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I am currently reading the book "An Essay on the Psychology of Invention in the Mathematical Field" by Jacques Hadamard, and one chapter in particular seem very interesting to me.

Hadamard discusses the role of imagination in mathematics, and he seems to have talked to a lot of mathematicians and scientists about this. According to him, a majority of them visualize the problems they are trying to solve as some vague image before formulating the solution using algebra and words.

He uses as an example Euclids proof of the infinitude of primes (https://primes.utm.edu/notes/proofs/infinite/euclids.html). He says he sees the finitely many primes as a confused mass, and their product $N$ as a point far away from the confused mass. $N+1$ is another point close to $N$. The prime that must divide $N+1$ is in the space between the confused mass and the point $N$ (how he visualizes the contradiction?).

Other mathematicians, like Polya, thought of the problems as words and puns before solving the problem in a rigorous manner.

This book was written in 1954, and thus, a lot of the psychology in the book is outdated (concepts like the subconscious were still controversial at the time Hadamard was writing the book). So, is there a more recent book that also covers the topic?

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  • $\begingroup$ I'm no mathematician. But I know a lot of mathematicians. It seems the situation varies from person to person. So...it is a useful imagination so long as it works for you... $\endgroup$ – Troy Woo Jul 20 '15 at 12:42
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    $\begingroup$ This sounds like an interesting conversation, but it's not well-defined enough, and too much of a discussion to be on-topic. By all means, take this to the math chat rooms and I'm sure you'll get a good discussion. $\endgroup$ – rschwieb Jul 20 '15 at 12:44
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    $\begingroup$ Asking about the existence of a modern treatment of the subject is a well-defined question. I don't agree that this should be closed. $\endgroup$ – Cheerful Parsnip Jul 20 '15 at 12:58
  • $\begingroup$ So, should I remove the final paragraph? $\endgroup$ – Avatrin Jul 20 '15 at 13:04
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    $\begingroup$ @Avatrin: I suggest you do edit the question to focus on the existence of a modern treatment of the subject. $\endgroup$ – Cheerful Parsnip Jul 20 '15 at 14:20
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Here are few references:

Books:

  1. Gian-Carlo Rota, Fabrizio Palombi, "Indiscrete thoughts" (Part II of the book). Birkhauser, 2010. (2nd edition, I think.)

  2. Philip Davis, Reuben Hersh, Elena Anne, "The Mathematical Experience," Study Edition, Birkhauser, 1995.

  3. E.W. Beth, J. Piaget, "Mathematical Epistemology and Psychology", Springer Verlag, 1974.

Articles:

  1. A. Borel, Mathematics: Art and science. The Mathematical Intelligencer, 5(4) (1983), pp. 9–17.

  2. T. Dreyfus, T. Eisenberg, On different facets of mathematical thinking. In R. J. Sternberg & T. Ben-Zeev (Eds.), The nature of mathematical thinking (pp. 253–284). Mahwah, NJ: Lawrence Erlbaum Associates. Inc. 1996.

  3. S. Papert, The mathematical unconscious. In J. Wechsler (Ed.), On aesthetics and science. Birkhäuser, 1978, pp. 105–120.

  4. W. Thurston, On proof and progress in mathematics. Bulletin of AMS, 30 (1994).

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  • $\begingroup$ Is there any particular reason why are you advising these readings, or you just found them on the web? $\endgroup$ – nbro Sep 13 '16 at 1:14
  • $\begingroup$ @Nuncameesquecideti I was aware of several of these earlier, before I saw the question and found them enlightening. They are as also written by excellent mathematicians. Do you know who Thurston, Borel and Rota are? Every single one was as good as Hadamard. Thurston was arguably better. Do you know Piaget' name ? $\endgroup$ – Moishe Kohan Sep 13 '16 at 3:01
  • $\begingroup$ I was not aware of those mathematicians, also because I'm not a mathematician, but I've just done a little search and they seem to have done great things in mathematics. I would give a brief description (at least) of what the books or papers are about or the perspective of the book with respect to what was asked in the question, if you have time and you know a little bit about them. $\endgroup$ – nbro Sep 13 '16 at 3:25
  • $\begingroup$ @Nuncameesquecideti sorry, but this is too much to ask. Take a look by yourself. If you find these sources interesting, read in detail, if not, then look for other sources. $\endgroup$ – Moishe Kohan Sep 13 '16 at 9:29

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