The Higgs fell out of math that was originally designed to model the world. It was a testable prediction of a physical theory. The math could have been wrong. The math probably is wrong, except about other things than the Higgs.

I'm not valuing anything. I'm saying that if I build a toy axiomatic system, there's no reason a priori to believe that it resembles anything in the physical world. Such a system (say, Peano arithmetic) doesn't actually make any predictions about the real world at all, and cannot be falsified scientifically.

It's like saying, I painted a picture of a dog, therefore I have discovered a dog, by proving that it could exist. Except your depiction is of a 100-foot-tall mecha-dog that definitely doesn't exist. Mathematics is sometimes (though by no means all or most of the time) more like painting than it is like science.

Ex: the Banach-Tarski paradox is almost certainly not describing anything physical.