Is free will a thing, actually?

If he doesn't have free will, who does?

I always read backreaction with pleasure and one of the latest articles attracted my attention.

It was about free will. Free will is cool, really. This is a slightly, but only slightly, religious concept, so people from all faiths will discuss about it without much arguing. But it also philosophical, so hard core atheists will join the discussion. It has something to physics, since somebody in the room always bring up determinism. Everybody has it (or nobody, depending on who you ask) so everybody is going to discuss it. In the end, it does not matter that much, so discussions do not become too heated. And finally, it is the main topic of The Devil's Advocate, which is a great movie. I love it. Both free will and Al Pacino, of course.

Free-will has nothing to with determinism

That said, I would like to offer my humble contributions to the topic. The first objection to free will which always arise is determinism. What is determinism? The belief that if I would know position, velocity and everything else of all particles in the universe, then you would be able to predict everything which is going to happen in the future. It looks like this strong form of determinism would destroy free will. The first clear formulation in western philosophy of this is due to Laplace. Luckily enough, we don't have to discuss this because it is probably non-true. First, it is non practical due to the amount of chaos available in the world: small perturbations lead to large changes.

It is plainly wrong in the standard interpretation of quantum mechanics, since wave functions collapse contains a randomness. So, no physical determinism. 

Also, if determinism has anything to do with inference, it has been shown that you cannot do complete inference in systems where you can perform standard logics. Interestingly, this mathematical proof requires Cantor's diagonal argument. Which one of the single most important pieces of maths, so make sure to follow the Wikipedia link, please. I will come back to this in the end.

Given that determinism is wrong, we can set it aside in discussing free will. Yes, in a deterministic world, there is no space for free will. In a non deterministic world, we do not know. Maybe we are free, maybe we are random. Maybe we are merely non-computable.

Free-will has nothing to do with predictability of human choices

In the later years, there has been a copious amount of papers about predicting human choices before the reach consciousness, due to continuous improvement in brain imaging methods. See here, for example. This has caused a lot of stir among people. I do not know exactly what about: the will can well be free even if it is not conscious. You can make a free, conscious decision with your own will, but your will could also make a free decision without notifying your consciousness organ in advance. Free will is not the same as rational choice, in the end.

What exactly is free in the free will?

This is the very tough question: I think one important point is the capacity of influence its own choices and the ability of (consciously or not) question one's first (or second or third) guess. In general, when we talk of something which is free, we mean that said something is able to initiate actions out of its own initiative and it is not completely determined by the external forces. We talk about freedom even in physics: how many degrees of freedom has a body? In how many way it can move freely without being constrained? I would claim that free in "free will" denotes the mind's ability of making conscious or non-conscious choices not completely determined by factors external to the mind itself. Similar to a body with a positive number of degrees of freedom can actually move, although it is influenced by external forces and the own inertia, a free-will can pick choices, although these choices are influenced, to large degree, by external factors.

Gödel, our old friend

We already have seen that Mr Wolpert used Cantor's diagonal argument to show that there is no mathematical free will. You cannot say Cantor without's saying Gödel. Gödel's first incompleteness theorem states that any axiomatic system which is has a model of arithmetics will contain a mathematical sentence which is neither true nor false. If you additionally impose that the arithmetics should be standard (in some very specific sense), this sentence will true but non provable. In other words, free will seems to be related to Gödel and to the very nature of natural numbers which is itself quite dubious. By the way, Gödel's theorem proof heavily uses Cantor's diagonal argument, so we are back to where we started.

Summing up

Summing up, we maybe have free will, but then again no. We'll see. In any case, if natural numbers are actually true, we will have to pick infinite number of axioms to avoid problems. And this means a lot of choices, and all of them are free, unless you use standard arithmetics. But you don't need to. Good evening to you.