June 10, 1991 Copyright 1991, Anthony Birch, Ph.D. Contact: tbirch@gis.net Note: This paper may be a useful starting place for those interested in Wittgenstein's philosophy of mathematics -- not because the paper is particularly insightful, but because Waismann's book is an excellent, often overlooked, resource. WAISMANN'S CRITIQUE OF WITTGENSTEIN Does Wittgenstein understand the language game of mathematics? Such a question almost seems ridiculous given the amount Wittgenstein wrote on mathematics, his own training in mathematics, and the use of mathematical examples throughout his works. But Friedrich Waismann, a onetime follower of Wittgenstein, would answer the question with a qualified "No." Waismann's criticisms of Wittgenstein rely to a great extent on Wittgenstein's own dicta that in language games the meaning of words, at least for a large number of classes, is the use of the words (PI, para. 43) and that we ought to "look and see" rather than theorize, if we want to understand a problem. Like Wittgenstein, Waismann was to remain uncommitted to prior theoretical constructions through which mathematics must be understood. But at the same time, Waismann draws heavily on the actual practice of mathematicians, their actual use of language, and their employment of philosophic terms such as "existence." It can be said that to a certain extent Waismann turned Wittgenstein's method back on itself. Waismann's views, therefore, provide an excellent case study for examining Wittgenstein's philosophy and developing a critical appraisal. The purpose of this paper is to examine some of Waismann's central criticisms of Wittgenstein. In the process, I will develop critical comments on Wittgenstein's philosophy of mathematics and examine the relation of his philosophy of mathematics to his philosophy of language and mind. I will argue that Waismann's criticisms of Wittgenstein are essentially correct and that in several important areas Wittgenstein makes an inaccurate appraisal of the content and nature of mathematics as a language game. It will be useful to characterize some of the basic elements of Wittgenstein's thought relevant to our topic. For Wittgenstein, mathematics is a practice, an intersubjective, anthropological phenomenon whose rules derive from basic human responses to training and instruction. Mathematics is a language game (or series of language games) whose rules are established by linguistic convention but whose basis is rooted in what is given to us by the way the world of ordinary experience, that is, our "form of life" actually works. But, as in the case of ordinary language, we are sometimes bewitched by the rules of our own making into thinking there are "essences" or "spiritual causes" or "abstract objects" that exist apart from human language and dictate or guide our thoughts. The example at the beginning of the Investigations illustrates how Wittgenstein seeks to circumvent the essentialism or Platonism that we use to explain behavior. The Platonist assumes that if we ask S to retrieve five red apples, S somehow knows that there is a "five," that is, a timeless, eternal "five" that can be instantiated by a group of objects. But Wittgenstein shows that the action can be adequately described in the following way: "he opens the drawer marked apples...then he says a series of cardinal numbers -- I assume he knows them by heart -- up to "five" and for each number he takes an apple...out of the drawer" (PI, para. 1). Hence, "five" is not an eternal object, not even a concept. "Five" is acted out through a procedure which someone has been trained to do without thinking (i.e., by heart). Finding that there are five apples is empirical and takes time; it is not instantaneous. But we are fooled, as it were, by our own training into thinking that when we say such things as 2 + 2 = 4 we are referring to eternal objects whose relations are predetermined in some timeless realm. Such simple truths as 2 + 2 = 4 must be acted out in the world and a social practice must reinforce their use before they can become automatic for us. Thus, mathematics has both empirical and normative elements. The basic behavioral, empirical human responses to the world account for the perceived inexorability of the rules of mathematics. At the same time, Wittgenstein shows us that it is ontologically necessary for there to be a community of language learners (not isolated individuals) that can set up, explore, and redefine the boundaries and norms of their own language games (cf. Wittgenstein's private language argument in the Investigations). Wittgenstein wants to undermine usual theories of the foundations of mathematics. This is consistent with his conception