For a long time I've been puzzled—and I still am—by this. Our physics is based on the real numbers (complex numbers, vectors, Banach spaces—all that is built out of real numbers). After all, there are non-standard numbers that can do everything real numbers can. So what reason do we have to think that "the" real numbers are what the world's physics is in fact based on?
I think one can use this to make a nice little argument against the possibility of us coming up with a complete physics—we have no way of telling which of the number fields is the one our world is based on.
Do you think that the question about the completeness of physics is one for physics, philosophy of physics, or ontology, or some other area?
ReplyDeleteA uniqueness objection against the completeness of physics, perhaps.
ReplyDeleteDear Alex,
ReplyDeleteWhy is this an argument against the incompleteness of physics? Might it not be the case that for most physical purposes, the non-standard numbers contain superfluous structure, and are perhaps genuinely helpful for modeling a few situations? I can't actually see what using non-standard analysis would add to most physical theories.
Historically, people have debated the question of whether QM can be formulated over the reals. Would you say that such a formalism (with all the same physical predictions) was physically distinct from standard QM?
Nic