I don’t think I agree with Allan’s critique that you have missed the mark on how science works. We indeed DO eschew the totally unpredictable, and do aim for things that seem potentially able to be predicted. So we are ignoring many well formed questions because we sense we cannot predict them.

But both in your number theory example, and in the real world there still seems there is structure that needs explanation. After all if digitally encoded the entire universe and chose a random next step function there would never be something like the reals to pull out of that mess. Indeed the larger the universe, the greater the chaos of that random function. Our universe is indeed big, and merely the repeatability of random physics in a position independent manner is already a MASSIVE amount of global structure, relative to some random next state function. I think (not sure) that one could even prove that randomly generated next state rules could never generate any global structures that regular. Things get sticky since one must presuppose a language for the expression of that structure, and you would argue that our choice of this language was already tainted by the universe we live in.

I am sympathetic, but it still seems a stretch to assume something like a term rewrite grammar (one of the least commitment mathematical models of computation) is derived from our universe. Well it is derived from a notion of time, state and persistence. I accept this level of circularity.

But starting from some random universe it still seems unlikely to me there would be universal patterns to be found in any subset. (e.g. a pattern that applied over an unbounded number of particles in an unbounded number of conformations.)

In the same way, if one started from random operators I don’t think the reals would pop out either. In both cases it appears that God started with some regularities then buried them among complexity. in both cases we ignore the complexity to find the regularity. But this leaves the central question: why those regularities?