Ontological Reduction in Mathematics. Part I

Abstract

The article is the first part of a series of papers devoted to the problem of ontological reductions in mathematics – in particular, of choosing the basic category of mathematical entities. The received view is that such a category is provided by set theory, which serves as the ontological framework for the whole of mathematics (as all mathematical entities can be represented as sets). However, from the point of view of "naive mathematical realism" we should rather think of the mathematical universe as populated by a variety of diverse mathematical objects, and the set-theoretic reduction seems to be rather unnatural. In the first (introductory) part I discuss the general problem of providing an ontological foundation for mathematics.