Math is an invention of man it doesn't exist in the real world. There is matter and energy and they interact. Quantities of stuff exist but there are no numbers. That is extrapolation and modeling with man-made tools. Math decides nothing as it is not a fundamental aspect of reality.
I see this from time to time, and certainly opinions can differ, but I tend to cringe when I see this idea pushed. From my perspective, math is not an invention of man, but a discovery. Math is an intrinsic part of the universe, that we continue to discover and piece together. Certainly we have invented the symbols and the terminology, but the underlying concepts...these laws, these truths exist whether we discover them or not, whether we give them names or not.
Math dances along the edge between discovery and invention.
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.
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.
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?