Over dinner just now I had the pleasure of reading MIT PhD student Alma Steingart’s wonderful essay *A group theory of group theory: collaborative mathematics and the ‘uninvention’ of a 1000-page proof*, which is about the classification theorem for finite groups of simple order.

Steingart is one of those writers who highlights the very *human* enterprise that is sense-making in mathematics, including what counts as proof, which changes not only in time but even across different subfields. It’s an insight that first came to my attention during the hullabaloo over whether Shinichi Mochizuki actually did prove the *abc* conjecture back in 2012. It got really hammered home, however, in reading (and rereading) the late Bill Thurston’s classic *On proof and progress in mathematics*, especially the parts where he talks about both his introduction as a graduate student to the notion of proof-in-practice, and (much later) how the community responded to his proof of the hyperbolization theorem for Haken manifolds, in particular his realization that they were not interested in the thing proved itself so much as how he did it, his way of thinking.

This is echoed in Steingart’s abstract:

The history of the Classification points to the importance of face-to-face interaction and close teaching relationships in the production and transformation of theoretical knowledge. The techniques and methods that governed much of the work in finite simple group theory circulated via personal, often informal, communication, rather than in published proofs.

Consequently, the printed proofs that would constitute the Classification Theorem functioned as a sort of shorthand for and formalization of proofs that had already been established during personal interactions among mathematicians. The proof of the Classification was at once both a material artifact and a crystallization of one community’s shared practices, values, histories, and expertise.

However, beginning in the 1980s, the original proof of the Classification faced the threat of ‘uninvention’. The papers that constituted it could still be found scattered throughout the mathematical literature, but no one other than the dwindling community of group theorists would know how to find them or how to piece them together.

Faced with this problem, finite group theorists resolved to produce a ‘second-generation proof’ to streamline and centralize the Classification. This project highlights that the proof and the community of finite simple groups theorists who produced it were co-constitutive–one formed and reformed by the other.

But I’ve harped on that at length elsewhere. Today I just want to collect some quotes on a monster mind.

Bill Thurston (*hi again!*) famously wrote that in his early years he inadvertently “killed the field” by being *too* good at proving results, driving others away, and that in retrospect this made him understand the sociology of mathematical proof better:

At that time, foliations had become a big center of attention among geometric topologists, dynamical systems people, and differential geometers. I fairly rapidly proved some dramatic theorems. I proved a classification theorem for foliations, giving a necessary and sufficient condition for a manifold to admit a foliation. I proved a number of other significant theorems. I wrote respectable papers and published at least the most important theorems. It was hard to find the time to write to keep up with what I could prove, and I built up a backlog.

An interesting phenomenon occurred. Within a couple of years, a dramatic evacuation of the field started to take place. I heard from a number of mathematicians that they were giving or receiving advice not to go into foliations—they were saying that Thurston was cleaning it out. People told me (not as a complaint, but as a compliment) that I was killing the field. Graduate students stopped studying foliations, and fairly soon, I turned to other interests as well.

I believe that two ecological effects were much more important in putting a damper on the subject than any exhaustion of intellectual resources that occurred.

First, the results I proved (as well as some important results of other people) were documented in a conventional, formidable mathematician’s style. They depended heavily on readers who shared certain background and certain insights. The theory of foliations was a young, opportunistic subfield, and the background was not standardized. I did not hesitate to draw on any of the mathematics I had learned from others. The papers I wrote did not (and could not) spend much time explaining the background culture. They documented top-level reasoning and conclusions that I often had achieved after much reflection and effort. I also threw out prize cryptic tidbits of insight, such as “the Godbillon-Vey invariant measures the helical wobble of a foliation”, that remained mysterious to most mathematicians who read them. This created a high entry barrier: I think many graduate students and mathematicians were discouraged that it was hard to learn and understand the proofs of key theorems.

Second is the issue of what is in it for other people in the subfield. When I started working on foliations, I had the conception that what people wanted was to know the answers. I thought that what they sought was a collection of powerful proven theorems that might be applied to answer further mathematical questions. But that’s only one part of the story. More than the knowledge, people want personal understanding. And in our credit-driven system, they also want and need theorem-credits.

