## Mathematical Structuralism, Essays 1 & 2

Here are some quick comments on the first two of the essays in The Pre-history of Mathematical Structuralism. Let me start, though, with a remark about the angle I’m coming from.
I have been wondering about getting back to work on my stalled project, Category Theory: A Gentle Introduction. And what I’d like to do is write some short preliminary chapters around and about the familiar pre-categorial idea that (lots of) mathematics is about abstract structures and their inter-relationships. What does that idea really come to? I sense that there is some disconnect between, on the one hand, what this amounts to in the nitty-gritty of ordinary mathematical practice and, on the other hand, some of the arm-wavingly generalities of philosophers with axes to grind. I’m interested then in seeing discussions of varieties of structuralism which are, perhaps, more grounded in the varieties of mathematical practice. Hence I hope that looking at some of the historical developments in maths that have led to where we are might prove to be illuminating. Let’s read on …
The book opens with an introductory essay by the editors, Erich Reck and Georg Schiemer. The first part of this essay is in effect a summary version of the same authors’ useful entry on ‘Structuralism in the Philosophy of Mathematics’ in the Stanford Encyclopedia of Philosophy. In this book, then, they touch again more briskly on the familiar varieties of structuralism as a philosophical position; perhaps too briskly? — a reader relatively new to the debates will find their longer SEP version significantly more helpful. But they also press a distinction between structuralism as a philosophical story (particularly a story about the ontology of mathematics) and what they call methodological or mathematical structuralism — a term which “is meant to capture a distinctive way of doing mathematics”. “Roughly,” we are told, “it consists of doing mathematics by ‘studying abstract structures’”, something a. . .

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