Do uncomputable functions actually "exist" in reality in any meaningful sense, or are they just mathematical abstractions?
BTW - The field trying to deal with defining what exist means is called ontology:
This seems unlikely to me. Functions are a mathematical abstraction. Almost all of those functions are uncomputable, but what does the universe care? In what manner could the universe be organized in such a way as to depend on an uncomputable function?
Perhaps a better question for my above post would have been:
Can we imagine any empirical experiment to distinguish between a universe that depends on uncomputable functions and one that does not?
Edit: And if not, in what sense can the universe actually depend on an uncomputable function if we can't distinguish?
This follows from the fact that set of all Turing machines is countable (you can write an algorithm that lists all Turing machines), but set of real numbers is uncountable. So, by Cantor's diagonal argument you can prove that there are real numbers that can not be computed.
My guess is that universe can not depend on real numbers (i.e. real numbers are not real) because real numbers introduce infinity related breakdowns in physical models. For example, if there exist a physical state can encode arbitrary real number then that would allow to store infinite amount of information. But it is not possible to prove that something does not exist, see arguments related to Russell's teapot.
https://en.wikipedia.org/wiki/Computable_number
Thanks for the interesting discussion!
Such concepts cannot appear in a finite universe.