Mathematics as a Science of Patterns: Ontology and Reference

Abstract

entities, that is, immaterial and nonmental things which do not exist in space and time. So I am a platonist.1 Many philosophers of mathematics who would like to be platonists are bothered by two rather deep problems. The first is that since platonic mathematical objects do not exist in space or time the very possibility of our acquiring knowledge and beliefs about them comes into question. The second arises from the fact that no mathematical theory can do more than determine its objects up to isomorphism. Thus the platonist seems to be in the paradoxical position of claiming that a given mathematical theory is about certain things and yet be unable to make any definitive statement of what these things are.2 I think that these problems arise in part because of a fundamental misconception of what mathematics is about. If we conceive of the numbers, say, as objects each one of which can be given to us in isolation from the others as we think of, say, chairs or automobiles, then it is difficult to. avoid conceiving of knowledge of a number as dependent upon some sort of interaction between us and that number. The same line of thought leads us to think that the identity of a number vis a vis any other object should be completely determined. For some time mathematicians have emphasized that mathematics is concerned with structures involving mathematical objects and not with the nature of the objects themselves. They have recognized that we are not given mathematical objects in isolation but rather in structures. That 13 is a prime number is not determined by some internal property of 13 but rather by its place in the structure of the natural numbers. Some