What is the philosophical basis of intuitionistic mathematics?

Logic, Methodology and Philosophy of Science IX D. Prawitz, B. Skyrms and D. Westerstlhl (Editors) 9 1994 Elsevier Science B.V. All rights reserved.

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W H A T IS T H E P H I L O S O P H I C A L BASIS OF INTUITIONISTIC MATHEMATICS?

RICHARD TIESZEN Department of Philosophy, San Josd State University, San Josd, USA

How should we understand the philosophical basis of intuitionistic mathematics late in the Twentieth Century, some 25 years after the death of Brouwer? I believe this is an important question because there are insights in intuitionism that are found nowhere else in the philosophy of mathematics, insights that ought to be preserved, clarified and extended. Chief among these is the idea that a proof is a mental construction. The idea that a proof is a mental construction already distinguishes intuitionism from other philosophical views of mathematics like (ontological) platonism, nominalism, and formalism. There are also many philosophical problems that can be raised for intuitionism and I intend to discuss some of them below. I shall first briefly consider some views on the question that are found in the literature on intuitionism. I shall then argue for what I take to be a good working answer to the question, an answer which I think is in the tradition of Brouwer and Heyting but which can be used to clarify their views and to defend some of the key philosophical insights of intuitionism. 1. A b r i e f s u r v e y of v i e w s o n t h e q u e s t i o n For Brouwer the philosophical basis of intuitionist mathematics was to be found in the concept of intuition. In particular, Brouwer portrayed intuitionism as abandoning Kant's apriority of space but adhering all the more resolutely to Kant's idea of time as an a priori form of intuition. Brouwer describes the basic intuition upon which mathematics is founded in a number of places in his writings. In "Intuitionism and Formalism" [Brouwer 1912], for example, he describes it as follows: This neo-intuitionism considers the falling apart of moments of life into qualitatively different parts, to be reunited only while re-

580 maining separated by time, as the fundamental phenomenon of the human intellect, passing by abstracting from its emotional content into the fundamental phenomenon of mathematical thinking, the intuition of bare two-oneness. Brouwer often notes the role of memory in retaining earlier life moments while the succession of life moments continues. He says that it is introspectively realized how this basic operation of mathematical construction, this intuition of two-oneness, successively generates the finite ordinal numbers, inasmuch as one of the elements of the two-oneness may be thought of as a new two-oneness, and the process may then be repeated indefinitely. Also important for Brouwer's view of the role of this basic intuition is the claim that (i) what has meaning in mathematics is derived from the basic intuition and (ii) that mathematics is a languageless activity of mind. Many of Brouwer's comments suggest a very strong separation of thought from language. This figures into Brouwer's conception of how the basic intuition of mathematics can provide a foundation for mathematics that is exact and free from error and misunderstanding. In "Weten, willen, and spreken" [Brouwer 1933], for example, Brouwer says that ... the languageless constructions originating by the self-unfolding of the primordial intuition are, by virtue of their presence in memory alone, exact and correct; ... the human power of memory, however, which has to survey these constructions, even when it summons the assistance of linguistic signs, b y its very nature is limited and fallible. For a human mind equipped with an unlimited memory, a pure mathematics which is practiced in solitude and without the use of linguistic signs would be exact; this exactness, however, would again be lost in an exchange between human beings with unlimited memory, since they remain committed to language as a means of communication. On the basis of comments like these it appears that the certainty that is supposed to be guaranteed by founding intuitionism on intuition is certainty for the ideal mathematician only. The actual, practicing mathematician does not have such certainty, nor does there appear to be any intersubjective certainty for Brouwer, since the expression of mathematical ideas needed for communication is always imperfect. The separation of thought from language leads to the charge that Brouwer's notion of intuition and the concept of meaning it supports is thoroughly solipsistic. Insofar as Brouwer's view is solipsistic I think it clearly deviates from (or is inconsistent with) the Kantian view of intuition that he invokes elsewhere, and to deleterious effect. I shall come back to this later and argue that Brouwerian solipsism is philosophically untenable. In any case, I think it