of philosophy in the Investigations. Philosophy consists of reminders of the actual use of terms and the rules of grammar. The philosopher's task is simply to describe. Once a description is available, the problems of academic philosophy -- and the skepticism to which is leads -- disappear. Similarly, once the epistemological framework is removed from mathematics, the "problem" of its foundations will disappear (TP, p.3). At this point, it is easy to see how Wittgenstein's views connect to constructivism, which emphasizes the primacy of counting as the foundation of mathematics, and conventionalism, which emphasizes the game-like and linguistic nature of mathematics. Generally, however, commentators have found it useful to list the specific schools Wittgensteins's examples seem to target (I owe the following four points to Klenk, see KLE, p. 1). (1) We have already seen how Wittgenstein critiques Platonism since, according to Wittgenstein, mathematics is not about timeless propositions but about forms of inference in a practice. (2) He undermines intuitionism because it relies on internal mental representations to establish proofs and it does not introduce intersubjectivity and communal norms. (3) He defuses formalism because it attempts to reduce a practice to the manipulation of meaningless symbols and ignores actual human use. (4) Although his position shares elements with conventionalism, Wittgenstein also opposes conventionalism because it leaves the creative and synthetic aspects of mathematical practice (i.e., the forming of new, useable concepts) unaccounted for. A few commentators have attempted to place Wittgenstein in one foundational camp or another. Dummett, for example, calls Wittgenstein "a full-blooded conventionalist" (DUM, p. 424). (Waismann, too, seems to have made this mistake.) The most consistent view, in my opinion, is that Wittgenstein's project in mathematics is the same as it is in the philosophy of mind and language; namely, that he will advance no theses, but will simply describe things as they are and assemble reminders that uncover the roots of our confusions. While Wittgenstein may be said to "borrow" certain themes from various foundational schemes, it is only to return them when it becomes clear that philosophy leaves mathematics untouched and its foundation will take care of itself. Mathematics does pose special problems for Wittgenstein. Unlike the terms of ordinary language, the boundaries of whose meanings are diffuse, the terms of mathematics seem definite and fixed. Unlike the propositions of science, which can be matched to a state of affairs or to empirically verifiable facts, many mathematical assertions seem to have no empirical conditions, or empirical but untestable conditions, that make them true. Further, if counting is to provide the ultimate basis for proof, then Wittgenstein has to face the following problem: Suppose I say "263 + 492 = 755." Do I need to verify this by counting? Is this, ultimately, the only true method of verification? Furthermore, what do I count--marks on the paper or physical items? What assures me that I have not missed a mark or that a physical item has not disappeared? If I solve the problem by limiting myself to cases where I could in principle count, does this mean only finite mathematics is well-founded? What type of training results in my being able to calculate correctly? These and similar problems for Wittgenstein will remain in the background as we compare his views to Waismann's. (In what follows, I will take the liberty of attributing "views" to Wittgenstein -- often as Waismann perceives them.) It is illuminating to compare and contrast Wittgenstein and Waismann on the issue of the meaning of the cardinal numbers and the expansion of infinite series. Elaborating on Wittgenstein's examples from the Investigations, Waismann shows how primitive language games using commands such as "Five bricks here!" in fact model the use of numbers in our own language quite well (LPM, pp.55-56). It is not the case that I observe, in the process of counting, some factual relation between the objects which are before me and the sounds "one," "two," "three" I utter. Rather, I count with the numbers and then compare them to the objects (LPM, p. 51). The supposition that numbers have any extra-lingual referent is superfluous. If we ask what numbers are, Waismann replies that numbers are ...the meanings of the number symbols. And the inquiry into these meanings is the inquiry into the use of these symbols. What we are in search of is not a definition of "number" but rather a clarification of the grammar of the word "number" and the numerals. Instead of trying to compress the meaning of