What can I tell a student I am mentoring who claims: "I want to do pure mathematics because it is superior to any other subject in the world"?

Question

This question is partly inspired by this thread.

I am mentoring a student who claims: "I want to do pure mathematics because it is superior to any other subject in the world like applied math, statistics, science, and engineering". On one hand, I like it when he finds the thing he really loves. But on the other hand, I really don't want him to have such a belief that what he wants to do is the most important thing in academia while other subjects are inferior. Such arrogance will hurt him in the long run as a scholar. But I cannot give him a convincing reason since I also do research in pure math myself and have to confess that it was not until I found it hard to find academic jobs in pure math did I realize that pure math may not be so "superb" compared to other subjects. 

However, I also do not want to tell him that he will find it hard to survive in academia if he does pure math. But if you believe I can say this to him you are welcome to explain in your comments and answers. I am happy to know!

It is not uncommon in the math community that people have the belief that what they do is sacred and pure math is the supreme of all subjects in the world. Even within pure math, I have seen the very hilarious hierarchy (this student is also aware of this without me telling him):

number theory, representation theory and algebraic geometry > topology and geometry > analysis and PDE

(although I cannot deny that Fields Medal and Annals of Math do favor people in number theory and algebraic geometry)

So, what can I tell him?

Answer 1

I see no real reason to tell him anything. If he’s being an obnoxious or abrasive jerk, you could advise him to knock that off. And of course, you can share your views and experiences if the right moment arises. But it’s probably not the case that “something must be done”: immature views and partially-informed opinions have a way of resolving themselves with age and experience.

Answer 2

Oh man, this question triggered me so much that I actually created an account.

So I'm a "pure" mathematics dropout; I finished a PhD in 2012 and did a couple of postdocs. It was fantastic. I had a great working relationship with my advisor. The community was great. Sure, there were politics between journal editors, referees, and authors, but not to the extent that I could see in physics or computer science. I genuinely loved it, but life shoved, and I opted for a career transition to better accommodate my wife's career.

So as someone who once thought like your mentee, here are some thoughts:

1. "Pure" math ain't a thing. Math all over the spectrum finds applications somewhere. Elliptic curves are applied to cryptography. Banach spaces show up in compressive sensing. Even category theory shows up in programming language design. "Applied math" has heretofore been restricted to PDEs and I think is a ploy by the PDE community to ensure they get a pile of NSF funding without competition, but there's plenty of math that has real world impact outside of PDEs.

2. The math research situation wasn't great 10 years ago, and has only gotten worse. I used to argue with my mom that my math professor aspirations were at least a safe career choice if I could get tenure. Then WVU cut their PhD program and laid off tenured faculty. I always said that a university ceases to be a university when they don't have a math department; WVU is officially a professional basketball team that oddly has some math teachers attached to it. They won't be the last to make such a move. Imagine being 50 and having been a research mathematician your whole life and hitting the open job market. I had a hard time transitioning at age 30.

3. The equation changes if you decide to be a normal person and get married. It's emotionally draining to postdoc thousands of miles away from your spouse; it's also unfair to expect them to just follow you around the world while you build a career. If you're not Fields Medalist material, you might have between 0 and 1 "good" tenure-track opportunities in a given year during your postdoc. So be ready to have the "Are you cool moving to Columbia, Missouri?" conversation with your spouse. And be prepared for them to say "Hard pass." (Nothing against UM... I've never been. But I was in a similar situation with a comparable place.)

4. Even when you love the subject, you'll end up joylessly writing a paper that you know doesn't need to be written and citing your buddy's paper that also didn't need to be written because it helps your careers. It'll go to some minor journal. It's not dishonest. It's just life in modern academia. Where's the "purity" in that? Sure, the goal is to write the paper that revolutionizes X and wins you a Fields Medal. Even if you don't do that, you'll write a couple of things that you're actually proud of. But in between, if you want to have a career, you'll write something that sucks a piece of your soul out. A little academic Horcrux.

If any of this depresses you, don't fret. I have nothing but great memories, great friends, and pride left from my time as an academic. I landed in a good-paying tech career where I work remotely and can spend lots of time with my family. I have a few regrets, but nothing I would change if I could do it again. I would never tell someone that they shouldn't try their hand at academic math, but just like to point out the difficulties, obvious and not with trying to live this life.

Answer 3

I would gently push back if he seeks validation of these beliefs, or brings them up on his own volition in casual conversation. If that happens, I would point out that yes, pure mathematics has a certain kind of beauty to it that is not found in other domains to the same extent, namely that which comes from results being in principle rigorously provable starting from whatever foundational system of axioms one subscribes to. I would then point out, however, that there are some caveats about this.

