Mathematical Structuralism, Essay 3

The third essay in The Pre-history of Mathematical Structuralism is by José Ferreirós and Erich H. Reck, on ‘Dedekind’s Mathematical Structuralism: From Galois Theory to Numbers, Sets, and Functions’. The title promises something rather more exciting than we get. Why? Let’s work backwards by quoting from (some of) their concluding summary. They write:

From … Dirichlet and Riemann, Dedekind inherited a conceptual way of doing mathematics. This involves replacing complicated calculations by more transparent deductions from basic concepts.

“Replacing”? That seems rather misleading to me: isn’t it more a matter of a change of focus to new, more abstract, general questions, rather than a replacement of methods for tackling the old questions that required “complicated calculations”?

Both Dedekind’s mainstream work in mathematics, such as his celebrated ideal theory, and his more foundational writings reflect that influence. Thus, he distilled out as central the concepts of group, field, continuity, infinity, and simple infinity. A related and constant aspect in his work is the attempt to characterize whole systems of objects through global properties.

From early on, Dedekind also pursued the program of the arithmetization of analysis … . A decisive triumph came in 1858, with Dedekind’s reductive treatment of the real numbers. From the 1870s on, he added a reduction of the natural numbers to a general theory of sets and mappings. This led to an early form of logicism, since he conceived of set theory as a central part of logic … [H]is attempt to execute a logicist program
had a decisive effect on the rise of axiomatic set theory in the 20th century.

Its conceptualist and set-theoretic aspects are central ingredients in Dedekind’s mathematical structuralism. But we emphasized another characteristic aspect that goes beyond both. This is the method of studying systems or structures with respect to their interrelations with other kinds of structures, and in particular, corresponding morphisms.

But again this seems slightly misleading to me. It isn’t that Dedekind and contemporaries “distilled out” the concepts of a group and of a field, for example, for the fun of it and then — as an optional further move — considered interrelations between structures. Weren’t the abstracting moves and their interrelations and their applications by relating them back to more concrete structures all tied together as a package from the start, as for example in the very case that Ferreirós and Reck consider:

A historically significant example, particularly for Dedekind, was Galois theory. As reconceived by him, in Galois theory we associate equations with certain field extensions, and we then study how to obtain those extensions in terms of the associated Galois group (introduced as a group of morphisms from the field to itself, i.e., automorphisms). …

Just a word more about this in a moment. Finally,

As we saw, Dedekind connected his mathematical or methodological structuralism with a structuralist conception of mathematical objects, i.e., a form of philosophical structuralism.

Now, in bald outline, all this is familiar enough (though of course in part because of the earlier work of writers like Reck and Ferreirós in helping to highlight for philosophers Dedekind’s importance in the development of mathematics). So does this essay add much to the familiar picture?

Not really, it seems to me. For it proceeds at too armwavingly general a level of description. There’s too much of the mathematical equivalent of name-dropping: ideas and results are mentioned, but with not enough content given to be usefully instructive.

Take the nice case of Galois theory again. If you are familiar with modern basic Galois theory from one of the standard textbooks like Ian Stewart’s or D.J.H. Garling’s, you won’t pick up much idea of just how Dedekind’s work related to the modern conception (well, I didn’t anyway). And if you aren’t already familiar with Galois theory, then you won’t really understand anything more about it from Reck and Ferreirós few paragraphs, other than it is something to do tackling questions about the roots of equations by using more abstract results about groups and fields — which isn’t exactly very helpful. By my lights, it would have been much more illuminating if our authors had devoted the whole essay to Dedekind’s work on Galois theory as a case study of what can be achieved by the “structuralist” turn.