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I did not mean to imply that there was a simple empirical connection between quantum mechanics and the axiom of choice. Suppose AOC is false, then all we can conclude is that there is at least one (infinite-dimensional) vector space V without a basis (good luck constructing it). We do not know, however, that V represents the configuration space of any quantum system. Even though we often assume a quantum system has a basis without displaying it explicitly, we have not contradicted anything unless we know that the "defective" space V in fact represents a real quantum system (which would force the dichotomy I mentioned above).

But as you mentioned, another "test" of mathematical results is their aesthetic value. Why do we place so much emphasis on the ring of integers? Of course it is because the integers are suitable and applicable, as you say. But there is not only one model of arithmetic, and so there remains the question of whether we can rule out various models based on empirical testing. Similarly, there is not only one model of set theory, so this leaves open the possibility that some of these models will be shown to be incompatible with empirical facts.

That all depends on what you mean by "test it." You alluded to validity above, and I agree that the validity of mathematical theorems are independent of any empirical fact. The theorems are all tautologies after all; although we often leave out the antecedent of theorems like "the square root of 2 is irrational" we could choose to make the hypothesis explicit if so desired.To put it another way: if you can test it experimentally, it's a natural science, not (pure) math.

But as you mentioned, another "test" of mathematical results is their aesthetic value. Why do we place so much emphasis on the ring of integers? Of course it is because the integers are suitable and applicable, as you say. But there is not only one model of arithmetic, and so there remains the question of whether we can rule out various models based on empirical testing. Similarly, there is not only one model of set theory, so this leaves open the possibility that some of these models will be shown to be incompatible with empirical facts.

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