First, hardly anybody really does this. Pure mathematicians produce things (natural language proofs using naive set theory) that could be turned into really rigorous proofs (doing so is called formal verification), but ask most mathematicians to actually do this and they will run, because it would involve a lot of mechanical work that is more like programming than the part of mathematics they love. Instead, they present their argument at the level of detail and rigour that will satisfy themselves and their peers - much like other scientists do as well.

Secondly, being rigorous means being able to really prove something (which, yes, is beautiful when it works), but it also means you can solve less. So the ability of pure mathematics to present clear, complete, absolute arguments is part of a trade-off, where we trade the ability to do this for ignoring a lot of interesting problems that people want answered. Applied mathematics and other sciences get around this problem by introducing modelling assumptions and having experimental grounding, both of which comes with a different type of beauty in thinking that pure mathematics lacks.

Third, if we order fields by purity, then normal pure mathematics does not come out on top. Depending on your viewpoint, I would say it is either formalized mathematics instead, or mathematics based on intuitionistic logic, or maybe doing both? In any case, normal mathematicians would view these I think as interesting subjects of study, but they would think anyone weird who would claim that actual, practical everyday pure mathematics should better be done this way. He can expect a similar reaction from most people outside and probably also inside pure mathematics if he claims that pure mathematics is in some way superior to other fields.

So in summary, if he brings it up, I'd validate his passion for pure mathematics while nudging him towards a more expansive and generous view, emphasizing the complementary beauty and importance of other fields, the perils of status hierarchies, and the shared culture of all scholarship.

Answer 4

I would first discuss with him what he means by "superior". I see at least two possibilities:

First is the one you seem to be hearing: He thinks pure math is really the best subject, in some intrinsic and objective sense, and other subjects are worse, implying that the people who study those subjects are also worse - maybe just dumber, or maybe worse as people.

Second, the notion that it is superior for him. That it brings him more joy and satisfaction than anything else.

The first is obviously pernicious and will lead to difficulty with the 99+% of people who don't do pure math. The second is fine.

In the first case (well, really both), I recommend a wonderful book: The Cruise of the Aardvark by Ogden Nash, about an aardvark who goes on a cruise to escape the endless rain. At the beginning of the book he wishes everyone was an aardvark because being an aardvark is wonderful. But, over the course of the 40 days, he learns that, while it is, indeed, wonderful to be an aardvark, the other animals are also necessary. Note that this book has nothing to do with math. Also, despite the obvious allusions, it's not remotely religious. But it's very funny.

I also recommend books by Carlo Rovelli. Rovelli is a physicist and a renowned one. But he writes about all sorts of things.

Answer 5

There are, unfortunately, plenty of people who subscribe to the xkcd ranking of fields in terms of purity: https://xkcd.com/435/

With my students, at least, I can guide them towards the POV that much of pure math, historically, was enriched by applied math problems, and there is a lot of benefit to paying attention to how one's pure math might be used someday. Additionally, it's very wise for a pure mathematician to keep some time free to interface with applied math problems, because these are often interesting and can reach a large audience.

Regarding it being hard to survive in academia in pure math: Don't despair! Plenty of people have great careers in pure math, whether or not informed by potential applications. I think, much more than the pure/applied choice, it's essential to do research in something one is passionate about.

Answer 6

Tell them - tongue in cheek - "available funding is inversely related to purity of a field". Of course, that's not strictly true, but then neither is the statement "pure mathematics is superior to any subject in the world".

Kernel machines are a very nice piece of mathematics across functional analysis and Linear Algebra, and are quite useful. Representation theory is supremely elegant and feeds large parts of physics. Number theory has become central in crypto methods. Topology is now used in Machine Learning. Category theory is relevant to data bases and programming language type theory.

All these fields have ("The horror, the horror") now real applications.

It is the privilege of young students to be cocky and overconfident. That's how they are ready to push towards the unknown. With time, they either will mellow, or they will succeed, or both.

My suggestion here is to take it with tolerance and benevolent humour and just hint to them how much more there is, so, when they/should they become more open, they know what to pay attention to.

Bottom line: If someone burns for a topic, that's a flame you do not wish to kill. Just help them open their eyes a bit further.

Answer 7

I believe that OP has a professional duty to advise caution and a more generous viewpoint, as someone above already said.

In relation to the student's ultimate success or failure, this will be - largely - decided by their ability and effort plus that little bit of luck.

Not having sat in on the original exchange, I can't say the student's remark was arrogance or the artificial reduction of horizons to make decisions easier, which is more common in students.

Either way you have, I think, a clear duty to remind such a student that education is supposed to prepare people for life, not just a career.

It's surely time for a relationship in this guy's life. Someone to show him another outlook, another process of relations and the joy of simple service to another. That way pure math may well remain his preferred occupation. But only in service to his primary human obligations.

