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Except from the goal of finding some new truths, proving some theorems, or new ideas. What is the career path for someone pursuing a PhD in pure mathematics?

Continuing in research and trying to land a professor's position somewhere ? Then a PhD student should also worry about landing a post-doc before finishing his PhD ?

Why do I keep hearing that : "the competitiveness in pure math is rather bad these days" and "an academic position in pure math nowadays is a tough choice" ?

Sriotchilism O'Zaic
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NotaChoice
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    Related question, why to attend grad school as becoming a professor is unlikely: https://academia.stackexchange.com/q/103645/78796 – usr1234567 Jun 20 '22 at 07:39
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    You're thinking backwards. Determine what you want to do with your life, then figure out what you need to do to do it. Don't be a degree looking for a job. – Scott Seidman Jun 20 '22 at 15:19
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    Because you're stuck with a calling in a subject you love, even if getting one of the few permanent academic positions in that subject also requires the political skills of running for public office and the sheer luck of winning millions of dollars in the lottery. It's nice while it lasts. – anomaly Jun 21 '22 at 02:42
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    @ScottSeidman Respectfully, must disagree. If you really know what you want, then of course, pursue that. But determining what to do with your life is pretty hard. Sometimes it's better to learn some things you are interested in first, just because you really like it, then figure out what to do with it later. Worked for me, anyway. There are many things you can do with a math degree. – Jair Taylor Jun 21 '22 at 03:14
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    I think the main prerequisite is that you must really, really like math. (and of course, have good enough grades etc. to get in.) – Jair Taylor Jun 21 '22 at 03:33
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    All I can say is my math teachers, instructors, and profs thought I should go in Engineering. Frankly, one can make more money in engineering. If you have talent in engineering, you should consider it. – JosephDoggie Jun 21 '22 at 13:45
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    @JairTaylor. It may be difficult, true, but even so, I suggest PhD enrollment is a pretty cruddy way to spend time if you're not sure what you want to do. There are better options. – Scott Seidman Jun 21 '22 at 14:59

5 Answers5

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For most of us who followed that path, if you have to ask the question then it may not be right for you and you should explore all your options before you jump.

There is a saying that "You don't choose mathematics. Mathematics chooses you."

For myself and many others we never thought of any other options.

Yes, at the moment the academic job market is terrible. Most pure math doctorates will need to deal with it, though there are a few other options, though mostly in applications.

Yes, post docs seem to be a necessary step. Not entirely pleasant but not entirely unpleasant if you meet new people with new ideas.

But a lot of academics in other fields are also "idea driven". It is a great life if you can manage to get in the door. People pay you to think. Not much could be better than that.

And, It is hard to predict what your career path might be several years in the future. Things change. When I started grad study the future looked extremely bright for PhD mathematicians. But the successful moon landing ended all of that and universities discovered they had already hired too many people and the faucet immediately closed. Overnight, essentially.

But the opposite can also happen. Some new challenge can open doors that nobody saw before. In CS, AI and data science has had something of that effect recently.

If you really love something, try to do that and try to avoid substitutes even if you need to compromise for a while as you build a reputation.

Let me add that I'm not implying that some people are "fit" for math and others are not. That isn't true, at least in my experience. It is a matter of desire and training as well as hard work. To repeat a comment made to another answer here (and elsewhere on this site): My teachers wanted to hold me back a year, primarily for poor performance in math at about age 10 (My mom saved me from that one). My first truly positive educational experience in any area was a high school geometry class. I really liked solving those problems and bought problem books to have more to solve. I finally learned my multiplication tables after earning a doctorate in math. There were other struggles along the way, but each defeat caused a renewed effort to succeed. Luckily I had good mentors.