581 is fair to say that Brouwer did not have a philosophically sophisticated conception of intuition. Heyting's work adds an interesting and important new dimension to the discussion of the foundations of intuitionism. His 1931 address on the intuitionistic foundations of mathematics is especially rich in philosophical content [Heyting 1931]. In the address Heyting explained and defended the intuitionistic viewpoint by suggesting that we view mathematical propositions as expressions of intentions, in the sense of Husserl's theory of intentionality. "Intentions" in this sense not only refer to states of affairs thought to exist independently of us but also to experiences thought to be possible. Heyting then identifies proofs, as mental constructions, with fulfillments of intentions. In the 1931 address he goes on to describe the meaning of the logical constants of the intuitionistic propositional calculus in these terms. Martin-LSf, in his lectures on the meanings of the logical constants [Martin-LSf 1983-84], has said that Heyting did not just borrow these terms from Husserl but that he also applied them appropriately. I agree, but in agreeing with Martin-LSf's remark I think we are at the same time committing ourselves to the need for the kind of clarification and extension of the philosophical views of Brouwer and Heyting that is called for by Heyting's identification. I discuss this below. Note, by the way, that Heyting does not use the term "intuition" in his description but anyone who knows Husserl's philosophy knows that the concept of intuition is defined in terms of the fulfillment of intentions. So in identifying proofs with fulfilled (or fulfillable) intentions Heyting too holds that intuitionism is founded on the evidence provided by intuition, only now Heyting has invoked a much more sophisticated and philosophically developed conception of intuition than had Brouwer. This concept of intuition also forms part of an elaborate theory of meaning, but one that is different from Brouwer's in several important respects. In particular, I shall argue for a theory of meaning that is not solipsistic. Martin-Lgf's views on the philosophical basis of intuitionistic mathematics are similar to Heyting's. In his 1983-84 lectures on the theory of meaning, for example, Martin-LSf says that ... the proof of a judgment is the evidence for it ... thus proof is the same as evidence ... the proof of a judgment is the very act of grasping, comprehending, understanding or seeing it. Thus a proof is, not an object, but an act. This is what Brouwer wanted to stress by saying that a proof is a mental construction, because what is mental, or psychic, is precisely our acts ... and the act is primarily the act as it is being performed, only secondarily, and irrevocably, does it become the act that has been performed.

582 Martin-Lhf's comments emphasize the intuitionistic view that a proof is a cognitive act or process before it is an object, an act or process in which we come to see or intuit something. In his discussions of Heyting's Husserlian interpretation [Martin-Lhf 1983-84, 1987, 199?] of the logical constants Martin-Lhf has been more careful than some writers to distinguish between proof as an act or process and proof as an object. We might say that proof as an object is constituted, in its most primitive form, in an act of proof by virtue of the kind of retention in memory that Brouwer emphasizes in his descriptions (see [Tieszen 1989], pp. 99111). Also, Martin-Lhf's system of intuitionistic type theory in [MartinLhf 1984] use four basic forms of judgment, among which are the two that "S is a proposition" and "a is a proof (construction) of the proposition S". Martin-Lhf notes that one can read these, equivalently, as "S is an intention (expectation)" and "a is a method of fulfilling (realizing) the intention (expectation) S", respectively. Thus, one can understand his system as a formalization of features of the informal concepts of intentionality, intuition and evidence. In a somewhat different vein, Troelstra and van Dalen, in their twovolume book Constructiyism in Mathematics [Troelstra and van Dalen 1988], have argued that since the perfect introspection that Brouwer postulates for the ideal mathematician is simply not accessible to us we must look elsewhere for the philosophical foundations of intuitionism. They suggest that 'informal rigor' (in Kreisel's sense) is the main source of mathematical knowledge in intuitionistic mathematics. As Kreisel described it [Kreisel 1967], the idea of informal rigor is that we obtain definitions, axioms or rules by analyzing intuitive notions as precisely as possible and putting down their properties. Kreisel thought the general idea applied equally well to realist or idealist conceptions of mathematics. In idealist conceptions one supposes that the intuitive notions are related to thought or cognition instead of to a mind-independent, external world. The general form of the view of Troelstra and van Dalen that intuitionistic mathematics is based on informal rigor therefore amounts to the idea that we attempt to rigorously analyze intuitive concepts concerning various cognitive acts, structures and abilities instead of analyzing intuitive concepts concerning a mind-independent, external world. Their view is, I think, clearly consistent with the views of Heyting and Martin-Lhf. In Heyting's work we already have the idea that in intuitionism we are to focus on intentions insofar as they refer to experiences thought to be possible, and this is distinguished from focusing on the reference of intentions to states of affairs which are thought to exist independently of us. The view of Troelstra and van Dalen also has the same form that is involved in