Answer 8

I would simply answer that humility and respect are qualities a researcher needs.

You can be humble and respectful and still love your field above everything else - it's not exclusive.

Answer 9

"Define superior?"

The student might notice what they are doing if you simply ask that question. What is the ordering of this?

1. Order by difficulty?

   Not having to think about whether the model fits reality reduces difficulty, you can just make the assumptions which are most convenient for the theory (look most general, regardless if you can practically check them). This frees the pure mathematician to do more difficult theoretical theory because they do not have to be bothered by whether it actually explains anything. But is it really easier?

2. Order by usefulness? Order by job opportunities?

   No application, uff...

3. Order by truth/rigor?

   At this point you could go into the direction of formal proof verification, proof assistants such as lean. But that will most likely feel like programming and fighting the proof checker.

4. Order by personal preference?

   "It is good you like your subject, but maybe you should highlight this personal prefence more to be viewed as less arrogant."

5. Order by ...?

"superior" is very ill defined. It is really just saying that there is some ordering relation in which something is greater than another thing but nothing about this ordering relation. Is the ordering even complete?

Answer 10

"I want to do pure mathematics because it is superior…”

Setting aside the deplorable arrogance and snobbery for a minute, here are a few thoughts.

Depends what he means by “do”. If he just means he wants to take more pure math courses, then fine.

If he wants to do pure math in his free time, as a hobby, also fine. It’s an exquisite art form (though with a tiny audience), and it can be very satisfying, for some people. That’s why I do pure math.

If he wants to do pure math to earn a living, he should understand that the road ahead may be rockier than he imagines. It might be hard to find a job in academia. If he’s lucky, and finds one, he might have to teach calculus courses, and spend lots of time applying for grants. Personally, I think it would be irresponsible to allow the guy to remain unaware of these difficulties. They’re not insurmountable, but statistically, they’re quite likely to arise.

I’d advise the guy to think a bit more broadly — consider how pure mathematics relates to his life goals. Does he want to be respected, fall in love, raise a family, become rich, win a Fields Medal? Doing pure mathematics as a job doesn’t leave much time for falling in love, is a somewhat shaky way to support a family, and it’s a terrible way to become rich. He might get some respect, but only from a tiny community of people who understand his work.

If he says that his life goal is to do mathematics, then in my opinion he’s seriously unbalanced. The proper balance might return sooner or later — if he falls in love, starts making mortgage payments, or his children need braces. A good mentor can (and should) help to accelerate the enlightenment.

Answer 11

Today I had lunch with a friend who told me that I should simply tell this student frankly how difficult the life of a pure mathematician is --- doing research, getting a PhD, publishing papers, applying for jobs/grants etc. --- and how many brilliant people I knew from pure math had to eventually quit academia and go to industry.

This is a test for him: If he is unmoved by all of such stories, then he may be a true mathematician --- no matter how arrogant he looks, and arrogance is not rare in the pure math community. Otherwise, pure math is not suitable for him, and he will find his own path of life.

I think what he said is reasonable, but before trying (or not trying) it on this student, I would be grateful to listen to your opinions, since this could change the path of a young person (even if you want to downvote this. I don't care about points).

Answer 12

In a sense, what the student is saying is true in that pure math is the best for them. But it's a little weird to say that it's just "the best". I work in econometrics. I don't think it's the best or most important research on the planet. Rather, it's perfect for me, and makes me happy (and money) doing it. So I just think some perspective may help, that not everyone feels as they do about pure math, and that's okay.

Answer 13

Galileo Galilei said that "Mathematics is the language with which God has written the universe." This quote reflects the idea that mathematics provides a universal framework for understanding and describing the patterns and relationships found in the natural world.

"Mathematical relations/equations" are the biggest gift of pure mathematics. Mathematics serves as a precise and universal language that allows scientists and researchers to express complex ideas, formulate theories in terms of mathematical relations/equations, and make predictions about the physical world. Just as language is used to communicate thoughts and ideas among humans, mathematics is used to communicate the fundamental laws of nature that underlie the workings of the universe.

Thus what can be more beautiful than pure mathematics! This sentence is just symbolic, indeed the universe is beautiful.

Answer 14

Just let him do pure mathematics. In some senses it is superior and in some it is not. Let him discover it on his own.

Answer 15

I'm sure it wouldn't hurt for your mentee to begin "soft searching" now to familiarize himself with the volume - and quality - of current (relevant) job postings in the field. I, for example, have been subscribed to HigherEdJobs since 2015, and I have a strong idea of what is - and equally importantly, what is not - available in my field/discipline. The reality check that ensues can/will be helpful as he continues to consider what his next steps should be going forward.

Answer 16

However, I also do not want to tell him that he will find it hard to survive in academia if he does pure math.

