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Assumptions:

  • The typical pure math PhD-bound undergraduate goes into a pure math PhD because they are [insert reasonable prerequisites for pure math PhD, (not the focus of this question)] and they want to end up in academia.
  • (With some exceptions) A pure math PhD is for people who want to enter academia.

Since most pure math PhD students won't end up in academia, we must wonder, given the above assumptions, the following question.

Question

  • Aside from the easy answer (see below), why should the typical pure math PhD-bound undergraduate enter a pure math PhD program if they will (1) probably not achieve their goal of entering academia and (2) probably have a few hard years after their PhD in chaotically bouncing around post-docs (never knowing when the end is in sight) and eventually readjusting their career goals to go into industry?

(1), (2) are based on observations one can easily make by browsing this site and talking to professors. This question may be abstracted to

  • Why should one go on a 5 year journey to achieve something they will probably not achieve, and upon failure, will subsequently face irreversible consequences (i.e., time lost, financial insecurity, etc.)?

The easy answer: The typical answer is: "Follow your heart, and do it if you can't imagine yourself doing something else." But this is an emotional answer which is automatically true for every undergraduate who is very serious about a pure math PhD. And just because you think something will make you happy doesn't mean you should always do it. Sure, a PhD is fun, but it seems like a PhD gets really not fun very quickly upon graduation. Overall, it's very easy for naive undergraduates to accept this "follow your heart" answer because us pure math PhD-bound undergraduates tend to think "Others have trouble, but I'm special." That's an extremely naive and unrealistic way to address the fact that most graduates from pure math PhD programs experience employment difficulties.

My background: Mathematics undergraduate applying to PhD programs. I like math, I think I'm mentally and academically prepared to enroll in a pure math PhD program, and I have aimed to enter academia for a while. I know that with the right institution I would love a pure math PhD, but I want realistic career goals and not some naive expectation of "grad school -> PhD -> post-doc -> tenure" in going into a pure math PhD.

Note: My background is not really a background. It is included to reinforce the fact that I'm your average pure math PhD-bound undergraduate, who would probably do just fine in PhD program, and therefore this question might apply to similar undergraduates who are also not trying to be so naive about their career goals.

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    Hmmm. Why do you say that most math PhDs won't wind up in academia? Is that a given?
    – Buffy
    Commented Nov 19, 2020 at 21:59
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    Where I am, doing a phd is "not yet fully commiting to academia" (in a positive way). You can spend three years in a phd and have fun and make contributions to academia. You don't neccessarily have to move if you don't want to. For a postdoc, you would have to move (and thus give more things up/make academia to the focus of your life). Why should the fact that you won't gind a place in academia imply that you shouldn't do the fun of a phd (if phd is fun for you)?
    – user111388
    Commented Nov 19, 2020 at 22:07
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    For me it was just like that: I loved my subject, but also other things (like friends and family). I didn't love academia enough to give them up (like a would have at an academic career). But I could do the phd without giving them up. Others like to go abroad for a phd, experience a different culture/language they like and go back home after their adventure. For staying in academia, they probably would constantly have to move to places they don't as much like.
    – user111388
    Commented Nov 19, 2020 at 22:11
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    There is an old saying among mathematicians. "You don't choose mathematics. Mathematics chooses you." If you don't have that level of commitment (obsession), don't bother. And if you do, then nothing will stand in your way. The same can probably be said for a career in academia, though maybe not with the same intensity.
    – Buffy
    Commented Nov 19, 2020 at 22:49
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    thinking. Otherwise step into the job market right after school. There are only few studies like medicine or law which are very tailored for specific jobs. But all the other jobs are somehow in most of the ways reachable without a study. If a company promotes you to a certain position based on your education is another question. But from pov only very few positions require a study at all, also a lot of my engineer colleagues could be replaced by motivated technicians. On the other hand, e.g. physicists conducting state of the art research need a well educated thinking. I don't believe you
    – Ben
    Commented Nov 20, 2020 at 9:27

5 Answers 5

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why should I enter a pure math PhD program with the goal of entering academia if I will (1) probably not achieve this goal, (2) probably have a few hard years after my math PhD, and (3) probably end up in industry in which case I could have taken a less stressful route and even prepared myself better with a PhD in another field?

With the mindset that you are showing by the very loaded way your question is phrased - including several rather misguided beliefs I will address below - consider the possibility that the answer is “you shouldn’t”.

What are these beliefs I’m referring to? They are:

  1. Something is not worth trying if the probability of success is lower than 50%

  2. The definition of “success” in pursuing a pure math PhD is to have a career in academia

  3. Any argument for pursuing a pure math PhD is not helpful or worth considering if it is “vague and subjective”.

In fact, these beliefs are not true in any universal sense. The decision to pursue a PhD in pure math is a personal and completely subjective one. It’s right for some people, and not right for others, and for each person it’s either right or not right for different reasons: e.g., for some people who do define success as an academic career it’s still the right decision because they are talented enough to defy the average not-so-great statistical odds you are referring to; others simply define success in completely different ways.

