Perhaps the fundamental logic behind pure math is "real" in the sense that it's "found in nature" (whatever that means, I don't see how we would prove that), an

Perhaps the fundamental logic behind pure math is "real" in the sense that it's "found in nature" (whatever that means, I don't see how we would prove that), and we're only applying this real thing to models that we've completely made up. e.g., modus ponens is real, but a Turing Machine is not. We can reason about it logically but we didn't discover it, we made it up. Maybe the same can be said about numbers.