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This is a atypical question for the forum. I'd like to get some advice on whether I should keep pursuing Math in the traditional route, i.e. get a PhD, do research & teach, etc.

Due to financial constraints, I worked almost full time and did not really explore much during my undergrad years. I got my BS in Finance in 2.5 years from one of the top schools in the country. Then I worked for a couple years in the investment industry, and decided it is not what I wanted - very flat learning curve and to be a leader in the industry, it's really not how much you know, but how well you can sell. So after I saved enough money to live frugally for a few years, I quit this industry and see if I can find another career path.

I then went to take various classes at community college to explore, while working part-time on and off. In about a year or so, I decided to do math. I liked the subject very much, and seemed capable in it. Later, I did almost two more years of Math at a state university, all the way to advanced real analysis and advanced linear algebra. I also participated in some research with professors. I struggled in some classes, but overall I got As and Bs. Everyone expects me to pursue a PhD.

This is my struggle: although I like the subject and mostly enjoy the thought-process, I hate the lifestyle. I was either studying or in the lab, because in order to do well, I have to do it all the time, including almost all weekends. I feel very very lonely. A few years ago, I had a very busy undergraduate schedule, but I could predict how long a homework/project would take and schedule time to make and keep friends accordingly. Now, I don't know if I can be done with a proof in 3 hours or 3 days! I feel bad if I abandon my unfinished work and just go out. I do try to talk to my current peers and professors, but they are mostly about the coursework or the projects we or they are working on. I look at my professors, they are almost always there to do research, day and night, weekday and weekends, even the ones with family. The more leisurely ones are the ones who either got their professorship or published some great papers a while ago. Otherwise, everyone else is very busy. I start to doubt that if everyone else is okay with it, it's me who don't love Math enough to infiltrate it into most parts of my life. After all, they are satisfied with seeing their friends once in a while, have a little chit-chat here and there, but I am very unhappy. I also realize that while I have to spend hours on a proof, some people get it very quickly. Maybe I am just incapable. People also work on a very irregular schedule - sometimes they come in at 2am and leave at noon, sometimes vice-versa. I don't mind when they work, but somehow the social activity, if any, is held very late at night, out of a spontaneous mode... "oh, my brain can't function, let's watch a movie/play frisbee/poker!" at midnight! And sadly, I don't want to be out at midnight. Again, other people seem to be fine with it as well.

Should I continue this path? If I do it, I don't want to do it in a rush way; so it'll take me another 2 years to do a master; and another 4 years to do a PhD, plus many more years in this research environment. (I also would like to learn how you cope with the financial challenges to support a family if you are pursuing the research route. I am afraid that my research is not good enough to get good funding; and positions at universities are very very competitive.)

Should I somehow engage it part-time only? I think my professors will welcome me to continue doing research for them, even just on the weekend. However, without formerly enrolling in school, my work will be mostly unpaid since their funding comes from NASA and other institutions, and I don't have a PhD to qualify me as a visiting scholar to let me access school's supercomputers legally. Also, I feel like I won't grow much in this part-time manner.

I had considered doing this on my own, at my own pace, but I am not at a level where I can do/learn on myself yet. I need someone to discuss and check if I understand something correctly. I much prefer doing it in person than digging through online forums.

This is long, and I appreciate your time in reading and I really look forward to some guidance or experience-sharing. Thank you.

[Technical comments:] Question asked by Flora. Originally tagged mp.mathematical-physics , ca.analysis-and-odes , ra.rings-and-algebras, to provide more focused mathematical context. [end technical comments, by quid].

