Contemporary mathematical theories are generally thought to be consistent. But it hasn't always been this way; there have been times when the consistency of mathematics has been called into question. Some theories, such as naïve set theory and (arguably) the early calculus, were shown to be inconsistent. In this chapter we will consider some of the philosophical issues associated with inconsistent mathematical theories.

Introducing inconsistency

A five-line proof of Fermat's Last Theorem

Fermat's Last Theorem states that there are no positive integers x, y, and z, and integer n > 2, such that xn + yn = zn . This theorem has a long and illustrious history but was finally proved in the 1990s by English mathematician Andrew Wiles (1953–). Despite the apparent simplicity of the theorem itself, the proof runs to over a hundred pages, invokes some very advanced mathematics (the theory of elliptic curves, among other things), and is understandable to only a handful of mathematicians. But consider the following proof of this theorem.

The applications of mathematics in the various branches of science give rise to an interesting philosophical problem: why should physical scientists find that they are unable to even state their theories without the resources of abstract mathematical theories? Moreover, the formulation of physical theories in the language of mathematics often leads to new physical predictions, which were quite unexpected on purely physical grounds. How can turning to the abstract representations of mathematics – far from physical applications – so often turn out to be just what is required in representing and understanding physical systems? In this chapter we will consider the problem of the applicability of mathematics and discuss the prospects for a solution.

The unreasonable effectiveness of mathematics

The Nobel-Prize-winning Hungarian physicist Eugene Wigner (1902–95) remarked in a famous paper that

This much-cited passage, however, does not make it clear exactly what the problem is. After all, in order for us to do science we need some language or other. Why shouldn't mathematics be that language? But what is worrying Wigner is something deeper than this. He is concerned about a mismatch between the methodologies of mathematics and physics – physics is driven by empirical evidence and mathematics is in some sense disconnected from the world – yet mathematics still turns out to be just what the physicist needs.