knowledge must be anchored in basic knowledge can be treated as question about the ideal of systematization, and, as such, independent of the questions at issue here. Suffice to say, our methodological holism is compatible with various forms and degrees of foundationalism. Perhaps the most interesting question here is one that I can do no more than touch on: whether the justification of formal knowledge is uniform. Are all cases of formal knowledge justified by excluding every possibility of falsehood, as are all cases of basic knowledge and many cases of transcendent knowledge? The question is not whether we can prove everything in a system, since we have recognized other ways of establishing the necessity of formal truths. The question is whether everything counted as knowledge at the level of transcendent knowledge is so counted because it can be shown, in some way or other, to be necessary. I think the answer has to be "not necessarily". We have to allow for the possibility that some principles counted as knowledge at the transcendent level are so counted, not because they can be shown to be necessary, but because they are essential for the best systematization of the truths that have been shown to be necessary. Church's thesis might belong to this category. Although conflicts with the principles is an impossibility in the space. Grammars, as I have argued (198 1 and forthcoming), can be conceived in a similar way, as theories of a class of abstract sentential structures whose principles express the possibilitiesof linguistic forms and grammatical relations within a language. In making a place for the notions of necessity and possibility in connection with pure geometries and pure grammars, we can bring geometric and grammatical knowledge under the scope of our rationalist epistemology. In Katz (forthcoming), I present an account of the distinction between pure and applied geometries, pure and applied grammars, etc., in terms of the different kinds of the objects they study.

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we cannot prove that recursiveness is effective computability, applications of the ideal of systematization to our body of transcendent logical knowledge force the thesis upon us. I will use the term "apodictic" refer to such principles. The point that it is important to make here is that knowledge of apodictic principles is a priori knowledge because it is established by reason alone on the basis of necessary truths established by reason alone. Hence, systems of a priori formal knowledge to which we add apodictic principles remain a priori. The importance of the point is that it permits us to claim that our rationalist account of basic formal knowledge can be appropriately extended to knowledge of formal laws and theories. Since knowledge of basic formal facts is a priori, since the step from that knowledge to transcendent knowledge of formal laws and theories is also a priori, and since, as a consequence, filling gaps and correcting errors is a priori, our rationalist account of formal knowledge does not depend at any point on causal contact with abstract objects. On our account, such knowledge always has an a priori warrant based on reason alone.

7. Have any questions been begged? Do we now have the "improved" concept of mathematical knowledge that remedies the defect Benacerraf noted in our understanding of mathematical truth and thus enables realism to meet his epistemic challenge? Or, having taken the unchelmite approach of searching the dark road, have we found only a counterfeit coin? The concern is that our "improved" concept begs the question because the reasoning it says justifies principles in mathematics and logic often rests on the very principles that it is suppose to justify.I5 Since the acceptability of such reasoning depends on the acceptability of the principles, we would already have to accept the principles to accept the grounds that the reasoning provides. This doubt about whether we have been successful raises a more general doubt which goes well beyond the question of our success. Let us l 5 Circularity would be the wrong way to put the alleged difficulty. An argument is circular when the very same proposition appearing as the conclusion appears as a premiss. On a realist view of mathematical and logical principles, this could never happen. Realism sharply separates the objects of formal knowledge themselves, i.e., the numbers, sets, logical principles, proofs, etc., from our inner epistemic states and processes which provide us with knowledge of them. The former are abstract, objective, and autonomously existing, while the latter are concrete, subjective, and mind-dependent. Thus, the reasoning that provides us with grounds for taking mathematical and logical principles to be true cannot contain those abstract principles.

