Figures, formulae, and functors

Abstract

This article suggests a novel way to advance a current debate in the philosophy of mathematics. The debate concerns the role of diagrams and visual reasoning in proofs-which I take to concern the criteria of legitimate representation of mathematical thought. Drawing on the so-called 'maverick' approach to philosophy of mathematics, I turn to mathematical practice itself to adjudicate in this debate, and in particular to category theory, because there (a) diagrams obviously play a major role, and (b) category theory itself addresses questions of representation and information preservation over mappings. We obtain a mathematical answer to a philosophical question: A good mathematical representation can be characterized as a category theoretic natural transformation. Assuming that this is not some reductio against the maverick approach to these issues, this in turn moots some of the disagreement in the philosophical debate and provides better questions with which to go on.