On Wittgenstein's notion of the objectivity of mathematical proofs

Abstract

This work analyses and defends Wittgenstein's definition of mathematical objectivity, looking particularly at his account of mathematical proofs, of what makes them normative, and what role mathematical and linguistic practices play in their establishment. It aims to provide a clearer view of Wittgenstein's idea that the objectivity of proofs is inextricably rooted in empirical regularities, detected within what he calls our 'form of life', and which are in turn constituted by both regularities in nature and regularities in human practices. To accomplish this, the thesis makes an exhaustive analysis of Wittgenstein distinction between proofs and experiments. Drawing from Wittgenstein's philosophy of language, this paper addresses important criticisms that have hindered a serious study of his remarks on mathematics, and vindicates his arguments as cogent, solid accounts of mathematical necessity and practice. This exploration will show that mathematical proofs do not have to be regarded as tools for the discovery of mathematical truths, of any sort, that depict mathematical facts, but can be correctly characterised as norms that produce new understanding, forming new concepts to deal with reality, and which ultimately affect the very limits of intelligibility, of what we can think, express and do. Thus, a philosophically relevant link between mathematical proofs and possibilities of action is set.