number into a fixed definition, we should describe the use of the term and follow the learning of a child. It is at this point that Waismann diverges from the Wittgensteinian account--for Waismann thinks Wittgenstein misdescribes the actual learning process. There comes a point when every child reaches a certain level of intelligence and sees that there is an infinite possibility of continuing the number series. The endlessness of the number series is not, as Wittgenstein would have it, a convention that "n is followed by n + 1" is a rule in a language game that might have been different. There is an intuition or, as Waismann prefers to call it, an "insight," non-tautologcial and non-analytic in character, that gives us a notion of what the number system is. Wittgenstein, of course, makes light of the notion of intuition, calling it "an unnecessary shuffle" and drawing attention to the example that if 1, 2, 3, 4... requires intuition in order to be continued, then 2, 2, 2, 2...would also require the same type of (here, more obviously unnecessary) intuition (PI, paras. 214-215). Waismann counters: There is nothing in the series itself which, when one stops somewhere, points beyond itself. In order to understand that a series is infinite we need not just a rule of "permission to play +1 indefinitely" (see RFM, II, para. 27), but a conception of infinity gained through insight. Waismann even toys with the idea that an "axiom of infinity" may be necessary to mathematics. In one sense, it adds nothing to our intuition, and the axiom would not guarantee the truth of infinity; on the other hand, it would provide a procedure for constructing the number system (See LPM, p. 122). Wittgenstein, as some critics have mentioned, often seems hostile to infinity and infinite sets. We are in danger, he says, when we think of infinite sets, such as the real numbers, as "facts of nature" (RFM, II, para. 19). We can make rules for infinity and even higher and lower order infinities, but if there are no coherent rules for the use of these conceptions, this is of dubious value. Even though, for example, members of an infinite set follow a "counting rule" and can be placed in correspondence with members of another infinite set, this does not legitimate the use of the symbol for infinity as a numeral. Such a use could, however, be invented (RFM, II, para. 38). Wittgenstein was not categorically against the use of the word "infinite" in mathematics, but only against using it to confer meaning where none exists (RFM, II, para. 58). Again, Waismann feels that such concerns are not well founded. Wittgenstein is right in thinking that we can not treat infinity as a numeral and we should avoid imagistic thinking and purely psychological apprehensions of infinity, but wrong in thinking the grammar of infinity is obscure. First, to prove a theorem holds for every number, one substitutes n + 1 for n. In this manner, a proof, say that a formula is invariant under substitution of n + 1, can be built up step by step with no need to refer to an infinite totality. This procedure makes use of mathematical induction, but there need be no ambiguity here or cause for doubt. (Wittgenstein himself probably had no doubts about the viability of induction, at least in the physical sciences. For a discussion of the viability of mathematical induction, see MK, pp. 93-108.) Second, there are clear cases where proposed definitions make improper use of conceptions of infinity. A definition such as "n is an even number, if a number x exists such that n = 2x" clearly evokes the conception of x as variable within a closed aggregate, rather than one in an open-ended series. Such uses should be eliminated from proper mathematical grammar. Third, "and so on" is perfectly intelligible when properly used -- "it is not merely a surrogate, an Ersatz, we reach out for because we are incapable of writing down all the numbers, rather it is a new sign with its own finite grammar" (LPM, p. 61). Finally, the insight into the nature of infinity based on the natural numbers places both finite and infinite mathematics on a sure footing. Waismann shows that insight, plus coupling the idea that equations provide a substitution license for equivalent quantities with the notion that simple algorithms can be substituted for counting, makes it unnecessary to see elementary arithmetic as a problem quite as vast as Wittgenstein envisioned (LPM, pp. 63-65). Waismann's position on the intuitive nature of the concept of infinity and its grammatical structure within mathematical language, is, on the whole, certainly more satisfying and, I believe, more true to the language game of mathematics as it is played by mathematicians. There