What's wrong with this information theoretic argument for free will? Forgive me if this is incredibly naive. I am an undergraduate studying mathematics and have studied almost no physics, but a friend of mine mentioned this argument and it's been bugging me since it seems to have some intuitive thrust, even if it's not rigorously set forth. In the setting of nonlinear dynamics, we know that particles in a many-body system that have a starting position even infinitesimally apart will eventually diverge in motion when subject to the laws of eg. gravitation, in classical physics. And so any predictions about the long term (asymptotic to infinitely far into the future) must have more and more detailed knowledge about their starting positions to be able to maintain accurate predictions. Mathematically, this means writing down or storing their positions to increasing degrees of numerical accuracy, manifesting as more digits to be stored in "memory" of any such system making such a prediction. To be able to use the laws of physics to predict the future in its entirety (as in Laplace's Demon), you require infinite precision, and therefore infinite amount of "memory" (in some abstract sense), which is not possible, given the universe is finite and the smallest thing that can serve as a quantum of information is a fundamental particle. "Therefore," there is structurally no way physics can be used to predict the future entirely correctly, and thus no way for any model of the universe to be deterministically correct. This was the thrust of his argument, and then he mentioned off-hand this suggests we have free will (I do not think that implication holds), which I found interesting.
My intuition says the trickery in the argument here is mostly semantic, since it's not stated formally, but I am sure physicists have something to say about this line of reasoning? Thanks!
 A: You seem to equate free will with a nondeterministic universe. This is reasonable, but there are more things to say about it.
First, we have no theory of the mind. Nobody know how it works. People are making the beginnings of such a theory. Come back in a hundred years and we may know more.
Given an initial state of the universe, the universe evolves in a way that is sensitive to initial conditions. So it is not possible to exactly predict a future state unless you know the initial state exactly and do an infinitely exact analysis. This is true. But in the short term, you can do a pretty good approximation. You can predict weather several days out.
While non-linear dynamics says the universe is deterministic, but sensitive to initial conditions, quantum mechanics predicts the universe is non-deterministic. The Uncertainty Principal says you cannot know the position and momentum of a particle precisely. Not that you cannot measure them, but that a precise position and momentum do not exist. From that, it follows you cannot predict the future state exactly.
Again it works out you can predict probabilities, the near future is more predictable than the far future, and big things are more predictable than small things.
So are you saying your mind is like this? Predictable in the short term? It is only mental states far in the future that are free? As the future gets closer, your mind gets more and more locked into a state that can be predicted from its current state? Many people mean something different when they think about free will. I am not telling you what you should think. Just suggesting thoughts.
Of course the mind is not a simple thing. The way it works is not straight forward, and neither is figuring out how it works. It isn't easy to make psychology an exact science. Here is an online course by Robert Sapolsky of Stanford. Note that as Dr. Sapolsky says, it is a topic full of politics and opinions. Never the less, I am finding it interesting.
A: I see two unwarranted logical jumps in this argument:

*

*Equating "unpredictable in practice" with "not deterministic", and

*Assuming that being "not deterministic" implies "free will".

The problem with the first, chaos-based jump is that, if we believe in our deterministic description of the universe (I'll get to quantum mechanics in a minute), then our ability to compute something or not doesn't change the deterministic nature of the description — the same way turning off the lights doesn't make the colors disappear from a picture in the room. So, chaos or not, in this description everything is (still) predetermined by the initial conditions.
As for the second part, and assuming for simplicity that quantum mechanics is intrinsically random, its problem is a fundamental one: It's hard to argue that randomness is the same as free will. When you trow dice (which we usually model as random, since it practically is) do you feel to be exercising your free will? Did you choose the 3 and the 5?
A: I agree with the excellent answers preceding mine. I would add the observation that a necessary requirement for free will is that thought is self-directing. Assuming thought is correlated in some way with the physical behaviour of the particles that comprise the brain, then free will requires that thought must be able to influence the physical behaviour and not simply arise from it. If all thought is a consequence of the behaviour of the physical constituents of the brain acting in accordance with the laws of physics, then it follows that thought is predetermined to the extent that any other physical process is.