To me, as an advisor, I think telling the truth about this is one of the responsibility of an advisor. After presenting the truth, it is up to the student to decide on his/her own.

As also mentioned by many other comments, no need to tell him anything to change his/her mind. It is his/her belief. He/She may have some other beliefs on other stuff besides maths that do not fit yours.

Unless his/her belief does not violate any rule within your institution, no need to intervene.

Answer 17

There are two issues here: potential arrogance and job prospects.

As to the first, when he says that pure math is "superior", how does he mean this? If he means simply, "I think this is the most interesting subject to study and that's where I want to devote my life", then hurray! It's great that a student is passionate about a subject.

If he means, "People who do anything else are lowly scum and I am a better person than them", well that's a problem.

If the latter, I'd try to gently tell him that, yes, pure math is a very worthwhile field of study, but other people have other preferences and your value as a person does not depend on your major. At some point this goes beyond your role as a mentor. Is it your job to guide the student's moral and social development, or just his academic career? I mean, if a student disdained anyone who didn't agree with him about some political or social issue, like gun control or abortion or trade with China or whatever, would that be a proper matter for you to get involved in? This is a question of an academic field so maybe it gets borderline.

Regarding job prospects, I would straight up tell him, "This may indeed be a very interesting and worthwhile field ... But have you considered what sort of job you will get after graduation?" Some majors lead to many job opportunities, like business administration and IT. Others not so much, like history and gender studies and ... pure math. Some majors, the only job prospects are to become a teacher or professor to teach them to others. Pure math is probably not the most limiting, I'm not a job recruiter but I'd guess it could lead to jobs in science and engineering as well as teaching. But yeah, not the same prospects as BA or IT.

Answer 18

Nice and Interesting!

In my understanding, Firstly, the responsibility of a mentor is just to guide the student by providing the required information and knowledge. Usually, sometimes, by answering the questions, sometimes by giving a task or asking answers and sometimes being silent.

Secondly, it is really nice when student decide what to do. Half work is already done. Then the mentor just need to show the road map. In this case being kind on him (understanding the level of knowledge of the student), a mentor just need to show the required path. For example, a possible answer could be, "You are right, mathematic is a major subject but we do not have any proof for the superiority of it above all subjects in the world. So, it is arguable, but if you want to choose mathematic than that is also a good choice." I hope this will be helpful.

Answer 19

I studied pure math as well.

I would pose the question, what good is studying something if the results are never used? And then provide an analogous subject. To the mentee propose, Suppose a you study cooking. You come up with new techniques, ideas for recipes, food and drinks pairings, delve into the chemistry of scents and flavors. And you never make a single dish. You surround yourself with like minded individuals who explore the theoretical. They expand on your theories, and vice versa. But no dish ever manifests your individual work nor the collective work. So while your fulfilled, is their any value in the work you do beyond that? Should there be? Is it possible le that some element of cooking knowledge is being missed because you didn't apply or experiment, or consult with those who do?

Like this analogous cooking g example pure math and other disciplines are supportive. Inspiration derived by application of math to other disciplines have spurred other examine object leading to new fields of Mathematics, line of Inquiry in existing fields, new techniques, ect.

I would put forth that nothing can be superior if it has been, is currently, or will be dependent on the things it claims superiority over, directly,or transitive.

Answer 20

Sounds like a good time to have the academic equivalent of "the talk" with him.

• The ultimate goal of studying is to put food on the table, both for oneself and for one's family
• To put food on the table, one needs to earn money
• To earn money, one (probably) needs to find a job
• To find a job, one needs to acquire skills that employers want
• Since jobs in pure mathematics are so hard to come by (and because it involves lots of travel that inconveniences one's family), it's a good idea to prepare a fallback plan

By all means study pure math and take it as far as one is able (or willing). But be ready for it not working out and prepare a backup plan, to avoid situations like this.

It's likely the student won't take you seriously (c.f. point #2 of this), but you should still at least tell him, since your comments here could easily have a major impact on their life.

By the way I don't consider this a "pure math vs. every other subject" issue, since once one starts preparing a backup plan one inevitably starts to appreciate other subjects.

Answer 21

Is he good enough? If he is potentially a Wiles, or a Ramanujan, or a Conway, or a Perelman, then let him go for it. Otherwise he may need to learn a little humility. Caution him gently about funding, and let him find out for himself. I expect that you live in a country where people aren't killed for making simple mistakes (very often), so he probably won't be harmed too much for his attitude.

Answer 22

What can I tell a student I am mentoring who claims: "I want to do pure mathematics because it is superior to any other subject in the world"?

I'd ask them to back this claim with a proof: if they take the challenge seriously, they should stumble upon Hume's law at some point, and you can point to it in any case after they struggle with the problem a bit