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There is a lot of talk about "hidden structures" in the realm of mathematics: hidden structures in the ZFC system, hidden structures in the natural number system, and so on. Saunders Mac Lane poignantly formulated:

Mathematics is a network of hidden structures. [Mac Lane 1980, p. 362].

I'd like to know how this manner of speaking is to be understood concretely:

How and why can a structure be hidden?

Can this be made comprehensible to a layman?

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There are lots of times when a set admits more structure than you might have originally noticed. A pretty elementary example is when a combinatorial or set-theoretical device turns out to admit an action by a group $G$, making it into a $G$-set. This is what underlines the unreasonable effectiveness of group theory in combinatorics. So a laymen's example might be something as simple as the 15-puzzle or the Rubik's cube. Or maybe you run into a group that itself admits a "hidden" action from some other group, further illuminating its structure. Or maybe your abelian group turns out to be an $R$-module for some ring $R$ of interest. All of these phenomena have occurred many times in the development of mathematics.

On a (only slightly) different front, maybe you find that the set of points on a geometric curve admits a shockingly important group structure. Ditto for other geometric objects like the space of line bundles on a projective variety. Or maybe a topological space (or a group) admits the structure of a Lie group. Or maybe a space parameterizing some interesting collection of objects turns out to actually be a fine moduli space for such objects.

The list goes on and on. In fact, I think it would be a reasonable stance (among many other valid such stances) that the search for hidden structures is the process of doing mathematics.

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Thanks! I do have to carefully think about each of your examples. Having deeply understood only one of them would mean a lot, wouldn't it? –  Hans Stricker Feb 23 '12 at 0:45
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"the search for hidden structures is the process of doing mathematics" -- nice! +1 –  Damian Sobota Feb 23 '12 at 1:23
    
@HansStricker: Well, I'll let you know once I deeply understand any of them. :) –  Cam McLeman Feb 23 '12 at 2:54
    
@HansStricker: Seriously, though, understanding the role of group theory in combinatorics is not particularly difficult (google, for example, applications of the orbit-stabilizer theorem). I think that'll give you a feel for what I was trying to convey without having to, e.g., learn sheaf cohomology. –  Cam McLeman Feb 23 '12 at 2:56

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