The idea that there is some "real" mathematical structure, and statements about other structures are nonsensical because "numbers don't work like that!"
So there *is* something that the real numbers map to in reality; our current model of physics, which is absurdly accurate and useful even if it turns out to be wrong. I don't see the "but it's not the base level" argument as having a point for the "reality" of real numbers.
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You're arguing that because X allows Y, where Y is known to be a simplified model that doesn't capture everything, therefore X is real? Yes, "real" numbers are conceptually useful and consistent. That doesn't imply they exist in the territory, it means they help with mapmaking.
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No numbers exist in the territory. If the "base" level of reality is fully described by integer mathematics, that doesn't mean integers are "real" and all the rest of the number systems are "not real". I think we have very different definitions of "real" when it comes to math.
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