Is arithmetic empirical?
One of the questions I've been wanting to think about (in part because of my interest in the Quine-Carnap relationship) but haven't really got around to yet is: Is there any important sense in which arithmetic is empirical? I know there is some good literature on the subject, but I've thus far only perused it without really digging into it.
For me, one consideration that makes me think it might not be crazy to think of arithmetic as empirical is what happened with geometry and general relativity. If Einstein can show that the space in which we live is non-Euclidean, isn't it at least imaginable that some future scientist will show us that the 'true' arithmetic of our physical world is non-classical (which I suppose means: it does not obey the Peano axioms). [There could still be a mathematical structure that obeys classical arithmetic, just as Euclidean space is still a mathematical object that obeys all five of Euclid's axioms.]
However, I've always had a hard time imagining what possible observation could cast doubt on classical arithmetic. In last week's Science, there's a report that at least might merit consideration as a candidate. Researchers found that if you add one photon to a light beam and then take one away, you observe a different end-state than if you reverse the order of operations, i.e., first remove one and then add one. In other words, x + 1 - 1 does not equal x - 1 + 1. Even stranger, the authors find that "under certain conditions, the removal of a photon from a light field can lead to an increase in the mean number of photons in that light field," that is, (roughly) that x-1>x. The summary and background for non-specialists is here, and the full technical report is here (both behind subscription walls).
Now, this effect depends on the failure of commutation relations ubiquitous in quantum mechanics, so it is quite possible that this in no sense makes arithmetic look empirical. But I'm not 100% sure about that. Any thoughts?