Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre (...) Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations. (shrink)

To be asked to provide a short paper on Wittgenstein's views on mathematical proof is to be given a tall order . 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. (shrink)

This chapter concentrates on two discussions, both of which enlist Wittgenstein's rule‐following considerations in support of radical and highly revisionary conclusions about the objectivity of meaning ‐ conclusions which may appear to entail, and have been taken to entail, consequences for the objectivity of truth and judgment which are no less radical and revisionary. There is widespread agreement that Wittgenstein advances, in the rule‐following sections of Philosophical Investigations and Remarks on the Foundations of Mathematics, considerations that are quite destructive of (...) certain conceptions of meaning, understanding, and rule‐following into which one may easily slide when attempts a general philosophical account of them. Saul Kripke interprets central sections of Philosophical Investigations as developing a 'skeptical paradox' about meaning. Kripke himself describes the skeptical conclusion as 'insane and intolerable'. (shrink)