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I am a set theorist. Since I began to study this subject, I became increasingly aware of negative attitudes about it. These were expressed both from an internal and an external perspective. By the “internal perspective,” I mean a constant expression of worry from set theorists and logicians about the relevance of their work to the broader community / “real world”, with these worries sometimes leading to career-defining decisions on the direction of research.

For me, this situation is unwanted. I studied set theory because I thought it was interesting, not because I wanted to be a soldier in some kind of movement. Furthermore, I don’t see why an area needs defending when it produces a lot of deep theorems. That part is hard enough.

To what degree does there exist, in the various areas of mathematics, a widespread feeling of pressure to defend the relevance of the whole subject? Are there some areas in which it is enough to pursue the research that is considered interesting, useful, or important by experts in the field? Of course there will always be a demand to explain “broader impacts” to funding agencies, but I am talking about situations where the pressure comes from one’s own colleagues or even one’s own internalized sense of what is proper research.

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – Todd Trimble Apr 28 '19 at 16:18
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    $\begingroup$ Rather relevant: Melvin Henriksen, There are too many B.A.D. mathematicians, Mathematical Intelligencer 15 #1 (December 1993), 6-9 freely available version. See especially the paragraph beginning with "I was shielded from mathematical bigotry until" and what follows. $\endgroup$ – Dave L Renfro Apr 28 '19 at 19:42
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    $\begingroup$ I am not sure if any of the answers (or the comments in chat) emphasise that this can vary significantly from country to country, and from (mathematical) generation to generation. I am seeing a lot of implicit universal quantifiers and this makes me rather uncomfortable $\endgroup$ – Yemon Choi Apr 28 '19 at 22:35
  • $\begingroup$ (I wrote my comment just before seeing Tim Campion's answer mathoverflow.net/a/330238 which scrupulously does include the caveats I referred to) $\endgroup$ – Yemon Choi Apr 28 '19 at 22:39
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    $\begingroup$ Late to the party, but I note that Davis and Hersh discuss "Hardyism" vs "mathematical Maoism": books.google.com/books?id=I1fj_B60fyoC&pg=PA95 $\endgroup$ – Steve Huntsman Apr 29 '19 at 12:53

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Overall, people in academia in general and mathematicians in particular are very lucky in being free to study (and being able to make a good living) according to the standards of their discipline, without feeling pressure to defend the relevance of their whole subject. In fact even within our disciplines we have a lot of freedom to pursue our individual visions and tastes. (To appreciate how lucky we are compare the situation with musicians, writers, artists, film directors, actors, ...)

Relations with other areas of mathematics or outside mathematics are nice but they are one (and not necessarily a major one) among variety of criteria to appreciate mathematical progress.

I think we do have some duty to try to explain what we are doing outside our community and even outside the mathematical community. (But also this task is easier in some areas and harder in others.)

Another thing that I found useful in similar contexts is the "sure thing principle". Given an unwanted situation that has no implication on your action why worry about it at all too much. For example, suppose a paper you wrote and regard as a good paper is rejected. If the rejection was unjust then the conclusion is: "Improve your paper", and if the rejection was just then the conclusion is "Improve your paper".

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    $\begingroup$ +1 but not sure about the last paragraph. The hope is that by discussing it, I can either get some feedback that allows me to understand and accept the situation, or move some logicians to be a little less self-deprecating. $\endgroup$ – Monroe Eskew Apr 28 '19 at 8:51
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    $\begingroup$ My point is that you should relax and do what’s interesting and important to the best of your judgement. $\endgroup$ – Gil Kalai Apr 28 '19 at 9:52
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    $\begingroup$ One more comment: With regard to the artists etc, we can be sure that no one will ask them if it has applications. What counts the most is whether people like it or not, and this can be affected by the amount of expertise or experience of the critic. The artist (or critics) can also add to a work's appeal by explaining why the work is especially original or was challenging to produce, or just situate in a historical or personal context in such a way that adds interest. Now these ways of adding to an artistic work's appeal are nearly the same kinds of things that may add to a mathematical $\endgroup$ – Monroe Eskew Apr 28 '19 at 15:31
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    $\begingroup$ If you write a paper you think is good, it gets rejected, and you're confident that the rejection was the result of (say) prejudice against your field or against some group the reviewer rightly or wrongly thinks you belong to -- then the conclusion may be not "improve your paper" but "submit it somewhere else that may use different reviewers". $\endgroup$ – Gareth McCaughan Apr 29 '19 at 9:08
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    $\begingroup$ When I was a student, applications were looked down upon. "Applied mathematics" was - almost - a dirty word. You wouldn't even dream of defending what you did, theorems should have a value in itself (I was then also interested in applications). It is almost ironic that since then the fashion pendulum has swung to the other side. Today, woe to whom would have no - even so harebrained - application scenario in mind. Today the dirty word is "pure"/"blu-sky" science. Lesson learnt? I sacrifice the occasional lamb(da) to the prevalent fashion, and ignore it as much as possible beyond that. $\endgroup$ – Captain Emacs Apr 29 '19 at 13:34
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These questions can be an opportunity for a bit of soul-searching; that's a not a bad thing once in a while.

