Abstract

On what basis can we justify the inclusion of mathematical concepts and theories as legitimate objects of philosophical study? One answer to this question is: mathematical concepts and theories ought to be included because they are indispensable for describing the physical world. But what if a particular mathematical concept or theory has no such application? Are its corresponding statements to be counted as meaningless? Are its corresponding objects to be taken as mere linguistic fictions? One way of avoiding these conclusions is provided by the set-theoretic foundationalist: Mathematical concepts, insofar as they reduce to set-theoretic concepts, are meaningful and hence are to be regarded as legitimate objects of philosophical study. The problem with this approach is that it leads to a conflict between truth and meaning: some mathematical statements are true for one interpretation of set theory yet false for another. ;I show that if we shift our philosophical focus from what mathematical theories are about to what mathematical theories say and if we take category theory as the language of mathematics , then we can justify the inclusion of mathematical concepts and theories as legitimate objects of philosophical study. Thus, I argue that the current debate over the status of mathematical objects and statements is resolved by adopting a linguistic approach to the philosophical analysis of mathematical concepts and theories. In particular, I show how category theory can be taken as the language used to specify and organize the common structural features of mathematical discourse. Category theory, then, provides the means for representing and justifying our talk of mathematical existence, meaning and truth from within a given mathematical theory.