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Before I begin, please note that I am not troubled by the votes to close of my question here, but I'm somewhat puzzled about them. The purpose of this meta thread is for someone to explain why the question isn't about research-level mathematics.

To save some of your time:

1) I realize that MO should be reserved for technical questions related to specific spots in research where people are stuck, or need a reference.

2) The question could be "morphed" into a general question about any area: what are some intuition pumps for e.g. Zoology.

3) It may not be desirable to provide answers that "oversimplify" technical subjects, as a little knowledge is a dangerous thing.

My justification for posting the question despite the above points is as follows: (1) If MO is only about technical questions and not about more general strategies for conducting one's life as a research mathematician, it loses some appeal in the community. (2) Even with the morphability of the question, an expert's effort to intuitively organize an important idea in a subject can be useful for awareness of ideas outside one's own research area and can be regarded as meaningful research work (these intuitions are shared during talks, but if you don't work in an area, you may never attend such talks). (3) Regarding oversimplification, it is always a danger, but the purpose here is to promote awareness across fields and to promote unity. We may assume that practicing researchers won't think they really understand a technical area when hearing a high level intuition pump like this.

Perhaps the problem is that the sheer number of these intuition pumps is too large. I argue that the level of idea I'm looking for should be sufficiently high that the number of such ideas will be relatively small. The deformation/rigidity example is pivotal in my subject and has had enormous influence, so having this high-level analogy gives immediate feeling of a central movement in a subject. I'm looking for such broad things, to get a general feeling of the movement of mathematics as a whole. Maybe the reason to close this question is that such analogies can be provided in a "What is?" article in the Notices. This said, I haven't found such descriptions there.

I also must say that some of my friends don't really like the example I cite given by Popa. I am also puzzled about that. I can't see how such attempts to make things very memorable and salient are not helpful to mathematicians.

I did ask the question as an exploration of how mathematicians attach "intuition" to their techniques, and wonder to what degree experts work to find intuitions like these, and so won't be offended if the question is closed. This said, I'd like to record a solid argument for why such questions are not of interest to the mathematics research community.

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  • $\begingroup$ It occurs to me that maybe I should have posted the original question on meta. :) $\endgroup$ – Jon Bannon Mar 21 '16 at 13:27
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    $\begingroup$ I find it unfortunate that there are four votes to close (with the most generic reason of all), yet not a single comment to explain the issue these users have. $\endgroup$ – user9072 Mar 21 '16 at 14:24
  • $\begingroup$ "It occurs to me that maybe I should have posted the original question on meta. :)" If I'd seen the question on meta I'd vote to close it and recommended to ask it on main, where it may or may not be on-topic. $\endgroup$ – user9072 Mar 21 '16 at 14:26
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    $\begingroup$ @quid possibly Jon was thinking of a meta post to vet the question, rather than to ask it here. $\endgroup$ – David Roberts Mar 22 '16 at 1:29
  • $\begingroup$ @David: Regarding questions of this type, would moderators prefer the vetting of such questions before posting? $\endgroup$ – Jon Bannon Mar 22 '16 at 14:10
  • $\begingroup$ @quid Actually, I thought Jon was kidding with this first comment, as indicated by the ":)". Jon: that's always an option. I'd like to add that I applaud your efforts to discern what the matter might be. $\endgroup$ – Todd Trimble Mar 23 '16 at 13:28
  • $\begingroup$ @ToddTrimble Maybe Jon should lift the tension what he meant with the comment. By now there are three interpretations. :) It would not have been the first time that this type of question (in a broad sense) is actually asked on meta, so I responded essentially at face value. Generally, I do not interpret that emoji as necessarily joking or ironic, it is just a smile, in contrast to :D or ;), say. $\endgroup$ – user9072 Mar 23 '16 at 14:57
  • $\begingroup$ @Todd: I had not thought of vetting the question here. I was, in fact, joking! $\endgroup$ – Jon Bannon Mar 23 '16 at 18:33
  • $\begingroup$ @DavidRoberts OP did not have this intention. See above. // Jon: Thanks for the clarification. And sorry for not getting the joke. I am afraid I still don't get it, but nevermind that. :-) $\endgroup$ – user9072 Mar 23 '16 at 20:02
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    $\begingroup$ @quid: I'm not a very good comedian. No apologies necessary! $\endgroup$ – Jon Bannon Mar 23 '16 at 20:26
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I didn't vote to close, and wouldn't have. But the premise seemed off base to me. Here is a purely personal reaction which might say more about me than anything else but might also explain why some people did vote to close.

1) I find the phrase "Mathematical Intuition Pumps" in the title off-base.

2) The question starts out with the claim: Experts have a store of high-level intuitive analogies that keep track of the various parts of an important argument or strategy. Some experts might, but a better questions might be "Is it common? In some fields more than others?"

3) Some of what you describe sounds to me like the way one might use metaphor to to convey the gist of a topic or technique to people not in one's field, but not necessarily part of the tool kit of "most" experts or the way they communicate with fellow experts.

You might look up the work of Hadamard.

You will find the word intuition mentioned there but it refers more to the leaps of insight which happen sometimes (with some people) after hard work (stretching over years in some cases) and unconscious "processing."

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  • $\begingroup$ Thank you for the answer! I am aware of the Hadamard essay on the nature of research in mathematics. I think your bullet (3) would be also very good, by the way (admittedly, aiming at the layperson was a stretch). I also like (2). Regarding (1), I didn't segue into that parenthetical in the introduction. You are probably aware, but I am referring to "intuition pump" as defined in philosophy: en.wikipedia.org/wiki/Intuition_pump. Perhaps you are right that this is off base, but I don't really understand why. $\endgroup$ – Jon Bannon Mar 23 '16 at 11:05
  • $\begingroup$ Would you mind if I incorporated some of this answer as a modification to the question? I don't really want to bump the question to the top again... $\endgroup$ – Jon Bannon Mar 23 '16 at 11:08
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    $\begingroup$ Yes, fine with me. I'll be interested to see the answers. I was just giving you my cranky first reaction to intuition pump. I'm not sure it is standard philosophy. It seems to be coined by Daniel Dennet in his 1991 book. Kind of a "thought experiment" which gives insight. $\endgroup$ – Aaron Meyerowitz Mar 23 '16 at 19:11
  • $\begingroup$ You are right. It may not be mainstream. I'll remove it (as its inclusion is a bit of a non sequitur.) $\endgroup$ – Jon Bannon Mar 23 '16 at 19:19

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