UC Davis

This dissertation considers the prospect of category theory (CT), or the frameworks thereof, as a philosophically satisfactory foundation of mathematics. Roughly speaking, CT is a theory of mathematical structures that studies their formal properties and relations. CT has also received attention as a promising candidate for the foundation of mathematics. This has made CT philosophically noteworthy against the backdrop of mathematical structuralism, a philosophical position maintaining that pure mathematics is primarily about mathematical structures. The dissertation is built upon the recent debates on CT-structuralism, which combines structuralism with CT. Each chapter of the dissertation supports a stand alone thesis. In Chapter I, I review structuralism in philosophy of mathematics and the philosophical debates around CT, explaining how these two lines of thought came together and led to the emergence of CT-structuralism. In Chapter II, I introduce the fine-grained notion of conception to the debates on the foundation of mathematics. Using orthodox set theory as an example, I show how the same foundational account can be conceived of in multiple ways, which leads to a significant difference in how we see the various foundational debates. I argue that the introduction of conception also gives us a response to the well-known critique of CT-structuralism known as “the problem of the home address.” In Chapter III, I consider the relationship between structuralism and inferentialism, which have been seldom considered together. Based on their common historical origin in the famous Frege-Hilbert debate, I argue that structuralists can, and should, be inferentialists. I also consider how this conclusion allows us to overcome another well-known critique of CT-structuralism known as “the mismatch objection.” In Chapter IV, I link the debates on CT-foundation with the growing scholarship on mathematical explanation. I argue that the term ‘foundation’ can be understood in an explanatory sense, which is backed up by both historical and philosophical considerations. Based on recent case studies about the explanation in Galois theory and CT, I suggest that the philosophy of mathematical practice can shed new light on more traditional, foundational debates. In Chapter V, I argue that the classic metaphor of CT as “the language of mathematics” can be explicated using the notion of metaphysical perspicuity. I provide a new account of metaphysical perspicuity, which has begun to receive philosophical interest in many subfields, challenging some common conceptions about the notion. This will allow us to better understand the appeal of CT as the language of mathematics.