(I’ve also been wondering for a few years whether it is possible to make a version of quantum mechanics which works with the wavefunctions taking on values from a (very large) finite field instead of the complex numbers. My impression so far is “probably not” because you need exponentials, and the multiplication group of a finite field doesn’t have a nontrivial group homomorphism to/from the additive group. Also, for the simple way of defining differentiation for functions over finite fields, has 0 as the only eigenvalue of differentiation, and I’m thinking that the other definition I made up for it also does. I think I saw some definition of differentiation for finite fields which maybe does have a non-zero eigenvalue, Yeah found it, https://arxiv.org/abs/1501.07502 but, this doesn’t have the “derivative” of a constant be 0, so I’m not sure that it would be suitable either.

So, my guess is that probably one can’t make quantum mechanics work with wave functions that have their values all from a finite field. Still going to keep looking a bit more though. )

Maybe your intuition provides an analogous but different conclusion that may be more suitable to novel conceptual models of reality. Pursue your intuition I’d say. I’m in a similar path.

Oddly I find music to be a good hint of reality lately. I wasn’t too interested at musical theory but now digging into it the geometrical spaces it shadows is inspiring. Projective geometry allows continuity. Axiomatically it also allows arithmetic among other things. Very intriguing.

The unfortunate truth is that once you pick apart the definitions of deep things like logic and math, you see that we don’t even all agree on them. We like to think that because we can describe something with math it’s inherently true, but the answer is pretty much always up for debate.

We found flaws in basic set theory and had to move to the updated Zermelo-Frankel set theory because of it. People try and create new foundations of math all the time for varying reasons.

Anyway, I see so many people speak with such conviction about truth and objectivity, that it’s refreshing to see someone show humility about what we actually know about our universe. Lots is still up for debate, is the sad truth.