I don't know how you define "determinism" but perhaps in the sense that future is determined by the past, right? The initial state of a system and the forces acting on it determine its state at some future time. Right?

But universe doesn't work tat way. Universe is non-deterministic because quantum effects happen randomly. They cannot be predicted, only their probabilities can.

That does depend on your interpretation of quantum mechanics: the wave-function is still entirely deterministic. Non-determinism is only induced by 'collapse', but it's not mandatory for this to ever happen: the many-worlds interpretation is basically the result of saying it doesn't, and is completely deterministic. (In fact 'collapse' interpretations are not particularly popular theoretically, because it's actually a blurry line of decoherance: it's just generally convenient for interpreting the outcome of a calculation, and you can assume it's a close enough approximation to experiment once your system has interacted with your detectors).

>the many-worlds interpretation is basically the result of saying it doesn't

Few problems with this, the state that a particle can be in after the collapse of the wave function follows a probability distribution, so if all states are equally real then why does it seem like some states are more likely than others? Why would probabilities be such a powerful mathematical tool, and how would it work in a MWI.

If all points of the universe branch out into several universes after each particle interaction, where does all this energy come from? If all branches are equally real why even have the wave function in the first place?

> if all states are equally real then why does it seem like some states are more likely than others?

Because some states are more probable than others. That is what the wave-function tells us. It does not give equal probability to every possible outcome.

> Universe is non-deterministic because quantum effects happen randomly.

One of Penrose's foundational arguments is that the random element of quantum collapse is deeply unsatisfactory; he thinks it's a defect in the theory that must eventually lead to its replacement by a better theory.

Einstein had a similar conviction. But from all we know now quantum description of the world is the best theory we have. Particles behave following probability distributions. There are laws (of QM) they follow, but that does not mean the outcome is deterministic. We cannot predict the outcome of quantum experiments, only their probability based on the previous state of the world. And that is not because our instruments are not sharp enough, it seems indeterminism is a fundamental part of quantum reality.

Now if the tiniest elements behave that way non-deterministically and we can assume they do have an effect on the world around them, then it seems to be the whole world is more or less indeterministic. Probability distributions can be observed that's all.

Further credence to the idea of non-deterministic world is given by Chaos Theory. A butterfly flapping its wings in Africa can cause a storm in Kansas. Therefore the indeterminism at the atomic level is amplified to the macro-level, as predicted by Chaos Theory.

And if that is the case then we can assume that what happens in our brains is also indeterministic. Thus you could say we have "free will", our wills and thoughts and likes are a product of something indeterministic. Thus you can say that "our will is free", more or less. I'm not sure what "Having a Free Will" would mean, but I can understand that our will is not wholly determined by history.

> from all we know now quantum description of the world is the best theory we have

Penrose would agree!

I think that in The Emperor's New Mind, he sets out a scheme for classifying the quality of different theories, e.g. quantum mechanics, darwinian evolution. He classifies quantum mechanics (quantum chromodynamics?) as a "Superb" theory, because of the accuracy and precision of the predictions it makes. But he still says it's fundamentally flawed.

> We cannot predict the outcome of quantum experiments, only their probability based on the previous state of the world. And that is not because our instruments are not sharp enough, it seems indeterminism is a fundamental part of quantum reality.

I know very little about QM, so I could be completely off here, but, not being able to predict the outcome of an experiment could be completely different to the outcome being deterministic.

Lay person point of view. Rewind the universe by an hour and replay. Does QM have anything to say about that? I'm inclined to think not, but I really don't know. In other words, does QM say:

1) The result would certainly be the same.

2) The result could certainly be different.

3) The theory doesn't tell us one way or another, meaning it doesn't exclude either 1 or 2 being true.

If it is 1, then we have determinism. If it is 2, then we have randomness. I don't think either of those are compatible with what people think of as free will (mysterious ability to choose outside of physics).

> Rewind the universe by an hour and replay.

I think it's impossible to rewind the universe, so no sensical theory should say anything about that. A theory that theorizes about what happens if something impossible happens would seem to be waste of time. :-)

But I think the answer is: The outcome of a quantum experiment is random. So if the same experiment is repeated, it should most probably give a different answer, unless the answer is always the same, or one of only a few possibilities.

And there's the rub: it's always about the beauty.