## July 18, 2018

### The Duties of a Mathematician

#### Posted by John Baez

What are the ethical responsibilities of a mathematician? I can think of many, some of which I even try to fulfill, but this document raises one that I have mixed feelings about:

Namely:

The ethical responsibility of mathematicians includes a certain duty, never precisely stated in any formal way, but of course felt by and known to serious researchers: to dedicate an appropriate amount of time to study each new groundbreaking theory or proof in one’s general area. Truly groundbreaking theories are rare, and this duty is not too cumbersome. This duty is especially applicable to researchers who are in the most active research period of their mathematical life and have already senior academic positions. In real life this informal duty can be taken to mean that a reasonable number of mathematicians in each major mathematical country studies such groundbreaking theories.

My first reaction to this claimed duty was quite personal: namely, that I couldn’t possibly meet it. My research is too thinly spread over too many fields to “study each new groundbreaking theory or proof” in my general area. While Fesenko says that “truly groundbreaking theories are rare, and this duty is not too cumbersome”, I feel the opposite. I’d really love to learn more about the Langlands program, and the amplitudohedron, and Connes’ work on the Riemann Hypothesis, and Lurie’s work on $(\infty,1)$-topoi, and homotopy type theory, and Monstrous Moonshine, and new developments in machine learning, and … many other things. But there’s not enough time!

More importantly, while it’s undeniably good to know what’s going on, that doesn’t make it a “duty”. I believe mathematicians should be free to study what they’re interested in.

But perhaps Fesenko has a specific kind of mathematician in mind, without mentioning it: not the larks who fly free, but the solid, established “gatekeepers” and “empire-builders”. These are the people who master a specific field, gain academic power, and strongly influence the field’s development, often by making pronouncements about what’s important and what’s not.

For such people to ignore promising developments in their self-proclaimed realm of expertise can indeed be damaging. Perhaps these people have a duty to spend a certain amount of time studying each new ground-breaking theory in their ambit. But I’m fundamentally suspicious of these people in the first place! So, I’m not eager to figure out their duties.

What do you think about “the duties of a mathematician”?

Of course I would be remiss not to mention the obvious, namely that Fesenko is complaining about the reception of Mochizuki’s work on inter-universal Teichmüller theory. If you read his whole article, that will be completely clear. But this is a controversial subject, and “hard cases make bad law”—so while it makes a fascinating read, I’d rather talk about the duties of a mathematician more generally. If you want to discuss what Fesenko has to say about inter-universal Teichmüller theory, Peter Woit’s blog might be a better place, since he’s jumped right into the middle of that conversation:

As for me, my joy is to learn new mathematics, figure things out, explain things, and talk to people about math. My duties include helping students who are having trouble, trying to make mathematics open-access, and coaxing mathematicians to turn their skills toward saving the planet. The difference is that joy makes me do things spontaneously, while duty taps me on the shoulder and says “don’t forget….”

Posted at July 18, 2018 9:59 PM UTC

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### Re: The Duties of a Mathematician

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.

Posted by: Chris Brav on July 19, 2018 10:03 AM | Permalink | Reply to this

### Re: The Duties of a Mathematician

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.

Posted by: Mike Shulman on July 19, 2018 10:49 PM | Permalink | Reply to this

### Re: The Duties of a Mathematician

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.

Posted by: David Roberts on July 20, 2018 12:34 AM | Permalink | Reply to this

### Re: The Duties of a Mathematician

I agree with all this, especially the counterproductive nature of telling mathematicians it’s their “duty” to keep up with particular developments: this drains the fun out of it.

Fesenko very clearly says that people who don’t know anabelian geometry can’t understand or pass judgement on Mochizuki’s work:

It is reasonable to be sceptical about a new fundamental development but only if one has or has acquired an expertise in the relevant area, which in the case of IUT is anabelian geometry and IUT itself. It should be stated clearly that to declare oneself a sceptic in relation to a theory, whose subject area one does not know and does not apply efforts to study, is unethical.

Furthermore:

Sometime, the author is led to present certain things in a way which is natural from the point of view of how the theory has been developing in his head, but remains unknown to the readers of his papers. There is much more about the theory, which is known to the author, but cannot be included in the author’s paper by various reasons. Good learners should reach this stage of knowing. In the case of IUT, this stage is not achievable without a solid knowledge of anabelian geometry. Some researchers may need several resolute attempts to proceed with the IUT papers. (The author of this text needed some 20 attempts.)

So, either his claimed “duty” to study Mochizuki’s work is limited to people who already understand anabelian geometry, or it includes learning anabelian geometry.

