What is Free Will

One blog article asked : What the Hell is "Free Will?"

"nobody ever seems to give me a straight definition".

I do have a straight definition of free will, maybe not containing its full nature (which I don't think can ever be formulated), but at least a negative definition already sufficient to make it rigorously clear that it differs from the case it had no real meaning:

Precise formulation of free will

Free will is the combination of determination and indetermination that is the very expression of the time order, that is : the actual outcome cannot be predicted in advance because "correct prediction" would require the full underlying stuff (the individual mind with all past experience in memory...) to be correctly produced in the way it is produced: by that real mind actually coming to really exist into that circumstance, all that stuff which can only come to exist and produce that behavior at the time and place where it indeed comes to produce that behavior in reality. That is the same situation as what has been demonstrated to be the fact of how things go in the universe of pure mathematics itself (so that such a seemingly paradoxical statement of “combination of determination and indetermination” cannot be dismissed as illogical nonsense): except that the actual content is moreover very different from these "random" contents of the facts of pure mathematics (as explored by Gregory Chaitin), by the fact that the mind with its free will (unpredictable behavior), is not only non-algorithmic (as Penrose considered) but not mathematical.

Short, logical positivist formulation

Precisely, this concept of non-algorithmicity, that conscious behavior cannot be well approximated by any algorithmically expressible probabilistic law, can be expressed by the following statement, clearly very meaningful for logical positivism (experimentally falsifiable)

Artificial intelligence cannot pass the Turing test

Some necessary details

Description of time in the foundations of mathematics (unpredictability)
About the non-algorithmicity of the mind: Gödelian arguments against mechanism : what was wrong and how to do instead, with my actual argument summed up in the bottom of the page, referring to the possibility to justify that ZF is consistent.
My pdf presentation of the mind/mathematics dualism gives some developments and mathematical definiteness to this concept of conscious (non-mathematical) "law" of free will w.r.t. the Turing test, in terms of information entropy.
The Mind Makes Collapse interpretation of quantum physics
The decay of materialism.
More comments on the experimental testing of free will

Reply to his "challenge":

"Across from you are what appear to be two identical twins.  They look the same, they act the same, and in all physical respects, they are as alike as two people can possibly be. However, there is one key distinction that exists between them: one of them has free will, while the other one does not. (...) Your only job is to tell me which is which. Which one has free will and which one does not?"

My answer is simple : if they are physically identical, and one has free will but the one hasn't (is a zombie), then they do not act in the same "human way". They can be distinguished by their behavior (as soon as they are put in such difference circumstances that prevent you from solving the problem of "programming the zombie" by the simple answer "let us program the zombie's behavior by the simple instruction of copying the behavior observed from the real person" in which case "observing the zombie" would be a mere indirect way of observing the real person). The zombie's behavior is best described as obeying the probability law given by quantum mechanics (something usually not observed; still I guess it sometimes happens and is then qualified as a sort of coma); the one with free will is visibly departing from this law.

What can indeterminism bring to free will

That is the misleading question that some philosophers ask, trying to argue from the lack of answers that indeterminism cannot help free will.
But in the course of the argument they usually make a fundamental mistake. Because, well, they are usually materialists, and that is such a prejudice rooted in them that they cannot stop making materialist assumptions when trying to justify materialism. In particular the present fallacy consists in implicitly starting from the materialistic assumption that consciousness is not at the root of reality but emerges from some blind unconscious laws of nature. So of course if you put some blind unconscious force at the foundation of reality, may it be deterministic or not, and you wonder what kind of "consciousness" can emerge from that void, then such a nonsensical "consciousness" will be automatically devoid of all the normal attributes of consciousness, among which is free will.
The true solution can be stated very simply:
Of course the indeterminism of a universe cannot induce/explain free will for consciousness that emerge from that universe, because consciousness cannot ever emerge from any universe anyway. Instead, universes emerge from consciousness, so that the fundamental free will character of consciousness can induce/explain the likeliness of the creation/emergence of indeterministic universes.
Even, the free will dimension of consciousness makes the indeterminism almost necessary for a universe to get any "physical" i.e. conscious existence beyond mere mathematical existence, by the fact deterministic universes would be rather inhospitable (too boring) for free minds, which aren't likely to stay there long (hmm... some may do that, like mathematicians finding interest to explore the necessary truths of mathematics, but...)

Does Free Will violate any physical law (Born's rule) ?

