Sets and Categories: What Foundational Approaches Tell Us About Mathematical Thought
Robin Martinot
Abstract:
This thesis explores the idea that set theory and category theory represent different ways of thinking. Adopting the perspectives of various foundational systems based in set theory and category theory, we investigate two common and informal conceptions about the distinction between set-theoretical and categorical thinking. One concerns the intuition that set theory and category theory respectively correspond to a bottom-up and a top-down approach to mathematics. The other captures the idea that category theory represents a higher level of abstraction than set theory. Our investigation brings us the two main results of this thesis. First, we argue that the bottom-up/top-down distinction is irrelevant to the distinction between set-theoretical and categorical thinking. Second, we claim that, while categorical foundations are generally characterized by a higher level of abstraction compared to set-theoretical foundations, this difference is more variable and more modest than generally thought.
In order to familiarize ourselves with the various foundational systems, we discuss set-theoretical foundations in Chapter 2, and categorical foundations in Chapter 3. Chapter 2 also treats the development of general category theory from set-theoretical foundations so as to better delineate the categorical way of thinking. The incorporation of a variety of systems in our approach is significant for the arguments leading to the main results in Chapter 4. Additionally, the arguments benefit from the new refinements we make to the bottom-up/top-down distinction and of a method of abstraction coming from computer science.