Category Theory as a Framework for an in re Interpretation of Mathematical Structuralism

Abstract

The aim of this paper is to present category theory as a framework for an in re interpretation of mathematical structuralism. The use of the term 'framework' is significant. On the one hand, it is used in distinction from the term 'foundation'. As such, what I propose is that we consider category theory as a philosophical tool that allows us to organize what we say about the shared structure of abstract kinds of mathematical systems. On the other hand, the term 'framework' is used in the sense of Carnap [1956]. That is, category theory is taken as a language used to frame what we say about the shared structure of abstract kinds of mathematical systems, as opposed to being a "background theory" which constitutes what a structure is.