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

TL;DR

This paper proposes a neo-Kantian framework based on Cassirer's philosophy to explain why mathematics effectively applies to physics, emphasizing cognitive synthesis and a priori principles that underpin scientific development.

Contribution

It introduces a novel neo-Kantian explanation for mathematical applicability in physics, integrating pragmatic and indispensability considerations through a cognitive synthesis perspective.

Findings

Cassirer's neo-Kantian view explains mathematical effectiveness in physics.

Invariance groups like Lie groups exemplify the proposed principles.

The framework unifies pragmatic and theoretical accounts of applicability.

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.