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What kind of things do you find that help you get the "creative juices flowing," to use a tired cliche, when you're stuck or burnt out on a problem? I've read about some studies that suggest listening and playing music can stimulate mathematical thinking. Any particular style that someone finds helpful?

Other things that help?

(In case you haven't figured it out, I've been stuck on some things lately.)

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Might want to community wiki this and add the usual request for a sorted list, i.e., one answer per post. –  Greg Stevenson Nov 7 '09 at 6:32
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I've converted this question to wiki. @Josh: please make such questions wiki when you ask them in the future (see mathoverflow.net/faq#communitywiki). @Greg (and those who upvoted Greg's comment): if you find a question that really should be community wiki, you can flag it for moderator attention by clicking the "flag" link. It's much easier to moderate the site if the community helps out in this way. I encourage you to try out the flag link right now. You'll see that you can enter a bit of text to explain the problem, and you can click cancel if you don't want to flag after all. –  Anton Geraschenko Nov 7 '09 at 15:06
    
But the comment flag says "please flag this comment as offensive, spam, or hate speech", while the question flag says "please flag this post for serious problems". If you want to use the flags this way, then they should both say "please flag this post/comment for moderator attention". –  Greg Kuperberg Nov 7 '09 at 16:03
    
@Greg Kuperberg: Sorry, I forgot to mention that comment flags and post flags differ. A comment can only be flagged offensive/hate-speech, but a post (question or answer) can be flagged offensive, spam, or "Requires moderator attention". Try clicking the link, and even try clicking the radio buttons to see what happens; you'll get a chance to cancel. You should flag a post whenever you want to call moderator attention to it, even if it's not a "serious problem". I'd like to leave the wording as is to discourage new users from flagging before they've learned the MO community standards. –  Anton Geraschenko Nov 7 '09 at 17:00
    
@Anton: So, generally speaking, which types of questions are should be make community wikis? Asking for advice and such? Anything where there's no one right answer? –  jd.r Nov 7 '09 at 17:02

12 Answers 12

Some blocks are caused by desiring a mathematical goal but being unable to achieve it. Some are caused by losing your desire to do mathematics.

I think the second type of block happens more often than most people admit. It needn't be cataclysmic: it might just show up in a feeling of being burnt-out and tired. You've lost your appetite, your energy, your fizz. Everything's a burden.

When that happens, it can be useful to remember that you don't have to spend your life doing mathematics. However great a mathematician you are, it will make next to no difference to the world at large if you drop math and become a postman. Seriously: you could do something else. You'd survive, and you might be perfectly happy.

Once you've really internalized that, the feeling of heaviness should lift. You're doing mathematics because you want to, not because you have to. It's not an obligation. And if your appetite for mathematics never returns, that's probably a signal that you should do something else.

This might sound like depressing advice, but it shouldn't be. I find it freeing and energizing.

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I think one of the problems I have right now is the pressure of having to do it. This is my last year as a grad student and I'm looking for jobs. I've got a paper coming out, but I'd really like one more submitted by the end of the year. –  jd.r Nov 7 '09 at 21:18
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One can also lose his desire when it seems that his hard work is not being rewarded: no promising stable job, no stable income, not much vacation. –  Yoo Nov 17 '09 at 19:45

I can personally endorse forms of the answers provided so far.

  1. Work on more than one problem/project. Open problems are hard. Instead of chasing after one fish that could be out past the horizon, cast a net or set out several poles. Or, if it is important to stay focused on one project, work on more than one piece of it at once.
  2. Make your work a one-person seminar and take notes. I have always kept pages of calculations when working on problems. At some point, actually in mid-career, I realized that I was not just writing down calculations, but also reasoning and Q&A. Thinking often amounts to "the voice inside your head". (This may say something about the nature of schizophrenia; it seems that we all hear voices.) I ask my PhD students to write notes about what they are doing, not mainly for me to read them, but as a research aid for them.
  3. Give talks, ask questions, and answer other people's questions, again just to keep moving. This site, which is among other things a reincarnation of the old sci.math and sci.math.research, can be very useful for this purpose!
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I can't agree strongly enough with all three of your points. In particular the second. At least for me it is really surprising how much writing down one's thoughts can help. –  Greg Stevenson Nov 17 '09 at 23:16

This is something I've been wondering about for a little while now. My research has stalled somewhat over the past couple of years and I'm trying to build up some momentum again, and to find something interesting to work on.

