>>29
You can't ``prove'' it. The scientific method can only speak of observations and facts. It can show what something isn't.
Quantum particle movements are statistically random.
Synapses? They may exhibit some random variation like any physical process, but they are not ``random'' (what would this even mean? synapses are formed by known biochemical processes and their function is understood, despite the fact that the system itself (brain, its regions, minicolumns) is a lot more complicated).
Now back to quantum particles, this depends on the physical realization of our 'laws of physics'. Do we have branching like in MWI where from a state you can get a large other number of states? In which case the observer will discern randomness by being aware of the "state" he is in (not possible for an observer except by interaction with the world). See my previous example for an explanation.
If you don't subscribe to MWI then there would be some underlying rule which explain why the particles are in the states that they are (a hidden variable theory), in which case there is no physical randomness, just pseudorandomness.
The question lies in wether we want to group pseudorandomness together with subjective randomness of perception of being in a particular universe. We may even consider our laws of physics of being such a random factor, "Why these particular laws? Why not other laws? Why not a cellular automaton? Why not some different physical constant? Why not a completly different mathematical structure (we know enough which are different enough)? It may be said that the answer to this question would be wether the structure could result in self-aware observers as ourselves contained within this structure and being aware of the structure and themselves, however with a bit of thought (and possibly one day experimental verification) one would realize that such structures are possible, thus our own shouldn't be the only structure capable of having self-aware substructures/observers.
Even in the case if MWI is false and the worst possible conclusion is true (no multiverse due to ours being the only consistent structure(why would this be the case baffles me)), there would be randomness in the conscious observers. The fact that you are conscious of a particular person with particular atributes is enough to qualify as random. Unless you believe that only you in the whole of existence is conscious and everybody else is a philosophical zombie (a full fledged solipsism) and nothing else exists outside of that, in which case, you could state genuine randomness doesn't exist, but how could you be so sure that your position is correct?
I wonder if any of this matters to us in any practical way. A good PRNG with a state hidden from the observers' eyes is as good for practical use as genuine randomness.