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I thought I remembered reading something along the lines of the following advice in Littlewood’s Miscellany, but it doesn't appear to be there.

Researchers spend the vast majority of their time feeling frustrated. To improve the ratio of time feeling fulfilled to time feeling frustrated, whenever you find a new result or succeed in completing a proof, take the time to enjoy it, preferably by taking a long walk. Definitely don’t dive into the next problem, or go back and check the proof. There is plenty of time for that later.

I'm offering a chocolate bar to the person who can supply me with the true source.

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This question and its answers might help, though originally when I asked it I was raked over hot coals here ... mathoverflow.net/questions/10419/depressed-graduate-student –  Anweshi May 23 '10 at 7:25
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I've taken that walk many times when I've discovered a new theorem, only to figure out that I made an error when I work it out more carefully later :( –  Ian Agol May 23 '10 at 7:28
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In my experience, if I feel like taking a walk after having a promising idea, instead of checking the idea and working it out... it's quite a suspicious sign. –  Pietro Majer May 23 '10 at 17:13
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I've definitely heard something to that effect before: "When you prove your big theorem, celebrate tonight; there will be time to find the mistake tomorrow morning." But I never heard an attribution for it. –  MTS May 23 '10 at 19:18
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2 Answers

I may have spent enough time feeling frustrated that I can't find anything that resembles this quote in Google Books. :-) The closest that I came was a quote from Halmos in the book "Paul Halmos celebrating 50 years of mathematics". Halmos explained that he started walking a lot on his doctor's advice, as an alternative to hypochondria. This wasn't about walking in relation to research.

Are you sure that it wasn't e-mail?

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Thanks for trying to find the quote, Greg. There are closer quotes from Littlewood's Miscellany than the Halmos one you found, but on pages not included in Google books. On p. 196 Littlewood comments that "Minor depressions will occur, and most of a mathematician's life is spent in frustration." At the bottom of the same page "I should stress the importance of giving the subconscious every chance. There should be relaxed periods during the working day, profitably, I say, spent in walking." That quote ties walking to research, and makes a point I agree with, but not the one I remember. –  Eleanor Rieffel May 24 '10 at 6:00
    
Greg - You ask if I'm sure it wasn't e-mail. That's hard to say. Before I looked for it, I was sure the quote was in Littlewood's Miscellany! My best guess is that the quote is in some other book I read at around the same time. I read Littlewood's Miscellany and a number of biographies and autobiographies of mathematicians my first year in graduate school. I remember reading the quote you found in Halmos then too. –  Eleanor Rieffel May 24 '10 at 6:03
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Now you make it sound like a kind of summary of different remarks in Littlewood's Miscellany. –  Greg Kuperberg May 24 '10 at 7:37
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I remember Alain Connes writing about working on problems whilst walking, but I can't remember where I read it and I don't think it had anything to do with enjoying fulfillment.

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Connes has a piece on advice to young mathematicians. It was online for a while and eventually it appeared in the Princeton Companion to Mathematics. The bit about taking walks is to "train the live memory" (his words), forcing yourself to do math without pencil and paper. Fwiw I agree, and would add that the act of walking, and natural or city sounds, keeps one from falling asleep. :) –  Pietro KC Jun 6 '10 at 22:57
    
Connes piece is online as part of the Princeton Companion to Mathematics section "Advice to a young mathematician." press.princeton.edu/chapters/gowers/gowers_VIII_6.pdf The section includes advice from Atiyah, Bollobas, McDuff, and Sarnak, all worth reading. I blogged about my some of my favorite quotes from these essays. palblog.fxpal.com/?p=536 –  Eleanor Rieffel Jun 7 '10 at 17:14
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