Posted by: John Baez on July 20, 2018 1:27 AM | Permalink | Reply to this

### Re: The Duties of a Mathematician

John said:

I agree with all this, especially the counterproductive nature of telling mathematicians it’s their “duty” to keep up with particular developments: this drains the fun out of it.

The AMS has some ethical guidelines which includes the following under the responsibilities of mathematicians with regard to mathematical research and its presentation.

• To endeavor to be knowledgeable in their field, especially about work related to their research;

Do you feel that’s different to what Fesenko is asking for?

The ethical guidelines perhaps give the AMS’s answer to your question about the duties of a mathematician.

Posted by: Simon Willerton on July 20, 2018 12:32 PM | Permalink | Reply to this

### Re: The Duties of a Mathematician

Perhaps the difference between this and what Fesenko is implicitly suggesting is that what the AMS means (or, at least, what I would interpret it to mean) is that a mathematician should keep up with developments that are directly related to their research? Put differently, if it’s something that you would have an obligation to cite in a paper that you write, then you have an obligation to know about it so that you can cite it. That seems pretty clear to me.

There are also different degrees of “knowledgeable”. For instance, I don’t know a whole lot about cubical type theory and cubical models of HoTT. But I try to maintain a passing familiarity with what they do and what they look like and what the main papers are, so that if I need to cite them in some paper then I can mostly figure out who I should cite, and so that I’ll know if part of my own research starts to seem related to it so that I can go read up on the relevant stuff at that point.

Posted by: Mike Shulman on July 20, 2018 4:21 PM | Permalink | Reply to this

### Re: The Duties of a Mathematician

I hadn’t known the AMS proposed ethical guidelines for mathematicians! Interesting.

Somehow

To endeavor to be knowledgeable in their field, especially about work related to their research

seems more of an aspirational goal, and more forgiving of failure, than

The ethical responsibility of mathematicians includes a certain duty, never precisely stated in any formal way, but of course felt by and known to serious researchers: to dedicate an appropriate amount of time to study each new groundbreaking theory or proof in one’s general area.

It seems hard to deny that one should endeavor to be knowledgeable about work related to ones research. That one should spend an “appropriate amount of time” (whatever that is—presumably not none) to each new groundbreaking theory or proof seems less flexible. For example, I would easily forgive Wiles for not keeping up with each new groundbreaking theory while he was busy trying to prove Fermat’s last theorem holed up in his attic. There are times when hermit-like isolation is called for.

Posted by: John Baez on July 20, 2018 9:01 PM | Permalink | Reply to this

### Re: The Duties of a Mathematician

I can not agree with an imperative “to study each new groundbreaking theory or proof in one’s general area” as our reasons to be mathematicians and to be paid as such might have emphasis somewhere else. Even more I can not agree that there is no imperative on following the fundamental events in our wider field at all: in most cases, the AMS guideline “endeavor to be knowledgeable in their field” is really good one. Namely, it is NOT that we only engage writing new papers and justifying just those, but we also engage in refereeing grants, journal editorship, we influence hiring and promotion of colleagues and so on. I have been so many times a witness that narrow minded or ignorant specialists not aware of where the most vigorous currents in their wider field point to, made damaging decisions. In role of committee members or grant panelists they bogusly reported on the issues, ill advised or informed voting staff, policy makers or wider audience and so on. Also, if we are paid with public money, we should be responsible to answer to orientation questions/demands from general public or individuals and how to do this resposibly without our homework beyond the purposes of our overspecialized papers ?

Overall, it is the real imperative to understand our personal purposes and also expectations from the funding community and reality of the totality of work and decision we do, to judge what level of breadth is reasonable to maintain in order to justify our role and justifying the expenses we create (within the system of science our salary is just a part of it; regard the office space, grants, even conference travel cost burning fossil fuel). Also it may be beneficial to think that we change jobs and academic levels and something what is less of our responsibility today may be or is even very likely to be our major responsibility tomorrow.

Posted by: Zoran Skoda on August 10, 2018 6:46 PM | Permalink | Reply to this

### Re: The Duties of a Mathematician

Sometimes, the author is led to present certain things in a way which is natural from the point of view of how the theory has been developing in his head, but remains unknown to the readers of his papers.

I would say that among the duties of a mathematician is not to do that.

