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 mathematician can do without explicitly considering metaphysical questions about the nature of stuctures. And then “One main goal of the present collection of essays is to clarify the origins, and with it the nature, of methodological/mathematical structuralism up to the rise of category theory”.

So what, in a little more detail, do Reck and Schiemer count as being involved in methodological structuralism? (1) The use of concepts like ‘group,’ ‘field,’ ‘3-dimensional Euclidean space’, which (2) are characterized axiomatically and “typically specify global or ‘structural’ properties”, and where (3) we importantly study systems falling under these concepts by relating them to each other by iso/homomorphisms, and (4) by considering ‘invariants’, and where (5) “there is the novel practice of ‘identifying’ isomorphic systems”.

[T]his can all be seen as culminating in the view that what really matters in mathematics is the ‘structure’ captured axiomatically, on the one hand, and preserved under relevant morphisms, on the other hand.

But having said this, Reck and Schiemer allow that not all of features (1) to (5) are required for what they call methodological structuralism: they say that we are dealing with “family resemblances” between cases.

Plainly, themes (1) to (4) did indeed emerge in nineteenth century mathematics. I rather discount (5), though, as the supposed novel practice of identifying isomorphic systems seems typically to amount to no more than ignoring specific differences between isomorphic Xs for certain purposes when doing the theory of Xs. For example, while focusing on the pure group-theoretic properties of some structures, we do ignore the non-group-theoretic differences between isomorphic groups. But of course, when we start to apply the group theory, e.g. in Galois theory, such differences become salient again; it is crucial, for example, that that certain groups are permutations of roots of equations, a property not shared with isomorphic groups.

There are various stripes of mathematical problem. To stick to nineteenth-century ones, at one end of the spectrum, there are very specific problems. For instance, we might be interested in finding a closed form solution for some integral: we tackle our problem using a rather specific bag of tricks we’ve developed (like substituting variables, integrating by parts, etc.). Again, we might be interested in finding an approximate solution to a specific application of the Navier-Stokes equation, and there’s another bag of tricks to develop and use. At the other end of the spectrum, there are considerably more abstract problems — is there a global solution in radicals for quintic equations? is the parallel postulate independent of the other Euclidean axioms? if a and k are co-prime, must there be an infinite number of primes in the arithmetic progression a, a + k, a + 2k, a + 3k, …? Not surprisingly, such abstract general questions often need abstract general ideas to answer them — and these ideas, being sufficiently abstract, will typically have other applications too, so may indeed well not need to be tied to a particular subject matter (i.e. they will be, in a sense, structural ideas).

So, at least those mathematicians interested in sufficiently general questions (far from all of all mathematicians, then) will naturally find themselves deploying abstract concepts as arm-wavingly gestured at by Reck and Schiemer’s (1) to (4). And the successes in doing this were impressive. But were (1) to (4) pursued with enough unity and clarity of purpose to constitute a methodological “ism”? We shall see.

The second part of Reck and Schiemer’s editorial introduction gives thumbnail sketches of the remaining fifteen essays in the book. So I won’t delay over that here, but will move on to the first contribution, Paola Cantù’s ‘Grassmann’s Concept Structuralism’.

I have to report, however, that I really got very little from this. Not knowing Grassmann’s work, I found Cantù’s sketched exposition of some his ideas quite opaque. So I imagine this essay will only be of interest to those who already know something of her topic; it certainly didn’t engage this reader. So I’ll skip over it.