By “studies” I of course meant philosophers—I thought His Honor the Rebbe would understand. According to Wikipedia, most of them think Lucas’s argument is unsuccessful, but I don’t know why. Hofstadter also deals with it in Gödel, Escher, Bach, but it seems that all he really has against it is his own deterministic personal outlook. I couldn’t find any actual argument there against Lucas.
“It is commonly thought that physics can be described by a machine”—is there anyone who disagrees with that? Seemingly it’s obvious that, given all the positions of the particles and their velocities at a given moment, one could in principle calculate the positions and velocities—or their distribution—at any other time.
If physics can be described by a machine, does that mean that every physical thing is a Turing machine? After all, even if human consciousness is somehow a result of physical activity, it’s obviously not the activity of the particles themselves. Even staying within the system, it’s hard to see, for example, how the activity of particles becomes the Pythagorean theorem in the mind. Couldn’t it be that the activity of the particles in the brain somehow gets translated into Gödel’s theorem?

Ah, I should have vocalized it differently: researchers.

Because of my sins, I thought you were talking about mathematicians. They usually treat what philosophers say and conclude from Gödel’s theorem with disdain. By the way, although quite a few philosophers do talk nonsense, especially on subjects they don’t understand, that dismissiveness is usually unjustified. Mathematicians think that a conclusion not proved logically is not a conclusion of Gödel’s theorem. And apparently their sins caused them not to understand that philosophers are also satisfied with conclusions that reasonably follow from the theorem, and not only with those logically-mathematically proved from it. That’s exactly what I was careful to write in my previous reply.
As for your question: John Searle, for example, argues that a physical system can generate, at the macroscopic level, behavior that is not “machine-like,” what later came to be called “emergence.” In my opinion, this reflects a misunderstanding of physics, as I explained in my book.
Precisely for that reason, it is impossible to translate the activity of particles in the brain into Gödel’s theorem. The Pythagorean theorem is a proof within the system—what is called in computer science a “computational operation”—and therefore there is no obstacle to its being a translation of physical activity, though of course the mechanism of translation is unknown to us, and we do not even have a language with which to describe and investigate it. Therefore the epiphenomenalist assumption is wild speculation. But even if one accepts epiphenomenalism for the sake of discussion, it still cannot be applied to the proof of Gödel’s theorem. That requires stepping outside the system; in other words, it is non-computational. Therefore, according to the accepted conceptions of physical systems, to the best of my understanding, it cannot be some mechanical translation of physical actions. So here, even if we adopt the assumption of epiphenomenalism—or emergence—we still fail. That is a refutation of it.
Let me just note one more point in your remarks. According to quantum theory, even if you give me the positions of all the particles—and then of course you won’t be able to give me their velocities, because of the uncertainty principle—that still does not necessarily predict the future state. Physics is apparently not deterministic. But quantum theory adds only a random component to physics, and therefore it too cannot explain mental phenomena, and especially not free will—for one who believes in it. In my book I explained that this is a third mechanism, different from determinism and different from randomness. That is why I set quantum theory aside in this discussion. This too is explained in my book.
As for the arguments against Lucas, that characterizes almost every argument in this field. They all beg the question, or show that the argument is not purely logical—that is, not deductively valid. I do not think there is a good argument against Lucas, and the matter remains for each person to judge according to his understanding and assumptions. As I said, in my view it is quite a decent argument. By the way, it already appears in my book Two Carts—without his name; I didn’t know of him then. On various “scientific” websites—such as The Reasonable Doubt, Lo Meduyak, Hayadan, and various atheist sites—you will mostly find expressions of contempt and dismissal toward Penrose and Lucas, but very few actual arguments, if any.

No. I actually read it correctly. I understood that that was what you meant; I just thought you were talking about mathematicians or brain researchers, not philosophers.

I wrote “distribution” precisely because of quantum theory.
And I already wrote that the argument appears—yes, under his name—in Two Carts and a Ballista in a footnote. Check the index.
By the way, Hofstadter writes that those who address Lucas’s argument really split into two groups: those who get excited by it and those who sneer at it. He says that he himself does not sneer, but takes it seriously and rejects it—and he presents an incomprehensible rejection that seems based only on begging the question, and says that more will follow later, but they don’t.