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look at logic where this doubt arises in a particularly strong form. Logicians, too, are in the situation of attempting to justify inferential principles on the basis of inferences whose validity depends explicitly on the principles they purport to justify. It is no coincidence that their situation is the same, since our account of reasoning in the formal sciences was developed as an explication of rational practices of formal scientists. Hence, if it is concluded that we beg the question in attempting to defend realism, it must also be concluded that logicians beg the question in attempting to justify theories of logical structure on the basis of inferences which depend on those principles. Hence, they cannot legitimately justify those logical principles, and cannot claim to have knowledge of them. Hence, there can be no science of logic. Furthermore, in so far as principles of logic are invariably used in the acquisition of other branches of scientific knowledge, both formal and empirical, no other science can exist either. Such extreme consequences undercut the doubt that we failed to provide the desired concept of knowledge. Their derivation is a kind of reductio of the supposition that such question begging, if that's what it is, is grounds for doubting the legitimacy of our defense of realism. Moreover, it is easy to see what is wrong with that supposition: the doubt is the doubt of the philosophical skeptic. Nothing less than the philosophical skeptic's challenge of the prevailing standards for knowledge claims in the formal sciences would be strong enough to undermine such claims across the board. To avoid begging the philosophical skeptic's question, we would have to provide a basis, independent of those standards, which explains why it is reasonable to use them rather than others that could have been used instead. From the perspective of philosophical skepticism, empiricists and rationalists are in the same boat. If for no other reason than that empiricists, too, rely on logical principles to infer general empirical laws and theories, if there are grounds for suspicion concerning a rationalist epistemology, there are the same grounds for suspicion concerning an empiricist epistemology. As a consequence of the fact that both sides are vulnerable to the charge of begging the skeptic's question, that charge is not one empiricists can use against our rationalist response to Benacerraf's epistemic challenge. It is clear what the Humean skeptic will say to Hermione. He will ask her how she can claim to know that what she is holding is a truffle when, for all the evidence she might have from causal contact, she could just as well be holding a truffrock as a truffle. Since every extrapolation consistent with all the evidence-including even extrapolations about mediumsized objects, since they involve the posit of permanence-have the same empirical ground for their truth, there is no reason to prefer the inductive

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extrapolation to a truffle over a counter-inductive extrapolation to a truffrock. It is question begging for Hermione to present the fact that the former extrapolation was obtained on the basis of the inductive rule. The skeptic can present the fact that the latter extrapolation was obtained on the basis of a counter-inductive rule and claim that one rule is no better than the other. Thus Hermione has failed to establish that she is more rational than the skeptic who thinks that the world is a gruesome place which will show its true colors after the year 5000. If empiricists have their Humean skeptic, rationalists have their Cartesian skeptic. As the Cartesian skeptic sees it, rationalists are in essentially the same hopeless epistemological position with respect to logical and mathematical knowledge. Descartes explains that it is possible for logical principles to have been false because "the power of God cannot have any limits" (1970, pp. 150-1 and 23&7). How can we rationally claim to know that non-contradiction is a necessary truth on the grounds that reason shows it could not be false when God could "make it not be true ... that contradictories could not be [true] together"? To the Cartesian skeptic who complains that we have no right to claim to know that laws of logic are necessary truths, we cannot answer that God could not make something conflict with the principle of noncontradiction because a contradiction cannot be true. To answer that a contradiction cannot be true begs the question. It is precisely the necessity of a contradiction's falsehood that the Cartesian skeptic is challenging us to establish when he introduces the idea of God's absolutely unrestricted omnipotence. Since the empiricist begs the Humean skeptic's question as much as a rationalist begs the Cartesian skeptic's, the empiricist is in no position to use the fact that we beg the philosophical skeptic's question against our reply to Benacerraf's epistemic challenge. Thus, we can admit that a rationalist epistemology cannot satisfy the philosophical skeptic, and, consistent with that admission, claim that the reasoning such an epistemology sanctions provides adequate grounds for formal principles.' Furthermore, we can argue that there is an appearance of inconsistency only l 6 It is a further question whether this admission might be only for the sake of argument or whether we have ultimately to concede that philosophical skepticism is right. But, as I see it, whether or not it turns out to be right is irrelevant to the present issue. Perhaps showing that realists can provide an epistemology which handles formal knowledge as well as the anti-realist's causal theory handles natural knowledge would in some sense be pointless if philosophical skepticism did turn out to be right. But as we have no reason to think it will, we have no reason to think my reply to Benacerraf is pointless. It is also worth point out that a refutation of Humean skepticism does not confer an advantage on empiricism, since what we now accept as mathematical knowledge is certainly not going to be taken as mere opinion unless something like Cartesian skepticism is established at the same time. However, even that would con-

