Introduction

As we have seen, Kant claims that we have innate intuitions of time and space, and that these produce in us knowledge about arithmetic and geometry. In Kant, there is no real distinction between geometry as a study of formal relations within a theory and the study of the structure of physical space. This distinction is more recent than Kant. It is, however, an important distinction. Pure mathematics is the study of abstract structures. The natural sciences apply mathematical theories to actual physical systems.

In this chapter I discuss the epistemology of pure mathematics. However, I also look briefly at the problem of applied mathematics. Mathematicians, scientists and philosophers have found it difficult to explain why mathematical theories are so easily applied to physical reality. One might think that the Aristotelian has an easy answer to this: mathematics is easily applied to physical reality because it is created through a process of abstraction from our experience of physical reality. This may be part of the answer, but it cannot be the whole answer to the problem. Modern physics, in particular, uses very difficult and abstract branches of mathematics (probability theory, vector space and so on). These are not related to experience in a straightforward way. Aristotelianism, even if we do accept it as part of our epistemology of mathematics, needs to be supplemented in some way. We shall begin with pure mathematics and leave the problem of its application until later.