[empath-nirvana]

Well, given that structuralism as a program didn't exist until basically the 19th century, you can look at the entire history of mathematics to see what mathematics without structuralism looks like. It's sort of hard to argue that the isomorphism between symmetries and permutations is "obvious" when group theory wasn't invented until the 18th century, and symmetry groups weren't formalized until the 19th century. Your comment is basically this:

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.

[klysm]

I think sometimes obvious concepts like symmetries are hard to distill into the appropriate mathematical language. I’m willing to bet the isomorphism there is obvious to most folks, but the expression of its mathematical essence is not.

[woopsn]

This is really an important point for mathematical philosophy. It is a single enterprise going back to ancient Babylon, China, India, Greece, Egypt, and before. The elementary meta-theory needs to be syntonic to mathematical activity and knowledge predating (or otherwise practiced without) formalism, structuralism, categoricity, platonism, etc.