The n-Category Café

Many people tell the story of arriving at IAS late in the evening and having some variation on the following conversation when picking up their keys from the porter.

Porter: Let me guess. You are a geometer.

Geometer: How did you know!?

Porter: You seem far too nice to be a number theorist.

I suspect that old porter would have some insight into l’affaire Mochizuki.

Maybe Fesenko’s proposed “duty” for such “gatekeepers” could be combined with your suspicion of them (which I think I share) to become a duty of all mathematicians: not to pronounce on the importance, or lack thereof, of an idea or theory without being fully informed about it. This might be taken to also include having a realistic appraisal of the limitations of one’s own knowledge, and giving other mathematicians and their theories the benefit of the doubt in cases when you don’t understand them.

Surely the truly groundbreaking advances will be spontaneously studied by people in that area, much like Scholze’s work was being studied in seminars around the world, or Ricci flow became a huge topic after Perelman’s 2003 papers, or Thurston’s work on 3-manifolds, or Zhang’s work on sieves and prime gaps, or homotopy type theory etc. People latch on the exciting new results for the sheer joy of discovery, or the desire to be in on the ground floor, or the capacity of advances to solve problems that had been resisting attack. It is true that sometimes great results have, in the past, languished, but the main examples I can think of are not that recent (of course, there’s a bias in this assessment…). When it is hard to see what a claimed breakthrough will ‘do’ for everyone else aside from merely give a truth value for a conjecture, then any barriers to learning, have a disproportionate impact, and only the truly dedicated people will fight through, the rest waiting until it becomes clear that the barriers are dropping, or the effort is worth it. For the case at hand, when in became apparent that the only known (and simplest) application of IUTT was the abc conjecture, people probably felt less inclined to put in the effort until the utility of doing so would be rewarded, be it with plain enjoyment of understanding, or the skills learned would be applicable to their own work, or by recognition of solving yet-unsolved problems. Telling people it’s their ‘duty’ to learn and understand it counterproductive. For instance, how does one decide who falls under that duty. Do all number theorists, say, have to study IUTT? Or just the algebraic number theorists? Or all arithmetic geometers? Anabelian geometers? All algebraic geometers? All pure mathematicians? The community as a whole has a responsibility to not suppress new results, but no one person should be forced to study this or that particular line of inquiry.

Although it is directed towards a different scenario (which is kind of at the opposite end of the spectrum, and even less likely to be perceived as a “duty”), this made me think of a paragraph from an essay by Gil Kalai on Robert Aumann:

How should we react to very unlikely or even absurd scientific claims? We had better ignore most, if not all of them. However, if we choose to react, what is the most appropriate way of doing so? Concerning the question of whether we should relate to unlikely claims at all, let me mention a mathematician who is one of the great number theorists of the twentieth century who has spent a considerable amount of time reading and finding the mistakes in proposed proofs (usually by laymen) for Fermat’s last theorem. I often wonder why he has taken the trouble all these years. I think the primary reason is his sense of responsibility as a scientist and perhaps also his sympathy towards people who share his dreams, if not his abilities. He perhaps also realizes that laymen occasionally propose useful mathematical ideas and, although it is a remote possibility, a layman may arrive at a valid proof for Fermat’s last theorem. (I don’t think he would have spent the same amount of time on proposals for trisecting an arbitrary angle into three equal parts using a compass and a ruler, which has been proven impossible.)

(This is a passage from a longer section where Kalai discusses Aumann’s early association with the “Bible Codes” controversy, and how that was resolved.)

In a mathoverflow post, Bill Thurston implied that duties of a mathematician include “spread[ing] understanding and breath[ing] life into ideas both old and new”.

The final paragraph says:

In short, mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification. The question of who is the first person to ever set foot on some square meter of land is really secondary. Revolutionary change does matter, but revolutions are few, and they are not self-sustaining — they depend very heavily on the community of mathematicians.