Buffy
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  • I appreciate your feedback, I like pure math but not only pure math, I like physics and CS, then are you saying I can do whatever from that if I do pure math? I do appreciate if we have this superiority reflected on the real world also, then my choice is definitely pure math – NotaChoice Jun 18 '22 at 21:16
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    "You don't choose mathematics. Mathematics chooses you.", very aptly said! – John_dydx Jun 19 '22 at 06:59
  • Following this top line logic is how one could remove anyone with a shred of pragmatism from the profession. –  Jun 19 '22 at 14:49
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    "at the moment the academic job market is terrible" I do not see how it can be any better. If every professor has at least two grad students, and we assume that we wanted each of them to get academic jobs, then the number of professors would have to double! If the purpose of PhDs were to get academic jobs, then it would be a pyramid scheme – Agnishom Chattopadhyay Jun 19 '22 at 15:26
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    @AgnishomChattopadhyay, you need to consider the educational system (along with the larger economic system) as a whole. Sometimes there are huge increases in demand for undergraduate education that universities work to fill. In the US, at least, most undergraduates are not at R1 doctoral granting institutions. Liberal arts colleges need math faculty also. – Buffy Jun 19 '22 at 15:34
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    @NotaChoice "are you saying I can do whatever from [physics, CS] if I do pure math?" - that's not what Buffy is saying at all (Buffy, please correct me if I'm wrong) – Filip Milovanović Jun 19 '22 at 15:59
  • @FilipMilovanović, actually the comment question confuses me. So I haven't responded. – Buffy Jun 19 '22 at 16:06
  • I think NotaChoice misinterpreted your remark that "a lot of academics in other fields are also 'idea driven'" to mean something like "don't worry, if you do pure math, you can just switch careers at your leisure, cause you're the abstract idea expert, you know it all [or something]" – Filip Milovanović Jun 19 '22 at 16:20
  • +100 for your last paragraph. While I'm not pursuing pure math, I am working on getting into a math-based research field at the age of 41, having discovered a love of math within the last year, and for the last 40 years not having considered myself good at math (I took advanced math classes including in grad school and did well, but there were always others who did much better than me at math problems, so I figured I wasn't a "math person". But I've since discovered I love the creative challenge of synthesizing math concepts to solve problems). So this is a very helpful perspective. – bob Jun 20 '22 at 13:17
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    @Agnishom Chattopadhyay because it is a pyramid scheme, we need to be honest on that. – Dani Jun 21 '22 at 14:26
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    @NotaChoice it is not superiority, although you may think so, by speaking with the median math student (and, statistically consequently, the median math practitioner). Pure math is the ability of describing and expressing the reality with a very powerful formal language. Unfortunately for you, there is a lot whole to do if your goal is to manipulate reality:and that's what humans do. If someone is extremely skilled in writing and in grammar of a language (let's say English) it doesn't make them superior in writing tehcnical instructions to repair a car or in writing Nobel acceptance speeches. – EarlGrey Jun 21 '22 at 15:19
  • @AgnishomChattopadhyay I just want to point out a mathematical glitch in your argument. Professors can have more than one grad student, in different years of study. If they do not graduate together, they do not necessarily compete for the same position, do they? That said, of course, the essence of what you're saying is clear. – stoic-santiago Jun 21 '22 at 19:13
  • @FilipMilovanović I asked this expecting the answer to be negative, since I know even when pure mathematicians claim this, they never did anything to reflect it in the real world in some concrete terms, which is kind of annoying/ frustrating! – NotaChoice Jun 24 '22 at 14:53
  • @bob I would like to know more please on how are you getting into a math-based research field ? I am 35 so I guess I am in the same position. – NotaChoice Jun 24 '22 at 14:59
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    @NotaChoice I work at a company (so not academia) that does research, and I'm just starting out in machine learning research. I haven't published yet but hope to within the next 2-5 years (hopefully 2 but I'm trying to be realistic with myself). It's math based in particular because I'm focusing on the theoretical underpinnings of parts of ML. – bob Jun 24 '22 at 16:37
  • I think if you find a company that does research and can get into a research or research adjacent role, that's probably going to be a good way to get a foot in the door as it'll give you access to journals. – bob Jun 24 '22 at 16:38
  • I realize now that you are the OP—in any case though a research organization (industry or government) is an option, and you don’t necessarily need a PhD—I just have a Masters. That said I’m just starting at this so I can’t claim success yet but it seems like a viable path. I’m not sure I’m up for a PhD at my age and life situation, and work life balance outside of academia can be a lot better at least based on what I’ve heard about the latter. – bob Jun 25 '22 at 03:52
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Let me try to provide a somewhat different though not entirely distinct perspective, namely that this is a totally reasonable series of questions to ask oneself and the mere act of considering them seriously isn't some type of heretical disqualification from pursuing a PhD. But you should think very carefully about your motives. The top line takeaway is that the opportunity cost of doing practically any PhD is massive.

Except from the goal of finding some new truths, proving some theorems, or new ideas. What is the career path for someone pursuing a PhD in pure mathematics? Continuing in research and trying to land a professor's position somewhere?

  • The major work one produces when doing a PhD in math is finding some new truths, proving some theorems, and refining some interesting new ideas. Suppose you do not think these will be satisfying for you or you are generally have different priorities than being "intrinsically satisfied" by your work. Then the pay, working conditions/job benefits, and future opportunities provided by the job are roundly beaten by a whole host of alternative vocations which require similar types and levels of training to enter.

  • There are vanishingly few jobs other than being a professor for which a PhD in mathematics is an essential qualification. For those where a PhD is helpful but not strictly necessary, there are usually other preparatory jobs you can take which will also qualify you and compensate more (except for the manner of work) compared to doing a PhD.

