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I am a master student in mathematics. For me a large part of doing mathematics is thinking about, reading and verifying the proof of theorems that I find them in my field of study. I can do this action in 3 ways:

  • When I see a theorem I get a paper and think to prove it: this action takes time a lot and maybe I couldn't prove it after thinking for a lot of time.
  • Finding the proof of the theorem in a book or in the internet and begin reading, going step by step with proof,understanding and verifying all steps: this action may takes time a lot and maybe it is not necessary that I read all steps and it's better that I jump from not important steps (but how I can find that a step is not important?).
  • Finding the proof of the theorem and just read it like reading a newspaper for finding the sketch of the proof: this action is good because of its speed but maybe there be some important details in the proof that I couldn't see them in this type of reading.

My questions:

  • What is the way that famous mathematicians like Fields medalists take for reading the proofs usually?
  • Which way is the the best for which proofs? (For example classifying proofs and saying that the first way is good for the first class and...)
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    $\begingroup$ This is a very difficult question. The problem is that the way you should read a proof may depend more on your personality than on the type of proof. I'm curious what suggestions you get. But I'm pretty sure that "the Way that famous mathematicians like fields medalist take it for reading the proofs usually" doesn't exist. $\endgroup$ – Helene Sigloch Mar 24 '15 at 8:21
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    $\begingroup$ I am not entirely sure that this question is on topic here (seems that Math Educators may be a better fit). But in any case, your first question was very kindly answered by Martin Hairer here (the question and answer are on the last page of the transcript of the interview). $\endgroup$ – Willie Wong Mar 24 '15 at 8:25
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    $\begingroup$ The OP is a master student in math, but I think the question (and the answer it gets) could be equally interesting for any mathematician $\endgroup$ – Qfwfq Mar 24 '15 at 13:01
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    $\begingroup$ I flagged to make CW since there is clearly no one correct answer. $\endgroup$ – Benjamin Steinberg Mar 24 '15 at 14:17
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    $\begingroup$ AFAIK, Faltings doesn't read proofs. He simply writes them. $\endgroup$ – Fan Zheng Apr 13 '15 at 16:29
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Here is a quote of Poincaré (one of the most accomplished mathematicians of all time) regarding the reading of mathematics:

I am used, when I read a memoir, to glance over first quickly so as to have a general impression, then come back to the points which seem to me obscure. I find it more convenient to do proofs over than to examine thoroughly those of the author. My proofs are generally far poorer, but they have for me the advantage that they are mine.

(Letter from Poincaré to Mittag-Leffier, 5 February 1889--IML; cited as per the article "The Poincaré-Mittag-Leffler Relationship" by Philippe Nabonnand, Mathematical Intelligencer, 1999.)

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My two cents on "proof-reading": It depends what you want to do with the proof.

  • Do you want to understand why the result is true? Then don't read to proof but try to find counterexamples and notice where you fail.
  • Do you try to prove something similar and want to know how some step can be made? Then jump to that step and see how it works.
  • Do you want to generalize the result? Then try to divide the proof into logical steps and analyze them separately and look for room for improvement. (Of course, this is just one way to find some generalization… More interesting would be to find a totally different proof that works in a more general framework.)
  • Do you just want to admire a clever proof? Then read from end to end.
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    $\begingroup$ I'd add "Do you want to be convinced that the theorem is true?" Even if the source is credible, I often read proofs to check that I have understood the theorem correctly (in other words, to check that the theorem I have in mind is actually true, which usually means that my interpretation of the theorem in text was correct). $\endgroup$ – Janne Kokkala Mar 24 '15 at 11:07
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    $\begingroup$ I like the distinctions that are drawn in this answer $\endgroup$ – Yemon Choi Mar 24 '15 at 16:54
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If one tries to read a sophisticated proof without prior thinking, it would be hard to absorb the main ideas. It is always helpful to try to solve the problem yourself (even if it is unsuccessful) then to read the proof. If reading a proof is like running, trying to prove it is like warm up.

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