are other points of contact between Wittgenstein's and Waismann's philosophy of mathematics that are worth mentioning. Waismann rejects the constructivist idea (which he attributes to Wittgenstein) that we can not understand a proposition unless we know the proof. This would mean that we could not understand Goldbach's conjecture, for example, until a method of proving it was discovered. Waismann objects that if this were true, it would be impossible for me to recognize the proof as a proof of the theorem in question. Whether or not Wittgenstein actually held this view is open to question (see in particular RFM VI, para. 13) where Wittgenstein admits mathematicians are not completely blank when confronted with Fermat's last theorem). It is perhaps more accurate to say that Wittgenstein thought proof fixes the sense or content of the proposition (LPM, p. 7). In any case, it is Waismann's method that is of interest here, for in the process of arguing against constructivist views, he uses the Wittgensteinian method of finding a family of related uses for the word "existence" in mathematics. Waismann finds no less than six related senses, each of which is embedded in a family of related mathematical concepts. Mathematical senses of an "existing" number include (1) what can be found by inspection; (2) what can be constructed according to a rule; (3) what is postulated as existing because the supposition of its non-existence involves a contradiction; (4) what exists as a requirement for consistency. No doubt Wittgenstein would raise questions as to the coherence of these uses, particularly the last two. But Waismann's analysis is neutral; he attempts to lay out the language before us. Like Wittgenstein, Waismann has reasons to reject the supremacy of the law of the excluded middle in mathematics (see RFM V, para. 9 for the problem of the existence of a sequence in an infinite expansion), but this does not impede his investigation of the concept of "existence." In a similar fashion, Waismann investigates the function of axioms in mathematics (cf. LPM pp. 131-167). He finds that axioms are neither meaningless terms that define only each other through a set of rules nor raw starting points of a strict deductive system. Since mathematics is a language with practical applications, axioms are fluid, indicating a set of admissible interpretations. Axioms provide a structure for interpretation of such terms as "point," "straight line," and "curve." As needs and applications change, terms can fall out of the suitable range of interpretations, and as a result new axioms can be created. These changes are made in accord with experience and in response to the realities of nature. For Waismann, the rules of syntax in mathematics can slide off into empirical statements, for example, when we define "straight" as analogous to the path of a light ray. Likewise, inference will meet requirements of form as well as the stipulations of meaning. Thus, Waismann acknowledges the creative and interpretive aspect of mathematics as well as the formal and practical requirements. Mathematical language is never meaningless, nor is it fully defined--it is in an in-between state (LPM, p. 135). Part of what Waismann says about axioms, coupled with the previous comments about Wittgenstein's notions on the number series (above, in our discussion of infinity) amounts to an implied critique of Wittgenstein's supposed conventionalism regarding truth. Dummett has given an excellent expression of what Wittgenstein's conventionalism is supposed to entail: it means that "all necessity is imposed on us not by reality but upon our language, a statement is necessary by virtue of our having chosen not to count anything as falsifying it. Our recognition of logical necessity thus becomes a particular case of our knowing our own intentions" (DUM, p. 424). It follows from this view that we are free to accept or reject any proof even though we have accepted the axioms, because each act of accepting something as true is separate. Waismann seems to have thought Wittgenstein held a view like this. But as Stroud has shown, Wittgenstein's examples are really intended to indicate not the arbitrariness of mathematics but how it is derivative of a form of life, the judgements about which we are not, ultimately, free to choose (STR, p.493). Wittgenstein's "conventionalism" ultimately boils down to our being responsible for the course of mathematics but also to our responding to what we objectively find (STR, p.496). Ironically, Waismann used nearly an identical formulation: "we both make and do not make mathematics." On the fundamental point that mathematics is a practice enmeshed in social reality