Why do I keep hearing that : "the competitiveness in pure math is rather bad these days" and "an academic position in pure math nowadays is a tough choice" ?

I've never been presented with a convincing framework that uses hard data and discusses comparative questions over time like this. By anecdote everything fluctuates wildly by sub-field and depends heavily on which two points in time you're comparing. If you are talking to someone more than 20 years older than you about the ease of getting jobs in their sub-field, it is almost a default that professorships are harder to get in it than when they got theirs because they obtained a job in the past and are fulfilling that potential demand in the present.

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Is you question WHY or SHOULD?

from the content of your question, it sounds like "should", but the title is too obvious. If you're not really asking, I will provide a lemma at the end.

Anyhow, welcome to the club of:

  • "I had bad grades at school... but look at me now!",
  • "You don't choose mathematics. Mathematics chooses you!",
  • "It's a fact that physicists and mathematicians have on average a higher IQ than the average STEM or humanities academic.".
  • ...

Actually, since STEM includes Mathematics, it reminded me of the joke that "everybody thinks he's better than average." Oh wait... it's not even a joke!

But here are some good parts too: Every mathematician here decries the statement: "Mathematics chooses you". Even the ones who wrote it!! :) But some people do believe it: otherwise it didn't exist! Assume you're gullible enough to believe this stereotype: "Mathematics chooses you". Then, following the logic of it, you will not pursue Math in the first place. What does that tell you about the "average IQ" of Math vs NOT Math set of students? Therefore: it's a self reinforcing idea! QED.

Corollary, the proof I leave for the reader as it's too trivial: Lemma: If you're a mathematician then you think that if someone doesn't choose to pursue a PhD in math then he/she has a low IQ.

Jokes aside, if you have no better alternatives, doing a degree in Applied Math or just Math is worth it! (Example: being from Moldova and no money). Later, you can easily switch from Math to any other field: Banking, Insurance, Telecom, Bio-Chemistry, Software Engineering etc.. Mobility: It provided many people I know a legal ticket to US/EU/other and an exit from abject poverty to a rather decent life.

Glorfindel
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moldovean
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  • well, you phrased the point of my question to question the need/motivation of a pure math PhD if you won't and can't become a math professor in a much funnier perspective, so +1 :-) – user48953094 Jun 20 '22 at 15:13
  • concerning the IQ. I don't google studies here for others I remember to have read, the IQ is higher on average, one can only question if IQ tests test more for mathematical and abstrac thinking capabiliites than e.g. language or emotional or social cognitive abilities which play in academic career also no minor role and many prodigies probably fail of a lack of such (see savant-syndrome, or asperger or autism-syndrome which is also more pronounced among phys/math) – user48953094 Jun 20 '22 at 15:17
  • The last paragraph provides a different perspective. – heretoinfinity Jul 05 '22 at 14:37
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While you could be born to have a mathematical mind and work as a mathematician as Buffy writes, or like someone with synesthesia has cognitive advantages in arts/music, remember a PhD in math from starting your studies to graduating with a PhD does often take 10 years alone. As a postdoc there will be maybe another 10 productive years, but at some point your job will become routine and even hinder you from inventive progress climbing up the ladder of responsibility. (It's really interesting from that perspective to read the Perelman case, who apparently had to leave Harvard to be able to focus on mathematical research or Einstein and when and where he made his biggest discoveries).

My strong advise to all my students is to try out as much and different opportunities as possible during their study/PhD career. If your primal goal is to get a PhD in mathematics and a follow-up academic position, then you wouldn't have asked this question here.

One would think in studies like pure math or astrophysics, why don't end mainly Perelman, Einstein, Wolfram like prodigies on professor positions, and nowadays maybe even less than a century ago? Partly and funnily because a PhD in such scientific branches is pretty much needless if you have theoretically groundbreaking ideas and can realize them (without a need of big funding sources). PhD and professort titles are side-effects in the careers of such researchers, they rather think in scientific projects before earning any degree while on academia.se one can read hundreds of questions on the ideal PhD and how to get it. (For me the best sign, academica offers to many PhD positions, but that is a side note)

If you are already reasoning before starting this path how much financial or job security it guarantees you, your path is already doomed. I had a similar decision during my physics studies to make, either pursuing astrophysics for PhD/professorship as the latter is the only way to work in astrophysics apart from becoming a ESA/NASA engineer or joining a branch of physics that gives more financial security, life planning but still offers many options to do fundamental and intellectually challenging research as a physicist.