You can say that you want to work on whatever you find interesting and no further justification or motivation is needed. And I agree with you and support you in that.

But others can ask why they should give funding and space to support you in doing so, and these are legitimate questions that deserve some answer, since money and research space are limited and desired by many people.

You can answer that, like an artist, you are exploring and creating and this is a valuable activity, and I agree with you there too. But it is more valuable the more people you share it with. Unfortunately, how receptive people are depends on tastes of the moment. So this can become a fashion industry. On the other hand, if you feel there is beauty and value in your area, it may be worth some work to demonstrate this and share with others (or you may call that politics).

You can also answer that your work will be useful to people now or in the future; your area may lead to understandings or breakthroughs with eventual impact on engineering, physics, etc. Given the history of mathematics, you can try to justify that in the long term unforeseen connections often happen.

So, it is not surprising that (to answer your stated question) areas with practical impact, or historically-accepted beauty and merit, find it easier to make these arguments and less pressured to defend their field.

And it can be valuable to ask how your work helps or inspires others, whether in practical or aesthetic senses. I'm not saying it has to do so, especially if your position and funding are already secure. Then do what you want! And I'm not saying one should have this existential crisis every day or even every year, but it can be healthy once in a while.

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First let me try to answer the question in an "ideal world", where (in particular) set theory is treated like any other branch of mathematics, and then let me discuss how we might fall short of it.

In an ideal world the question of "What is the importance of your field to the rest of mathematics?" would not get asked so often (unless the field is so small that one has never heard of it before) but the question "What is the importance of your work to the rest of mathematics?" would. The reason for this is that the web of connections between different fields is one of the things that makes mathematics most beautiful.

Of course, this does not mean that if someone's work is not important to other fields of mathematics, mathematicians should reject all their grant proposals and deny their job applications. It just means that connections to other fields of mathematics are one possible very strong selling point. There are a number of reasons that a mathematical theorem can be interesting, of which applications are one.

In my preferred subfield (etale cohomology theory, one of the most fashionable topics of the 1960s), and I suspect in many other fields of mathematics, the way I see it is that there is a technical core of the subject, consisting of works that address the most difficult technical problems or advance the methods of the field, and a surrounding periphery consisting of works which attack specific examples or make only small variations on existing techniques. It is these peripheral works that need applications. (Let me clarify that I am not trying to criticize other mathematicians' works - I am thinking primarily of my own work when I think of work on the periphery, though I am proud of some of my applications.) To convince other mathematicians of the importance of works without applications, one must lean on other points (How long a problem has been open, who has tried and failed, the way it connects different subfields of your field).

An additional advantage for those whose work has applications to other areas of mathematics is that mathematicians in those areas will better be able to understand when they talk about the importance of their work. When someone explains to you work in your field, or that has relevance to your field, you are able to more accurately judge its value. For work in another field, even if it's equally good, you may not be able to verify this yourself. For this problem it seems the only recourse is to try harder to explain your work, for instance using analogies with other work.

Now what may be the problem with set theory?

It seems to me that non-set theorists may have a skewed perception of set theory for a number of reasons:

Some mathematicians view math entirely through the lens of finitary mathematical objects.

One reason is that, because set theory is the foundation of mathematics, most mathematicians work with sets at some point in the research, but typically with sets that are almost entirely dead (i.e. devoid of intrinsic set-theoretic interest). Most mathematicians probably work with only finite sets and sets of zero, one, or two different infinite cardinalities. So if you work with sets you work with something that the mathematician has seen, but not that they have seen be interesting.

Some mathematicians are skeptical of both platonist and formalist philosophical views in mathematics, and take comfort in the fact that their research (and most mathematical research) can in principle be reduced to statements about concrete, finitary objects. For set theory this usually can't be done except in a purely formal way.

Perhaps only because I am optimistic, I think the biggest reason is that mathematicians simply do not understand set theory very much. It is not so transparent to outsiders how a zoo of exotic mathematical fauna arises from the simple concept of size. I think that a lot of mathematicans who are opposed to set theory would be receptive if it were explained to them in an appropriate way. One merely has to examine the top reputation of all time scoreboard on mathoverflow to demonstrate that it is possible for a set theorist, by explaining set theory well, to be popular among mathematicians of all stripes.

Probably the best advice for how to do this is to find the mathematicians in your field who have the most professional success, and try to see how they present themselves to a broader mathematical audience. I attended a beautiful colloquium talk by Hugh Woodin which I thought did a great job of explaining the importance of set theory to non-set theorists without, I think, ever mentioning an application to other branches of mathematics.

