De Haro, Sebastian (2025) The Geometric View of Theories. [Preprint]
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Abstract
Recent critiques of the semantic conception of scientific theories suggest that a theory is not best formulated as a collection of models satisfying some set of kinematical or dynamical conditions. Thus it has been argued that additional structure on the set of models is required. Furthermore, there are calls for developing a 'theory of theories', where what was formerly a 'theory' is seen as a 'model' within a larger theoretical structure. This paper makes a two-pronged proposal for the "shape" that physical theories should take, based on recent insights on dualities and quasi-dualities in physics. First, I develop a geometric view of theories, according to which a physical theory is a set of models with topological and geometric structure on it. This general view is briefly illustrated in an example from quantum cosmology. Second, I make a more specific proposal for a natural structure that can encompass various 'theories' as its models, with topological and algebraic-geometric structure on them. I call the latter more specific structure a 'model bundle', where the models are in the fibres and there is a moduli space in the base. I illustrate my second proposal in the Seiberg-Witten theory (Seiberg and Witten, 1994a,b), where the moduli space is the complex plane with three punctures, the states and quantities are in the fibres, and the modular group is the structure group that acts on the fibres. This view highlights the important role of quasi-dualities as local transition functions between fibres; dualities are recovered as global transition functions when the bundle is trivial. I discuss some philosophical issues that this geometric view of physical theories opens up, such as its realist interpretation.
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