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.

Reuben Hersh

I think aka "∅" therefore I know I thought aka "{∅}" therefore I know I knew I have thought aka "{{∅}}"

and ...

boom! The entire Math system is imported. (BTW, limitations known as "computation theory" is also introduced)

So maybe we are actually mathematical being on a manifold named as "real world". To me it is more concise and profound than "philosophy". As we are real, so do all mathematical objects.

One fish says to another fish: "The water's nice today." and swims off, the other fish says "What's water?". Your entire mathematical world view is so permeated with the language of structuralism that you can't see it any more.

https://en.wikipedia.org/wiki/Representation_theory_of_finit...

"The rough idea is to define an algebraic theory as a category with finite products and possessing a “generic algebra” (e.g., a generic group), and then define a model of that theory (e.g., a group) as a product-preserving functor out of that category."

https://ncatlab.org/nlab/show/Lawvere+theory#the_theory_of_g...

You can read Gower's paper on "The two cultures of Mathematics" at: https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf