Great question, Trent!
This is what some people at the end of the 19th century thought. If we had Newton's laws and all the initial conditions, we could predict everything! We could (in principle, not in practice) run our computations backwards and see where we came from and how the universe started!
Advances in physics in the twentieth century showed that nothing could be further from the truth. Quantum mechanics predicts that the outcomes of experiments are truly random, but that the distributions of possible outcomes can be predicted precisely. Quantum fluctuations even have an impact on large scale structures in the universe -- fluctuations at very early times after the Big Bang created local density distributions which were different in some places than others. These are left over in the cosmic microwave background radiation and also in the deposition of galaxy clusters throughout the universe.
The sots of things that quantum mechanics allows to be specified- quantum states- don’t even have precise values for all the variables (position, momentum, etc.) that would go into a classical calculation. The perceived outcomes of quantum processes are not the same as the outcomes of the deterministic quantum equations, so somewhere in the process there’s a chance element that enters. Even if you say, as some modern interpreters of quantum mechanics do, that the whole ensemble of non-random outcomes really exists, any particular experience is just a random-feeling piece of that whole. (mbw)
Another feature which became apparent in the 20th century (starting with the work of Poincare, and notably that of Lorenz), was that even classical systems obeying Newton’s laws exhibit chaotic behavior. That is, if the initial positions and velocities are known with some uncertainties, then as time goes along, the uncertainties get magnified. The same system with even a tiny change in its initial position or velocity may end up in a wildly different state after some time. This kind of behavior is seen in mixing fluids, the weather, and in some other kinds of mechanical systems.
Couple this with the impossibility of specifying initial conditions and I’m afraid the plan of predicting everything from scratch becomes hopeless. We can try to approximate as best as we can, in order to make practical predictions about the weather, the climate, and how things behave.
p.s. On your other question, about ’free will’, we don’t know what to say. Pure quantum randomness wouldn’t help because whether or not it’s ’free’, it ain’t ’will’. Maybe free will is a sensation that makes sense within the physical world but doesn’t really have anything to do with ultimate causal patterns.
(published on 10/22/2007)