The Geometric View of Theories
Sebastian De Haro
TL;DR
This paper argues that the traditional semantic and combined views of scientific theories are insufficient to capture the structure revealed by dualities in physics. It introduces a geometric program that (i) treats a theory as a single structured object with topological and geometric structure on its model space, and (ii) organizes multiple theories as a model bundle over a moduli space, with quasi-dualities as local fibre transitions and dualities as global bundle trivializations. The central contributions are (a) the formalization of a model bundle framework in which models reside in fibres over a moduli base, (b) the identification of quasi-dualities as the bundle’s structure group and (c) demonstrations via Seiberg–Witten theory and quantum cosmology that this approach yields rich geometric structure with physical interpretation (e.g., monodromies, moduli-space metrics). The framework offers a natural realist reading of geometric structures on moduli spaces and provides a unifying language for inter-theoretic relations, with potential implications for quantum field theory and quantum gravity. Overall, the geometric view advances a principled, mathematically structured understanding of theories and their interrelations, moving beyond a static collection of models toward a dynamic, parameterized geometric landscape.
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 an example from quantum field theory. 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.