  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.