583 understanding Martin-L5f's intuitionistic type theory as a formalization of features of the informal concepts of evidence, intuition and intentionality. Thus, I would argue that founding intuitionistic mathematics on the idea of informal, rigorous concept analysis in the sense of Troelstra and van Dalen is not at all incompatible with the above-mentioned views of Heyting and Martin-LSf. In fact, the general form of all these views is remarkably similar to what is called for in parts of phenomenological analysis. Let us now turn briefly to Dummett's views. Dummett has written more on the philosophical basis of intuitionism than anyone in recent times [Dummett 1975, 1976, 1977, 1991]. Prawitz, Sundholm and many others have discussed and elaborated on Dummett's arguments [see especially Prawitz 1977, 1978, 1980, and Sundholm 1983, 1986]. In his argument for rejecting classical logic in favor of intuitionistic logic Dummet takes the philosophical basis of intuitionism to lie in considerations involving the philosophy of language and, in particular, the theory of meaning. Indeed, Dummet argues that there is no way to approach these questions independently of or prior to investigations in the philosophy of language. Dummett's view is that the theory of meaning underlying intuitionism is, roughly, Wittgenstein's theory that meaning is determined by use. To say the meaning of a mathematical statement is exhaustively determined by its use is to say that the meaning of a statement cannot contain anything which is not fully manifest in the use of the statement. If two people agree completely about the use to be made of a statement they agree about its meaning. 'Undertanding' consists of knowledge of meaning in this sense. Prawitz, it should be noted, has modified the claim that meaning must be fully determined by observable uses of sentences. He suggests instead that "the samples of use with which we are presented never completely determine the meaning but only enable us to form some theories or hypotheses about the meaning" [Prawitz 1977]. This leads him to formulate an adequacy condition on meaning theories that is weaker than Dummett's requirement that implicit knowledge is to be fully manifest in behaviour. Dummett contrasts his view with the view that the meaning of a proposition is determined by its truth conditions. The problem with the latter view of meaning, which is essentially the view embodied in classical twovalued logic and also in platonism, is that it gives us a notion of meaning which is not recognizable by us, or which transcends our knowledge or understanding. It cannot be a view on which meaning is fully determined by use. The argument for this claim is based on the fact that, in general, truth is not decidable. In particular, a platonist theory of meaning re-

584 quires us to have an understanding of quantification over infinite domains but this transcends our capacity to recognize statements which quantify over infinite domains as true. Suppose the meaning of an undecidable statement ~ is given by its truth-conditions and we know the meaning of (I). How could this knowledge be manifested? Not by giving a proof or disproof of ~. The best we could do is to paraphrase or restate (I), but that does not give the meaning of (I) except to someone who already knows it. Knowledge of the meaning of a mathematical statement could not, on pain of an infinite regress, consist solely of explicit verbalizable knowledge, of the ability to state or to paraphrase the meaning of a statement, for then it would be impossible for anyone to learn a language who was not already equipped with a fairly extensive language. Knowledge of meaning must ultimately be implicit, and the ascription of implicit knowledge requires saying in what the full manifestation of the knowledge consists. There must be observable differences between the behaviour or capacities of someone who is said to have such knowledge and someone who is said to lack it. Thus, the truth-conditional view of meaning underlying classical logic and platonism cannot give substance to the idea of having implicit knowledge of what the condition for the truth of a mathematical statement is, since there can be nothing which constitutes a manifestation of such knowledge. Dummett's argument is framed by views about how language would not be learnable if meaning were not fully determined by use, for if there were some kind of meaning that transcended the use made of an expression then we would have to say that someone might have learned a language and behaves in every way as if he had learned and yet does not understand, or understands incorrectly. Such a view would make meaning private, ineffable. It would be inconsistent with the idea that meaning is communicable and with mathematics as a social enterprise. Dummett also does not want to be understood as a radical conventionalist, as was Wittgenstein, about what counts as correct use of mathematical statements. Another important component of Dummett's argument is his rejection of meaning holism, i.e., of the view that nothing less than the total use of language determines the meaning of an individual statement. Dummett argues that the theory that meaning is determined by use would rule out revisionism in logic and mathematics if meaning holism were correct, because in that case the question of justifying deductive practices cannot really arise. Much more could be said about these matters than we have space for here.