Sometimes, researchers hop on the bandwagon of the latest hot topic de jour even if it doesn’t directly relate to their field but the history of science is replete with examples of unexpected connections between different fields. It is worth it is try to relate a new discovery is another field to an intractable problem within your own field. We have known about dark matter since Fritz Zwicky in 1933, but it’s only recently that people started using the phrase “dark sector” to refer to a whole set of dark matter particles and mediator particles. It does immediately seem obvious how this could explain the primordial magnetic field of the universe, but Kamada, Tsai, and Vachaspati invoke the dark sector to attempt to explain primordial magnetogenesis.

Magnetic Field Transfer From a Hidden Sector

https://journals.aps.org/prd/pdf/10.1103/PhysRevD.98.043501

This is despite the fact that the proposed dark sector is still very mysterious, even from the point of physicists studying dark matter. Physicists proposing models of dark matter intentionally leave the identity of the dark matter particle unspecified, so their model will be as general as possible.

So, go ahead and see if you can apply the latest results or ideas from another field to your own field.

From what I understand, Mochizuki failed in the duty to communicate his ideas properly. (I could be more specific about what I mean by “properly” but I think most people reading this will know what I mean.) In my opinion, that absolves other mathematicians of any duty they might have had to get to grips with Mochizuki’s work, whatever principles one might want to adopt in general.

Fascinating discussion. I loved John’s comments and Thurston’s quoted above especially. I have to say completely disagree with Fesenko’s quoted sentiment “The ethical responsibility of mathematicians includes a certain duty…” and also with the AMS directive “To endeavor to be knowledgeable in their field, especially about work related to their research”. Each of us is (very) different, and each of us clearly should do her or his best, whatever that means. I have known in my area some people who keep up with the main journals, even with ArXiv. But why? Because they want to and are able to, not because they “should”. I know others who almost studiously avoid that, either for practical reasons (like too many teaching or family “duties”) or because they are constitutionally unable to do so. I have known a few brilliant people who have created revolutionary new ideas that have changed the field, but who themselves virtually never read anything, either learning by personal conversation, from conferences, or rediscovering proofs in their own way. Thurston is one who rediscovered countless pieces of mathematics, often coming up with a new proof. He really didn’t care, as he was focussed on the mathematics: both creating it, explaining its beauty and inspiring others. He was willing to be criticised for not citing correctly, apologising in that case and trying to correct the record, but no one should really be offended because he genuinely didn’t know.

What I myself think is the “duty” of a mathematician includes trying to have fun with mathematics, discover new and exciting things (at least new and exciting to me!) and then to try to communicate any enthusiasm I may feel to students in a course, participants in a seminar, readers of my paper. My own standard of “duty” does include that a paper should be interesting, new, inspiring, without errors, and well explained. And it’s frustrating to me when I encounter work (all too often) which doesn’t meet that level. (But I personally have far too little energy or time to read much of anything these days, it seems…!!!)

I have tremendous respect for colleagues who have an encyclopedic knowledge of the area, especially if they can combine that with great research. When I have a question, that’s who I will go to! Thank goodness they exist, and are generous with their knowledge. But for the best of them, that is not their “duty”, and they are not doing that out of any sense of obligation. Rather, they have the capacity (which I don’t) to do that, enjoy it, and thereby provide a great service to the community.

But I am also very grateful to those who may come up with quirky new ideas and who don’t perhaps have that encyclopedic reach.

In reference to the AMS or Fesenko dictums, frankly I find them both insulting and suppressive of the sense of freedom and joy we need, to both create fresh new mathematics and to be members of a healthy, happy mathematical community. We need all types, and each of us is different.

Lastly, about the Mochizuki question, I have no opinion (it’s far from my own area) except to be grateful to those who are able and willing to devote the considerable time and energy to make things more accessible to a wider community. I don’t doubt that Mochizuki himself is doing his own best to communicate his ideas. If his reception falls short of what he had hoped for, then he has to either accept this, or strive even harder to find ways to express and explain himself. It is no one else’s “duty” to do this for him, unfortunately. Dogen wrote his difficult zen texts in the 1200s knowing full well that almost no one at the time could or would understand him; but he did his best, knowing that “some time, some where, some one” would get it, and now, hundreds of years later, that is finally starting to happen. I doubt it will take that long for Mochizuki, but it’s not a bad attitude to take: do your best, then sit back and be patient, as that’s all you can do. The truth will out, and great mathematics will eventually be recognized.