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."
Best Answer
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.