1 Chihara Charles S., ‘Wittgenstein's Analysis of the Paradoxes in his 1939 Lectures on the Foundations of Mathematics’, Philosophical Review 86 (07 1977).
2 In references to Wittgenstein's work I employ the standard abbreviations: WWK—Ludwig Wittgenstein und der Wiener Kreis (Oxford: Blackwell, 1967), PR—Philosophical Remarks (Oxford: Blackwell, 1975), PG—Philosophical Grammar (Oxford: Blackwell, 1974), LFM—Wittgenstein's Lectures on the Foundations of Mathematics, Cambridge, 1939, Diamond (ed.) (Hassocks: Harvester, 1976), RFM—Remarks on the Foundations of Mathematics, 2nd edn (Oxford: Blackwell, 1964), LC—Lectures and Conversations on Aesthetics, Psychology and Religious Belief, Barrett (ed.) (Oxford: Blackwell, 1966), BB—The Blue and Brown Books (Oxford: Blackwell, 1958), PI—Philosophical Investigations (Oxford: Blackwell, 1953), and Z—Zettel (Oxford: Blackwell, 1967).
3 Chihara , op. cit., 381 n.
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.
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) 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), 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), 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. See Wright Crispin, Wittgenstein on the Foundations of Mathematics (London: Duckworth, 1979), Ch. 18, for a detailed criticism of Bennett's defence.
9 E.g. Dummett , op. cit. But see Wright , op. cit., for a very full defence of radical conventionalism against the objections of Dummett and others.
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). 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.
14 Chibara , op. cit., 378–379.
15 See Hacker , op. cit., Ch. 5, 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). 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).
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.