I was thinking about what Thurston said above when I read about Michael Aschbacher in Steingart’s essay.

Michael Aschbacher (henceforth MA) was the second-most prolific contributor to the Classification effort, despite joining only towards the very end, with 21 papers to Daniel Gorenstein’s 25 (out of the 416 included in Gorenstein’s bibliography); DG, for context, was the closest the effort had to a leader (this also shows in the fact that he coauthored 15 papers with 6 others, over twice as many as the next-highest).

Here’s where the MA quotes start:

Though Michael Aschbacher became interested in finite simple groups only as a postdoc (having written his dissertation on combinatorics), his early contribution to the field took the community by surprise. Gorenstein described Aschbacher’s entrance into the field as ‘dramatic’ (Gorenstein, 1989: 469), and Solomon remarked that following Aschbacher’s work in 1974 and 1975, ‘belief mounted that nothing could stand between him and the completion of the Classification’ (Solomon, 2001: 338). In recognition of his many accomplishments, Aschbacher won the prestigious Frank Nelson Cole Prize in Algebra in 1980.

However, as Aschbacher’s esteem within the community grew, his

papers became notoriously difficult to comprehend. When Aschbacher’s first important paper in the field appeared in print, Solomon was an instructor in the Department of Mathematics at the University of Chicago. Together with George Glauberman, he tried to read the proof, but found it incredibly difficult to follow Aschbacher’s arguments. Specifically, Solomon complained that ‘there were these clever counting arguments, but he never bothered to write it [them] down’. Aschbacher’s style made it notoriously difficult for other mathematicians to follow his proofs, and as his papers grew longer, the difficulties were exacerbated.Group theorists no longer judged a proof as a self-contained document, but took into account its author, his previous work, and his standing within the community. When asked about the process of refereeing in the finite simple group community, Gorenstein characterized Aschbacher’s papers as ‘extremely difficult’. He recalled, ‘Aschbacher is so smart … . Richard Lyons and I and Aschbacher had just written a joint paper … . I wrote a draft of this paper, Aschbacher rewrote it – I had enormous difficulty reading my own goddamn paper … . He can see things so much better than anybody else.’ He added, ‘he’s so quick that he misses things’.

If Gorenstein took on a leadership role as far as organization was concerned, Michael Aschbacher was the driving force behind completing the Classification. Beginning in 1973, Aschbacher proved one result after another, at least one of which had previously resisted sustained assaults by other mathematicians, including Thompson. When Aschbacher announced in Duluth that he had solved yet another problem, the group theorists in attendance were convinced that the end was in sight.

The Duluth conference had one more unexpected outcome. At the height of their excitement, group theorists began to realize that as their quest was coming to an end, so too was the field that had grown around it and the community that had sustained it. Mathematicians, especially the younger ones, started reevaluating whether they should stay in the field or move on to a new one. Duluth represented a turning point in the growth of the field. The sort of excitement that had characterized the field since the early 1960s was dampened as the end loomed on the horizon. Students were no longer drawn to the field, and while a complete Classification required several more years of sustained work, mathematicians had already laid claim to the major remaining problems.

From an interview with Ron Solomon:

Sometime around spring in ’75, Janko wrote to Danny and said, I give up. I can’t do the thin group problem … . Thompson who was visiting at Yale at the time started working on the thin group problem after Janko threw in the towel. He worked on it for several months around that spring and got stuck … . And I think Danny was seriously worried that if Janko couldn’t do it and if Thompson didn’t see at least after a few months of thinking about it how to finish it off, maybe there was some major intractable problem that the methods that we had developed just couldn’t handle. And Aschbacher picked up that problem over the summer and by the end of the summer finished it. And I think that was the psychological turning point of the

classification.

These quotes remind me of yet another monster mind — the late Jean Bourgain. This obituary by Terry Tao relates some of his experiences as a grad student struggling greatly with Bourgain’s papers, an experience common to most: https://terrytao.wordpress.com/2018/12/29/jean-bourgain/