Likewise, the people for whom the decision does not seem like the right one also differ in the reasons why they reach that conclusion. Some are passionate about math but don’t think the effort, stress and opportunity cost are worth the risk of not achieving the dream of an academic career. Others may be so mathematically talented that they can know with a fairly high degree of certainty that they can have an academic career, and yet nonetheless find other opportunities more alluring.

Your question suggests that you want someone to come here and persuade you to do a pure math PhD by giving you some objective, scientific reason to do it. That’s not how it works. You’ll have to do your own thinking and make your own decision based on your own ideas of what matters most in life and what kind of life and career you want. I said at the beginning of the answer that it sounds like a math PhD may not be right for you, but of course I don’t know you and may well be wrong about that impression, so don’t let it prevent you from making the choice that’s right for you. Good luck!

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  • Thanks for the response! I just wanted to point out--ending up in academia is not just less than a 50% chance, it's really much lower than that, right? I guess it doesn't really change your argument. Also, the phrasing of the question is not really for me personally, but for undergraduates who may have shed the naive "grad school -> PhD -> post-doc -> tenure" idea. I assume they may wonder this question. Otherwise, they're entering grad school thinking "sure, others have trouble with employment, but I'm different! So I'll be fine," which I think is naive (it just can't be true for everyone).
    – trujello
    Commented Nov 20, 2020 at 1:57
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    @trujello you used the word “probably” so I simplified that to something on the order of slightly better than 50% chance. If you had said “almost certainly” that would change my argument (but I don’t believe it’s the case that people who get a math PhD “almost certainly” fail to have an academic career if they want to have one). But my argument isn’t specific to the exact number 50% of course, if it were, say, 70% then what I said still applies.
    – Dan Romik
    Commented Nov 20, 2020 at 3:07
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    @trujello you phrased the question in the first person and gave background information about yourself so I assumed you were asking about yourself specifically. Anyway, others who read this discussion can interpret it from their own perspective.
    – Dan Romik
    Commented Nov 20, 2020 at 3:10
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    @trujello: I think the statistics still say that well more than half of pure math PhDs end up in academic jobs. Of course, most of these jobs are at four year colleges where there might not be much support or expectation for research. Many PhDs also find themselves teaching at community colleges and enjoying it. Commented Nov 20, 2020 at 6:34
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Look - for a homo economicus, a PhD program in (pure) math is clearly a bad idea.

I quit a $60K/yr job back in 1999 to do a math PhD. My current salary as a tenured associate professor is a little more than $70K/yr - less than my salary 20 years ago adjusted for inflation. I'm probably out most of a million dollars in lost earnings. I faced a much friendlier job market (though I still had to be lucky to get a job - it wasn't that friendly) than anyone doing a PhD now will face.

But - I really didn't like tech consulting, and in fact when I left I vowed to never work in a position of any responsibility in a for-profit company ever again.

A PhD doesn't make your job prospects worse. If you don't get an academic job after a PhD and maybe some postdoc, then you'll be able to get a slightly better version of whatever job you could get now. Would it be worth the lost earnings and lost time if you don't get an academic job? In large part that will depend on how much you enjoy actually doing the PhD. I can't answer that for you.

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First, a math PhD is meant to give you training to learn how to do research in mathematics, which is different than training you only for academia. There are different reasons to pursue a math PhD, and in my experience a lot of (most?) people who start don't even know what they want to do career-wise after they finish.

It's true a math PhD is the best preparation you can have for becoming a math professor, but it can also prepare you for many other jobs that require a high level of mathematical expertise, and often involve some aspects of research. At least in the US, if you look at the AMS annual survey, you'll see many people who get PhDs do stay in academia (for research, or teaching, or some combination of both). See also the AMS career page for some non-academic job options for math PhDs.

Second, at the undergraduate level, it's too early to tell whether it will be difficult for you to succeed in academia or not. While it's true that it's not easy to get a job in academia nowadays, it's certainly far from impossible (e.g., see the AMS data), and there are a wide range of jobs in academia from teaching to research.

Math PhDs are still very marketable, and going through a program will help you figure out exactly what you want to do (be it research or teaching, academia or industry). So if you want to do a math PhD, my suggestion is to apply and see what happens. After a year or two in the program, you might decide it's not for you and leave with a master's, which is also marketable. If you like, you could also apply for some master's programs and some jobs to see what you options are; sometimes being confronted with decisions helps you figure out what you want. (And taking a job for a couple of years, and then deciding on a math PhD is perfectly fine too.)

Finally, most paths in life involve "anxiety, stress, and uncertainty". Your attitude to how you approach your endeavors makes a big difference on how stressful they are. This blogpost is written a later academic stage, but maybe it will help give you some perspective on how to enjoy your time in academia, however long it lasts.

