Abstract

Wittgenstein's views on mathematics in the period before the Tractatus are examined and it is argued that at this time he endorsed the logicist view of mathematics propounded by Frege and Russell. His view of mathematics in the Tractatus itself is then considered and it is shown that he now rejects logicism. It is further argued that despite the brevity and sketchiness of the Tractatus discussion there are good reasons for believing that Wittgenstein intended it to be the basis of a comprehensive account of mathematics intended to replace Principia Mathematica. It is argued that the core of this new account of mathematics is based on Wittgenstein's concept of a 'calculus'. Some central aspects of this account are described. Attention then turns to Wittgenstein's views on mathematics in the period immediately after his return to philosophy in 1929. It is argued that the idea that his account of mathematics is now based on an 'anti-realist' theory of meaning inspired by his contact with Brouwer is mistaken. It is shown that far from making such a completely fresh start the most fundamental ideas of Wittgenstein's view of mathematics at this time stem from the calculus picture which he had arrived at in the Tractatus. The question of what aspects of Wittgenstein's views had changed is examined. It is argued that the crucial change concerns his view of the infinite. Whereas he had previously accepted the concept of the completed infinite as unproblematic he now rejects it emphatically. It is shown that almost all of the other prominent changes in Wittgenstein's view of mathematics derived from this single new idea. How Wittgenstein came to change his view of the infinite is explained. It is argued that his encounter with the ideas of Brouwer did have some role to play in bringing this change about, but that Brouwer's influence was much less fundamental and was exerted in a completely different way than is often claimed