vanhees71 said:
According to the physical laws we know today, quantum theory can provide "true random numbers", i.e., numbers that are really indetermined. The most simple case to produce a (discrete) sequence of random numbers is to prepare a polarization entangled photon pair in a Bellstate (like the singlet state), which nowadays is easy using parametric downconversion. You can use one of the photons as "trigger, heralding the presence of the other photon". Then you are sure to get a truly random outcome determining the other photon's polarization (encoding, say, horizontal representation with 1 and vertical with 0, you get a "truely random" sequence of 0s and 1s).
Which ones, many laws of physics are known to be non-truths (e.g. Newtons laws). And what does it mean to say a theory can provide "true random numbers". It is the physical thing itself that does that. You can argue that the theory (or the randomness in the theory) cannot be replaced, but that in itself doesn't tell us about the physical thing itself. Unpredictability, or uncertainty, and randomness are not the same thing. Heck, even simple, stationary, bounded deterministic dynamical systems (e.g. the logistic map) are mathematically impossible to predict. You would need at least to (1) store infinite amounts of information, (2) process infinite amounts of information, (3) and compute iterative solutions at infinite temporal frequency; All that just to even get a time-invariant bound on accuracy at all, or even a reasonable bound on accuracy over a given fixed length of time. We could go deeper analyzing what we know about the thermodynamic costs of information storage, processing, and so forth.
If that isn't enough, even simple discrete computational problems cannot be solved, like a method to determine the outcome of the game of life for arbitrary initial conditions. The inability for us to make predictions (notwithstanding even bigger surprises than wrong laws of physics) is logically deducible from even our best non-physical axioms.
That said, what goes on at the quantum level, or beyond, is a mystery, and we cannot rule out weirdness that changes the limitations we assume based on our limited models of reality.
More interesting facts that are relevant to the discussion might be that (under classical models of computation) it is possible for a system to simulate itself, but not without an increase in time complexity. That is, supposing somehow we can get around all of the other obstacles and simulate our reality, we could not use it to predict the future, because the predictions will always come later and the system is open (we need the whole state to do the prediction). We could do a hypothetical simulation far in advance (assuming we have all of this power), but then it would still be impossible to know when/if a general hypothetical situation will ever arise. And all of this is true even if we are talking about systems with finite unbounded states. Again, I'm not sure to what extent quantum weirdness could changes these things. But it also is an interesting fact under-looked in the simulation hypothesis (which I won't go into because its off subject).
My main thought is that we cannot in general make assumption proof claims about reality (and what is really going on with photons and electrons and so forth). It's just a mystery. Perhaps, it will be possible to whittle away at what we see as randomness, through new models and assumptions, indirect measurements, logical deduction, and so forth, but (notwithstanding big big surprises) we will never be able to reach the bottom with any predictive model, and this is an issue that is independent of whether or not our perceived randomness is actually deterministic or not. Even supposing we could measure outcomes perfectly, and something seemed perfectly random, it would still be infeasible to tell the difference between randomness and deterministic chaos. At some point further intellectual inquiry into the matter isn't physics anymore.
