¹ Chihara, Charles S., ‘Wittgenstein's Analysis of the Paradoxes in his 1939 Lectures on the Foundations of Mathematics’, Philosophical Review 86 (07 1977).CrossRefGoogle Scholar

² In references to Wittgenstein's work I employ the standard abbreviations: WWK—Ludwig Wittgenstein und der Wiener Kreis (Oxford: Blackwell, 1967)Google Scholar, PR—Philosophical Remarks (Oxford: Blackwell, 1975)Google Scholar, PG—Philosophical Grammar (Oxford: Blackwell, 1974)Google Scholar, LFM—Wittgenstein's Lectures on the Foundations of Mathematics, Cambridge, 1939, Diamond, (ed.) (Hassocks: Harvester, 1976)Google Scholar, RFM—Remarks on the Foundations of Mathematics, 2nd edn (Oxford: Blackwell, 1964)Google Scholar, LC—Lectures and Conversations on Aesthetics, Psychology and Religious Belief, Barrett, (ed.) (Oxford: Blackwell, 1966)Google Scholar, BB—The Blue and Brown Books (Oxford: Blackwell, 1958)Google Scholar, PI—Philosophical Investigations (Oxford: Blackwell, 1953)Google Scholar, and Z—Zettel (Oxford: Blackwell, 1967).Google Scholar

³ Chihara, , op. cit., 381 n.Google Scholar

⁴ This claim is not uncontroversial. I have defended it in my paper ‘Wittgenstein and Constructivism’ (forthcoming).

⁵ Hubert, , ‘On the Infinite’, Philosophy of Mathematics: Selected Readings, Benacerraf, and Putnam, (eds) (Oxford: Blackwell, 1964), 135.Google Scholar

⁶ As noted by Hacker, P. M. S. (in his Insight and Illusion: Wittgenstein on Philosophy and the Metaphysics of Experience (Oxford: Clarendon Press, 1972), 140)Google Scholar Wittgenstein is not unique in this. Hacker cites Nietzsche as another philosopher whose unsystematic style of exposition disguises the underlying system of his thought.

⁷ See, e.g. Ayer, A. J., Language, Truth and Logic (Harmondsworth: Penguin, 1971)Google Scholar, Ch. 4, for an account of this theory of necessity.

⁸ See Dummett's, Michael ‘Wittgenstein's Philosophy of Mathematics’ reprinted in Dummett, Truth and Other Enigmas (London: Duckworth, 1978)Google Scholar, for further discussion of this objection. Jonathan Bennett has tried to defend moderate conventionalism against Dummett in ‘On Being Forced to a Conclusion’, Proc. Arist. Soc. Suppl. Vol. 1961.Google Scholar See Wright, Crispin, Wittgenstein on the Foundations of Mathematics (London: Duckworth, 1979), Ch. 18Google Scholar, for a detailed criticism of Bennett's defence.

⁹ E.g. Dummett, , op. cit.Google Scholar But see Wright, , op. cit.Google Scholar, for a very full defence of radical conventionalism against the objections of Dummett and others.

¹⁰ Wrigley, , op. cit.Google Scholar

¹¹ Despite all this strong evidence of Wittgenstein's radical conventionalism this interpretation has been challenged, notably by Barry Stroud in his ‘Wittgenstein and Logical Necessity’, Philosophical Review 74 (1965).Google Scholar In Wrigley, op. cit., I have defended this interpretation against Stroud.

¹² Logically determinate that is. It may for all we know be causally determinate what we shall derive but until we actually do so we have yet to set up the convention which will decide whether this is the logically correct thing to derive.

¹³ Chibara, , op. cit., 377–378.Google Scholar

¹⁴ Chibara, , op. cit., 378–379.Google Scholar

¹⁵ See Hacker, , op. cit., Ch. 5Google Scholar, for an illuminating discussion of Wittgenstein's later conception of philosophy, and in particular of the tension between it and other aspects of his work.

¹⁶ Cf. Hardy's talk about ‘What Littlewood and I call gas, rhetorical flourishes designed to affect psychology, pictures on the board in the lecture, devices to stimulate the imagination of pupils’ (‘Mathematical Proof’, Mind 38 (1929).Google Scholar Cited by Cora Diamond LFM 13 n.).

¹⁷ Kreisel writes, ‘We note in passing an interesting aspect of Hubert's idea of a paradise: a characteristic of Cantor's set theory… is the abundance of transfinite machinery which Hubert regarded… as “ideal” elements to be used as gadgets to make life smoother’ (‘Hubert's Programme’, Philosophy of Mathematics: Selected Readings, Benacerraf, and Putnam, (eds) (Oxford: Blackwell, 1964), 159).Google Scholar

¹⁸ I say Wittgenstein's views need not have any such effect because the question of how a practising mathematician might react to them is ultimately an empirical one. For example, someone who became convinced of a conventionalist view of mathematics such as Wittgenstein's might well be moved to give up mathematics because the depth which he thought the subject had might now seem to him to be illusory. But there is no reason why he must do this, and it would certainly not be irrational to continue with mathematics whilst subscribing to a Wittgen-steinian viewpoint. Nothing Wittgenstein says compels anyone to abandon any part of mathematics, in contrast, for example, to intuitionism. So Wittgenstein's claim that he leaves mathematics as it is is quite justified.

¹⁹ I am grateful to Gordon Baker and Crispin Wright for valuable comments on an earlier draft of this paper.