The Certainty of Mathematics: A Philosophical Investigation

This doctoral thesis is a conceptual investigation of mathematical certainty. In philosophy, the concept of certainty has been given much attention, and mathematics is often regarded as a branch of knowledge that gives us certain knowledge. The beginning of the twentieth century saw several attempts to prove that mathematics is indeed certain. Considering the foundational crisis in mathematics, this is a reasonable reaction. In the present thesis, however, the idea is not to show that mathematics is certain (nor uncertain), but rather to investigate what the certainty of mathematics amounts to. This is a concept that must be understood more clearly before one embarks on a project to prove or disprove the certainty of mathematics.

It is argued that the contemporary philosophical understanding of mathematics (and thereby its certainty) is shaped by a certain picture of mathematical knowledge that is tacitly presupposed in many discussions. This picture portrays mathematics as a body of true propositions about a mathematical subject matter. While capturing some important aspects of mathematics, this picture is potentially misleading in philosophy as it tends to conceal the fact that mathematical knowledge is a skill, a practical ability to use the concepts of mathematics. It is, furthermore, argued that this concealment invites many of the problems of contemporary philosophy of mathematics, e.g. the problem of the ontological status of mathematical objects that is at the centre of the realism–anti-realism debate.

The emphasis on skill and practical ability in the thesis nuances the picture of mathematics and – through an analysis of the concepts ‘formality’ and ‘proof’ – lends support to the analogy between mathematical propositions and rules, stressed by Wittgenstein. Viewing mathematical propositions as rules for our dealings with mathematical concepts instead of as descriptions of mathematical states of affairs shows that the certainty of mathematics is a different form of certainty than the certainty of empirical facts.