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Perhaps this comment really belongs to Meta according to the current standards of MO, but I want to react (quickly, for lack of time) to the closing of this question and the comments already posted. I find them depressing. I do agree that this would not belong to a forum "focused on solving math problems" but, judging (for instance) by the thread "Ideas on how to prevent a department from being shut down" being the third most-upvoted one in the whole history of MO, I cannot see clearly that MO is devoted to "solving math problems" only. Unless preventing a department from being shut down (ctd) –  Jonathan Chiche Oct 19 '12 at 8:48
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(ctd) belongs to the realm of math problem solving. But then, why do struggling with loneliness or financial difficulties not belong to it? I would also point out that I have seen Grothendieck's famous quote on loneliness cited dozens of times by mathematicians as one of the most inspiring quotes they know, including on MO. Besides, it seems to me that the OP has stated clearly her situation and motivated her question, and made it clear that she had discussed the subject with people around her. If, as I think, MO appears to her as the only place where she could receive sensible advice (ctd) –  Jonathan Chiche Oct 19 '12 at 9:01
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(ctd) a terse answer like "this forum is not the right place" or "see your university occupational therapist" (I'm unsure what this is) may not prove very helpful (I hope I'm wrong). I also wonder as to how feel people when they see their question closed as off-topic while others get many upvotes in spite of their not clearly being (to me at least) more on-topic. I wonder whether people who have closed this question would close questions such as "How to deal with loneliness in research?" or "What can we do so as less mathematicians struggle financially?" when they are asked by (ctd) –  Jonathan Chiche Oct 19 '12 at 9:19
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Dear Flora, In my experience, not every mathematics department has the culture you describe among its grad students and faculty, at least not exclusively. When I was in grad school, and in my post-doc years, there were many grad students who had lively interests and relationships outside the math department, and many social activities occurred at reasonable hours. If you apply for graduate programs, one thing you should try to do is visit the various places you might go, and try to find out about the grad student culture. Perhaps you can find a department whose atmosphere is a better ... –  Emerton Oct 20 '12 at 5:14
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... fit for you than the one you are currently studying in. I think it is the case that grad school (and not just in math) can be emotionally difficult at times; after all, it is hard work, and one is also facing a fairly uncertain career future. But I don't think there is any reason for it to be unrelentingly lonely and isolating in the way you describe. Regards, –  Emerton Oct 20 '12 at 5:18
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7 Answers

One famous mathematician answered to all such questions from young people: No, you should not do math. You will do it successfully only in the case that you cannot do anything else. In which case you do not ask these questions.

Your observations are correct: mathematicians work overtime, at irregular hours, think about their problems continously, even at the time which should belong to their families, and so on. And sometimes you cannot solve a problem for months and years. And the way to a permanent academic position is long and hard. And the reward in terms of money is not really so great.

I understand that there can be various opinions, but I am of the same opinion as this mathematician. I know many people who quit mathematics on various stages of their careers, because of various obstacles. Usually they make more money than they could make if they pursued an academic math career. And have more time for all kinds of activities not related to their jobs.

On the other hand, if you really like mathematics, you can always do it as a hobby, without making it your profession. This hobby, unlike many other hobbies, costs you almost nothing, and you can dedicate as much of your time to it as you wish.

EDIT. Let me expand a little, and I stress that this is only my personal point of view. Doing research in mathematics is not a profession:-) It is a hobby. My profession is teaching (I am a professor). Of course, to qualify for this position, I had to show some ability to do research. But in the beginning, when I had to choose my career, I did not know whether I am able to do research or not. And I believe nobody knows, with possibly very rare exceptions. So when I made my choice (to pursue a degree in "pure" mathematics), I knew that in any case I can be a high school teacher, possibly a college teacher, and I did research as a hobby. The results eventually permitted me to become a professor at a university. In my youth, I even had for some time a "research position", with no teaching duties. But this caused discomfort: I never wanted my living (salary) to depend only on research results. Because when you do real research, you are never sure what (and when) will come out of it:-) It is nice of course when they give you some extra money (I mean grants) for your hobby, but I would not like to depend on this for my living.

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I strongly disagree with the first sentence of the second paragraph as it is written so far. –  Jonathan Chiche Oct 19 '12 at 9:47
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I second this answer. "Cannot do anything else" is one way to put it, "cannot refrain from doing mathematics (all the time)" is another way. But basically some kind of unrestricted devotion to the subject is what drives successful mathematicians (with the possible rare exceptions of some kind of universal geniuses who can and want to succeed in everything). The prevailing lifestyle is the corollary of this devotion. Not that I like it, just never learned to live in any other way. It's is pity that the OP seems to have nobody to talk to about such things in her department, though. –  Leonid Positselski Oct 19 '12 at 20:40
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"Doing research in mathematics is not a profession:-) It is a hobby." ? C'mon! –  Andres Caicedo Oct 20 '12 at 0:45
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The idea that research is a hobby and that teaching is the real purpose of your profession is demonstrably false. If this were true, then teaching proficiency would be considered much more important than your list of publications when you apply for a job. –  Dan Petersen Oct 20 '12 at 15:16
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"You will do it successfully only in the case that you cannot do anything else." I cannot disagree more. I will seem overly pretentious by saying this, but I, for one, could do almost anything else. Just, those things are often too easy to be as fun as maths. –  Joël Oct 20 '12 at 19:40
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I still think this question is too personal for MathOverflow and that advice from complete strangers should only be taken with a very serious grain of salt -- evaluated as to whether it really rings true to your actual situation. But given the series of comments so far, including my own, and that the question was reopened, I guess I'll add a few points.

