Isn't it likely that the better fit of probability to quantum phenomena is not due to some actual reality of said phenomena fitting probability but due to lack of knowledge and ability to experiment in the same way with them?
To be more specific/clear: I think that because math is not cosmic it should follow that even quantum phenomena aren't "really" having to be different than non quantum ones. They just are picked up as different by human observers, due to math (and other human mental stuff).
No. A century ago this was a valid view, but now we have the knowledge and ability to experiment with quantum phenomena very precisely (and arguably more precisely than we ever could with dice). The increased accuracy has only led to more precise confirmation that there is something strange going on with quantum mechanics that does not fit very well into a deterministic world view.
I don't know quite know what it has to do with Copenhagen vs Many worlds, but this first quoted sentence is exactly what Einstein meant when he said "god does not play with dice" as far as I know. Einstein believed that further discoveries would lead the probabilistic elements in contemporary quantum theory to fall away revealing an underlying, deterministic reality.
Exactly. And he was wrong. Unfortunately, it is a bit more complicated than that, so I cannot straight out confirm or deny the following:
@uppi my impression was that Kyriakos is basically rehashing Einstein's old claim that "god does not play with dice" and that experiments have since more-or-less proven that quantum events are inherently probabilistic, they do not merely appear so due to deficiencies in theory or instrument, can you confirm or deny?
There is a thin line between "proven by experiments" and "reasonable conclusion but not actually proven", so excuse me if I get too technical:
As you said, Einstein postulated that the probabilities in quantum mechanics must be due to some underlying unknown theory - there are some hidden variables, which determine the outcome and which we cannot measure (at least for now). And due to relativity, this theory must be local, i.e. constricted by the speed of light. Much later (after Einstein's death), John Bell showed that any theory with local hidden variables must have a different result than quantum mechanics in very specific situations. The nice thing about this, that if you could create such situations and measure the outcome, you could rule out either quantum mechanics or all theories with local hidden variables. And for the next 50 years, physicists sought to make such experiments and started to come up with results in line with predictions of quantum mechanics. The first ones had some obvious deficiencies, but in the last decade they were improved to the point where theories with local hidden variables can be ruled out beyond reasonable doubt (for my definition of reasonable).
However, this does not completely rule out deterministic theories for which quantum mechanics is just an approximation, because they could be non-local. This would require faster-than-light effects, violate relativity, lead to causality problems, but cannot be ruled out conclusively. So far, all experiments testing quantum mechanics and relativity have confirmed these theories and we have not found a way to enable faster-than-light communication (and it is proven to be impossible within quantum mechanics), so arguing for non-local theories is not very convincing - but not impossible. For example, an outside entity knowing or even manipulating the outcome of quantum experiments, or the universe just being a simulation on a very powerful computer cannot be ruled out (and might be impossible to rule out in principle).
The experiments also don't tell us how exactly we should interpret quantum mechanics. There are many possibilities to form a world view, which is compatible with quantum mechanics and cannot be ruled out. However most of them contain some component of probability (except for those that try to build a non-local theory, but I don't think those are very convincing).