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because of a failure to distinguish two notion of acceptability, "acceptable as a response to the philosophical skeptic's challenge to the prevailing standards" and "acceptable on the prevailing standards". The only question that arises about Hermione's claim to know that what she is holding is a truffle is whether she has an adequate basis, relative to prevailing botanical standards, for claiming to have such knowledge. If Hermione's botany teacher asks her how she knows she is holding a truffle, and she produces evidence that it is a fungus, grew underground, has a warty surface, is blackish in colour, exudes an earthy aroma, and so on, she has justified her claim to know she is holding a truffle. It would be absurd for her botany teacher to fail her because the evidence does not exclude every possibility that what she is holding is not a truffle. Similarly, the only question that arises about Hermione's claim to know that two is the only even prime is whether she has an adequate basis, relative to the prevailing mathematical standards, for claiming to have this knowledge. Suppose Hermione's mathematics teacher ask her how she knows that two is the only even prime, and she argues as in (A). "There", she says, "that's how I know that there couldn't be another even prime!". It would be equally absurd for her mathematics teacher to fail her because the reasoning does not exclude the supposition that there are other even primes. Hermione's demonstration that she fully understands the proof shows she has an adequate basis for claiming to know that two is the only even prime. Having met the prevailing standards, she has to be credited with the knowledge. The same goes for Hermione's claim to know logical principles on the basis of reasoning which depends on the principles that the reasoning is intended to justify. If, for instance, her reasoning depends on the logical principle that contradictories cannot be true together, then, of course, the reasoning would not be acceptable were it not the case that contradictories cannot be true together. But it would be absurd for Hermione's logic teacher to fail her because she hasn't excluded Descartes's supposition about God's omnipotence. Once philosophical skepticism is out of the picture, the only remaining question is whether one or another piece of reasoning measures up to the prevailing standards. Neither our rationalist nor Benacerraf's empiricist epistemology has any worry on this score. Each sanctions reasoning fer no real advantage on empiricism in the present controversy. If something like Cartesian skepticism were established, there would be no rationalist epistemology, but that would only be because there would no formal knowledge for it to explain. The question whether realism can meet the epistemic challenge would not arise.

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which will meet the prevailing epistemic standards in its respective area of science because each was developed as an explication of those standards. No doubt, as they stand, both epistemologies are less than fully adequate explications. But, as they can be corrected without affecting the issues here, inadequacies are incidental shortcomings which raise no question about whether relevant justificatory reasoning meets the prevailing standards.

8. The difference between natural and formal knowledge A full distinction between the formal and the natural sciences would have prevented basing an epistemology for realism on causal contact. To be sure, traditional realism distinguishes them as being about ontologically different kinds of objects. But, as history shows, this aspect of the distinction by itself does not prevent anti-realists from thinking that realists must subscribe to a causal contact epistemology or prevent realists from subscribing to one. What is missing is what the discussion in the previous sections tell us about how investigation in the formal sciences differs from investigation in the natural sciences. Possibly gorillas like bananas, but possibly they don't. Natural science asks what their tastes actually are. Supposably there is one even prime, but supposably there are more than one. Formal science asks which supposition is necessary. Painting the picture in broad brush strokes, we can say that investigation in the natural sciences seeks to prune down the possible to the actual, while investigation in the formal sciences seek to prune down the supposable to the necessary." Not only the nature of the objects studied but also the character of investigation differs in the formal and natural sciences. Given that the aim of investigation in the natural sciences is to determine what possibilities are actualities while the aim in the formal sciences is to determine what supposabilities are necessities, natural science and formal science require appropriately different epistemic methods and produce different kinds of knowledge. Pruning down the supposable to the necessary requires reason, the whole of reason, and nothing but reason. Pruning down the possible to the actual requires perceptual contact as well. Since pruning down the supposable to the necessary requires only reason, formal knowledge is a priori knowledge. Since pruning down the possible to the actual l 7 One qualification that might be required is that the supposable may in certain special cases be pruned down to the apodictic rather than the necessary. Whether a qualification is required for the possible is a question that involves linguistic and metaphysical issues beyond the scope of this paper.