  1. I completely agree with Henry Cohn's (hidden from sight) comments that working on math itself can be very social -- it's good to find collaborators and mentors who are people you like to be around, and it is good to find study groups in school for learning material. People vary a lot in how collaborative they are, probably in part because as you observe people vary in how introverted/extroverted they are. Some people do almost all of their work in collaborations and develop close friendships with their collaborators in the process.

  2. I also very much agree with Emerton's discussion of how cultures in graduate programs can vary a lot and that it's important to find one that is a good fit for you.

  3. Mainly I wanted to say something about your comments that others seem to get things more easily, while you have to study proofs for many hours. I've seen it happen many times that students believe the other students all understand things better and more easily than they do when in fact this just isn't the case at all -- some people have a lot of bravado, throwing fancy words around and pretending they understand things they don't really understand, while others tend to focus all their attention on the things they don't yet understand and ignore all the things they do understand, figuring they don't need to worry about those. Of course I don't know if this is your situation, but it is a possibility to consider as to what might be going on. I think honesty is really valuable in making genuine headway in math -- including working with people where you are comfortable enough that you can be honest with each other about what you do/don't understand. If students around you are acting like they understand proofs right away, one distinct possibility is that they haven't actually thought through all the subtleties and don't actually yet have a full understanding; another possibility is that in fact they are secretly working very hard, only acting like it is coming easily. One excellent mathematician said to me a couple years ago that he thinks hard work plays at least as big a role in success as raw talent. I think there's something to that, along with the importance of finding a good niche for oneself.

I have no idea whether you should continue in math, since I don't know you, but those are a few general thoughts, for whatever they are worth. I hope people will refrain from asking very personal questions on MathOverflow, especially considering what a huge readership it seems to have, but I do wish you good luck.

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Hi Flora,

it seems that many people have answered your question assuming that you're asking about becoming a research mathematician. But you can do a PhD in math and then go on to a different career, where you use your skills but don't necessarily go on to do original research. Many companies are hiring math PhDs and it seems to me that there are opportunities in very diverse fields. I'm not in the best position to discuss them ($p$-adic analysis doesn't seem to be very much in demand in the real world these days), but you should be able to find such information easily, maybe starting with the AMS?

And don't let yourself be discouraged by the behavior of your colleagues. It's true that some mathematicians spend all their waking hours thinking and talking about maths, and a few of them are very successful but these are really a minority. It's my opinion that most of those who display this behavior would really be better off if they had a more balanced lifestyle. There may be cultural factors here: in the protestant anglo-saxon world, it's important to be seen working all the time, and bad form for people to enjoy themselves.

Just do your own thing at your own pace. It's normal that you spend hours on a proof. Maths is not about being the first, but about having a deep understanding of what is going on. It certainly helps if you have flashes of insight, but there's nothing wrong about really understanding the situation first and finding the answer afterwards.

Concerning the funding, it's true that being a PhD student does not pay much and that can be a real problem. I don't know about the country where you are but in France, some companies have programs where you work for them part-time and do a PhD in the rest of your time, the PhD being in some applied domain that is then useful for the company.

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As regards work ethic, the line I've often heard is that successful mathematicians do tend to spend large amounts of time working on problems in their heads - BUT most of that time it's only a background/subconscious task, while doing other things or even while asleep. A person's mental resources are far greater than those he/she is directly aware of. –  Colin Reid Oct 21 '12 at 7:39
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+1 for making the point that working more can be counter-productive. –  quid Oct 21 '12 at 11:52
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I look at my professors, they are almost always there to do research, day and night, weekday and weekends, even the ones with family. The more leisurely ones are the ones who either got their professorship or published some great papers a while ago. Otherwise, everyone else is very busy.

First of all, thank you for realizing that. I'm serious. Most people seem to believe nowadays that math. professors are the laziest people in the world...

Now back to the question.

