This dissertation answers how mathematical representations enable knowledge of physical systems. Contemporary responses rely on matching the properties of physical systems to properties in mathematical models, arguing that such matching allows scientists to successfully draw conclusions about physical systems through the inspection of their models. We argue that such “matching accounts” cannot adapt to the routine mismatching pervasive in physical theories. These mismatching problems arise both when idealized models match some “similar” but better behaved potential physical system, and in cases we classify as pathological idealization, where the models employed must satisfy constraints that could not possibly be matched by realistic physical systems (e.g. requiring an infinite particle number or infinite density). In the latter cases such pathological constraints can also lead to incompatibilities with the governing laws of the physical theory. Despite such pathologies, conclusions drawn with these representations seem to enable improved understanding and empirically confirmable knowledge of the studied physical systems.
To address this dichotomy, we develop a novel condition of successful mathematical representation, called epsilon-fidelity, under which mismatched models may facilitate knowledge of realistic physical systems. Arguing against direct matching, we propose that representations can meet the conditions of epsilon-fidelity by establishing a manifold of associations between topological neighborhoods of mathematical models and clusters of relevantly similar physical systems. We then demonstrate that this shift in the scope of representation relationships explains how suitably similar models entail conclusions about the relevant systems while avoiding the problems of individual model to system mismatching.
As a signature case study, we investigate Einstein’s canonical interpretation of the geodesic principle, originally proposed to govern how gravitating bodies travel according to general relativity theory. We argue that under the canonical interpretation models of bodies must either meet unrealistic assumptions or violate the theory’s fundamental field equations, marking them as pathological idealizations. To recover the principle, we reinterpret geodesic dynamics as a universality thesis about the collective behavior of certain classes of systems, explaining how this reinterpretation satisfies the epsilon-fidelity criteria and can be used to gain knowledge about the observable motion of actual classes of gravitating bodies