In the article Is Quantum Indeterminism Relevant to Free Will? by Michael Esfeld (2000)
it is stated that the idea of free will is physically unacceptable as it would be a violation of one of the laws of physics : Born's probability rule.
It is compared to some ancient speculations (by Descartes) on how a non-physical mind would act on matter by a "non-physical force" that would violate the conservation laws of classical mechanics:

"Indeterminism in physics in the described sense opens up the possibility of an interactionism which does not have to assume a force over and above those force that are acknowledged in physics. It thereby reduces the ontological burden of interactionism.
(...) instead of having to endorse an additional force for metaphysical reasons, we have to endorse a change to the probabilities that a physical theory indicates for metaphysical reasons.
Within the framework of the positions considered in this paper, the conclusion is as follows: taking everything into account and given the current state of the art, quantum physics does not reduce the price which one has to pay for interactionism."

Visibly, he contradicts himself on the question whether the "burden" or "price" is reduced or not by taking quantum randomness as the place of operation of free will. In his mind, the diverse "laws of physics" seem to be assumed as being all at "the same level" of necessity.
These are quite naive and far from any kind of well-assessed comparisons "given the current state of the art", I would say. By the way, whose "state of the art" is he referring to ? Probably, the one of science philosophers trained with their usual bullshit kind of "reasoning", who pretend to discuss science without having any decent understanding of the needed concepts and theories.

Actually, Born's rule is very far from having a status of "physical law" with a necessity for things to obey it, that can stand any comparison of strength with the the conservation laws of mechanics. These conservation laws (of energy, momentum...) are not mere "postulates" like arbitrary assumptions, axioms or speculations, but come as theorems in either framework of General Relativity and QM. Thus, there is absolutely no way, even for any God, to violate them, as this would contradict a mathematical theorem. It is absolutely impossible, logically unconceivable (the only "conceivable" process that would look like this would be a transformation of ordinary matter into dark matter, but that is still so unlikely....).

On the other hand, the "strict respect of Born's rule" is not really a law of physics, because:

Explanation by Robert Lipschitz

quoted from here (answer to the Quora question "Does quantum physics end the free will debate? Is Michio Kaku correct in saying that Heisenberg’s uncertainty principle shows us that we have some kind of free will?")
The "no" answers are wrong. In the free will versus determinism debate, you have to understand context.
Historically (pre industrial revolution), western societies believed in free will. More importantly, intuitively (based on simple reflection), our starting assumption is also that we have free will. After all, if I knew not a whiff about science, I would say that you, right now, have absolute control over whether to move your hand up or down. I can chose absolutely whether my next key will be an A or a Z:.......Z!
Enter science. It says: don't buy this notion about free will and inherent responsibility; everything is deterministic and every direction of every molecule (including those in your brain) were in fact predetermined at the moment of the big bang. Throw in the apparent success of Newton and Darwin, and it seemed convincing. If materialism is true, then everything was determined at big bang including that I would type...Z!
Many argued against these base assumptions. They refused to accept that materialism (notion that there is nothing non-physical) was true without evidence. Core to this position was the historical religious basis to our societies, as well as our relentless experience of free will, which seemed so...obvious that it could not be doubted (that's certainly my view.....A! Ha!)
Then came quantum physics and showed that determinism was false. So that means that the main argument against free will fell away. That makes the argument for free will as stronger, much much stronger. Why not resort back to our intuitive observations?