However, the years I’ve spent trying to read mathematics papers suggest to me that a typical practitioner sees their duty as presenting neither how the theory developed in their head, nor how it could be appreciated by their readers. Instead, the professional obligation seems to be the ritual erasure of all traces of humanity. As David and Hersh said of how “the Ideal Mathematician” writes,

Three pages of definitions are followed by seven lemmas and, finally, a theorem whose hypotheses take half a page to state, while its proof reduces essentially to “Apply Lemmas 1–7 to definitions A–H”. […] To read his proofs, one must be privy to a whole subculture of motivations, standard arguments and examples, habits of thought and agreed-upon modes of reasoning. The intended readers (all twelve of them) can decode the formal presentation, detect the new idea hidden in lemma 4, ignore the routine and uninteresting calculations of lemmas 1, 2, 3, 5, 6, 7, and see what the author is doing and why he does it. But for the noninitiate, this is a cipher that will never yield its secret. If (heaven forbid) the fraternity of non-Riemannian hypersquarers should ever die out, our hero’s writings would become less translatable than those of the Maya.

Posted by: Blake Stacey on July 20, 2018 2:11 AM | Permalink | Reply to this

### Re: The Duties of a Mathematician

I would say that among the duties of a mathematician is not to do that.

Huh, I would have said the exact opposite!

The computer scientist Alan Kay once remarked “A change of perspective is worth 80 IQ points”, and so I’ve always felt that if you’ve got a new private view of a subject, it’s a mathematical good deed to publicize it.

The caveat (which I have to thank my PhD supervisor for pointing out to me) is that attention is finite – if you’ve got a private view on an idea that’s a supporting actor for the main result, then forcing readers to divide their attention can make it harder for them to understand the main point.

So I always feel a bit of tension when writing, because there are often competing mathematical goods, and the trade-off can be difficult.

Posted by: Neel Krishnaswami on July 24, 2018 10:23 AM | Permalink | Reply to this

### Re: The Duties of a Mathematician

I think the two of you are reading the quote in different ways. Neel is saying that if you have a personal view of a subject you should explain it to the reader; whereas I think Blake is reading the quote as suggesting a license to present a subject according to one’s personal view without explaining that view to the reader. I think the context of the quote supports the latter reading.

Posted by: Mike Shulman on July 24, 2018 2:22 PM | Permalink | Reply to this

### Re: The Duties of a Mathematician

Yes, the latter reading is how it came across to me.

Posted by: Blake Stacey on July 24, 2018 3:54 PM | Permalink | Reply to this

### Re: The Duties of a Mathematician

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.)

Posted by: Noam Zeilberger on July 20, 2018 9:59 AM | Permalink | Reply to this

### Re: The Duties of a Mathematician

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.

Posted by: Simon Willerton on July 20, 2018 12:37 PM | Permalink | Reply to this

### Re: The Duties of a Mathematician

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.

Posted by: Jeffery Winkler on August 1, 2018 6:12 PM | Permalink | Reply to this

### Re: The Duties of a Mathematician

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.

Posted by: Timothy Gowers on August 31, 2018 4:37 PM | Permalink | Reply to this

### Re: The Duties of a Mathematician

I was very hesitant to comment on this, but I feel someone should. I find the sentiments expressed here to be depressing. The principal myth propounded by mathematicians is that what counts is proof. Mochizuki has written down his work carefully, in detail, and conventionally (in form). It is therefore astonishing to me that a prominent and influential mathematician, who as far as I know is not an expert in the area, should assert that Mochizuki has ‘failed in his duty to communicate his ideas properly’.

It is one thing that few people understand Mochizuki’s work today. One can debate the reasons for that. But regardless of what those reasons are, let us be honest that the ‘duty’ that the comment refers to appears to be one which has nothing to do with the Mochizuki’s work in itself, but rather means something like: the ‘duty’ to follow the behaviour prescribed at this point in time by a significant proportion of mathematicians, including a highly influential minority that hold much of the ‘power’ in the community; that prescribed behaviour being sufficiently strongly imposed that one may as a non-expert (as far as I know) judge Mochizuki’s ideas to be worthy of being ignored if it is not followed.

For what it is worth, I, though entirely a non-expert myself, have had a little contact over email with Mochizuki, and found him polite, humble, and willing to discuss his ideas. It is clear that there are highly novel ideas in his work, regardless of whether one can determine that everything is correct (I am not able to judge that either way). Whatever happened to having a simple, humble interest in understanding a novel piece of mathematics for its own sake?

Posted by: Richard Williamson on September 2, 2018 11:59 PM | Permalink | Reply to this

### Re: The Duties of a Mathematician

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.

Posted by: Al on October 6, 2018 8:13 PM | Permalink | Reply to this