Matthew Cobb called my attention to an interview with John Horton Conway in Notices of the American Mathematical Society. Conway is a famous mathematician at Princeton who says he’s proven that free will exists.
I haven’t seen his formal treatment of the Free Will Theorem, so I can’t say I can evaluate it—much less understand it. From the interview it sounds simply like a refutation of pure physical determinism, which most of us who accept quantum mechanics don’t see as problematic. The question is whether our behaviors and “choices” can be influenced by quantum dynamics, but even if that were true it wouldn’t prove “free will” exists in any meaningful sense. But the proof of “free will” is also connected with the bizarre phenomenon of quantum entanglement.
Talking to interviewer Dierk Schleicher, Conway explains his proof:
My friend Simon Kochen taught me one thing about quantum mechanics which I understood, and I ﬁnd that many physicists don’t understand that one thing (of course, they understand many things that I don’t). And that one thing we were able to pursue until we had this great theorem. If we make reasonable assumptions, including the assumption of free will, this one thing tells us that the little elementary particles are doing their own thing all over the universe. One atom is deciding to move a little bit leftwards and another to move a little bit rightwards. And it all very nearly cancels out, but not quite. And here [points to Schleicher] is what we call a life. You might be a robot, but I doubt it. I rather suspect you to have the same kind of consciousness as I have. And that is probably a manifestation of the freedom of the particles inside you: they do their own thing.
. . . Schleicher: Could you make a simple statement about what exactly, or intuitively, the Free Will Theorem says?
Conway: Yes. [Throws a piece of paper.] I just decided to throw that piece of paper on the ﬂoor. I don’t believe that that was determined at the start of the big bang, 14 billion years ago. I think it’s ludicrous to imagine that the entire development of the universe, including, say, this interview, was predetermined. For the Free Will Theorem, I assume that some of my actions are not given by predetermined functions of the past history of the universe. A rather big assumption to make, but most of us clearly make it. Now, what Simon and I proved is, if that is indeed true, then the same is true for elementary particles: some of their actions are not predetermined by the entire past history
of the universe. That is a rather remarkable thing.
Newton’s theory was deterministic. In the 1920s, Einstein had difficulties believing that quantum mechanics was not deterministic. That was regarded as a defect of quantum mechanics. Certainly when I tried to learn quantum mechanics and didn’t succeed, I thought it was a defect. It’s not a defect. If the theory could predict what one of those particles could do, then that theory would be wrong, because, according to the Free Will Theorem—supposing we do have free will—a particle doesn’t make up its mind what it’s going to do until it does it or until shortly before it does it.
Let me describe the theorem this way. Suppose there is only a very tiny amount of free will in humans: you can press either button A or button B in a manner that is not predetermined. That is a very tiny part of what we normally consider free will for humans. And if we have that tiny amount of free will, so do the elementary particles, in a sense that a particle in response to some experiment can choose which path, C or D, that it follows. It has free action. It chooses C or D in a manner that is not a predetermined function of all the information in the past history of the universe.
Schleicher: You believe that humans have free will.
Conway: I do. Strict determinism tells us that all of our actions are predetermined by the past history of the universe. I don’t know, maybe it is. I can’t disprove it. I can prove that I can’t disprove it. I can prove that you [points to Schleicher] can’t disprove it either. But I believe anyway that humans have free will.
It seems, then, that because particles have free will (i.e., purely indeterminate behavior, for they certainly don’t have minds), we must too. But what makes me think that I don’t understand Conway’s proof of free will comes from the way it’s characterized in, say, Wikipedia:
The free will theorem of John H. Conway and Simon B. Kochen states that, if we have a certain amount of “free will”, then, subject to certain assumptions, so must some elementary particles. Conway and Kochen’s paper was published in Foundations of Physics in 2006. . .
The proof of the theorem relies on three axioms, which Conway and Kochen call “fin”, “spin”, and “twin”. The spin and twin axioms can be verified experimentally.
- Fin: There is a maximum speed for propagation of information (not necessarily the speed of light). This assumption rests upon causality.
- Spin: The squared spin component of certain elementary particles of spin one, taken in three orthogonal directions, will be a permutation of (1,1,0).
- Twin: It is possible to “entangle” two elementary particles, and separate them by a significant distance, so that they have the same squared spin results if measured in parallel directions. This is a consequence of (but more limited than) quantum entanglement.
In their later paper, “The Strong Free Will Theorem,” Conway and Kochen weaken the Fin axiom (thereby strengthening the theorem) to a new axiom called Min, which asserts only that two experimenters separated in a space-like way can make choices of measurements independently of each other. In particular, they are not asserting that all information must travel finitely fast; only the particular information about choices of measurements.
The theorem states that, given the axioms, if the two experimenters in question are free to make choices about what measurements to take, then the results of the measurements cannot be determined by anything previous to the experiments. Since the theorem applies to any arbitrary physical theory consistent with the axioms, it would not even be possible to place the information into the universe’s past in an ad hoc way. The argument proceeds from the Kochen-Specker theorem, which shows that the result of any individual measurement of spin was not fixed independently of the choice of measurements.
My view that this is all about determinism, and not really “free will” in the only meaningful sense it can be taken in such a context—that is, dualistic or libertarian free will of the human mind—is buttressed by a critical assessment, also on Wikipedia:
Conway and Kochen do not prove that free will does exist. The definition of “free will” used in the proof of this theorem is simply that an outcome is “not determined” by prior conditions, and some philosophers strongly dispute the equivalence of “not determined” with free will. Some critics argue that the theorem only applies to deterministic models.Others have argued that the indeterminism that Conway and Kochen claim to have established was already assumed in the premises of their proof.
I warmly invite philosophers, mathematicians, physicists, or anyone who thinks they really understand the Free Will Theorem to explain it in the comments, but be aware that this will certainly be a very hard thing to do. I’m looking for clarity here—not just for readers, but for myself. In the end, I can’t believe that quantum mechanics can prove that we have libertarian free will. And if we’re compatibilists and believe in a kind of free will that isn’t libertarian or dualistic, then we don’t need quantum mechanics or mathematics to show it.