And there are many and it's actually the majority of physicists that started physics for particle or astrophysics, graduated with a PhD in such, but ended doing in job life completely different things due to becoming bored by the necessary strong focusing after 10 years of studying or not hitting a job in academia.

user48953094
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    "born to have a mathematical mind" reinforces the harmful negative stereotype of mathematics as something one either has or doesn't have; this fixed mindset pervades our culture and causes people to give up on math when it gets nontrivially hard. I prefer language that emphasizes a more accurate and helpful growth mindset. – Greg Martin Jun 19 '22 at 04:55
  • @GregMartin well, it's a fact that physicists and mathematicians have on average a higher IQ than the average STEM or humanities academian. In many other scientific branches ambition and hard work are enough for an academic career. I doubt this in phys/math. Waiting till "math chooses you" is not an option for every student and my answer describes that there are maybe much more important decision factors. Funnily, my impression is that Buffy's answer much more pronounces the mathematical mind, math surely doesn't choose candidates that lack the ability to success in it well vs. other branches – user48953094 Jun 19 '22 at 12:11
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    @GregMartin, indeed. I wasn't "born to it" in any sense. My teachers wanted to hold me back a year, primarily for poor performance in math at about age 10. But my first truly positive educational experience was a high school geometry class. I really liked solving those problems and bought problem books to have more to solve. I finally learned my multiplication tables after earning a doctorate in math. There were other struggles along the way, but each defeat caused a renewed effort to succeed. Luckily I had good mentors. – Buffy Jun 19 '22 at 14:08
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    @user48953094 "In many other scientific branches ambition and hard work are enough for an academic career. I doubt this in phys/math." Is there evidence for the claim that ambition and hard work are enough for an academic career in scientific branches that are not phys/math? Is there evidence for the claim that in phys/math ambition and hard work are not enough for an academic career? – user505117 Jun 19 '22 at 14:19
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    "it's a fact that physicists and mathematicians have on average a higher IQ than the average STEM or humanities academian." Please remember that extraordinary claims require extraordinary evidence. So far you have not provided any evidence for your claim at all. – Jochen Glueck Jun 19 '22 at 15:15
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    @GregMartin How is the average IQ-test score of physicists/mathematicians relevant here? There are many IQ-test prodigies who failed in academia and vice versa. Even if your claim is true, and I tend to agree it is, it doesn't show that having innate cognitive abilities is a necessary or sufficient condition for academic success. Well, it simply isn't. – Dilworth Jun 19 '22 at 15:36
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Ofcourse pursuing PhD in pure mathematics and applying all the innovative, creative and analytical power solving unsolved problems is very difficult task. One should not choose mathematics just for profesionality but one should choose mathematics with both affection towards it plus targetting a job. But then looking at the competitiveness market in academia might demotivate one morally. However competition is everywhere and most of all succesful researchers (PhD+Post docs) get academia job in time. The world need mathematicians more and more in every academic institutions because the language of pure mathematics as well as applied mathematics enlarge the idea of advanced world and natural sciencies are devloping continuously based on mathematical language developped by pure mathematicians. I am myself final year PhD in number theory and I am optimistic because I can see most of the seniors got academia job. So don't lose hope if you really love pure mathematics with heart not in terms of professionalism only.

learner
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    " most of all succesful researchers (PhD+Post docs) get academia job in time." This is survivorship bias. Without an academia job, you cannot be a succesful researcher. Good luck! – EarlGrey Jun 21 '22 at 15:27
  • @EarlGrey, here succesful means they completes PhD and post-docs succesfully. I never meant all have to be outstanding performer in research. – learner Jun 21 '22 at 16:43
  • What do you mean with "completing a Post-doc succesfully"? – EarlGrey Jun 22 '22 at 06:37
  • @EarlGrey, I don't know the definition of "successful" but one can feel it only. – learner Jun 22 '22 at 07:12
  • all the succesful researchers get academia job in time;
  • the succesful researcher definition is not clear and one can feel it only.

    • To me it follows logically that you define succesful Phds and Postdoc the ones that get academia job in time. This is either delusional (you aspire you will be one of them) or tautological. Well, at least your degree is in number theory, not in formal logic :D (I am just kidding, but beware of the gap between your feelings and reality)

    – EarlGrey Jun 22 '22 at 08:21
  • @EarlGrey, it is not delusional, it is my experience. I mentioned in the post that most of the seniors with successful PhD and post-doc got job in academia, in fact permanent position in govt institutes. I am not talking about industrial placement here, which would give extra scope to math phd holders. That motivated me to answer this question. One should not be so skeptical I think. – learner Jun 22 '22 at 08:40
  • then define "successful PhD and post-doc". If success is "they got permanent position" or "they got a position in the industry", that is not helpful! – EarlGrey Jun 22 '22 at 08:42
  • @EarlGrey, see my 2nd comment. I am interested in real solution of the OP rather than digressing the topic. I believe the word "successful" can be felt successfully. I am not here to claim my mastery here but I wrote whatever I have experienced so far in my surroundings. – learner Jun 22 '22 at 09:00