Introduction

The relationship between mathematics and science is clearly of fundamental concern in both the philosophy of mathematics and the philosophy of science. How this relationship should be represented is a crucial issue in this area. One possibility is to employ a model-theoretic framework in which ‘physical’ structures are regarded as embedded in ‘mathematical’ ones. In section 2 I will briefly outline a form of this type of account which offers a function space analysis of theories (Redhead 1975). This function space analysis is then used to represent the relationship between theoretical and mathematical structures. In subsequent sections I will consider the role of group theory in physics from within this meta-theoretical framework and then draw some conclusions for realism in the philosophy of science.

Function spaces and the model-theoretic approach

According to Redhead, it is an ‘empirical-historical fact’ that theories in physics can be represented as mathematical structures (Redhead 1975). This then allows the possibility of representing the relation of mathematics to physics in terms of embedding a theory T in a mathematical structure M′, in the usual set-theoretic sense of there existing an isomorphism between T and a sub-structure M of M′. M′ is then taken to be a non-simple conservative extension of M. There is an immediate question regarding the nature of T. To be embedded in M′ it must already be ‘mathematized’ in some form or other. Thus, the issue here is not so much Wigner's inexplicable utility of mathematics in science, in the sense of its being the indispensable language in which theories are expressed, but rather the way in which new theoretical structure can be generated via this embedding of a theory, which is already mathematized, into a mathematical structure.