Barry Arrington at Uncommon Descent offers a succinct rebuttal of compatibilism, i.e. the view that our choices are fully determined and yet at the same time free. As Arrington points out, this certainly sounds like a contradiction.
The compatibilist defines freedom, however, as the lack of coercion, so as long as nothing or no one is compelling your behavior it's completely free even though at the moment you make your decision there's in fact only one possible choice you could make. Your choice is determined by the influence of your past experiences, your environment and your genetic make-up. The feeling you have that you could have chosen something other than what you did choose is simply an illusion, a trick played on us by our brains.
Compatibilism, however, doesn't solve the controversy between determinism and libertarianism (the belief we have free-will). It simply uses a philosophical sleight-of-hand to define it away. As long as it is the case that at any given moment there's just one possible future then our choices are determined by factors beyond our control, and if they're determined it's very difficult to see how we could be responsible for them. Whether we are being compelled by external forces to make a particular choice or not, we are still being compelled by internal factors that make our choice inevitable.
The temptation for the materialist (i.e. one who allows no non-material entities in his ontology) is to simply accept determinism, but not only does this view strip us of any moral responsibility, it seems to be based on a circularity: The determinist says that our choices are the inevitable products of our strongest motives, but if questioned about how we can know what our strongest motives are he would invite us to examine the choices we make. Our actions reveal our strongest motives and our strongest motives are whichever ones we act upon. But, if so, the claim that we always act upon our strongest motives reduces to the tautology that we always act upon the motives we act upon. This is certainly true, but it's not very edifying.
On the other hand, it's also difficult to pin down exactly what a free choice is. It can't be a choice that's completely uncaused because then it wouldn't be a consequence of our character and in what sense would we be responsible for it? But if the choice is a product of our character, and our character is the result of our past experiences, environment, and our genetic make-up, then ultimately our choice is determined by factors over which we have no control and we're back to determinism.
It seems to me that if materialism is true and all we are is a material, physical being, and all of our choices are simply the product of chemical reactions occurring in the brain, then determinism must be true as well, and moral responsibility and human dignity are illusions, and no punishment or reward could ever be justified on grounds of desert.
This all seems completely counter-intuitive so most people hold on to libertarianism even if they can't explain what a free choice actually is. However, they can only hold on to a belief in free will if they give up their belief materialism. Only if we have a non-physical, immaterial mind that somehow functions in human volition can there be free will and thus moral responsibility and human dignity.