585 2. P u t t i n g i n t u i t i o n b a c k i n t o i n t u i t i o n i s m The view of the philosophical basis of intuitionism that I shall argue for is different from D u m m e t t ' s view in some important respects, although it can incorporate parts of D u m m e t t ' s argument. It is, I believe, consistent with the views of Heyting, Martin-LSf, and Troelstra and van Dalen. I would like to say that it is also different from Brouwer's views in some ways, although it is still in the Kantian spirit of founding intuitionism on intuition. Dummett of course points out that in his argument for rejecting classical logic in favor of intuitionistic logic he is not concerned with the exegesis of intuitionistic writings or with how well his account jibes with the views of the intuitionists themselves. I think this goes without saying since elements of the Wittgensteinian theory of meaning that Dummett takes as the philosophical basis of intuitionism are really quite alien to intuitionism as it has traditionally been expounded. What happens on D u m m e t t ' s account, for example, to the distinctive idea in intuitionism that a proof is a mental act in which we can come to see something, or to have evidence for a judgment? Not too surprisingly, it disappears. While Dummett distances his position from classical behaviorism it is nonetheless true that on his account of intuitionism the entire vocabulary of cognitive acts, processes and abilities in fact disappears, or in some cases is reinterpreted, after Wittgenstein, in terms of observable practices and abilities. The distinction between inner and outer phenomena vanishes, along with the very basic distinction between act and object. So much then for MartinLSf's description, cited above, of a proof of a judgment as the mental act of 'grasping', 'comprehending', 'understanding' or 'seeing' the judgment. I think this shift is also a source of the objection raised by Troelstra and van Dalen that they do not see how the formulation of axioms based on the process of informal rigor, such as their own formulation of axioms for lawless sequences, fits into D u m m e t t ' s theory ([Troelstra and van Dalen 1988], p. 851). Of course one of the things that disappears along with the idea of mental acts and processes on D u m m e t t ' s approach is any philosophical objection to intuitionism based on solipsism or subjective idealism. But I shall argue that Dummett goes too far here, that we can perfectly well keep the distinction between inner and outer phenomena without succumbing to the pitfalls of solipsism. D u m m e t t ' s account of intuitionism, on the other hand, contains no theory of intentionality, fulfilled intentions and evidence of the sort that Heyting appeals to. It has no theory of mathematicians as cognitive information processors, of the structure of cognition, of mental

586 acts and meaning, of mental representation, of the content of mental acts, of implicit or qualitative content, of consciousness, and the like. I think, however, that if proof is really to be understood as either a cognitive act or as an object of an act then these notions must figure into our understanding of the philosophical basis of intuitionism. Thus, I will argue that the philosophical basis of intuitionistic mathematics is best understood along the lines suggested by Heyting's 1931 address. I shall distinguish several key components of this view and then indicate how they can be used to enrich the philosophical understanding of intuitionism and also to defend intuitionism as a philosophy of mathematics. The first component of the view is that mathematical propositions are to be understood as expressions of intentions, in the sense of a theory of intentionality [Tieszen 1989, 1991]. Intentionality is the characteristic of "aboutness" or "directedness" possessed by various kinds of mental acts. The intentions of acts can be determined by "that"-clauses in attributions of beliefs and other cognitive acts to persons, as in propositions of the form "M believes that ~". M's intention here is expressed by ~. Note that this view of expression entails a philosophy of language and also, as we shall see in a moment, a theory of meaning. Thus, I do not wish to suggest, as against Dummett, that a philosophy of cognition is independent of and anterior to a philosophy of language. The two are intertwined and may stand or fall together. However, there is not only one kind of philosophy of language or meaning. Witness, for example, the subtle interactions between the study of language, meaning and cognition in treatments of transformational grammars, in the semantics of propositional attitudes, and in what has come to be called propositional attitude psychology. The second component of the view is that mathematical intentions can either be fulfilled or not, or even partially fulfilled, by additional acts carried out through time. Intentions can also be understood as expectations that can either be realized or not. When an (empty) intention is fulfilled then the object intended in an act is actually seen. That is, we have direct evidence for it. If the intention is fulfillable then it is possible to find the object intended. So, following Heyting, a proof or a construction in intuitionistic mathematics is a fulfilled (or fulfillable) mathematical intention. A fulfilled intention is an intention for which we possess evidence. The intention/fulfillment relation can also be understood in terms of Kolmogorov's interpretation [Kolmogorov 1932] of propositions as problems or tasks and proofs as solutions, and in terms of Martin-LSf's suggestion that we view empty intentions as specifications and fulfillments as programs that satisfy those specifications [Martin-LSf 1982]. I want to argue, however, that something crucial to the intuitionistic view that mathemat-

