A neo-Kantian explanation of the applicability of mathematics to physics

Abstract

Various 'optimistic' attempts have been made to reasonably explain the undeniable effectiveness of mathematics in its application to physics. They range over retrospective, historical accounts of mathematical applicability based on pragmatic considerations, on the one side, and prospective accounts based on indispensability considerations, on the other. In view of some objections that I will raise against these accounts, I would like to propose a third alternative based on Ernst Cassirer's neo-Kantian view which can overcome these objections and embrace both pragmatic and indispensability considerations. According to this view, mathematics and physics are seen as different modes of a basic process of cognitive synthesis that are essentially applied to each other according to a priori principles of theory development inherently incorporated into scientists' minds. As emphasised by Cassirer, these principles have a constitutive role (i.e., they explain the relevant phenomena across scientific change) and a regulative role (i.e., they incorporate the ideal of unity, permanence and generality at each stage of knowledge), both of which have been captured through a functional (i.e., structural) understanding of concepts. As a particular case study, these principles shall be instantiated by invariance groups responsible for the effectiveness of applying Lie group theory to physics.