Structural accounts of mathematical representation

Attempts to solve the problems of the applicability of mathematics have gen- erally originated from the acceptance of a particular mathematical ontology. In this thesis I argue that a proper approach to solving these problems comes from an ‘application first’ approach. If one attempts to form the problems and answer them from a position that is agnostic towards mathematical ontology, the difficulties surrounding these problems fall away. I argue that there are nine problems that require answering, and that the problems of representation are the most interesting questions to answer. The applied metaphysical problem can be answered by structural relations, which are adopted as the starting point for accounts of representation. The majority of the thesis concerns arguing in favour of structural accounts of representation, in particular deciding between the Inferential Conception of the Applicability of Mathematics and Pincock’s Mapping Account. Through the case study of the rainbow, I argue that the Inferential Conception is the more viable account. It is capable of answering all of the problems of the applicability of mathematics, while the methodology adopted by Pincock trivialises the answer it can supply to the vital question of how the faithfulness and usefulness of representations are related.