Math is in language. In a similar way that we can have fiction and non-fiction, with different uses of language, you can have 'pure' and 'applied' uses of math. Using 'real' as the goal kind of misses the point. It's not imaginative, if that's the antithesis of real in this whole conceptual discussion. But, like fiction to non-fiction, it's still a public phenomenon that "touches" the world in language as a shared, communal, element in a form of life. Math is remarkably useful as a representation of the world and for us solving problems, but you can also give directions to a location to someone with just hand signals and barking like a dog. In a way, math is not special. But, the degree of accuracy that we've developed in math, however, is what makes it remarkable.

> Using 'real' as the goal kind of misses the point

It misses the point of the question "is math real"?

Tbf, i am actually sympathetic to the position that "realness" is not really relavent or a well defined term when it comes to what is basically a descriptive language of patterns and relations. But "is it real" is the question we started the thread on.

The study of math in language is called semiotic, and thanks to a bunch of french poseurs it’s become a joke word among the genuinely intellectually curious. Nevertheless, attempts have been made to tie it all together, most notably by the brilliant and mostly forgotten Charles S. Peirce.

Whence the pragmatic maxim:

  Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object.

Pragmatically speaking, math is no more or less real than that.

Okay, here is precisely where the Simulation Hypothesis becomes useful and interesting. I know, I know. But bear with me for a minute...

The universe, as we know it, is simulatable. In other words, it can apparently be reduced to mathematics in precisely those three senses I outlined above. (If it helps, you can imagine electrons and other subatomic particles as bits of information in a coordinate space, which are constrained to operate in accordance with rules that govern logical relations between things.)

What mathematics is real? Anything that would relate to that universe or any of its constituent parts, in _any_ meaningful sense. We don't need to have found a use for it, and surely there's a great deal that is still undiscovered.

What mathematics is unreal? Whatever is, a priori, absolutely unnecessary to the existence of such a universe -- or, even worse, would break a universe described in terms of logic were it somehow made manifest.

Given that logics (of all sorts) are subsets of mathematics, I'm not sure one can convincingly hoist (any) logic above all other mathematics to make arguments.