In the progression from Newtonian physics to general relativity, the structural feature of absolute rest was abandoned because it was not necessary to account for the empirical validity of Newtonian physics. Ockham's razor-type arguments like this one, which appeal to a desire for minimal ontologies and more unified physical laws are often invoked in favor of one theory or model over another. But how do we distinguish between essential “structure” in a theory, and inessential contingencies of a particular description? Is there a precise way to “compare structure” across theories expressed with different language and mathematical constructs?
I adopt and adapt a method of comparing the structure of different formalisms that I call “theories as categories of models” (TCM). The motivating idea is that information about relationships between formal theoretical models provide crucial insight into the way in which these models are intended to represent real-world systems. Incorporating this so-called “functorial” information into the presentation of a theory yields a mathematical object called a category. Formal methods from category theory can then be used to enrich our understanding of the nature of these models and the systems they represent.
While I aim to develop a rich and rigorous account of TCM, I focus primarily on the ways in which it has been and can be productively employed by scientists and philosophers, rather than merely considering this method in the abstract. I couch my analysis in three primary
case studies in which TCM has produced novel insights into physical theories. Two of these applications, in general relativity and Yang-Mills theory are based on original theorems establishing that formalisms that many theorists consider meaningfully distinct are in fact equivalent in a precise, category theoretic sense. In the dissertation, I present these examples with a closer eye towards explicating the role that TCM plays in scaffolding the arguments in these papers. In the second case, I demonstrate how TCM opens the door to a larger and richer space of possible Yang-Mills formalisms, and indicate how the category theoretic structure in this space reveals the relationships between quantum field theories based in different classical formalisms.
I also consider topological data science (TDA), a popular method of analyzing the “shape” of large data sets. This represents a departure from other philosophical work on TCM, which has focused on theoretical foundations of theories from physics, whereas TDA is employed by a variety of researchers in multiple fields at a less theoretical, more practical level. TDA is a promising candidate for TCM because category theory is invoked by data scientists themselves to justify the use of its core methods. This case study reveals how scientists are and can be motivated by category theoretic considerations, and the ways in which these motivations do and do not align with those of philosophers of science employing the same tools. This chapter points towards new ways philosophers of physics might enhance TCM by analogy with TDA. In the other direction, the philosophical framework of TCM enhances the story data scientists want to tell about how TDA gets at the underlying “structure” of data.
