[Math] Which way for reading the proofs

gm.general-mathematicssoft-question

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…)

Best Answer

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.)

Related Question