There's a bit in one of Richard Feynman's books (probably Surely You're Joking, Mr Feynman) where he talks about suffering from some sort of research block early in his career. A colleague advised him just to play with stuff, without worrying whether it might lead anywhere. So a few days later he saw someone spinning a plate in the cafeteria and decided to work out the mechanics of what was going on, ended up really getting into the problem - which turned out to be more complicated than he was expecting it to, and ultimately gave him some insight into a serious research problem in quantum mechanics.

I've also just read How to Write a Lot by Paul Silvia, which has much useful advice in it. It's nominally aimed at academic psychologists, but most of what he talks about applies to researchers in other disciplines too. One of the things he recommends is to schedule regular writing times (he recommends at least an hour a day, ideally more) and to jealously guard that time from all but actual emergencies. He cites a study in which some academic researchers were divided into three groups: the first group were instructed to abstain from all non-urgent writing, the second to write when inspiration struck, and the third to write regularly, even if they felt they had nothing worth writing about. What the study found was that the third group not only (as might be expected) produced more pages of output than the other groups, but also tended to have far more creative ideas.

So I'm going to give this a go. (It was pretty conclusively scuppered this week by the catastrophic failure of a computer I'm responsible for, but hopefully next week will be more successful.)

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I find two near-orthogonal ways to be of help:

  1. For the tired or saturated brain. Relax your mind BUT doing something related with the processing of information: read a good scifi book (or even a jokes book!), watch some episodes of a smart TV series, improvise a piece in your piano or your favorite musical instrument... in other words, put other somewhat-related-to-creativity-and-logic-brain circuits to work, while avoiding to think directly on your problems and your math area. The block sensation should be gone in a couple of hours or days (if not before, it will happen when your are sick of the other things!).

  2. For the uninspired brain. Fill the nest of your "unconscious" ideas by looking at some other, not-very-related-to-your-work sources of good ideas: read a paper on another subject, read a general science magazine (for example, Nature or even Scientific American), read a book of hard-scifi stories, watch a scientific series like Junk Wars, Time Warp or Myth Busters, watch a scientific or engineering documentary... In other words, bombard your brain with a storm of ideas that are new and interesting to you for a little time: your own brainstorming should follow.

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Thanks for this. I often find myself in the first situation - after I've spent an hour teaching first-year undergraduates about analysis or linear algebra, I'm too tired to focus on any research. Maybe the solution is to go somewhere quiet and read a non-mathematical book or pop home for half an hour to do some piano practice (conveniently, I live about five minutes' cycle ride from campus). –  Nicholas Jackson Nov 7 '09 at 12:21
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That's funny.. as when I used to be a new graduate student.. I found the exact opposite effects. I didn't had any teaching load, and I wished to hell I had some as this could motivate me new ideas. I thought (still think) teaching even boring math subjects everything with some new perspective would help :) –  Jose Capco Nov 7 '09 at 17:40
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It's not the teaching itself that I find tiring, or being forced to rethink stuff that I learned several years ago - indeed, I find both of those aspects quite rewarding. However, I do find actually standing up in front of a load of people and talking for an hour both physically and mentally tiring, and I usually need to have a bit of a rest before I'm able to focus on mathematics again. –  Nicholas Jackson Nov 7 '09 at 23:21

I like to get out and do some physical activity, even if it's just walking, to clear my mind. That sometimes seems to help.

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It helps me to (attempt to) explain my mathematical thoughts to another person. It somehow helps even if the conversation itself isn't fruitful: the act of putting math explicitly into words can be revelatory. I think that's partially why new ideas often pop into my head right as I'm walking away from a meeting; the simple act of walking is probably also responsible. (Better late than never!) Even if another person isn't available, just trying to write things out formally (in good English) can have a similar effect.

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My usual solution is to work on a different project. Sometimes ideas need to time to marinate.

Talking to other people is also really good. I'm pretty often on both ends of conversations of conversations of the form:

Mathematician 1: "trying to do X is driving me crazy"
Mathematician 2: "isn't that in paper Y?"/"can't you just Z?"

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I knew I should have tried Z. –  Richard Kent Nov 7 '09 at 18:00

Collaborate! And, if you dare, enter into a long term relationship. The vast majority of mathematicians I know do most of their work alone and, when they collaborate, the relationships are often short term temporary ones. But it can be extraordinarily fruitful to commit to a long term collaboration. I recommend it highly.