587 ics is the precise part of human thinking would be lost if these alternative explications were not understood in terms of a theory of intentionality. This means, for example, that we ought not to immediately identify fulfillments, understood as programs, with programs computable by Turing machines. There might be a difference between machine computability and human computability and we do not want to ignore the problematic status of Church's Thesis in intuitionistic mathematics. The third component is that intentional acts are responsible for meaning or interpretation in the sense that strings of signs, noises, and so on would not be taken to have meaning, value, or significance if there were not intentional systems in the universe. This does not mean, as its detractors sometimes claim, that a person must always consciously, as it were, perform some mental act in order to understand a string of signs. It is a crude caricature of the view to suppose that there is first some completely uninterpreted sign configuration, and then someone performs a mental act which bestows sense upon it, whereupon it is understood. Rather, we normally understand the meaning of expressions quite automatically and prereflectively. The point is rather that it is a condition for the possibility of meaningfulness that there be individuals in the universe that have cognitive states that are characterized by intentionality. That is all that is meant by saying that mental acts are involved in meaning or understanding. The fourth component I would like to mention is that mathematical statements can be meaningful even if they are not fulfilled or are not fultillable. We have constructions for some mathematical intentions but not for others. But surely we can understand the meaning of a statement independenntly of knowing its truth value, for as Frege and Husserl remind us, and as Brouwer may have failed to recognize, we must not confuse lack of (intuition of) reference with meaninglessness, nor even logical inconsistency with meaninglessness. But the kind of distinction between meaning and reference implied by these remarks is not part of the Wittgensteinian theory of meaning that Dummett takes as the basis of intuitionism. In saying this I do not of course wish to deny that fulfilled mathematical intentions have a more determinate or explicit meaning than empty intentions. Fulfilled mathematical intentions provide more information about the object or state of affairs in question, including specific numerical or computational meaning that is otherwise lacking. Now let us fill in somewhat the view of the philosophical basis of intuitionism that is associated with these points. Of particular concern, visg-vis Dummet, is whether the intuitionistic emphasis on proof as mental construction can be preserved and defended. D u m m e t t has argued that

588 if meaning were not fully determined by observable uses of sentences we would have to say that someone might have learned a language and behaved as if she had learned and yet does not understand. Language would not be learnable if meaning were not fully determined by observable use. Meaning would be private, and ineffable, which is inconsistent with the possibility of communication and with the social character of mathematics. These arguments, however, are far from being decisive. First, I do not see any problem with saying that someone can appear from observable uses of sentences to have learned a language but in fact does not understand the language. Perhaps the simplest way to see this in recent times is through the type of argument John Searle gives about Chinese syntax manipulators [Searle 1980]. The person or machine in Searle's argument interacts with others in Chinese and passes the test for understanding Chinese based on the criterion of considering all possible observable uses of sentences, but does not understand a word of Chinese. What is missing? Intrinsic intentionality. Hilary P u t n a m has made arguments about the evolution of "perfect actors" to also show that observable linguistic behavior does not suffice to determine understanding or meaning [Putnam 1965]. It can be argued that observable behavior or practice generally underdetermines what we know or understand. We see this in linguistics, perception, mathematics, and elsewhere. Compare, for example, our linguistic performance and our linguistic competence. Now, because observable practice in using sentences underdetermines our knowledge or understanding, and does not suffice to explain it, we must make an inference to unobservable, inner processes or structures to fill in the explanation. This is a pattern of reasoning that goes back to Kant and is now used widely in linguistics, cognitive science and artificial intelligence studies. The role of informal rigor in intuitionism, if it is to be the source of mathematical knowledge, must evidently be to unfold and clarify our knowledge of these cognitive processes or structures. On this view there is no reason to expect a direct correlation between a set of observable linguistic behaviors and the structures of a semantic theory, where these structures may be cognitively real. This is not, of course, to say that there is no correlation of any kind. Surely there is some relationship between our internal cognitive states and our observable linguistic behavior, but it would not do to suppose that we know enough about the relationship at this point in time to simply substitute the latter for the former. This seems to be especially true in the case of mathematics, where our thinking appears to have much more complexity, subtlety or nuance than it does in some of our other cognitive or practical endeavors. Observable linguistic behavior in mathematics, one might believe, is just