A lot depends on what you want to achieve. If you want to stay in the "low middle tier" as a mathematician, you can perfectly well combine mathematics with social life. Moreover, you'll have excellent opportunities to travel, to design your own schedule, etc. You'll still need to work 6-10 hours a day to be worth anything and you are perfectly correct when saying that the outcome is not directly proportional to the effort in mathematics on the daily basis (though, on a very long run, it still is) but it is actually a much better life than that of a factory worker or an office clerk. If you want to move up from there, it'll take more effort. Slightly above the middle tier, you have to "run as fast as you can just to stay where you are". Then your social life may get screwed up quite a bit.

Money has always been a problem and will always be. Short of marrying a rich girl or becoming a single heir of a millionaire uncle, nothing will solve that problem immediately, so prepare for quite a few years of "frugal life". What may help is tutoring, occasional consulting, taking some odd jobs, etc. See what you can do with the skills you already have. Many (now) famous mathematicians survived that way when they were students.

However, without formerly enrolling in school, my work will be mostly unpaid since their funding comes from NASA and other institutions, and I don't have a PhD to qualify me as a visiting scholar to let me access school's supercomputers legally.

This tells me that you are more into the computer science than "math. proper". There is nothing wrong with that. What I said above applies more or less to any "skilled trade".

About job opportunities. Be careful what field you choose. There are two extremes: to choose whatever you like most and bet on your becoming so good that they'll have to notice you no matter what and do things the way you want whether they like it or not. The problem is that you may be not strong enough and then this approach will fail quite dramatically. The other extreme is to follow the current fashions and be in some "hot area". Then you will have to follow the trends whether you like them or not but you are almost guaranteed a decent job. The problem, of course, is that fashions change with time. Most people end up somewhere in between (except the lucky guys for which the two choices coincide).

About student years. Yes, those are most demanding. You've got to put a lot of effort into your training to be able to pass the threshold above which you become a "professional". Prepare to face the fact that "you are nobody until you prove otherwise". You'll, most likely, be treated with respect, given help, etc., but if the aforementioned proof is missing by the end of your student years, your future won't be very bright. I'm not talking about the formalities here: more or less everyone can get a degree. However, to build a lasting reputation is much harder and it is your reputation, not your formal degree, which determines your opportunities after you graduate. There are, of course, technical things like going places, giving talks, letting people know you, etc. but, to be frank, to arrange your meeting with any mathematician of any rank is not too hard and most advisers are good at that. The problem arises when you are there and it is time to show something.

A small comment on what Alexandre said. Math. research is a hobby in only one sense: you aren't paid for it directly. However, in every other respect it is like any other art or craft: as Paganini used to say "If I do not play one day, I hear it. If I do not play two days, experts hear it. And if I do not play three days, the public hears it". Of course, "day" means some longer (but not much longer) unit of time here, but the message is clear: it is not something you can do on an occasional basis when you feel like it (unless you always feel like it, which is the best case scenario as far as the motivation goes).

So, that's more or less what it takes (from my point of view). What are the benefits? Well, the main one is that you become a dual nature creature that is not bound to this world alone, but to explain what I mean by that will take too much time. Let's stick to a decent salary, great work freedom, meeting a lot of interesting people, etc. Of course, you still have an option to turn it all into hell (The best way to achieve that is to quarrel with your colleagues over some trifle issue of this world and the quickest way to achieve the latter is to look for the general fairness and ethics with concentration on what is unfair with respect to you personally. The second best way is to start worrying about your prestige and priority.) But if you avoid these traps, you'll like what you get in the end.

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Regarding "going for the money": There is buzz that (in the U.S.A. at least) doing data science a.k.a. "big data" is on the rise, as are job opportunities for such. You may find it not only a challenge but that there will be many friends you can make along the way. Gerhard "It Is Something To Consider" Paseman, 2012.10.20 –  Gerhard Paseman Oct 20 '12 at 19:09
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"a dual nature creature that is not bound to this world alone" That's a nice formulation (and notion)! :) –  quid Oct 20 '12 at 19:22
    
I second that, a very nice notion :). I think I know what fedja is referring to, but it'd still be great to get some more explanation. –  Michal Kotowski Oct 20 '12 at 21:22
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Dear Flora,

Your question hits so close home. I am now an old math professor and I had to answer this question so often!

When I was young, younger students would approach me with this question. I would say "If you have to ask, don't." This answer was given at the tone of saying "If you are not cut for it, don't even think about it." The arrogance was intended. I was young.

