Reading about Computable numbers I wondered if there is any physical experiment that returns non-computable numbers or if there is any physical theory that needs non-computable numbers. Because if that would be the case, we would have prove that the universe is not "simply" a simulation inside a Turing machine.
Bonus question: Could it be that classical mechanics is computable on a Turing machine but quantum mechanics is not?
Reading about Computable numbers I wondered if there is any physical experiment that returns non-computable numbers or if there is any physical theory that needs non-computable numbers. Because if that would be the case, we would have prove that the universe is not "simply" a simulation inside a Turing machine.
Measurements and experiments result in rational numbers (because we record finite precision decimal numbers as results), which are all "computable".
Physics theories use real numbers and their continuity in formulation of differential equations. However, this does not prove that the universe is not a discrete state simulation or reverse. Continuity in physics theories has little direct implication on whether that is how world really is, because with sufficiently short steps, this distinction makes no testable difference.
EDIT I should add that reproducing all properties of physical laws using a computer model is very difficult. Any discretization always introduces some behaviour that in some way does not respect physical laws. For example, discretized space, howsoever dense, cannot be exactly isotropic. Physical laws such as Maxwell's equations for EM field or Einstein equations for gravity, if we take them to be valid exactly, can't be reproduced exactly with discretized models. This does not mean universe can't be discrete, only that such discreteness should manifest in some ways that would contradict those physical laws. If that is so, those manifestations are so far too small to detect, or elude us.
As Sebastian Riese points out, quantum mechanics is computable. Interestingly, classical mechanics is known to be non-computable. If classical mechanics were valid on all length and time scales, then you could construct a so-called "rapidly accelerating computer", which is a computer that accelerates such that the next clock cycle takes half the time to execute as its previous clock cycle. This means that an infinite amount of computations can be done in a finite time. One can then verify the truth of theorems and also verify whether that theorem then known to be true or false is actually provably true or false.
E.g. the Riemann hypothesis can be false, in which case it is provably false (just point of that zero that is not on the critical line), or it is true in which case there may or may not exist a proof for it. A proof is just an argument of finite length that demonstrates that it is true and such a proof may not exist.
The rapidly accelerating computer can simply check out all the zeros one by one and be done with the countably infinite number of zeros in a finite time and then return the result of whether or not they are all found to be on the critical line. Also, it can generate all proofs of theorems using Hilbert's proof checkers algorithm and then check if it ever encounters a proof of a theorem demonstrating that the Riemann hypothesis is true.
But of course, we know that classical mechanics is false. But while quantum mechanics is computable, this is only when you keep track of the unitary evolution of an isolated system. If you perform measurements, then in no-collapse interpretations, one assumes that all possible measurement outcomes are realized, and it's this entire set of measurement results that is computable. What is not computable are the individual results you observe in some particular sector. So, if you repeatedly measure the z-component of a spin polarized in the x-direction, you'll get a random set of measurement result. If spin down is replaced by 0 and spin up by 1, and you put a decimal point ( or is this called "binary point"?) in front then the number between 0 and 1 you get is non-computable.
Classical Newtonian mechanics is uncomputable because the expression for gravitational force, $Gm_1m_2/r^2$, contains the uncomputable operation $/r^2$. If you cannot bound $r$ away from 0, then you cannot compute division.
General Relativity, by contrast, is computable, because it contains no such division.
See "Church’s thesis meets the N-body problem" by Warren D. Smith for more details.
I am not sure about quantum mechanics. I too would like to know if time evolution in the $n$-body problem is computable. (This blog post and moreover "Lazy Functional Algorithms for Exact Real Functionals" by Alex K. Simpson describe how to compute exact integration of computable real-valued functions. I'm not sure if there's a similar solution to differential equations...)
Edit: the same paper claims that Quantum Mechanics is in fact computable, and cites "Church’s thesis meets quantum mechanics" by Warren D. Smith. I haven't looked at the explanation yet though.