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requires interaction with natural objects as well reason, natural knowledge is a posteriori knowledge. Accordingly, naturalists and realists who insist that mathematical knowledge is based on perceptual contact misconstrued the epistemic task in the formal sciences. If they had seen that the methods of the formal sciences are designed for pruning down the supposable to the necessary rather than for pruning down the possible to the actual, they would have seen that perception is useless for acquiring formal knowledge. But, believing that the epistemic task in the formal science is the same as the epistemic task in the natural sciences, they thought that the investigation of formal facts must be based on the same empirical methods as the investigation of natural facts. Given that perception must play a role in the formal sciences, either mathematical knowledge cannot be knowledge of abstract objects, as many naturalists think, or we can have perceptual contact with abstract objects, as many realists think. It is not surprising, therefore, that many naturalists think that all realists must end up as mystics, and that many realists do end up that way.

9. A tenable dualism Our combined realism and rationalism provides a new position with which to oppose naturalistlempiricist views of scientific knowledge, particularly, the Quinean view that has enjoyed hegemony in Anglo-American philosophy over these last three decades. Our position is the very antithesis of Quine's monistic view (1961) of "total science". Given the difference between the natural and the formal sciences just set out, our position on total science is dualistic to the core. This dualism, which extends from the ontological status of the objects of study to the questions asked about them, and to the methods used to obtain knowledge, explains, contra Quine, why knowledge in the formal sciences is a priori. Elsewhere I have set out my criticisms of the arguments about meaning and translation on which Quine's position rests (1990a, pp. 175-202 and 1994, pp. 157-61). Here I want only to make the point that the dualism of the abstract and the concrete does not pay the price for positing causally incommensurable objects that the dualism of the mental and the physical pays. The reason is that realism is under no pressure to explain how causally incommensurable objects can causally effect one another. Cartesianism is under such pressure because our experience is replete with prima facie signs of causal interaction between the mental and the physical. But, in the case of realism, there is nothing corresponding to psycho-physical correlations and, hence, there is no pressure on realists corresponding to

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the pressure on Cartesians to explain or explain away the appearance of interaction between causally incommensurable objects. It has, of course, been assumed that the obligatoriness of a causal epistemology exerts such pressure on realists. But, as we have seen, that assumption rests on a Chelmite mistake.'* JERROLD J. KATZ

The Graduate Center CUNY 33 West 42 Street New York NY 10036-8099 USA

REFERENCES Benacerraf, P. 1965: "What Numbers Could Not Be". The Philosophical Review, 74, pp. 47-73. Reprinted in Benacerraf and Putnam. -1973: "Mathematical Truth". The Journal of Philosophy, 70, pp. 661-79. Reprinted in Benacerraf and Putnam (eds.), pp. 403-20. Benacerraf, P. and Putnam, H. (eds.) 1964: The Philosophy of Mathematics: Selected Essays. Englewood Cliffs: Prentice-Hall. Brouwer, L.E.J. 19 13: "Intuitionism and Formalism". Bulletin o f t h e American Mathematical Society, 20, pp. 81-96. Reprinted in Benacerraf and Putnam (eds.), pp. 77-89. Burgess, J.P. 1988: "Why I Am Not a Nominalist". Notre Dame Journal of Formal Logic, 24, pp. 93-1 05. -1990: "Epistemology and Nominalism", in Physicalism in Mathematics. Dordrecht: Kluwer Academic Publishers, pp. 1-15. Chihara, C. 1982: "A Godelian Thesis Regarding Mathematical Objects: Do They Exist? And Can We See Them?". The Philosophical Review, 91, pp. 211-27. Chomsky, N. 1965: Aspects of the Theory of Syntax. Cambridge: MIT Press. Descartes, R. 1970: Descartes: Philosophical Letters. A. Kenny (trans. & ed.). Oxford: Clarendon Press. l 8 I want to thank Jody Azzouni, Paul Boghossian, Arthur Collins, Hartry Field, Dan Isaacson, Juliette Kennedy, Sid Morgenbesser, Tom Nagel, Paul Postal, David Rosenthal, Mark Sainsbury, Robert Tragesser, Palle Yourgrau, Linda Wetzel, and an anonymous reader for Mind for helpful comments on earlier drafts. I also want to thank the members of my Spring 1993 and Spring 1994 seminars at the Graduate Center, the members of my 1994 Michaelmas term seminar at King's College, University of London, and the audience at my 1994 Oxford University lectures for stimulating discussions of earlier drafts. Special thanks to Arnold Koslow and David Pitt for very usehl comments from draft to draft.