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    $\begingroup$ I think it’s fair to ask: what computations are part of your work? I used to ask this when I worked on Wall Street and interviewed mathematicians or physicists, to get an indication for one of the academic skills most directly transferable to a corporate setting. So to Monroe’s students I would say: try the Friedman-esque exercise of determining which sentences with few quantifiers are true in various set theories, or try the Hamkins-esque exercise of illustrating set-theoretic phenomena with tikz — then you can do set theory and pick up some useful transferable skills too. $\endgroup$ – Matt F. Apr 28 '19 at 15:46
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    $\begingroup$ I've heard stories where some mathematicians, in hiring committees, would be opposed to promoting set theorists and other people in logic on the basis that "It's not actual math." That appears to be a problem for people working in logic. $\endgroup$ – Mehta Apr 28 '19 at 16:31
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    $\begingroup$ @WillSawin Yes. See for example the recent work of Ilijas Farah. $\endgroup$ – Monroe Eskew Apr 29 '19 at 11:20
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    $\begingroup$ @MonroeEskew That is precisely the wrong way of looking at it. The relationships and connections between different fields of mathematics are interesting in themselves, but to properly exploit them you need knowledge and good taste on both sides. Unless you have both, you need to borrow them from someone else. It's not about seeking outside approval, but rather about (informally) working together to create something greater than you could individually. $\endgroup$ – Will Sawin Apr 29 '19 at 12:08
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    $\begingroup$ @MonroeEskew That said, there is no requirement to do this, especially not if your work is interesting in itself. If your work is not directly applicable to other fields then you should feel greater pressure not to avoid the most technical parts of your fields, to work at the greatest possible level of generality, and to do other things that help your theorems fit better into the mathematical tragedy - and this should be a reflection of your personal (pure, uncompromising) mathematical aesthetic. $\endgroup$ – Will Sawin Apr 29 '19 at 12:14
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My impression is that the disdain for set theory among other mathematicians is a relatively recent phenomenon. Lauded mathematicians like Erdos, Gödel, Hausdorff, Kuratowski, Steinhaus, von Neumann, or Luzin could be considered set theorists in a broad sense. Certainly, one would expect that Hilbert would have had a deep respect for the modern development of set theory: it's no coincidence that the CH was the first problem.

One thing I think leads people to a negative outlook on set theory is that textbooks in every subject have "set theory introductions" or appendices which clumsily explain some very basic foundations which is then totally irrelevant to the rest of the book, leading people to believe that set theory is just shallow quibbling about axioms or de Morgan's law keeping them away from the truly interesting topic at hand. This tradition seems to trace back to Bourbaki, who funnily enough don't seem to suffer any tarnish for it. I have heard my teachers in other mathematical fields make some spontaneous remark about proper classes--"which you might care about, if you were a set theorist." Like in the same way I might care about numbers if I was a number theorist?

And of course, this bad sociology is not unique to set theory or logic. From what I have read, combinatorics used to be perceived as a second-class subject. This mindset is ingrained in a lot of the writings of Gian-Carlo Rota. And I think the situation has changed, since the subject has gained a lot of prestige in recent years and there have been many connections realized with other branches.

You can see this kind of thing happening all the time in applied mathematics, where fashions change in fast-forward, and the funding with it.

In homotopy theory, Clark Barwick posted a now-removed piece where he said the achievements of the field were being overlooked because of sociological tendencies, such as top researchers content not to publish in prestige generalist journals.

And to give a bit of perspective, I heard a very well-respected researcher recently complain that his field was fading and that his perception was that set theory has all the best people and most exciting progress these days.

I personally find set theory very easy to justify. It has hard problems of great intrinsic interest not too far from the surface (like number theory, in some respects), powerful non-elementary techniques, amazing elegance, and connections with other areas (and probably even more yet to be discovered because of sociological problems). The progress has been enormous but you can still see the chain connecting the classical and modern theory. I think if someone put their mind to it, a set theorist could explain the general picture quite nicely to other kinds of mathematicians, where even people who are informed sometimes have ideas like we are just trying to discover larger and larger cardinals.

In Alain Connes's advice in the Companion, he echoes a quote of Feynman--"why do you care what others think!" Of course, we care about what funding agencies and other evaluators think, but my take-away lesson is that even Connes faced detractors in his career. At the end of the day, it's not worth feeling like we have to continually defend ourselves from the vagaries of the opinions of those who don't (and choose not to!) understand.

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    $\begingroup$ When I was starting graduate studies, Robert Greene told me that he really liked Solovay's result that there is a model of ZF in which all sets of reals are Lebesgue measurable. Although this result requires an inaccessible cardinal, Greene said that he liked it because it means that any set of reals which you construct "by hand" must be measurable. So maybe set theory is useful after all :) $\endgroup$ – Bill Chen May 1 '19 at 4:34
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    $\begingroup$ Can you give an example of an open and accessible problem in set theory that you like? $\endgroup$ – Matt F. May 1 '19 at 11:56
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    $\begingroup$ Kunen's question: Can there be an ultrafilter on $\omega_1$ generated by fewer than the maximum number ($2^{\aleph_1}$) of sets? $\endgroup$ – Bill Chen May 2 '19 at 1:28
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    $\begingroup$ There are variety of results in set theory that are admired by mathematician outside the field (and should be). E.g., There was a recent result by Maryanthe Malliaris and Saharon Shelah that was rather widely discussed (e.g. by by Gowers gowers.wordpress.com/2017/09/19/… and by Baez (alas over Google+) and in Quanta Magazine) . $\endgroup$ – Gil Kalai May 2 '19 at 19:49
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    $\begingroup$ @MattF.: Assume ZF and "If there is a surjection from $A$ onto $B$, then there is an injection from $B$ into $A$". Does the Axiom of Choice hold? (Note we require just an injection, not a splitting of the surjection!) $\endgroup$ – Asaf Karagila May 3 '19 at 7:26
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Let me address the very final part of your question, concerning "one’s own internalized sense of what is proper research". I have definitely encountered this form of internal doubt during my years as a researcher, which by the way now lie behind me, since my doubt has gotten the better of me in the end.

