Did not Bell, who was a pretty accomplished mathematician as well as physicist, pretty much totally prove that there was no possible hidden variable theory that was compatible with the observations of QM?
You know, a fascinating question arises from this whole non-locality/Belle’s Theorem business: Since it states that quantum entities (photons, etc) which were once ‘entangled’ with each other at the subatomic level, then later separated, theoretically to unlimited distances, somehow remain in a state of instantaneous quantum connection with each other, then, since the entire universe was once quantumly entangled (within the big bang singularity), might the universe today be in some sort of instantaneous communication with itself? A single quantum entity? I once asked this question to Roger Penrose (Hawking’s mentor) at a lecture he gave a few years ago at Columbia University. He really loved the question which seemed to have never crossed his mind. Unfortunately, he went off into some related discussions which I didn’t quite follow.
In the physical world, all events are determined, and what appears as “undetermined” is just our ignorance of the minutiae of physical influences involved. The principle is this: every physical entity will be have in the same way in the same context.
That's where QM indeterminism is unique, and it's not exactly the kind of indeterminism people think it is, and includes an element of ignorance, but not in the sense that we just do not know, but cannot know.
The indeterminism, in the case of QM, happens to be very precise. We know exactly the degree of indeterminism in quantum events—it is always a factor of Planck's constant. It has nothing to do with what we usually call randomness.
While I happen to think the math is right, and that mathematically it will never be possible to “measure” a particle’s position and it's momentum simultaneously, there is a mistake in assuming that all characteristics can be described exactly in terms of mathematics. For example, we know there is an exact ratio between the legs and hypotenuse of an isosceles triangle, or between the radius and circumference of a circle, but all such ratios cannot be expressed exactly, mathematically. There is also the little matter of Planck's constant. As important as it is, no one asks why it is. It is just assumed because it, “fits the phenomena,” but no one knows (or even asks apparently) the explanation for why that number and not another.
Interesting, too, is the fact all this "indeterminism" is very determinisitic--such as the half-life of racioactive substances.
Mostly thinking out loud here. Forgive my rambling.