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Dummett, M. 1978: "Platonism" in Truth and Other Enigmas. Cambridge: Harvard University Press, pp. 202-14. Field, H. 1980: Science Without Numbers. Princeton: Princeton University Press. -1989: Realism, Mathematics, and Modality. Oxford: Basil Blackwell. Fodor, J.A. 1980: "Fixation of Belief and Concept Acquisition", in M. Piatelli-Palmarini (ed.) Language and Learning: The Debate between Jean Piaget and Noam Chomsky. Cambridge: Harvard University Press, pp. 143-9. 198 1: "The Present Status of the Innateness Controversy", in Representation. Cambridge: MIT Press, pp. 257-3 16. Godel, K. 1964: "What is Cantor's Continuum Problem?", in P. Benacerraf and H. Putnam (eds.), pp. 258-73. Goldman, A. 1967: "A Causal Theory of Knowing". The Journal of Philosophy, 64, pp. 357-72. Gottlieb, D. 1980: Ontological Economy: Substitutional Quantijkation and Mathematics. Oxford: Oxford University Press. Hale, B. 1987: Abstract Objects. Oxford: Basil Blackwell. Hilbert, D. 1926: "On the Infinite". Mathematische Annalen, 95. Reprinted in Benacerraf and Putnam (eds.) pp. 183-201. Katz, J.J. 1972: Semantic Theory. New York: Harper and Row. 1-

977: Propositional Structure and Illocutionary Force. Cambridge: Harvard University Press. -1979: "Semantics and Conceptual Change". The Philosophical Review, 88, pp. 3 2 7 4 5 . 198 1: Language and Other Abstract Objects. Ottawa: Rowman and Littlefield. 1990a: The Metaphysics of Meaning. Cambridge: MIT Press. 990b: "Has the Description Theory of Names been Refuted?", in G. 1Boolos (ed.) Meaning and Method: Essays in Honor of Hilary Putnam. Cambridge: Cambridge University Press, pp. 3 1 4 1 -1992: "The New Intensionalism". Mind, 101, pp. 1-3 1. 994: "Names Without Bearers". The Philosophical Review, 103, 1, 1pp. 1-39. -forthcoming: The Knowledge and Nature of Abstract Objects. -in preparation: "Analyticity, Necessity, and the Epistemology of Semantics". Kitcher, P. 1984: The Nature of Mathematical Knowledge. Oxford: Oxford University Press. Kripke, S. 1972: Naming and Necessity. Cambridge: Harvard University Press.

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Lewis, D. 1986: On The Plurality of Worlds. Oxford: Basil Blackwell. 1973: Counterfactuals. Cambridge: Harvard University Press. Pitt, D. 1994: "Decompositional Semantics: A Defense of Semantic Decomposition, with Applications in the Semantics of Modification and the Philosophy of Mind". PhD Dissertation, CUNY Graduate Center, New York. Putnam, H. 1975: "Explanation and Reference", in Mind, Language, and Reality: Philosophical Papers, Volume 2. Cambridge: Cambridge University Press, pp. 1 9 6 2 14. Quine, W.V. 1961: "Two Dogmas of Empiricism", in From a Logical Point of View. Cambridge: Harvard University Press, pp. 2 W 6 . Smith, G.E. and Katz, J.J. in preparation: Supposable Worlds.