John von Neumann wrote (in an essay entitled The Mathematician):

I think that it is a relatively good approximation to truth — which is much too complicated to allow anything but approximations — that mathematical ideas originate in empirics. But, once they are conceived, the subject begins to live a peculiar life of its own and is … governed by almost entirely aesthetical motivations. In other words, at a great distance from its empirical source, or after much "abstract" inbreeding, a mathematical subject is in danger of degeneration. Whenever this stage is reached the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas.

I know that there are probably many mathematicians who would disagree with the above, but at least Von Neumann openly considered the fact that not all fields of research are inherently valuable. One could argue whether this is solely due to a connection or lack thereof with empirical reality, or whether an inborn sense of mental aesthetics plays a role as well, but this is really beside the point. The point is that there is such a thing as the "value" of a mathematical theorem, or a research direction, or perhaps even an entire research field. Mathematicians do not like such questions, I suspect because they do not like to think about concepts that cannot be clearly defined. And therefore the issue is mostly shoved under the rug, only to be felt by people who are too philosophical in temperament not to be bothered by it.

My area of research was number theory (rational points on varieties), so I was pretty far removed from empirics indeed. So then there was the possibility to justify my research by an appeal to mental aesthetics. (Let's say this is the "Platonic" viewpoint, since we might say that a piece of mathematics is certainly valuable when there is palpable sense that it has an existence of its own, that it was merely waiting to be discovered.) But the problem with an appeal to mental aesthetics is that a big part of it is in the eye of the beholder: I can find my research programme interesting, but even when this is the case, it might change the next year. So this can't count for much. So perhaps we are back at Von Neumann's idea that all mathematics is eventually justified by its connection to empirics.

Unless, of course, one's passion or sense of direction is so overwhelming, that questioning the usefulness of your endeavour feels completely beside the point. You do it because, to you, it is obviously the right thing to do, and you would do it no matter what, and you would rather run the risk of spending 8,000 years in Purgatory for wasting taxpayers' money than not pursue your research.

For me though, it was never like that. I constantly felt the pressure to defend the usefulness of my work, and this in turn led me to doubt whether I was actually enthusiastic enough about mathematics generally to be involved in academic research. But I still think yes, I was enthusiastic enough about mathematics generally, but I wasn't sufficiently excited about the questions I was working on, or the answers I was getting. So here probably also your talent comes into the equation. If you are hugely talented, then the mere opportunity of exercising your talent must be a joy in itself. As Edgar Allan Poe writes in The Murders in the Rue Morgue:

As the strong man exults in his physical ability, delighting in such exercises as call his muscles into action, so glories the analyst in that moral activity which disentangles.

Or so I have always believed. (Now I am neither strong nor an analyst, so I can only accept Poe's authority for this.) Anyway, this is all I have to say on the subject. Your question stirred me, so I wanted to give you my thoughts. All the best, René

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    $\begingroup$ My issue with your answer is that I don't think set theory has come to this problem. $\endgroup$ – Bill Chen May 1 '19 at 4:19
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    $\begingroup$ Not set theory as a whole, sure. But I was mainly talking about the crises of conscience that can befall the individual researcher. The field of research of any given set-theorist is usually a lot smaller than the entirety of set theory, so that the relevance of set theory as a whole is in itself not enough to justify one's own particular existence as a researcher. Otherwise you could, by the same argument, justify any type of research activity just by appealing to the relevance of mathematics itself. $\endgroup$ – RP_ May 1 '19 at 8:54
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It's not just math, I've seen similar stuff in the software and data science industry. Functional programming, test coverage, Bayesian statistics, and deep learning are just a few areas I've encountered in my short industrial career that bring up strong feelings and flame wars (both across the Internet and within a single company).


However, it might be stronger in mathematics, and logic in particular. I think this is a difficult situation that is driven by two things. (1) The first is finite resources. There are only finitely many available positions for graduate students and faculty. There is also only so much grant money. This scarcity causes conflict since one must constantly fight for a piece of the pie. (2) The second is aesthetics. There are more available problems and research areas out there than one could ever work on. A lone researcher's work will be lost into the archives, and an overly saturated topic (especially an easy one) could lead to lots of trivial research, much of it being small modifications on the known theory. Many mathematicians have a personal aesthetic about what is good mathematics, and IMHO, if used correctly, this is very useful for guiding one's field to the problems and tools which are maximally interesting and useful. But if used wrongly and mixed with lots of ego it will lead to resentment and full-blown public fights.

