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I rather suspect that this must have come up here on MO already, but my handful of searches didn't turn up the thread, so...

I'm curious about examples of mathematical structure that seems to arise "from nothing." The example that motivates this is one that I was teaching today, namely, the central limit theorem.

I was trying to convey to my (business math) students how astounding it is that the sampling distributions of the mean of a uniformly distributed random variable approach a normal distribution as the sample size increases.

Out of complete randomness, very specific and rather subtle structure arises (if in the limit).

I'd be amused to see other examples of this perceived phenomenon in different areas of mathematics. Not just structure where it wasn't expected (which is quite cool, but ubiquitous), but structure that seems to "arise from a vacuum."

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I'm not sure I understand the distinction between structure where it wasn't expected and structure that seems to "arise from a vacuum." Could you give some more examples or non-examples? –  Qiaochu Yuan Apr 28 '11 at 6:27
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It seems to me that the unexpected is often a hallmark of good mathematics. Obvious (and technically easy) results are called exercices. –  Roland Bacher Apr 28 '11 at 8:00
    
It looks like you might be interested in universality phenomena. T. Tao has written a nontechnical article: terrytao.wordpress.com/2010/09/14/… –  S. Carnahan Apr 28 '11 at 9:00
    
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This may fall under the Ramsey theory answer, but the fact that a random, countably infinite graph has its isomorphism class determined with probability one (the Rado graph) seems like it might qualify. –  Charles Staats Apr 28 '11 at 14:29
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6 Answers

A nice example of seemingly trivial structure that hides highly nontrivial structure is that of a projective space. Such a space consists of "points', "lines", and "planes" with the obvious properties: there is a unique line through any two points, any two planes meet in a unique line, three points not on a line lie on a unique plane, and so on.

Surprisingly, any such space has an underlying skew field which coordinatizes the space so that lines and planes have linear equations. This due to the fact that the Desargues theorem holds in any projective space. Hilbert (1899) showed (in a highly roundabout way) that one can then define sum and product of points, and use the Desargues theorem to prove their skew field properties.

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This is correct as stated, but it's worth pointing out explicitly that it depends on interpreting "projective space" as having three (or more) dimensions, as explained in the first paragraph. Projective planes can be non-Desarguesian, in which case they cannot be coordinatized using a skew field. –  Henry Cohn Mar 11 '12 at 16:30
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I'm not exactly sure what you're after, but one could think of Ramsey theory as saying that any large enough structure will necessarily contain an orderly substructure. Or, even more loosely, that order is unavoidable in a large enough chaos. So I suppose the following would be examples of answers to the question:

  • Ramsey's theorem;
  • van der Waerden's theorem;
  • the Hales-Jewett theorem;
  • Szemeredi's theorem;
  • the Green-Tao theorem.

And there are many more in this vein. Especially infinite versions of such theorems seem to match nicely with your example of the central limit theorem.

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I was about to suggest Ramsey Theory, a highly appropriate answer, given the asker. –  Gerry Myerson Apr 28 '11 at 9:03
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Thurston's geometrisation conjecture might be a good example here. If you start with a few simple 3-manifolds, you can combine them to create any 3-manifold you want.

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The universal friend theorem:

Suppose that in a finite graph, every pair of vertices have exactly one neighbor in common. Then the graph must a windmill, i.e. there is a hub vertex that neighbors every other vertex, and the remaining vertices are connected to each other in pairs.

Also sometimes phrased as, "in a party, every two people have a unique friend in common. Then there is a person at the party who is friends with everyone."

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