### Dialectical Realism in Utrecht

#### Posted by David Corfield

Last Friday saw me in Utrecht to deliver a couple of talks for the From Plato to Predicativity seminar. I spent a very pleasant morning in a typical Dutch café chatting with Klaas Landsman about mathematics, physics and philosophy.

At the seminar in the afternoon I spoke first on Lakatos and then on Lautman, uniting them under the banner of ‘dialectical realists’. One obvious difference between the two is how contemporary were Lautman’s case studies. In the 1930s he’s talking about class field theory, where Lakatos’s main work from the late 1950s and early 1960s concerned early to mid-nineteenth century mathematics. His aesthetic antenna was so finely tuned to detect a certain kind of structure similarity that, had his life and work not been curtailed by the 1939-45 war, I wonder whether Lautman might have prompted his Bourbaki friends to take up category theory more rapidly.

The talk I gave on Lautman was similar to an earlier version, which includes the glorious example of the commonality between Galois theory and deck transformations. Jean Dieudonné said of his work on this:

La ‘montée vers l’absolu’ qu’il y discerne, et où il voit une tendance générale, a pris en effet, grâce au langage des catégories, une forme applicable à toutes les parties des mathematiqures: c’est la notion de ‘foncteur représentable’ qui joue aujourd’hui un rôle considérable, tant dans le découverte que dans la structuration d’une théorie.

I presented the algebraic number theoretic and topological manifestations side by side, with the ‘imperfect’ rationals $\mathbb{Q}$ finding their perfected form in the algebraic closure, and the ‘imperfect’ circle finding its perfected form in the universal covering space, the real line sitting as a helix above the circle. Klaas wondered aloud why in the algebraic number theory case the object appears to become more complicated as we move up from $\mathbb{Q}$ to its algebraic closure, while the real line seems simpler than the circle.

What should one say to that? That the covering space is not just the line, but the line equipped with a map to the circle, so that it possesses symmetries unseen by the circle?

## Re: Dialectical Realism in Utrecht

Well, the analogy between universal covering spaces and algebraic (or say separable) closures isn’t perfect. Basically, a separable closure is just a way to package all the finite separable extensions together, so that a closer topological analog would (morally) be a suitable inverse limit over finite covering spaces – and this will be a pretty complicated space, even when you’re starting with the circle.

The passage from the universal cover to this more complicated guy amounts to only remembering the profinite completion of the fundamental group – which is all the algebra can see in general (c.f. etale homotopy theory).

Also worth noting is that, from this perspective, the circle is much closer to a finite field than the rationals: the profinite fundamental group is, in both cases, Z hat.

While I’m making a post, I’ll go off-topic a bit and say that I really appreciate your philosophical contributions to this blog, David, even if I have nothing to say about them.