It is not just set theory, I've seen this in many areas of logic (and I imagine it is also happens in other areas of mathematics). Here is another example:

Before leaving academics I used to work in algorithmic randomness, a field which gave a rigorous definition of a type of randomness. I have encountered a lot of open hostility from those who have considered this a sort of "bastardized probability theory". This wasn't helped by a loud minority (which I very much disagree with) who both think that this actually is both the right canonical definition of "randomness" and that probability theory should be based on this theory. I also have to admit I have my own internal biases against the field as well. I think it is focusing too much on the wrong types of problems and I unsuccessfully tried to move the field in a different direction.

I've also seen big fights or snide remarks related to set theory vs category theory, homotopy type theory vs ZFC, reverse math vs constructive math, and the merits of TTE-style computable analysis and proof theory.


My best advice is to try to stay above the fray, and don't get sucked into the flame wars. But also listen openly to the criticisms on both sides and use this to guide your personal mathematical aesthetic.

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This kind of issues plagues the whole of mathematics and theoretical computer science (TCS), where short-term fads (one can only get grants in "impactful" areas - this is certainly the case in many European countries) and internal politics (why would we hire that person, we'd rather increase our influence by hiring someone very close to our "own" area) dictate funding decisions.

It's basically a fashion industry now, unfortunately.

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Here are a few more data points. Caveat: I think this is an inherently subjective question, and partly also regionalized. I don't presume that my impressions are necessarily accurate or representative. Actually, I'm kind of hoping that someone might speak up if their impressions contradict my own, because I'd love to adjust my perspective!

As a category theorist, I have often felt as though I need to justify the idea that what I do "is real math". I think this is less the case in many places outside the USA.

I think I've observed a similar anxiety among some model theorists.

As somebody who dabbles in algebraic topology, I have gotten the sense that algebraic topologists by and large don't feel this pressure in the same way.

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    $\begingroup$ Oh, I could tell horror stories about a general math audience comments about homotopy theory :). I feel this is something that happens pretty much to everyone... $\endgroup$ – Denis Nardin Apr 28 '19 at 22:44
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As others have mentioned, "why should society give you money to do the research you're doing" is a legitimate question. And, it's legitimate whether the source is internal (other mathematicians) or external (the university), as long as the person doing the asking is genuinely interested in hearing you defend yourself rather than just being snarky.

The trouble, as I see it, is that many of the standard answers do not apply to set theory, at least not directly.

Those who are working on new foundations for mathematics, or trying to unify math and programming in hitherto unimagined ways, can explain that the benefits of their work include better software for doing mathematics, better programming languages, and better tools for automating mundane mathematical tasks. Case in point: I saw a website a few weeks ago where the author had formalized a proof that $3 < \pi < 4$ in Lean; apparently, it was quite a task. And, apparently, to get an extra decimal point required several more days of work. This is totally impractical, and such tools need a lot of work. The trouble is, that's not really what set-theorists do, so this style of answer won't work.

Some have mentioned the connectedness of mathematics. The connectedness is certainly wondrous, but set theory seems to be in a league all of its own. It asks its own questions, develops its own tools, and interacts with the rest of math relatively infrequently. So this style of answer also won't work, at least not directly.

With that in mind, I would answer such questions in terms of the long-run connectedness. I would say something like this:

In the 1940's, number theory did not have many known applications. Yet it was interesting, so people studied it in their spare time. Some of the people studying it were so brilliant, they managed to convince society to fund their research, because sure, it had no known applications, but they were obviously having tremendous success solving apparently intractable problems, and there was a feeling that this was important. So universities supported them and they were consequently able to solve difficult and beautiful (though apparently pointless) problems.

The thing is, math seems pointless until you need a hard result that someone else spent their entire career to solve. This only needs to happen once before the natural contempt for people who are wasting their time on overly-abstract problems is replaced by a certain grudging respect. You begin to understand that all mathematicians stand on the shoulders of giants one way or another, and it's hard to know ahead of time whose shoulder you'll need to shift your foot onto to go where you're going.

The kind of set theory I do has few known applications, but the problems are very deep, very hard, and very beautiful. Some of us, those who have a track record of success against such problems, have managed to convince universities to support them so they can carry on their research. This isn't such a crime; there's billions of people in the world, and only a small handful of them are set theorists. The cost to the world of our existence is pretty small. And yes, most of us aren't that clear about what the long-term implications of our work are, but there's a sense that this stuff is important, and we seek to be the shoulder that someone will eventually stand on to go where they're going.

In the 1940's, number theory was believed to have little real utility, yet only 30 years later it had emerged as a central theme in public key cryptography, and the internet would be nearly useless without it. I don't know if the same thing will happen with set theory, but maybe it will, and nobody knows ahead of time what ultimate role it will play in our civilization. I do know that if you removed the support from the small handful of people who are working on such problems, there's a real risk that progress in mathematics and technology would ultimately be stifled in a very non-abstract way.

I can't tell you the ultimate role set theory will play in our civilization, in the same way that even the great G.H.Hardy could not predict that number theory would end up playing a fundamental role in the world economy only a handful of decades after his apology for pursuing beauty for beauty's sake. But the cost is tiny, and the cost of removing this small amount of support is potentially huge. Set theory seems disconnected from the rest of mathematics now, true, but the long-run connectedness of things is a difficult thing to comprehend, let us alone predict.