                          I'm am alive, I'm in the forest, I'm thinking, I'm choosing

But that could be an illusion. Yes, it could be. So could the fact that I'm in the forest. Or that Im typing. Those type of sceptical arguments are boring and unconvincing 
So then the die hard materialists come back and say, well QM may not show determinism but it does show Randomness and Randomness does not admit free will.
"That's all, Randomness does not admit Free Will, so on your way, nothing to see here... "
What! What the heck is Randomness? How does Randomness actually "choose" in any given instance for object to go up or down.
"It does not choose, it just happens, and nothing causes it to happen (nothing material anyway), it just happens. Thank you, please move on."
It just happens! Nothing causes it to happen!
"Yes, nothing. It just 'pop' does it, no explanation, no cause at all - certainly not free will, certainly not the free will choice of God or man. Please now, move along."
All we know from QM experiments is that what material objects do cannot be determined, but we can come up with reasonable odds whether it will go up or down. Then these guys come and say that a magical thing called Mr Randomness comes and makes the "choice" in any given instance. Only that Mr Random does not chose. Nothing chooses. And nothing causes it to happen. It just happens.
So here is the rub. There is NO evidence that Randomness as opposed to Free Will is responsible for movements of physical molecules. Both ask you to come up with an interpretation of what is going on - and if you ask me, Randomness is a far, far greater stretch: a magical force that causes things to happen without any cause at all.
Its far simpler to say that a force called Free Will causes it to happen as I can attest to every single moment....Z! Simpler, that is, if you don't have any metaphysical pre-commitments (e.g. that everything is physical and there cannot, simple cannot be anything non-physical...except of course our dear Mr Random who happens to determine the direction of every single object in the universe!)
But finally, and critically, even if you don't like the free will explanation, nothing about science precludes it now that determinism has been debunked. Certainly not unfounded calls to mysterious things called randomness - forces which causes things to happen without actually causing anything at all.
(Note, randomness used in statistics is a epistemological (knowledge based) stance; it never relied on things being random in actuality)
My answers employs the classical interpretation of quantum physics, the Copenhagen Interpretation. While there are many interpretations for many different things, that does not make them all equally valid. In any event, I do not want to get into that debate here. Suffice to say, I was relying on what is (or at least what has been) the most scientifically accepted interpretation.
Now, you indicated that you did not understand my critique of Randomness (as employed in Copenhagen interpretation). The critique is simple. It says that physical processes obey a probability distribution, say: the atom goes left 50% of time and right 50% of time. However, and here is the rub, on any given occasion, nothing - ABSOLUTELY NOTHING - determines which way the ball will go. That is, even if you knew all the facts in the universe the second before the ball moved, you could not predict which way it was going to go (with certainty) BECAUSE nothing IN the physical universe causes it to go in either direction. It just happens. Like magic.
You were wondering how it was possible for the actions of free will to be instantiated into the world. You suggest that, if free will actions are not determined or random, it is difficult to imagine how they could could connect to the physical world.
But that is exactly the point. The Copenhagen interpretation posits a force - ontological randomness - which, by definition, "causes" objects to go in one direction or another (i.e. they are "randomly" assigned) but nothing IN the physical world is responsible for the cause - if you search for it, you will never find it. Thus, randomness is itself "outside of the physical world" and you may ask your question: but how does it interact with the physical world? Well, says science, it just does.
My last point is that free will is a better explanation than Randomness. Both operate from outside of the physical world (in my view, free will originates from our non-physical spiritual selves as is the view of most free will proponents). But Free Will beats Randomness in that at least we say that there is a source X (= free will) that causes Y (=an action) through process Z (= act of choosing). Randomness makes less sense because there is no posited process (Z) by which the source (X) produces the action (Y). In Randomness, it just happens. Moreover, free will is intuitively far better, because we all experience that we have radical free will (to chose to lift our hand up or down - just try it!). Further, although this MAY be an illusion, that is a claim which requires argument and faces counter-arguments, and so it does not upset the fact that free will is intuitively a superior solution to Randomness. Furthermore, there is nothing in science that says free will CANNOT be correct as determinism can be overturned by using one or more of the plausible interpretations of QM (and replacing the world Random with Free Will).

A quote from Chalmers

in 'Consciousnes and It's Place in Nature' (2002):
In fact, one might argue that if one was to design elegant laws of physics that allow a role for the conscious mind, one could not do much better than the bipartite dynamics of standard quantum mechanics (...) There is some irony in the fact that philosophers reject interactionism on largely physical grounds (it is incompatible with physical theory), while physicists reject an interactionist interpretation of quantum mechanics on largely philosophical grounds (it is dualistic). Taken conjointly, these reasons carry little force, especially in light of the arguments against materialism elsewhere in this paper.
See also his video

Is quantum randomness "at the right place" to be able to control behavior ?

This objection appears in diverse articles on the subject : "The kinds of indeterminacies discoverable at the quantum level may not correspond in any useful way to our ordinary idea of mental causes."
The answer is that quantum randomness is much more pervasive in reality than many people naively imagine when thinking that many phenomena, such as classical chaos and the randomness of statistical mechanics, can be understood without reference to quantum physics. I mean that while, admittedly, the place of not-yet-decohered forms of quantum superposition, that require the quantum mechanical concepts to be understood, seems limited to molecular scales irrelevant for consciousness, quantum randomness is anyway the real source of a large flow of effectively random data (classically probabilistic superpositions produced by decoherence), where free will has a wide margin for possible intervention. Namely at least, all the randomness that can be described as thermic actually has a quantum source, thus is an "absolute randomness" when analyzed in the framework of quantum physics, regardless the fact that, in the 19th century when quantum physics was not discovered yet, it was possible to describe this randomness as explainable in a deterministic framework. This explainability from pseudo-randomness is a pedagogical heritage of from the history of science, which many philosophers mistake as if it still kept nowadays the status of a scientific fact on the real source of this thermic randomness. It doesn't. More comments on this fact in the text on Bohmian mechanics.

See also

Seth's story of creation : an amazing report of how free will came to be, as a long waited solution of a "cosmic problem"

Quora questions

Is there any relationship between quantum mechanics and free will?

Related pages

A call to clarify the debate on the links between quantum physics and consciousness
Specifications for a Mind Makes Collapse interpretation of quantum physics
A mind/mathematics dualistic foundation of physical reality
Introduction to quantum physics (notions of states and measurements)
Main page of arguments on quantum physics interpretations
On materialism and its pathological pseudo-arguments far from science
my reply on quantum idealism and science