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I found this quote by Stephen Wolfram on page 1177 of his book A New Kind of Science.

Yet of the limited set of people exposed to higher mathematics, different ones often seem to think in bizarrely different ways. Some think symbolically, presumably applying linguistic capabilities to algebraic or other representations. Some think more visually, using mechanical expirience or visual memory. Others seem to think in terms of abstract patterns, perhaps sometimes with implicit analogies to musical harmony. And still others—including some of the purest mathematicians—seem to think directly in terms of constraints, perhaps using some kind of abstraction of everyday geometrical reasoning.

I understand what he means by algerberic and visual, but the other two are less clear to me, especially the part about "directly in terms of constraints". Can anyone shed some light on what he means by this?

MJD
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mwanxe
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    As it seems to me, the only one who can shed light on Stephen Wolfram's meaning is Stephen Wolfram. Unfortunately, I'm not sure he reads the questions in this forum regularly. Why not ask him directly? – Amitai Yuval Dec 18 '14 at 20:00
  • See constraint. – Lucian Dec 18 '14 at 20:10
  • @Lucian I know what constraints are, but I don't really understand how one can "think directly in terms of constraints" and how thinking in constraints can be ones primary way of looking at math. I try to think about math visually whenever possible, because that way it makes so much more sense than thinking purely in terms of symbolic manipulation, and I always assumed that the best mathematicans also thought visually, but according to this quote there are 2 other ways of thinking about math that I was not even aware of. – mwanxe Dec 18 '14 at 20:25
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    http://vserver1.cscs.lsa.umich.edu/~crshalizi/reviews/wolfram/ – Will Jagy Dec 19 '14 at 01:01
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    @WillJagy Ouch, that is a blistering review. – user_of_math Dec 19 '14 at 02:17
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    @WillJagy 's link seems to have moved to http://bactra.org/reviews/wolfram/ –  Jul 24 '16 at 22:15

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