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  • $\begingroup$ Do you have a reference for saying that the internet would be nearly useless without public key cryptography? $\endgroup$ – Matt F. Apr 29 '19 at 14:47
  • $\begingroup$ @MattF., no, and perhaps I phrased that too strongly. But w/o public key crypto, you can't communicate securely unless you've pre-swapped secret keys, which is impractical. In practice, this means you can't send a password to a website without have it stolen. So websites involving "logging in" don't work. Online banking wouldn't work for this reason. Also, purchasing things online wouldn't work (because someone could pretend to be you and spend your money). Indeed, the only uses of the internet would be fully public uses, like exchanging already-published scientific papers or streaming video. $\endgroup$ – goblin Apr 29 '19 at 14:52
  • $\begingroup$ @MattF., indeed, even with fully public things, there's a lot of scope for foul play. If I download a document in such a way that there's no crypto involved, anyone in the path between the web server supplying the information and me can modify this document and I'd have no way of knowing. Similarly with downloaded software. W/o crypto, you really can't rely on the validity of the documents you've downloaded or the safety of the software you want to install, and public key crypto is AFAIK needed to make crypto practical - otherwise, you need to pre-swap secret keys, which is impractical. $\endgroup$ – goblin Apr 29 '19 at 15:02
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    $\begingroup$ If the utility of set theory rests on public-key cryptography, it rests on Solovay too! The original RSA paper (people.csail.mit.edu/rivest/Rsapaper.pdf) cites the Solovay-Strassen algorithm for finding large primes. $\endgroup$ – Matt F. Apr 29 '19 at 15:18
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    $\begingroup$ It is also worth keeping in mind that the level of number theory used in RSA is hundreds of years old. The work done in number theory in the 1940s has nothing to do with RSA, but your long-term connectedness essay seems to suggest to the non-expert that it is part of what matters in RSA. The first time 20th-century math mattered in cryptography was the introduction of elliptic curve cryptography in the 1980s, which uses elliptic curves over finite fields (not $\mathbf C$). The development of algebraic geometry over finite fields beyond individual examples goes back to the 1920s and 1930s. $\endgroup$ – KConrad Apr 30 '19 at 8:12
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Back when I was an undergraduate, I took two ‘special study’ semester classes in surreal analysis that were structured as two-on-one meetings once a week in my (excellent) analysis professors office, with a great algebra teacher present and adding much needed insights.

We were working through Alling’s book on the subject, which mentions various different set theories and some small large cardinals, so I grabbed some relevant books from the library before one of our meetings — one of the books was ‘The Higher Infinite’ by Kanamori.

My analysis professor saw me reading it in the waiting room outside his office before the meeting and immediately stopped and asked me if I’d ever read any H. P. Lovecraft — I hadn’t, but I was a fiction fan and asked him what prompted the question. He remarked that Lovecraft incorporated a tome into many of his works called the necronomicon, a book containing secrets about reality not meant for mortal eyes. Any person who cracked this book open was rewarded with insanity and subjected to forces beyond our mortal perception, inevitably resulting in their destruction/bodily consumption etc.

He said (with a laugh) that Kanamori’s book had always reminded him of a real life version of the necronomicon, and to be wary. My algebra professor had joined us at this point and heartily agreed — both of these professors were very bright and successful in their own fields, with my analysis professor having tenure and good clout within the department. That both of them responded this way to ‘real’ set theory left a striking impression on me, that it was viewed in some sense as ‘gibberish containing inscrutable truths’ to other intelligent mathematicians in much the same way that all of mathematics is viewed in this way by the general public, and I think this is related the the stigma that exists towards the subject.

(I don’t mean to cast aspersions on either professor should they read this post, I just thought it was a relevant anecdote :)

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    $\begingroup$ Isn't this rather an illustration of the fact that some people think that set theory, or just a specific book in set theory, is difficult to understand? I think I have had this conversation with brilliant mathematicians about other topics, including homological algebra, category theory and some topics in algebraic geometry, analytic number theory... On the other hand,I have had other conversations where the relevance of studying some problems in set theory was questioned, but I dare say this was mostly out of unfamiliarity with problems in set theory and the beauty of set theoretic venture. $\endgroup$ – nombre May 16 '19 at 8:10
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    $\begingroup$ @nombre I can see getting that impression from the above exchange, but after our meeting that day we talked at some length about his views on logic and set theory. He said that ever since grad school (which was in the 60s for him) he had gotten the impression that logicians and set theorists were striving towards some greater sense of 'truth' than the rest of mathematics, in a way that came across as overreaching and off-putting -- I think that this 'philosophical disagreement' with the purpose of an entire discipline was limited to logic and set theory. $\endgroup$ – Alec Rhea May 17 '19 at 1:22
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Outside of obviously-applicable subjects,

The complete absence of pressure to defend the value of one's work is a sign of at least one of the following.