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People enter pure math PhD programs because they enjoy doing mathematics and will have a stable career afterwards. That career may or may not involve doing mathematics, but who cares? If you can get a programming job now, or after spending five years obtaining a math PhD, which would you rather do? You can leave the PhD at any time if your preferences change.

Now if there is another more applied area that you're also passionate about but has better career prospects, then that is a difficult question which would require more personalized advice.

This is an important question. I think the other answers are somewhat dismissive of the fact that those from less privileged backgrounds need to care about these things. Getting a CS PhD (with summer internships) and a pure math PhD (without) could be the difference between affording a yearly flight back to see your family and not. My main point is that math is lucrative enough that you'll still be able to make it, and you can change your mind at any time.

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Unless you are already financially free, ultimately the most important reason for education is to enable get you a job. Note an academic job is also a job. It is something that pays the bills and puts food on the table. In this sense it is no different from working as a cashier at your local supermarket.

Why study at all if you can just work at your local supermarket? Because there are some jobs that require technical skills that are imparted only by education. For example if you want to work as an electrical engineer, you will need to be familiar with things like Ohm's law. You need to be able to handle algebra, trigonometry, calculus, statistics, and so on. Humans are not born with the ability to do these things - you learn them via education.

In a similar way, a PhD in pure math will equip you with skills to handle certain jobs. One way to see what they are is to check your local jobs portal and search for jobs that require a math PhD degree. Here's an example.

Overview

SIG is looking for someone with an understanding and interest in sports analytics to join our Quantitative Research team. Quantitative Researchers at SIG use research to better understand markets and to identify, develop and improve strategies for the firm.

This is a research, development and modeling role in which you will use your background in mathematics and statistics to seek out reliable, timely sports data to clean, analyze, and transform into algorithms and strategies. This position will likely require intermittent international travel.

What we're looking for

  • Graduate degree in STEM concentration; PHD preferred
  • Interest in sports data/analytics with a strong understanding of US Sports-NFL, NBA, MLB, NHL
  • Data mining and modeling experience
  • Practical programming experience in Python, C++, Matlab, and/or R

If you want to do this job, a math PhD degree will be helpful.

You ask:

Why should I go on a 5 year journey to achieve something I am probably not going to achieve, and upon failure, will subsequently face irreversible consequences (i.e., time lost, financial insecurity, etc.)?

Because you think the rewards that come with getting the degree outweighs the losses if you fail. This is no different from many other decisions we make in life. Here's a similar decision you might already have made, reworded:

Why should I go on a lifelong journey with my significant other if it's possible they might cheat on me or fall in love with someone else, and subsequently force me to face irreversible consequences (i.e. time lost, financial insecurity, divorce lawsuits, etc.)?

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    Actually, I think the job posting you quoted conveys the exact opposite message you think it does. The reason there are finance jobs aimed specifically at PhDs is because they want to directly advertise to the giant pool of PhDs that need employment. The job itself doesn't actually need a PhD, nobody ever does a PhD with the express goal of getting such jobs, and you don't get paid more either (i.e. a fresh PhD will be paid a bit more than a fresh university graduate, but far less than a university graduate who spent their 20s instead getting 4-6 years of job experience).
    – knzhou
    Commented Nov 20, 2020 at 1:29
  • The case that these kinds of jobs provide economic security to STEM PhDs is very strong, but they are absolutely not a reason to do a PhD in the first place. Finance jobs tend to require nothing more than undergraduate math, if that.
    – knzhou
    Commented Nov 20, 2020 at 1:31
  • @knzhou I disagree - if that were the case, they wouldn't hire PhDs. After all, as you wrote, the fresh PhD is paid more than a fresh university graduate. They are asking for PhDs because having a STEM PhD is strongly correlated with the ability to do the job. It's true that the day-to-day job might only require undergraduate math, but hiring at that level also opens you to hiring the not-so-good undergraduates who wouldn't have qualified for PhD studies in the first place. Put differently, you can be sure a math PhD can do the job, but can't say the same for undergraduates.
    – Allure
    Commented Nov 20, 2020 at 1:34
  • That's simply not true -- SIG recruits extensively at the university level and does the selection through high-pressure interviews, which PhDs also go through. They happen to have job postings for PhDs, but they also have comparable postings for university graduates. (Being a US company, they might also recruit a bit in Canada, but that isn't a good reason to move from the US to Canada.) You will never meet anybody working in finance that wishes they had done a PhD for the career options.
    – knzhou
    Commented Nov 20, 2020 at 1:44
  • @knzhou no, I do not agree. SIG recruits at university level the same way any university recruits at high school level, there are different positions requiring different degrees. If you join with a PhD you will be assigned different responsibilities from someone joining with a Bachelor's. You will never meet anybody working in finance that wishes they had done a PhD for the career options might be true, but it'd also be misleading, because one cannot tell what one would have achieved if they had gone for the PhD.
    – Allure
    Commented Nov 20, 2020 at 2:18

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