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I am towards the end of my Phd (with some difficultues to overcome, I can say I am really satisfied about it) and I was wondering about what to do next. There are basically two paths: academia or industry. Now, I am not 100% sure which one to take but this is a different story, and maybe a different post (in AcademiaStackExchange I guess).

I just wondered: what is that keeps one motivated after 10, 20, 30+ years in math research? With up and downs I liked my Phd, but I just realized that if one asks me why I do math, I cannot say much more than: it's fun, i like it and it always amazes me how well it can describe reality. But the main reason is that it's very satisfying to think about a problem, try to solve it and finally succeed. And maybe this excitement fades over time.

Being almost over with my Phd I can say that I'm quite beat up. I mean, it had its fun parts, but I went through much more struggle. The idea to start over again with a post doc and then face the same hardships again and again is a bit scary to be honest, and the reason why I "do what i do" seems a bit shallow.

As many of you are professional mathematicians, what does keep you "in the ring", fighting? And does the fight become somehow easier? Because right now, maybe because of the fatigue in part, I cannot picture myself keep this fight for, like, a whole life.

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    $\begingroup$ what hardships? math is way more chill than jobs $\endgroup$
    – mathworker21
    Commented Sep 19 at 20:16
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    $\begingroup$ I think love is the answer; as someone else once said, don’t do math unless you can’t not do it (and in this situation the way forward is clear: do more math). $\endgroup$
    – Alec Rhea
    Commented Sep 19 at 20:21
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    $\begingroup$ @mathworker21 I would say that "but you get tenure" is a non trivial statement, it very much depends on many factors (geographical for example) $\endgroup$
    – tommy1996q
    Commented Sep 19 at 20:50
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    $\begingroup$ Not knowing your background and sub-field, it is hard to give advice/encouragements. When I finished my PhD I had quite a few work ideas which were solvable given enough time. With more time, you'll find that there are so many subjects to work on that you need to choose. For me it's not 'fighting'. It's a constant learning experience. Learn, experiment, sometimes solve, then repeat. There are important/difficult questions: keep them in mind until you gather enough tools to tackle some of them. In the meantime, always have some 'feasible' problems at hand to work on. $\endgroup$
    – Beni Bogosel
    Commented Sep 19 at 21:17
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    $\begingroup$ A friend of mine likes to say, "All jobs suck. That's why they're jobs." He says it somewhat tongue in cheek, but he's mostly serious, and I think he has a good point. The most obvious reason to hold down a job is to earn money. If you don't need the money and you don't like working, then don't work. If you do need the money, then just find the job that gives you enough money and sucks the least. With this perspective, I find that math rates pretty well. Hopefully you don't need to resort to this line of reasoning too often, but when the chips are down, I think it's a good POV to take. $\endgroup$
    – Timothy Chow
    Commented Sep 20 at 1:47

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This question seems to assume that after doing a PhD and continuing your career in mathematics you will keep struggling with the same things. I don't think this is quite right.

In a PhD the main thing one usually struggles with is trying really hard to do research for a long time and not making research progress. This is because you are just learning to do research, and it takes a long time to go from not being able to do research at all to completing a first serious research project.

After a PhD, most mathematicians will not struggle with this exact thing. This is because they either get more efficient at doing research, or devote less time to doing research and more to other responsibilities, or, most commonly, a mix of both. Of course research is not easy but unless you intentionally work on super hard problems you will not go as long without making progress as you did in grad school, or if you do it's because you're busy with other things you care about, which hopefully you are actually successfully doing so you are still making progress on something.

Of course after a PhD most mathematicians will struggle with new things, most commonly, I think, balancing a bunch of different responsibilities. But if you really hate one struggle you might not really hate the other, and in any case the coping mechanisms you need are different.

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It is very simple. If you have in sight some questions which, in your opinion, are worth answering, and you feel capable of answering them, you just try to do it. Not for the sake of your fame, which for me has never been a true motivation, but since the answers would clarify something previously hidden. I would not say for me it is a source of excitement, neither is this some form of altruism. It just happens that the knowledge we are after adds to our understanding of things, that's it. And there are so many such questions you encounter in books, papers, at conferences, or here on MO, that I don't think you will ever run out of them.

In other words, for those who do not feel motivated enough by just trying to know more than is known now, probably research is not their thing. If they do, then your question is not relevant for them.

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I agree w/ most of the other answers so far. I just wanted to chime in about this part of the question:

main reason is that it's very satisfying to think about a problem, try to solve it and finally succeed. And maybe this excitement fades over time.

I guess for me, it hasn't really faded!

But to get excitement, the problem had to have some difficulty for me. And problems I solved 10-15 years ago are no longer difficult for me. (As an extreme example: I used to get excited when I would solve an undergrad problem set, but solving such problems now is usually too easy for me and not exciting.) But when I crack a really tough problem I've been thinking about for a long time, I still get (I think) that same thrill of excitement.

Would that excitement alone have been enough to sustain me over the course of my career so far? I don't know. Feels like maybe a necessary but not sufficient condition for me. YMMV.

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I'll just say that on average it's easier to go from academia to industry than it is to go from industry to academia. So it may make sense to try academia first and see if you like it. If not, industry I is always happy to find a mathematician M who will apply M's skills toward solving I's problems.

Conversely, it's not easy to keep up with a field of pure math while also solving problems for industry. There's always some mathematics that is interesting, unsolved, and on one's wavelength that one can move to, but this direction contains more obstacles than switching in the other direction.

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I just want to pop in here to point out that industry is not the only path for mathematicians who do not want to work on pure math professionally. Governments big and small, non-profits, and think-tanks all have roles that use mathematical or programming skill sets, to greater or lesser degrees of technicality, for the public interest. Multiple programs exist to help folks with the transition, such as the AAAS fellowship in the US.

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Why does anyone do anything at all? Should you continue in the field of mathematics or go have a cup of coffee?

Practically, do what makes sense in the near future, which may mean continuing in math for awhile until you've figured this question out.

More deeply, I don't really think humans are meant to do the same thing for ten, twenty, years, be it career, marriage, etc., but we often have to even so. For me, it amounts to choosing the path that gives you the most freedom NOT to have to do anything (which gives you the option to solve a math problem or read a comic book). This may mean working hard and intense for 10+ years and retiring early or if you can't wait that long, lowering your needs now. If you don't need a lot of money, you can volunteer in Africa for little or no money, or work part time, and spend the rest of your day playing soccer.

What's really interesting--what do you do once you really have that free time? Questions about the meaning of life only then become interesting. And I wonder if people chain themselves up because they're really afraid of this question.

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