### What to Make of Mathematical Difficulties

#### Posted by David Corfield

T. R. drew to our attention a conference dedicated to Grothendieck. One of the papers there is Mathematics and Creativity (presumably written by Leila Schneps), which contains the passage:

Pierre Cartier observed that when Grothendieck took interest in some mathematical domain that he had not considered up till then, finding a whole collection of theorems, results and concepts already developed by others, he would continue building on this work ‘by turning it upside down’. Michel Demazure described his approach as ‘turning the problem into its own solution’. In fact, Grothendieck’s spontaneous reaction to whatever appeared to be causing a difficulty - nilpotent elements when taking spectra or rings, curve automorphisms for construction of moduli spaces - was to adopt and embrace the very phenomenon that was problematic, weaving it in as an integral feature of the structure he was studying, and thus transforming it from a difficulty into a clarifying feature of the situation. (p. 8)

This brings to mind the idea of Lakatos in Proofs and Refutations that one should be open to difficult cases, and not just exclude them by *monster-barring*.

In his long case study concerning the Euler formula,

$V - E + F = 2,$

he advises that if someone presents you with a ‘polyhedron’ which doesn’t fit, such as the picture frame, don’t limit your theorem to apply only to ‘proper’ polyhedra, that is, ones without holes. Rather, find out what it is about picture frames, and ultimately tori, which leads to their not satisfying the formula. You will arrive at a more comprehensive theory.

Lakatos presents a rather limited methodology for dealing with such ‘counterexamples’ via proof analysis, but he was pointing us in a useful direction. Also rather important is the ability to sense when a difficulty is likely to be a fruitful one.

## Re: What to Make of Mathematical Difficulties

When Grothendieck was `pursuing stacks’ in his letters to Ronnie Brown and myself, one of the main points that came across was his search for `concepts’ to handle the mathematical `difficulties’ that arose. Getting the definition right for a concept that was to encompass new difficult cases and examples as well as extending and relating to known ones is central to a lot of that sort of Methodology of Mathematics.

‘The proof of the pudding is in the eating’and the concepts arising in this blog show some good eating! but even here we can see people inching towards concepts as, perhaps, incomplete ‘categorification’ processes turn out to be not quite good enough to do what seems to be needed. (I feel we have not yet a thorough idea of what that process of categorification really involves, although there are insights on several different types of process involved.)

Perhaps Lakatos’ description is not adequate to describe all the processes involved but no really in depth study of methodology seems to have been undertaken by mathematicians and mathematically inspired philosophers (David, do you detect a hint!?)