  1. Great luxury (you can do what you want without answering to anyone)

  2. Total irrelevance (nobody believes a good answer exists, so they don't ask)

  3. Institutional corruption (the community knows it is open to severe legitimate challenge, and would like to keep things quiet)

  4. Intimidation (e.g. students or subordinates afraid to question the professor on why X is being done/taught)

A healthy research program should welcome inquiries and have convincing answers.

Inability to answer "why this?" questions is a sign of problems.

The desire to never be confronted with such questions is often a means of denial and postponing change. Of course, avoiding change from within leads to change being imposed from outside, either bluntly or through slow attrition.

As for set theory, it is clearly an outlier field by objective standards as well as in the more sociological judgements of fashion that were noted in the question. But if the sociological position doesn't improve from decade to decade, is it really correct to call the perception mere "fashion"? The reputation of subject X within mathematics is sometimes much different than the honestly defensible value of X. Still, there is quite a bit of correlation, and these are the persistent opinions of people who are not expert but not ignorant either (they are mathematician after all), and would certainly have been able to read and appreciate a substantial (i.e. engaging realistically with criticisms) account of the value of set theory had such a thing been published in the past 50 years.

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    $\begingroup$ This does not look like most first posts to MO...if it is an experienced user posting under a new name, it may indicate that feelings of intimidation are widespread. $\endgroup$ – Matt F. Apr 30 '19 at 18:52
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    $\begingroup$ @schematic_boi .... There is a broad range of standards of value from the stringent "does it cure cancer by next Wednesday" to the lax "is it of interest to the 50 people worldwide who publish on it". There are many such criteria, some of them totally objective or even numerical, that GL, GT and other mainstream research areas of mathematics would pass and set theory (of the same vintage) would fail, and no such tests that I can think of where the reverse would be true. In that sense set theory it is an outlier. $\endgroup$ – Not a Set Theorist Apr 30 '19 at 20:41
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    $\begingroup$ Regarding "and these are the persistent opinions of people who are not expert but not ignorant either", are we talking about people who studied forcing? inner model theory? pcf theory? any post-1960s set theory? I heard distinguished mathematicians (the kind that have multiple Annals publications) ask questions that would embarrass an undergrad who took a first course in axiomatic set theory. With all due respect, why should we take their judgement seriously? $\endgroup$ – Haim Apr 30 '19 at 21:10
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    $\begingroup$ @Haim, I think that for people who mathematically matured in other fields, and then take some time to read books and articles in set theory, it often (and legitimately) reinforces their feeling that set theory is relatively disconnected from normal mathematics. My own experience was the strong impression that the sort of arguments typically given for relevance of set theory are largely dependent on intellectual intimidation (such as the idea that significant expertise is necessary to confirm or deny the initial intuition of disconnection) or are simply false when examined in detail. $\endgroup$ – Not a Set Theorist Apr 30 '19 at 22:48
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    $\begingroup$ @Haim, the typical situation when one subject has connections to others, is that there are many points of contact at different levels of accessibility. If someone asks the connection of algebraic geometry to other math and the only answers are "we can't even begin to tell you until you read EGA" that would arouse justified suspicion. In fact, there are many down to earth explanations that do not require either EGA or comprehension of the specific proof techniques, because math.AG overlaps heavily with math.nonAG in its nontrivial objects of interest. Very different situation for math.ST. $\endgroup$ – Not a Set Theorist May 1 '19 at 1:26
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Timothy Gowers' essay,

Gowers, William Timothy. "The two cultures of mathematics." Mathematics: Frontiers and Perspectives 65 (2000): 65. PDF download

seems relevantly analogous:

"Loosely speaking, I mean the distinction between mathematicians who regard their central aim as being to solve problems, and those who are more concerned with building and understanding theories."

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    $\begingroup$ I'm confused. Which one is set theory? I feel that it's both. $\endgroup$ – Asaf Karagila Apr 27 '19 at 23:56
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    $\begingroup$ @schematic_boi If there were no set theory, then you would have no way of knowing whether a given mathematical statement was independent of ZFC. You’d spend your time trying to do the impossible, and never reach the point of declaring it to be not real mathematics. $\endgroup$ – Monroe Eskew Apr 28 '19 at 9:15
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    $\begingroup$ I don't think this is relevant to the question at hand - just as other areas of maths, set theory has room for both "cultures". $\endgroup$ – Dima Pasechnik Apr 28 '19 at 10:41
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    $\begingroup$ @schematic_boi: Kind of like defining a language as "something humans use to communicate", and then claim that humans are the only animal which have a language. $\endgroup$ – Asaf Karagila Apr 28 '19 at 13:39
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    $\begingroup$ @schematic_boi: But it's not like that. It's more like saying that some historians study medieval times, and other historians study ancient Roman times, and that medieval historian claim that Roman historians are not historians, because they deal with the prelude to the medieval era, rather than with the medieval era. $\endgroup$ – Asaf Karagila Apr 28 '19 at 17:04
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Personally, when I started studying Calculus of Finite Differences, and wrote some work related to that and Abel Functions, my professor said "I think we should get you working on some open problems, do you like Number Theory?" He didn't really see the point to Super functions and fractional iteration, and what it can tell us about difference operators on holomorphic functions.

