Category Theory in Philosophy of Science

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The proposed topic is the application of category-theoretic methods to the analysis of relations between scientific theories. Understanding such relations is a longstanding topic of interest in the philosophy of science: philosophers have been interested in questions such as when two theories are equivalent, i.e., when they say the same thing about the world; when two theories stand in a relation of reduction, i.e., when it is that one theory can be ``represented'' within another; and when one theory is a limit of another, i.e., when one theory may be regarded as an approximation to the other in certain physical regimes. These questions are not only of interest in themselves, but also have direct bearing on issues in philosophy of science, and indeed on scientific practice itself. For instance, understanding conditions for equivalence is important for determining the extent to which one could have distinct (i.e. non-equivalent) theories which are consistent with all the same empirical evidence—that is, on the extent to which realists should be concerned about the underdetermination of a theory by data. Or, for another example, understanding the extent to which the content of a theory is preserved in a theory of which it is a limit is directly relevant to assessing whether scientific knowledge is robust across theory change, since one often finds that an earlier theory emerges as a limiting case of its successor. These questions bear directly on how to evaluate new theories, and how we should interpret the results of experiments: without appropriate tools for comparing theories to one another, we can struggle to characterise what is novel about new theories, or how alternative research programs differ from one another.Recently, researchers in these areas have come to realise that categorical methods offer a powerful tool for addressing such questions. Roughly speaking, categorical methods involve characterising certain classes of mathematical structures (e.g. groups, smooth manifolds, or vector spaces) in terms of the structure-preserving maps between them (e.g. group homomorphisms, diffeomorphisms, or linear transformations). It has recently come to be appreciated that treating a theory as a collection of such structures (the models or solutions of the theory), together with an appropriate class of morphisms, gives a great deal of insight into the structure of the theory. This project will draw together a community of researchers interested in the application of these tools, and examine three specific issues: how can category-theoretic methods be used to make judgments of equivalence between theories? What kinds of category-theoretic resources are required for a proper analysis of theories? And what insights can category-theoretic tools yield into understanding reduction and limits?