589 too coarse to do justice to this complexity, except perhaps at the level of pebble arithmetic. The pattern of reasoning we are invoking establishes a distinction between inner and outer phenomena, but not in a way that makes meaning private, non-learnable, non-communicable or non-implicit. Why not? The answer is straightforward: because human beings are so constituted as to have at least some isomorphic cognitive structure, which is what makes learnability and communicability possible. In other words, the intuitionistic idea that a proof is a mental act of construction can be defended against the charge of incoherence on the following grounds, which are basically Kantian. We start by taking the science of mathematics as a social enterprise as given and then attempt to deduce the kinds of cognitive structures that are necessary to make it possible. On such a view there can be no philosophical defense of a solipsistic notion of proof, if solipsism is the position that there could be proofs that are in principle understandable to only one person. The rejection of solipsism does not, however, entail that there has to be intersubjective agreement at all times on all mathematical statements. Nonetheless, we have an explanation of how it is possible for there to be intersubjective agreement in at least elementary parts of mathematics and the explanation implies that the concept of a proof, as a fulfilled intention, is not solipsistic, and that it need not involve introspection. Thus, I argue that speaking of a proof as the fulfillment of an intention for a particular mathematician depends on the possibility of fulfillment of the same intention for other mathematicians. I freely admit that this view of proof contrasts sharply with some of Brouwer's remarks. In the early Leyen, Kunst en Mystiek, for example, Brouwer says that even in logic and mathematics "no two persons will think the same thing in the case of the fundamental notions" [Brouwer 1905]. Brouwer's viewpoint, however, fails to do justice to the fact that the science of mathematics, and intuitionism itself, exists as a social enterprise, that different people make contributions to it at different times and places. So I agree with Dummett insofar as he is pointing out that Brouwer's solipsism, so construed, is philosophically indefensible but, unlike Dummet, I do not jettison the-'idea of proof as an act of mental construction. I do not want to dispose of the act/object distinction, nor of some version of the meaning/reference distinction and of the view of epistemology that goes with it. Several logicians have suggested that perhaps D u m m e t t ' s view of the philosophical basis of intuitionism is, after all, consistent with the view I have presented thus far, or that D u m m e t t ' s is a complementary view, for Dummett is simply emphasizing the external or observable aspects

590 of acts while I am emphasizing the internal aspects. On the other hand, some other logicians have felt that the views expressed above are definitely inconsistent with Dummett's views. I have been inclined to agree with the latter position, but perhaps further philosophical analysis is needed. The other main point I want to discuss is one which bothers many logicians and mathematicians when the subject of intuitionism arises, and that is the question of the relation of intuitionistic to classical mathematics. In particular, I wish to ask how we should understand classical mathematics from an intuitionistic standpoint. In The Elements of Intuitionism and "The Philosophical Basis of Intuitionistic Logic" D u m m e t t is interested in developing an argument for showing that the classical way of construing mathematics is "incoherent and illegitimate", that it is "unintelligible". He is concerned to find grounds for the revision of mathematical practice, and his argument is not favorably disposed toward an eclectic position on this issue. Now I shall argue that there is a sense in which classical mathematics need not be construed as incoherent, illegitimate or unintelligible for an intuitionist, although it may be so construed if it is taken to do justice to mathematical knowledge. In order to grasp this let us first recall a response Dummett has made to a defense of platonism. Dummett has argued that human practice is simply limited and there is no extension of it, by analogy, that will give us an understanding of the capacity to run through an infinite totality. Meaning must be derived from our capacities. It cannot be derived from a hypothetical conception of capacities we do not have. To think otherwise only shows the extent to which illusions are involved in understanding our own language. It has of course been pointed out by Crispin Wright and others that one of the problems with this argument is that intuitionism is committed to some of its own rather strong idealizations of human practice, so that someone who took the limitations of our capacities seriously, like a strict finitist, could direct a similar line of reasoning to Dummett's own position [Wright 1982]. Thus, a strict finitist might argue that there is no extension of our practice which, by analogy, will give us an understanding of an effective procedure which is not feasible, according to some measure of computational complexity. Meaning cannot be derived from a conception of hypothetical capacities that transcends feasibility. Does this show that intuitionism is incoherent and unintelligible? I believe it no more shows this than Dummett's argument shows that classical mathematics is incoherent and unintelligible. But it does show us that something is amiss in Dummett's conception of how meaning is connected with idealizations of practice, especially as this is supposed to figure into the difference between basing meaning on use

