I didn’t say anything about my own talk at the Prospects in Mathematics meeting. But soon afterwards I was re-reading Lee Smolin’s 2006 book The Trouble with Physics, and it resonated with some of the things I said.
My topic was the Classification of Finite Simple Groups, and its impact on research in many parts of mathematics and computer science.
I began with one of my favourite examples. In 1872, Jordan showed that a transitive permutation group on a finite set of size n > 1 contains a derangement (an element with no fixed points). The proof is a simple counting argument which I have discussed here. In 1981, Fein, Kantor and Schacher added the innocent-looking postscript that the derangement can be chosen to have prime power order; but the proof of this requires the Classification of Finite Simple Groups.
In outline, easy reductions show that we may assume that the group is simple and the stabiliser of a point is a maximal subgroup; so we have to show that, if G is a simple group and H a maximal subgroup of G, then there is a conjugacy class of elements of prime power order disjoint from H. This has to be done by taking the simple groups one family at a time (or one group at a time, for the sporadic groups).
The subtext of my talk was: the proof of CFSG is very long, and is the work of many hands; it is certain that the published proof contains mistakes. (The proportion of mathematics papers which are error-free is surprisingly small, especially when possible errors in cited works are taken into account.) So can we trust it, and if not, should we be using it? Mathematicians have always had as a guiding principle that we should take nobody’s word for anything, but check it ourselves; CFSG makes that principle almost impossible to follow.
I told the students that, when they arrived at university from school, they were probably told that school mathematics is not “real” mathematics, and that now they would see the real thing, with an emphasis on proofs and building on secure foundations. Now that they are about to embark on a PhD, they have to be told something similar. In a university mathematics course, they are given the statement of a famous theorem, say Cauchy’s, with an elegant proof polished by generations of mathematicians. Now they are entering territory where proofs don’t exist; they will have to build proofs themselves, and sometimes they might make mistakes.
Smolin, in his book, has some hard words to say about the sociology of string theory (a subject in which he himself has worked). He was asked to write a survey paper about quantum gravity, and wanted to include the result that string theory is a “finite theory”. If this seems a little odd, string theory (like quantum electrodynamics) is a perturbative theory, where the answer to a calculation has to be found by summing infinitely many terms. (I remember my feeling of shock when I learned this from Mike Green quite a long time ago.) In the case of QED, it is well established that the sum converges, and according to Smolin, string theorists accept that the same is true for string theory. But when he went looking for a proof of this, he found that everyone referred to a paper of Stanley Mandelstam which showed only that the first approximation was finite. It seems that the assertion that finiteness had been proved was never checked by the people who quoted it; according to Smolin, the ethos of the field was to believe that such a statement must be true.
I do believe that finite group theory avoided this horror. First, it was never “the only game in town”, monopolising grants and postdoc appointments the way that string theory did. Second, everyone knew that the proof was not complete; many people hoped it could be completed (as it eventually was – the delay was partly caused by the fact that it was a big job, and experts were reluctant to commit themselves to it), but anyone who used it noted that it was being used. There were many such papers based on the assumption of CFSG, and had it proved to be false, revising them all would have been a huge job; but at least we had a good idea where to look.
However, it is true that the proof of CFSG is so long that it is unreasonable to expect a mathematician who uses it to have read and checked the proof. This is especially true for the many uses of the theorem outside group theory (in the theory of other algebraic structures such as semigroups and loops, in number theory, in computational complexity, and so on). This is probably the biggest change in a subject which has always taken the statement “Take nobody’s word for it” as a guiding principle. (In the paper of Fein, Kantor and Schacher, the assertion about derangements of prime power order is really a lemma in the proof of the theorem that the relative Brauer group of a finite extension of a global field is infinite – whatever that means!)
What about the mistakes, which undoubtedly occur in the published proof?
We have to hope that mathematicians will continue to recognise the importance of CFSG, and will continue to apply it, and even (in the case of an honorable few) to revise and improve it. How do errors in published mathematics come to light? By several methods. People read the papers and notice a problem; or they apply the result and are led to a contradiction; or they discover something which conflicts with the statement of the theorem. We are more likely to do this if the theorem in question remains an active part of our mathematical practice than if we put it on a dusty shelf somewhere and ignore this.
I believe that CFSG is too important to be put on a dusty shelf. So I am hopeful that it will stand up to the test.