The Problem of Mathematical Objects

Abstract

In seeking a foundation for mathematics, one may be looking for what may be called a foundation in the logical sense: a single, unified set of principles—perhaps unified by their jointly constituting an acceptable axiomatization of some concept or concepts plausibly taken as fundamental—from which all, or at least a very large part of, mathematics can be derived. In this sense, some version of set theory is plausibly taken as a foundational. But one may also be interested in an epistemological foundation—roughly, an account which explains how we can know standard mathematical theories to be true, or at least justifiably believe them. A foundation in this sense could not be provided by any mathematical theory, however powerful and general, by itself. Indeed, the more general and powerful a mathematical theory is, the more problematic it must be, from an epistemological point of view. What is called for is a philosophical account of how we know, or what entitles us to accept, the mathematical theories we do accept. Since such an account cannot very well be attempted without adopting some view about the nature of the entities of which the mathematical theories treat, this is likely to involve broadly metaphysical questions as well as epistemological ones. It is not certain either that we are right to demand such a foundation, or that one can be given, but I shall proceed on the assumption that it is reasonable to seek one.