# Hidden structures

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.