For what it's worth, I agree. I can't get myself to care about any math that cares about the detailed properties of reals because I care about understanding rea

For what it's worth, I agree. I can't get myself to care about any math that cares about the detailed properties of reals because I care about understanding reality with math and that can't be relevant. It seems to me that it's not so much that physics uses the rationals, though, as that it uses something like "fuzzy reals", the reals up to a certain resolution. IMO it should be impossible for physics to ever use any sort of infinitely-precise property of the number system it's using, simply because the number system is arbitrary. The argument for this (besides "it's obvious") is that the only ways physics can ever make numbers is (a) counting things, (b) measuring things with something that acts like a yardstick, or (c) apply formulas to numbers you already have. The first only creates integers. The second only creates fuzzy real numbers up to the resolution of your measurement. The third propagates the discreteness of the first and the uncertainty of the second through whatever calculation you do. Hence, physics cannot deal with infinitely precise descriptions of reality. One might argue that, once you have a model, you can get infinitely precise within the model (e.g. computing infinitely rich Feynman diagrams in QFT). I have no problem with that. I just don't believe the precise differentiability properties of physical objects like fields should be allowed to be used in the model, because they seem unphysical.