An Introduction to the Philosophy of Mathematics

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.