Bsrgvty

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".

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.

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.

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.

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.

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.

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

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.

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.

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.

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 :)

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."

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.

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.

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.

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.