Mathematics Without Content

Abstract

This chapter presents the position of logical positivism, which assumes that mathematics and logic play a particular role in science. This line of thought also holds the key to understanding the formalist interpretation of mathematics that can be considered a further development of the meta-mathematical programme.

A particular input to logical positivism is provided by Wittgenstein’s Tractatus. But what are the messages in the Tractatus? In his own preface, Wittgenstein summarises them in two points: first, that philosophical problems are caused by misunderstandings of the logic of language; and secondly, what is possible to say can be said clearly. In this way, a limit for human knowledge becomes drawn. This limit coincides with the limits of language. We do not possess epistemological tools that reach any further. Outside language exists, epistemologically speaking, dark-nightly nothing. When talking about language, Wittgenstein has in mind the formal language provided by mathematics and logic. This leads to the slogan that mathematics is the language of science. Wittgenstein also makes the observation that mathematics is composed of tautologies, which points towards the formalist programme, according to which mathematics can be identified with pure formal structures. Within formalism, all ontological issues with respect to mathematics becomes eliminated. Mathematics is not about anything. Mathematics is a game with symbols without content.