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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. 457.

  1. For an account of Ramanjuan and his ignorance of proofs, see Kanigel (1991, p. 92, and see p. 207 for the comment quoted from the great Cambridge mathematician, G.H. Hardy). With a finite number of finite models, we can always in principle determine whether an inference is valid. Mathematicians, however, have to think about the positive integers, 1, 2, 3, . . . , and there are infinitely many of them. How they reason about infinite sets is a mystery, which psychologists have yet to elucidate.
  1. Is the emboldened phrase true? If yes, how and why? Mathematicians have been doing this for millenia?

  2. If it isn't, please recommend books or papers?

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    $\begingroup$ I'm voting to close this question as off-topic because it doesn't seem to be about mathematics education. $\endgroup$ Jul 12 '18 at 1:24
  • $\begingroup$ Be aware that even some theorems of "finite mathematics" may require a detour through the infinite, e.g. see this answer to the MSE question "Math without infinity". $\endgroup$ Jul 12 '18 at 1:31
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    $\begingroup$ @JoelReyesNoche But a good answer clearly could be of great use in math eudcation, so one shouldn't be so quick to close. Why not be more constructive and help to reformulate the question to emphasize its pedagogical aspects? $\endgroup$ Jul 12 '18 at 1:38
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    $\begingroup$ A good answer to the question "Does convergence in measure imply almost everywhere convergence?" might also be useful to educators. The problem is that, as stated, it isn't a question about mathematics education. If you want to rephrase the question so that it actually focuses on the educational or pedagogical issues involved, I'll be the first to upvote. Until then, I've voting to close. $\endgroup$ Jul 12 '18 at 2:24
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    $\begingroup$ I suspect that, when posed by psychologist, the words "How do we reason about the infinite?" is in fact asking a different question than when asked by a mathematician. It may very well be a pedagogically valuable question, and probably won't be answered by "induction". $\endgroup$
    – Adam
    Jul 12 '18 at 15:05

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