The Geometric View of Theories

TL;DR

This paper proposes a geometric framework for physical theories, viewing them as structured sets of models with topological and algebraic features, emphasizing dualities and the concept of a 'model bundle' to unify various theories.

Contribution

It introduces a novel geometric perspective on theories, including the idea of a 'model bundle' with topological and algebraic structure, to better understand dualities and the organization of models.

Findings

Models can be structured as topological and geometric spaces.

Dualities are interpreted as transition functions in a model bundle.

The approach offers new philosophical insights into the nature of physical theories.

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.