and constrained by facts as well as form, Waismann and Wittgenstein are in agreement. Waismann's critique of Wittgenstein and his appropriation of Wittgenstein's methods show us how to approach the problem of the foundations of mathematics. We can utilize the basic approach of Wittgenstein, but we need to address the following critical considerations developed by Waismann. (1) Wittgenstein's dismissal of the role of intuition or insight in the actual practice of mathematics (or at least in consideration of its foundations) seems misplaced or at least to require greater defense. Waismann's appeal to insight is not made in order to support prior Platonist or hidden mystical agendas. He sees it not only as an objective phenomenon but as integral to the acquisition of the understanding of mathematical terms. Perhaps this error on Wittgenstein's part is due to a carry over from the philosophy of language and mind developed in the Investigations. In paragraph 214, of the Investigations, Wittgenstein does let some light under the door: it might be possible that I could teach someone a way of harkening or being receptive. However, Wittgenstein rejects this, for how could I be sure one was harkening in the right way? In light of our discussion, this seems only to impose a superficial difficulty. Couldn't insight be the natural outcome of ordinary training? Does insight automatically commit one to Platonism and infallibility? (For an account of non-Platonist intuition as a result of training see ME, pp. 375-399.) (2) Waismann shows that the mathematical grammar of terms such as "infinity," which seemed to give Wittgenstein so much trouble, can be investigated and shown to cohere in a way that does justice to both our common sense ideas and the practice of mathematics. This is not to say that definitions of terms are at hand. Rather, we become aware of the evolving nature of mathematics and take self-conscious control of its development in the light of needs and other practices. Mathematicians constantly adjust the meanings of their axioms to encompass new needs -- but this is not to say they have no meaning at all. This awareness of creative activity within a language game ought to mitigate the Wittgensteinian fear that mathematicians will always believe their own story and read back their own results as "the real and the true." (3) There may be a severe methodological mistake in Wittgenstein's attempt to investigate the foundations of mathematics staying primarily at the "lowest" level ("philosophy has to face temptations to misunderstand at this level of knowledge", RFM VI, para. 12). By staying at a "low" level I do not mean that Wittgenstein is to be criticized for using examples from simple arithmetic, but that he does not examine the levels of expert knowledge that are part of the account of the practice of these same activities. The expert approaches problems in a way different from the novice. In addition, the expert uses, as Waismann shows, a more extensive vocabulary and often more precise definitions (e.g., recursive definitions) that in his mind, at least, dissolve many dilemmas of the beginner. Waismann's discussion of how we use and ought to use "infinity" is more persuasive than Wittgenstein's because of his expert knowledge. But even if we are not impressed by experts (indeed, following Wittgenstein, we should be wary of them), it behooves us to investigate their understanding of their practice or we shall not be fully aware of what the practice is. BIBLIOGRAPHY REF SYMBOL Bloor, David, Wittgenstein: A Social Theory of Knowledge, New York, Columbia Univeristy Press, 1983. Canfield, John, ed., The Philosophy of Wittgenstein, Vol. 6, Garland Publishing, 1986. ME Davis, Philip, and Hersh Reuben, The Mathematical Experience, Boston, Birkhauser, 1981. DUM Dummett, Michael, "Wittgenstein's Philosophy of Mathematics" in Pitcher, George, ed., Wittgenstein: The Philosophical Investigations, University of Notre Dame, 1966. KLE Klenk, V. H., Wittgenstein's Philosophy of Mathematics, The Hague, Marinus Nighoff, 1966. TP Shanker, S. G., Wittgenstein and the Turning Point in the Philosophy of Mathematics, Albany, SUNY Press, 1987. MK Steiner, Mark, Mathematical Knowledge, Ithaca, Cornell University Press, 1975. STR Stroud, Barry, "Wittgenstein and Logical Necessity", in Pitcher, George, ed., Wittgenstein: The Philosophical Investigations, University of Notre Dame, 1966. LPM Waismann, Friedrich, Lectures on the Philosophy of Mathematics, Amsterdam, Editions Rodopi, B. V., 1982. PI Wittgenstein, Ludwig, Philosophical Investigations, New York, Macmillian, 1968. RFM Wittgenstein, Ludwig, Remarks on the Foundations of Mathematics, Cambridge, MIT Press, 1991.