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From: Philip Johnson-Laird BA PhD Psychology (UCL), Stuart Professor of Psychology Emeritus at Princeton. (Author isn't a logician.) How We Reason (1st edn 2008). p. 112.

  You might suppose that logic does at least underlie the mental abilities of mathematicians and logicians. Some of them have thought so too, but the disciplines demand more than just deduction. The Indian genius Srinivasa Ramanujan, who died in 1920, left behind a body of work that continues to occupy mathematicians, but proof was not crucial to his many conjectures. As another great mathematician, G.H. Hardy wrote about him, “[he] combined a power of generalization, a feeling for form, and a capacity for rapid modification of his hypotheses, that were often really startling, and made him, in his own peculiar field, without a rival in his day.

Is this psychologist correct that mathematical discoveries demand more than deduction? If yes:

  1. What's the term for this creativity?

  2. How can undergraduate students learn it?

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    $\begingroup$ There are already questions on this site about creativity; try using the search feature. Your previous post on MESE also included an excerpt about deductive reasoning from this same book [and the same author, for whom listing the degree of "BA" strikes me as odd...] and was closed; I am not sure that this site is ideal for the questions you are asking. That said, maybe Tall's The psychology of advanced mathematical thinking or the cited text of Hadamard [or its inspiration by Poincare...] would be of interest. $\endgroup$ – Benjamin Dickman Jul 10 '18 at 0:51
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    $\begingroup$ I think this question is about mathematics education, broadly interpreted, and is an acceptable question for this website. The "how can learn" part is a tad broad, though. $\endgroup$ – Tommi Jul 10 '18 at 12:34
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    $\begingroup$ Likely there's no way to learn Ramanujan's level of insight. $\endgroup$ – Daniel R. Collins Jul 10 '18 at 14:00
  • $\begingroup$ @DanielR.Collins, maybe not get to Ramanujan's wizardry, but learn some. Just like Pólya's book on problem solving is a help. $\endgroup$ – vonbrand Jul 10 '18 at 18:00
  • $\begingroup$ @TommiBrander I agree that the question is "about mathematics education, broadly interpreted," but it asks (1) whether mathematical discovery involves more than deduction and (2) if so, how undergrads can... "learn it." I believe this is way too broad - more than just a "tad"! $\endgroup$ – Benjamin Dickman Jul 11 '18 at 2:27
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Polya's classic was already mentioned by @vonbrand. It is full of insights.

Polya, George. How to solve it: A new aspect of mathematical method. No. 246. Princeton university press, 2004.


         


I hesitate to recommend the following as I have not read it, but it is in the same vein, albeit more algorithmically oriented:

Michalewicz, Zbigniew, and David B. Fogel. How to solve it: modern heuristics. Springer Science & Business Media, 2013. (Springer link.)


         


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