I recall reading this article that was written to explain how Descartes read philosophy effectively. I am wondering if such analogous tips have been made by past mathematicians?
3$\begingroup$ One of the most famous self-reports is Poincaré's Mathematical Creation. $\endgroup$– ConifoldJun 14, 2019 at 23:15
Mathematicians rarely describe the process which led them to their discoveries. One notable exception was Euler. Some books on the subject written by great mathematicians are:
J. Hadamard, The Mathematician's Mind. The Psychology of Invention in the Mathematical Field (English translation).
G. Polya, a) Mathematics and plausible reasoning. b) Mathematical discovery.
$\begingroup$ Wasn't it Newton who said something like "After the house is built I remove the scaffolding". $\endgroup$– nwrJun 14, 2019 at 16:41
1$\begingroup$ @NickR Gauss wrote something similar to that. $\endgroup$ Jun 16, 2019 at 9:27
1$\begingroup$ @JoséCarlosSantos Gauss, was it. I could not quite recall. It was much grander than a house. Thanks. $\endgroup$– nwrJun 16, 2019 at 17:13
Cédric Villani's Birth of a Theorem does exactly that. A great read for scientists, even non-mathematicians.
Edit: The English title of the book is Birth of a Theorem and not Living Theorem (the French title is Théorème Vivant). Thanks to Torsten for pointing it out.
1$\begingroup$ I assume you mean the book which is published in English translation as Birth of a Theorem, goodreads.com/en/book/show/23257776-birth-of-a-theorem. $\endgroup$ Aug 3, 2021 at 18:40
$\begingroup$ Indeed I do, my bad ! I translated the French title literally. I do stand by my recommendation, though. $\endgroup$ Aug 4, 2021 at 8:21
1$\begingroup$ I second that. I recommended that same book in the related question math.stackexchange.com/q/4033541/96384 on math.stackexchange. $\endgroup$ Aug 4, 2021 at 17:12
The first mathematician who did that was Archimedes, in his The Method of Mechanical Theorems (usually known as The Method).
Alexander Grothendieck, without a doubt one of the most creative mathematicians in the last 100 years, wrote a voluminous manuscript Récoltes et Semailles about (among other things) his approach to mathematics. Since he later requested that it be withdrawn from the public, it is hard (but probably not impossible) to find the full text on the internet.
One little snippet of this which has gained some popularity among mathematicians is the "Rising Sea" metaphor. There is an article by Colin McLarty which discusses this, quoting at length translated passages from Récoltes et Semailles.