Over the years however I observed through my students that "being cut for it" is mostly about having the motivation for it. Talent is a fertile soil which nourishes any seed you plant. Math is no exception.

Nowadays when faced with this question, I again give the same answer but my message is now more human. I talk about wanting to do it, will power and passion.

I also ask "Where do you want to see yourself in ten years from now?" Admittedly this is difficult to answer. But the actual answer to your question will probably be guided by your answer to this particular question. Finding out your life's passion is the most difficult thing in life, in my opinion. Once that is settled, the rest is easy.

So, in summary, my answer to your question is this: re-evaluate your passion and follow where it leads you.

Best of luck with your decision.

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Dear Flora,

I found your question very interesting. It partly reminds me of my own dilemas some years ago. You seem to like math and research but not to the point of devoting your entire life to it. You also wonder wether your research is good enough to get funding.

Probably the conventional career path in research, the Phd, tenure track, etc, competitive as it is will somehow condition your lifestyle goals, if you are to achieve the level of success you probably dream of.

The fact that you already made a career change move, and that you want to have a life besides work suggests you are a work-independent person who would eventually find happiness in a non-conventional career path.

I believe there is wisdom in the possibility of seeing mathematics as a hobby rather than as a profession. When you're not under the pressure of having to publish for career progression or to get funding, you have more freedom to pursue an idea of your own, at your own pace, taking risks (wasting years with it if necessary). This freedom is priceless. Even if you're not in academia you can still network with other researchers (like people do in MO) to get feedback on your work and share knowledge with others. Today, with everything the Internet has to offer, its a lot easier to be an amateur researcher.

If you follow this route I believe you will find that having a life will be of great help to your research. It will help you filter and pursue your best ideas (maybe just one), because you won't be able to follow them all.

As an amateur, you work when you can (when your family and your day time job lets you), not when you have to, because the paper delivery schedule is due and everything else stays behind. This means more freedom for your personal life.

These kind of generalizations are surely wrong somehow but I believe academia is sort of tailored for the efficient people, who can produce output without having to condemn their personal lifes.

Less eficient people aren't necessarily less bright. They may, for example, take the lead in creativity and use it to freely pursue new ideas.

As an amateur I have been very happy in doing research and raising a happy family besides my daytime job. And I am almost sure (but this is just my case) I would never have had the research progresses (quite promissing for my expectations) and the joy I have had if I turned into academia. Because I thought of my family (my life) first, the research gained momentum. It came as a gift and not as something I was desperate for.

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I don't think a "my condolences" or "I wish you the best" will cut it. Your question is well-thought out and is of general interest to aspiring and active Mathematicians.

Questions about professional development in Mathematics have a place here on MathOverflow - at least until a better place for it comes along. But have you tried quora? Maybe this is a Philosophy of Mathematics question.

Mathematics, in its logical rigor, is a counter-intuitive way of thinking. Most people are not that precise. How does that fit into the professional world?

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This looks more like a comment than an answer to the question. In any case, the argument for asking a question on MO that there is no better place to ask a question has never carried much weight, really! –  Mariano Suárez-Alvarez Oct 20 '12 at 17:15
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+1 Mariano, and John Mangual if you really care for these questions on MO I think it is in your best interest to be extra careful how you contribute to them. That they tend to create such 'answers' as yours is one (though not a main one) reason why I do not like them so much. –  quid Oct 20 '12 at 17:56
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Where did these question go before September 28th, 2009? [To avoid being too cryptic and as a safety-net if I misremember the date, this is/shoud be the day MO went live.] –  quid Oct 20 '12 at 18:37
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John Mangual: First, I really do not think that this is a 'Philosophy of Mathematics' question. Second, incidentally, there is a site academia.stackexchange.com whose expressed purpose (or at least on of them) is questions on "Life as a graduate student, postdoctoral researcher, university professor", which seems about as good a fit one can get. I am surprised you never heard of it, it gets mentioned quite frequently on this type of soft questions. –  quid Oct 20 '12 at 19:42
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Well, yes and no. We cannot 'move' anything there. We can close it here, and possibly in addition suggest to reask there. Yet OP said they want to ask it specifically here on MO. Of course, we can still close it. Actually, we did. But then some (not me!) seem to be against it. [Except that due to specific circumstances in this case the closing was so much underminded that I now think, as an extreme exception, it should indeed be open and stay for a while.] –  quid Oct 20 '12 at 20:18
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