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this notion that there is X and then there is rest of mathematics is a very very strange view of mathematics. mathematics is a fluid subject, ideas float around yielding results in many different places. partitioning mathematics into groups and putting names on them certainly has a value and purpose, but neither its value nor its purpose is to classify some of them as "normal" and some as "not normal".

for me personally, it is incredibly difficult to say what set theory is. often i have difficulty saying whether a paper is set theory or not. perhaps it is possible to make a distinction between computational math and more abstract math, like calculus vs abstract algebra, but it doesn't work. the boundaries are blurred. there is an example above where a user says that you cannot claim that the notion of being countable is set theoretic. very well! exactly the point! mathematical concepts are fluid concepts that appear in many forms and many places. in the case of the notion of "being countable", if you are not giving this to set theorists then you are admitting that the very action of splitting math into such boxes is silly, it is much better to think that "being countable" is a mathematical notion rather than set theoretic.

of course, there are concepts and notions that fall into groups. it will be hard to argue that triangles are not geometric. but the point is that the boundaries of these subjects are so confused and meshed that you will ultimately run into problems. geometry has been connected to algebra more times than you can count. is a group a geometric concept or an algebraic concept? we can debate this for the sake of debating mathematical points of views, but not for the sake of defining subjects, like drawing a boundary around geometry and algebra with no intersections, such an endeavor is ultimately divisive, meaningless and above all pointless.

there is no set theory or number theory, there is mathematical thought process, and people respond to that thought process differently when they are subjected to it, some pursue very concretely defined objects such as numbers, some pursue less concretely defined geometric shapes and some pursue the mysteries of infinity. more people pursue number theory, why? i don't know, i suspect it is because it is easier to grow in the subject, but if you are like me then you would leave number theory because number theory never made me feel satisfied with myself, i was never happy with myself while growing up in number theory, it was like i was living somebody else's life, set theoretic ideas and its mysteries did make me feel fulfilled and happy. i believe what we study has a lot to do with our personalities. i do not like the concrete, it bores me, i like what i don't understand, i like that there is independence and vagueness and my job is to clear it. i have noticed this exact characteristics in myself in many aspects of life, in my choice of music, films, books and etc.

there are different personalities. some mathematicians are problem solvers, and they want problems you can state in a few minutes and that they spend a month thinking about it. set theory of course has many of them of various difficulty. some were mentioned earlier by others. some people like theories and stories. there are people in between.

i cannot say that i am doing math so that i can then tell it to others, i believe this is more like a personal quest, at least that is how i got into it. however, this is a profession, and we need to be professionals. while hearing things like "X is disconnected from the rest of math" is stressful and frustrating, there are many situations where i enjoy defending myself and explaining the meaning of what i do. i enjoy giving general audience talks or write introductions to my papers. i think this is part of the profession and is important, given that the society ultimately pays our salary.

i do agree, though, that there is too much out there claiming that set theory is somehow irrelevant. but as i hinted above, most cases of this that i have encountered were expressed by people who didn't really know what they were claiming. already the meaning of the word "relevance" is problematic for me.

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This maybe doesn't apply to set theory per se, but logic is tremendously important in theoretical computer science, which transfers to programming language theory and from there to the real world. Feferman's book "In the Light of Logic" also does a good job showing why it's a wonderful subject.

As for set theory: should it be wiped off the map? I hope everyone would say heck no, even if they had differing views of its importance. So somebody has to do it. Why shouldn't the somebody be you? The physicist Shevek in Ursula LeGuin's "The Dispossessed" got asked something like that, and replied "if there's physics to be done, I claim to the right to do it". I read that in high school and that answer has been good enough for me ever since.

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    $\begingroup$ "In the Light of Logic" also asks "Is Cantor necessary?", and answers that "higher set theory is dispensable in scientifically applicable mathematics". The book is an argument against the importance of set theory. $\endgroup$ – Matt F. Apr 30 '19 at 9:16
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You've defined the internal perspective, but not the external one. Is it mathematicians who specialise in other areas, or academics who aren't mathematicians, or non-academics ... ?

I can see why the latter might say that about almost any academic endeavour that doesn't cure cancer or make things go faster or otherwise make the evening news. For the rest I'd say is just tribalism within and between disciplines. Put any two academics in a room and they'll disagree, unless the subject is economics. Then you only need one.

tl;dr Ignore it. You don't need to.

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Just point at our current high-technological society and say:

None of this would be possible without extensive work on mathematics.

And at that point in time they had no idea just exactly which branch or result would be applicable where. We need to go for all of it, because we never know in advance where the fruit is going to fall.

Done.

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    $\begingroup$ This sort of argument will in no way help (for instance) the chances of an NSF grant application to be funded. It seems self serving and empty of actual content. It also does not seem to be addressed at mathematicians in fields other than your own but rather at "the general public". $\endgroup$ – Andrés E. Caicedo Apr 29 '19 at 21:20
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    $\begingroup$ Ah it was about grants. Yeah, no maybe not. But its true. $\endgroup$ – mathreadler Apr 30 '19 at 5:21

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