Published online by Cambridge University Press: 08 January 2010
To be asked to provide a short paper on Wittgenstein's views on mathematical proof is to be given a tall order (especially if little or no familiarity either with mathematics or with Wittgenstein's philosophy is to be presupposed!). Close to one half of Wittgenstein's writings after 1929 concerned mathematics, and the roots of his discussions, which contain a bewildering variety of underdeveloped and sometimes conflicting suggestions, go deep to some of the most basic and difficult ideas in his later philosophy. So my aims in what follows are forced to be modest. I shall sketch an intuitively attractive philosophy of mathematics and illustrate Wittgenstein's opposition to it. I shall explain why, contrary to what is often supposed, that opposition cannot be fully satisfactorily explained by tracing it back to the discussions of following a rule in the Philosophical Investigations and Remarks on the Foundations of Mathematics. Finally, I shall try to indicate very briefly something of the real motivation for Wittgenstein's more strikingly deflationary suggestions about mathematical proof, and canvass a reason why it may not in the end be possible to uphold them.
1 For n = 2, of course, there are “Pythagorean” solutions—for instance, 3, 4 and 5; 5, 12 and 13; and so on.
2 Kurt Gödel is widely regarded as endorsing the ancient answer in his (1947), pp. 483–4. Penelope Maddy is also a staunch champion of it; see her (1980).
3 For the record let me say that Kripke's ideas about these issues and mine seem to have developed in complete isolation from each other. Kripke's interpretation originated, as he recounts, in graduate seminars given in Princeton as early as the spring of 1965 and was subsequently developed through a series of conferences and colloquia from 1976 onwards. I first proposed such an interpretation of aspects of Wittgenstein's later thought on mathematics in my (1968); and the material that constitutes the first six chapters of Wittgenstein on the Foundations of Mathematics (Wright, 1980)Google Scholar was first written up for graduate seminars given in All Souls College, Oxford in the summer of 1974. Kripke and I were, indeed, colleagues for several months at All Souls in the academic year 1977–8, when he held a Visiting Fellowship there. But we never discussed the interpretation of Wittgenstein.
4 It might be said, similarly, by an opponent of moral realism that the role of the sentence ‘Lying is wrong’ is not to describe a moral fact, but to express a condition compliance with which is a necessary condition for avoiding moral disapprobation.
6 And are further discussed in Wright, Crispin and Boghossian, Paul, ‘Meaning-irrealism and Global Irrealism’, forthcoming.Google Scholar
7 For further discussion of these ideas, and of the perplexities they generate, see my (1989a, b).
8 This point was perhaps the most important factor determining the overall direction of the argument of my (1980).
9 Provided, of course, that no other proof of the same conclusion is known to one.
10 A selection of relevant passages from RFM (second edition) would include part I, sections 36–57, 75–103 and 156–64; part II, sections 55 and 65–76; part III, sections 46–53; and part V, sections 6, 14–15, 17 and 40. Germane material published for the first time in the third, revised, edition includes Appendix II, part VI, sections 1–10, 15–16 and 36; and part VII, sections 25–6. Cora Diamond's edition of the (1939) Lectures on the Foundations of Mathematics (LFM) touches on the issues in lectures iii (36–9), vii (71 and following), x and xi passim, and xiii (128–30).
11 What I have elsewhere called the corresponding descriptive conditional. See my (1980), p. 452; also (1986c), pp. 203–4; and (1989c), pp. 231–2.
12 This is what Wittgenstein means when he says—RFM, III, 41–that causality must play no part in a proof.
13 I am here by-passing complications to do with rules of inference, like velintro., which permit more than one conclusion from given premises. For further discussion, see my (1989c), pp. 234–5.
14 For a somewhat fuller account see my (1989c), section IV.