EDIT: Inserted "I know" after "vast majority of mathematicians" to account for Ben's comment, which I have no reason to doubt.

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"The vast majority of mathematicians do most of their work alone..." This is a pretty bold claim, and I don't think a very accurate one; ten years ago, more than half of papers indexed by MathSciNet were coauthored, and the number was trending upward sharply. ams.org/notices/200501/fea-grossman.pdf –  Ben Webster Nov 7 '09 at 21:55
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It may become a bit difficult finding someone to collaborate with, especially if you are in a very specific field. Maybe we should start a new community wiki on ways mathematicians find collaborating partners. –  Jose Capco Nov 7 '09 at 23:01
    
@Ben: this doesn't mean that the authors did not work alone - it just means they pooled their results. –  Emil Dec 27 '09 at 13:44
    
Or perhaps the $1\%$ who did collaborate were able to write about $100$ times as many papers, on average. –  Douglas Zare Sep 3 '12 at 3:25

This is what I used to do or still do:

  1. Coffee, coffee coffee

  2. Sleep, sleep, sleep.

  3. Play some computer games

  4. Be more active in math forum than usual

  5. Totally forget math and play some chess. I'm a professional chess player, and funnily my math block and my chess block are inversely proportional.. whenever I excel in math, my chess gets extremely poor and whenever I'm bad in math the chess suddenly improves.

  6. Sports

  7. TV

  8. Read math instead of do math (and like the other poster said, do a lot of not-your-research-area readings as well).

In other words.. when I get my "block".. I procrastinate and try not to feel guilty. Because I know that this type of procrastinating is actually the good type of procrastinating :P

and if you want me to state the obvious

9. Socialize and tend to your lovelife. Some people can do magic
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this is funny.. I just discovered a bug in *overflow.net .. I was able to have an item 9. by putting lots of spaces before 9. .. otherwise they system renumbers it to 1. using some MS-Word-logic –  Jose Capco Nov 7 '09 at 17:49
    
It's not a bug -- you've turned it into code. The code automatically colors keywords from something like C++, like and and do. –  Ilya Nikokoshev Nov 7 '09 at 21:13
    
The lists are numbered starting from 1.; you can do 9) if you want non-consequtive numbering. –  Ilya Nikokoshev Nov 7 '09 at 21:14

As others have noted, there are different sorts of stuck. One that hasn't been addressed much by the other answers is the sort where you want to work on a problem, and you have time and energy, you just don't have an idea on how to make progress. In this situation, in addition to all the great advice in "How to Solve It", I can add two techniques that've worked for me: reading (or re-reading) related papers, and writing code.

Often, in the process of writing code to actually compute a nice example, or verify one, or automate some aspect of what's happening, I get into a different head-space that helps me to understand the nature of the difficulty. Also, other interesting closely related problems are forced in front of me, so that I have more things to think about.

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Yes I'd second this wholeheartedly! Of course it won't help you with certain types of problems which aren't code-able, but I think if you have programming abilities then you can get a great feel for a problem by coding it. It's not just the conjectural output of the program. It's having to spend hours thinking about how to explain the problem to your computer; it gives you another outlet to think about the problem. –  David Jordan Nov 9 '09 at 15:32

Here are a few that I haven't seen above yet:

1) Go for long walks (with a bit of math in mind). Don't worry about thinking about the math, just put it in your head and then walk. This is distinct from walking to clear your mind, as putting the math in there tends to get the juices flowing whether you like it or not!

2) Force yourself to write down questions. Lots and lots of questions...about whatever mathematics comes to mind.

An advisor of mine said that there are uncountably many open math questions out there, some of which you have the tools to solve, in contrast to the finite list of `big fish' questions we usually focus on. Every hundred or so questions, you may be able to answer something or will at least feel a bit energized. If you asked the question yourself, chances are it is connected to something you've worked hard on...perhaps indirectly.

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Parametric Answer …

  • When I was an undergrad, I usually just changed my major.

  • When I was in grad school, I usually wasn't that smart.

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No, seriously. True story. –  Jon Awbrey Nov 17 '09 at 17:59
    
Don't know why someone voted you down on that one. It's a community wiki question. And the answer is hysterical. –  Ian Durham Feb 3 '10 at 12:42

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