I’ve recently finished a draft of the first chapter for my masters thesis, on the topic of mathematical structuralism. I argue that the philosophical discussion of mathematical structuralism has, by and large, misrepresented mathematical attitudes on the subject. This dialogue, beginning with Benacerraf’s classic What Numbers Could Not Be, has focused on the ontology of structures; such an attitude fails to do justice to a circle of ideas designed to explain and exploit the application of certain mathematical ideas across a wide range of domains. This oversight is compounded by an inattention to mathematical practice, which we find already in Benecerraf’s treatment of rival ordinal systems in ZFC.

At the same time mathematical ideas on structuralism, especially as de veloped in category theory, may offer philosophical dividends by helping us to understand the mathematical value judgements neglected by Benacerraf and his successors. This is because the mathematician’s structuralism addresses the “How?” rather than the “What?” of mathematics. Rather than invalidating alternative approaches, structuralists argues that theirs is a better way of doing mathematics. In the case of the ordinal systems in ZFC, Von Neumann’s ordinals provide a better reduction of the natural numbers than Zermelo’s because they require shorter and simpler definitions and proofs. This is because we are able to use the existing structural machinery of set theory (i.e., products, disjoint unions, etc.) both in building the internal theory of N and in the extension
to transfinite ordinals.

The Red Herring of Ontology (right click & save)