I'd say that's debatable. My admittedly limited understanding of mathematics is that the underlying truths that emerge depend on the axioms in play. My understanding of axioms is that of 'rules' to the mathematical games we play, and you can change which rules you follow and get different results.

https://en.m.wikipedia.org/wiki/Axiom

https://en.m.wikipedia.org/wiki/List_of_axioms

If we accept that in mathematics we're finding out about the outcomes of the rules we choose to follow, then I have no problem in thinking of mathematical discoveries as truths, but I'd say they are conditional truths rather than absolute truths. What are your thoughts on this?

Right, but there are things like surreal numbers that feel more like constructs- useful, interesting and so forth, but not discovered in the real world so much as a tool built out of available materials.

Math dances along the edge between discovery and invention.

> there are things like surreal numbers that feel more like constructs

The fabric of the physical realm usually doesn't care about anyone's feelings.

> but not discovered in the real world so much as a tool built out of available materials

The fallacy here is that you assign different valuation between discovering a thing through observation, and discovery through thinking.

For example, the Higgs boson was conjectured many years before it was first observed. Following your argument, I could argue that the Higgs boson was invented by us.

What's true is that we made up the Higgs boson, but that didn't change the unobserved existence of it. In the same way, we may have made up surreal numbers, but that doesn't discard the possibility of their unobserved existence out there somewhere.

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.

Amen. Well-stated.