### Albert Lautman

#### Posted by David Corfield

Sifting through some old papers yesterday, I was reminded of my first forays into philosophy of mathematics while living in Paris. One very important influence on me was Albert Lautman (1908-1944). (The French Wikipedia has a little more on him than the English.) He’s often paired with Jean Cavaillès, understandably as they were friends, philosophers of mathematics, and both died as members of the French Resistance. Somehow I never got on with Cavaillès, even though he’s found greater favour in the Anglophone world. In Lautman, on the other hand, I recognised the kind of aesthetic sensibility I was later to enjoy in Mac Lane’s *Mathematics: Form and Function* and in *This Week’s Finds*.

I haven’t read Lautman in around 18 years, aside from a stolen hour spent in the Bodleian in Oxford, so went to see what was available on the Web. Not much it appears, but there is at least a translation of the introduction to *Essai sur les notions de structure et d’existence en mathématiques: Les schémas de structure*. Here are a couple of excerpts:

The structural design and the dynamic design of mathematics first of all seem to oppose themselves: the one indeed tends to regard a mathematical theory as a completed whole, independent of time, the other on the contrary does not separate the temporal stages from its development; for the former, the theories are like beings qualitatively distinct from each other, while the latter sees in each one an infinite power of expansion beyond its limits and connection with the others, because it affirms itself as the unity of the intelligence. We would however like, in the following pages, to try to develop a design of mathematical reality where the fixity of logical notions is combined with the movement that lives through these theories.

A dialectical action is constantly played out in the background and it is in order to clarify this that our six chapters will converge on this point. The first three chapters deal especially with de-structured mathematical notions. We study in chapter I (the local and the global) the almost organic solidarity which pushes the parts to be organized in a whole and the whole to be reflected in them; we examine then in chapter II (Intrinsic properties and inductive properties) if it is possible to bring back for the relations that a mathematical being supports with an ambient milieu, in the characteristic inherent properties of this being. We show in chapter III (rise towards completion) how the structure of an imperfect being can sometimes preform the existence of a perfect being in which any imperfection has disappeared. Then the three chapters relating to the concept of existence come. We try to develop in chapter IV (Essence and Existence) a new theory of the relations of essence and existence where one sees the structure of a being to be interpreted in terms of existence for beings other than the being of which one studies the structure. Chapter V (the Mixed) describes certain intermediate Mixtures between different kinds of Beings and whose consideration is often necessary to operate the passage of one kind of being to another kind of being; our final chapter (Of the exceptional character of existence) finally describes the processes by which a being can be distinguished within an infinity from the others.

What I remember, and I will have surely got some of the details wrong, of that third chapter ‘rise towards completion’ is a Galoisian comparison of covering spaces and field extensions, the ridding of the ‘imperfections’ of nontrivial first homotopy and algebraic nonclosure, respectively.

But the really bold part of Lautman’s chapter was to liken this situation to the one Descartes finds himself in when, having established that he exists, he wishes then to establish the existence of a Perfect Being, or God. Descartes argues that he recognises that he is imperfect, but that he has an idea of perfection. This idea must have arisen in him through a perfect Being, as one might say that the failure of factorisability of polynomials in $\mathbb{Q}$ prefigure its algebraic completion. There may have been an additional parallel between intermediate fields and angelic beings who lack some of our imperfections and some of God’s perfections.

Even if you find this philosophical analogy too fanciful for you taste, Lautman clearly reveal mathematical perspicacity. I remember that Jean Dieudonné in a preface to one of Lautman’s books praised him for his perception of structural similarity across fields, and claimed that he’d just about discovered the concept of a representable functor (many years, note, before category theory saw the light of day).

As a small act of homage, and because I’d really like to discover the answer, let’s see what we know about about taking Galois up the $n$-category ladder. So how does the deck transformation-simply connected covering space story run for higher homotopy? E.g., if we take $CP^{\infty}$, as $K(\mathbb{Z}, 2)$, is there a deck transformation story, featuring a covering space with trivial second homotopy?

What’s the closest thing we have to 2-Galois for factorising polynomials? Is it needed for species/structure types?

Of course, we’d like a very general account. As we suspect that practitioners of modal logic have already stumbled on categorified predicate logic, we need a 2-Galois theory to help us categorify what Todd told us about here. In particular,

any relation $R \subseteq A \times B$ whatsoever can be used to set up a Galois connection between subsets $A'$ of $A$ and subsets $B'$ of $B$: $\frac{A' \subseteq \{a \in A: \forall_{b: B'} (a, b) \in R\} }{B' \subseteq \{b \in B: \forall_{a: A'} (a, b) \in R\} } iff$ Indeed, each of the conditions above and below the bar is equivalent to the condition $A' \times B' \subseteq R$.

Can that be so hard to categorify? Just think, if we get it right then we’ll be able to work out a categorifed cartesian theology!

## Re: Albert Lautman

To make a start on categorifying what Todd said, we could note that a relation between $A$ and $B$ gives rise to order preserving maps between $P(A)$ and $P(B)^{op}$ and

vice versawhich are adjoint.Now you might hope that a permutation representation of the category $C \times D^{op}$ would similarly give rise to adjoint functors between the categories $C$-Set and $(D-Set)^{op}$.

Restricting to groups, does a representation of $G \times H$ allow such mappings between $G$-Set and $(H-Set)^{op}$?