[nicklecompte]

The point is that if you are interested in the structure of finite sets, then there are structure-preserving isomorphisms between the vertices of a triangle and the set {A, B, C} so that (for example) permutations of the set and reflections/rotations of the triangle are coincident.

But if you're interested in the structure of plane geometry then there is no such structure-preserving isomorphism because any map into a finite set will "delete" information about side length and angles.

The structuralist idea is that interesting mathematics can be "most easily" found by considering structure-preserving isomorphisms and not the structures themselves. In particular dealing with the structures directly can obscure the mathematics you are trying to discover, e.g. dealing with a full Euclidean group when the dihedral group is all the problem requires.

[082349872349872]

If you want, it's even possible to fully commit to the isomorphism point of view by saying that you'll represent the structures themselves by their identity isomorphisms, but then if you "delete" the little tag telling you which particular endo was the identity (would a physicist say "up to phase"?), you might discover other interesting things...