Abstract

The main objective of the paper is to show that Ludwig Wittgenstein's later writings on mathematics have been misinterpreted as just one more position competing for supremacy in the philosophy of mathematics. The way to make sense of Wittgenstein's works is not to wrap them up within the confines of some new '-ism' but rather to consider his remarks as a motley of criticisms. ;Wittgenstein's criticisms are aimed at the traditional philosophies of mathematics, i.e., Platonism, Logicism, Formalism, Intuitionism, etc., as well as various interpretations of well-known results in mathematics, viz. Cantor's work on infinity, Dedekind cuts, and Godel's incompleteness theorems. The bulk of the dissertation consists of an examination of Wittgenstein's remarks, separated by means of their presumed target, viz. some major position or major result of Wittgenstein's day. The guiding principle of my work is that there is no overarching position to be found behind the criticisms. Thus the criticisms must be taken at face-value and the attempt to make them cohere must be resisted. The remainder of my work consists of showing that Wittgenstein is neither radical conventionalist nor rule-sceptic nor strict finitist. I show that none of these positions represents accurately Wittgenstein's later remarks. This serves to support my thesis that Wittgenstein's remarks on the foundations of mathematics are best seen as a motley of criticisms aimed at specific targets. ;I focus on Wittgenstein's "descriptive enterprise". I show how his various criticisms are to be read as descriptions of the practice of mathematicians, both those who work in universities and those who do not. One of Wittgenstein's key criticisms of the major philosophies of mathematics is that they consist of explanations of mathematics rather than descriptions. Descriptions, for the later Wittgenstein, are the means for gaining a clear view of the way out language works. Once we gain a clear view philosophical puzzles will be dissolved. We will no longer want to be Platonists or Formalists, etc. Indeed, we will not want to do foundational work in mathematics at all