





So in short my question is:






Also, I apologize for the anonymity.

Thank you for your time!


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    $\begingroup$ I vote against closing. No soft question has a definite answer. $\endgroup$ – Steve Huntsman Aug 17 '10 at 15:38
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    $\begingroup$ I dislike most soft MO questions as much as anybody, but I think this one is very good. Partly, it is well-written, and partly I can imagine someone years from now finding this question on Google and benefiting from it. (Incidentally: there are many mathematicians who have websites dedicated to this type of advice. Maybe I or someone else will dig some up and post them.) So if this question is closed, I will vote to reopen. $\endgroup$ – Theo Johnson-Freyd Aug 18 '10 at 2:52
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    $\begingroup$ I wanna know what happened to the person that wrote this... $\endgroup$ – Jonathan Beardsley Aug 7 '18 at 7:45

I think that, for the majority of students, your advisor's advice is correct. You need to focus on a particular problem, otherwise you won't solve it, and you can't expect to learn everything from text-books in advance, since trying to do so will lead you to being bogged down in books forever.



One other thing: reading papers in the same field as your problem, as a clue to techniques for solving your problem, is often a good thing to do, and may be a compromise between working solely on your problem and reading for general knowledge.

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    $\begingroup$ Excellent remarks. I liked especially the third paragraph. $\endgroup$ – Claudio Gorodski Jun 15 '11 at 2:22


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    $\begingroup$ Pólya: "If you can't solve a problem, then there is an easier problem you can solve: find it." Great advice! $\endgroup$ – Joseph O'Rourke Aug 18 '10 at 16:21
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    $\begingroup$ I've always (mis?)remembered that quote as "If you can't solve a problem, then there is an easier problem you can't solve". There is a small but crucial difference there. $\endgroup$ – Victor Protsak Aug 18 '10 at 18:23
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    $\begingroup$ +1 to Ben and Joseph. Just do it! De Giorgi's version of Polya's method was even more explicit and easier to follow: "If you can't prove your theorem, ok, then, take a piece of the conclusion and move it to the assumptions. Keep doing that until you can prove it. Then, maybe, do the reverse..." $\endgroup$ – Piero D'Ancona Aug 18 '10 at 20:38
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    $\begingroup$ I have a copy of of "How to Solve It" Handy and it says "...there is an easier problem you can't solve: Find it." $\endgroup$ – Josiah Sugarman Mar 19 '15 at 21:38
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    $\begingroup$ See my question on whether Pólya said "can" or "cannot": math.stackexchange.com/q/2086285/413. It seems he said neither, and rather, it was John Conway who said, "cannot," but attributing it to Pólya. $\endgroup$ – Joel David Hamkins Aug 6 '18 at 14:48



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    $\begingroup$ Why on earth would you want to get over your "bred desire"? Because you want to graduate! $\endgroup$ – Kevin H. Lin Aug 17 '10 at 19:04
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    $\begingroup$ Kevin, a dissertation need not be restricted to one topic. $\endgroup$ – Bill Johnson Aug 17 '10 at 20:17
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    $\begingroup$ Bill, not every advisor shares this POV. $\endgroup$ – Steve Huntsman Aug 18 '10 at 2:00
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    $\begingroup$ @Bill A dissertation usually requires an actual contribution to a subject, something new. Not a laundry list of general nonsense across many fields. $\endgroup$ – Ryan Budney Aug 18 '10 at 3:57
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    $\begingroup$ There is also a question of interpretation over what counts as a collection of disparate topics, and what counts as a single theme with several facets that are explored. I don't think Bill is suggesting that one should write a shallow PhD dissertation. $\endgroup$ – Yemon Choi Aug 18 '10 at 7:19





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    $\begingroup$ In particular, are there really research active mathematicians without the "problem-solving itch"? $\endgroup$ – Pete L. Clark Oct 15 '10 at 2:28
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    $\begingroup$ Yeah, I agree with both of your comments. The analogy that I was tempted to give was, "How do you work up the desire to have sex?" But that reads better in a footnote (like this one). $\endgroup$ – Greg Kuperberg Oct 15 '10 at 4:30
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    $\begingroup$ So, @GregKuperberg, are you implying Erdős is the sluttiest mathematician in history? $\endgroup$ – Adam P. Goucher Jan 18 '15 at 17:58





Dear Anon:






What's the goal?




Gerhard "Ask Me About System Design" Paseman, 2010.08.17


One thing that might help is to think about this advice as saying you need to develop new skills rather than being about the one true way to do mathematics. That is, you have a lot of experience and skill at learning math systematically from the ground up like you did in your classes, but that's only one skill out of many that are necessary to be an effective researcher. You also need to hone your problem solving skills, you need to learn how to identify appropriate interesting problems and research programs, you need to learn how to efficiently wade through the literature to find what you need, etc. You have less practice at these things, so naturally you're not as good at them and so it's going to feel uncomfortable, but that's normal when you're learning new skills.

In other words, you're focusing too much on being well-rounded in terms of subject matter, but you're missing out on being well-rounded in terms of your skills.

That said, I do think it's worth following your interests, provided you still have enough time for your main research program. There's more time in a day than my brain can handle working hard on my main research program, and the more of the rest of the time that was spent learning other mathematics, filling in gaps in my background, or working on a side project, rather than say playing 2048, the better. So keep learning broadly and deeply, but don't let it eat into the time you're getting your main work done.

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    $\begingroup$ +1 This is a great answer, I had been pondering similar thoughts since this question bounced up to the top earlier today. When explaining these ideas to my students, I sometimes use analogies of hunting or mining: abstract understanding of the exotic creature or brilliant gem you are trying to find is important, but you still have to learn to stalk quietly, or to use your pick with dexterity. $\endgroup$ – Lee Mosher Aug 6 '18 at 19:56

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