591 and basing meaning on t r u t h conditions. On the Husserlian view of intentionality and meaning invoked by Heyting, we are to view m a t h e m a t i c a l statements as expressions of intentions, as expectations, or as problems. It is just that, as intuitionistic (weak) counterexamples show us, we have reason to believe t h a t some of our expectations, understood in their full generality, will never be realized. But I do not see why intuitionists or even strict finitists need to deny t h a t general logical principles like P V -~P, or universal quantifications over infinite domains, can function in our experience as regulative ideals in a Kantian sense. T h a t is, P V ~ P can be viewed as an expectation of what should be the case at a research point lying at infinity, a kind of postulation of reason t h a t reflects a natural tendency of h u m a n cognition, no m a t t e r how much we may try to suppress it. Then we can think of intuitionistic m a t h e m a t i c s as an expression of the view t h a t we k n o w far less about objects than we can r e a s o n about on a classical model of reasoning. We are inclined in our reasoning to postulate certain closure conditions, forms of completeness or of "perfection" which cannot be verified in intuition. We try to complete the incomplete. But, at the same time, this can be useful because in the process we come to grasp and measure the degree and defects of the incomplete. If Kant's view is correct then regulative ideals drive scientific research and problem solving. For example, they induce mathematicians, including intuitionists, to work toward the solution of open problems with the expectation t h a t a solution is to be found, although the source of the expectation is now taken to be immanent to cognition and is not derived from the idea of a mind-independent realm of truths, as it might be for an ontological platonist. For P V ~ P to have this kind of meaning is of course not the same thing as having an intuitionistic proof t h a t P V --P. On the other hand, it does not follow t h a t classical mathematics, with its a t t e n d a n t notion of "meaning as determined by t r u t h conditions", is unintelligible or incoherent, provided we now view it as postulating ' t r u t h ' as an absolute or regulative ideal, analogous to the abstraction from a finite bound on c o m p u t a t i o n involved in the intuitionist's own conception of acceptable m a t h e m a t i c a l reasoning. Intuitionists owe us at least an explanation of the origins of classical mathematics, if not of its status and significance, and on the view just described we have such an explanation. An intuitionist can ask about the conditions for the possibility of classical mathematics, and the answer will come in terms of some aspect of our cognitive makeup, some function involving the effort to complete the incomplete, to a t t a i n a kind of "cognitive closure". Parts of m a t h e m a t i c a l practice will be a product of this cognitive makeup, and in those parts where our idealizations are especially

592 far-flung lie the possibilities of antinomies, paradoxes, or illusions. Just as traditional rationalistic metaphysics existed, so parts of mathematical practice that cannot be constructively justified actually exist. There is, nonetheless, a foundation in our cognitive structure for classical mathematics and classical mathematics cannot be meaningless to us. In this way we can explain classical mathematics as a part of human practice to which different mathematicians in different times and places make contributions. The non-constructive meaning of mathematical statements also need not be construed as private, non-learnable, non-communicable, or non-implicit because, as I am construing intuitionism, humans are so constituted as to have at least some isomorphic cognitive structure, which is what makes learnability and communicability possible. Humans are bound to project their knowledge beyond their actual, even possible experience, but there is intersubjective agreement in doing this, even if the specific views that result from doing it are sometimes different. Thus, intuitionists can say of classical mathematics that it constitutes an illegitimate and perhaps even a dangerous extension of what we can be said to know about objects, but that it is cognitively inevitable and does serve some purpose in human affairs. It is just that intuitionism calls for a kind of experiential verifiability not found in classical mathematics. This boils down to the fact that the kind of "grounding in experience" called for in constructive mathematics generally gives us a foothold in reality, a standard, and a common, "objective" basis for mathematics. There is a core of elementary mathematics on which the views of mathematicians of quite different philosophical persuasions overlap, and this core is constructivist. Intuitionism loses none of its substance in making the point that we need to be careful in saying that we "know" classical mathematics, in making the point that we do not really "know" something that results from striving for cognitive closure when doing so could lead to illusions. 1

lI would like to thank the many LMPS IX participants with whom I discussed this paper for their comments, and especially Susan Hale, Geoffrey Hellman, Per MartinLSf, Dag Prawitz, Hilary Putnam, Michael Resnik, SSren Stenlund, GSran Sundholm, and Dirk van Dalen.

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