Wow, it’s been about four months since I’ve posted here! Anyway, I’ll try not to continue that pattern in the future.

In the February issue of the Notices of the American Mathematical Society, John Conway and Simon Kochen have a paper explaining their “free will theorem”, which I believe strengthens it slightly from earlier versions. I had heard rumors of this theorem for a year or two, but had never seen more than an abstract, or a discussion in the popular media, so I couldn’t see the actual content of the theorem in order to see what it really says. So this paper was nice to see.

It’s important to see the actual statement, because the way it’s been summarized is basically as they put it, “It asserts, roughly, that if indeed we humans have free will, then elementary particles already have their own small share of this valuable commodity.” Which is a tendentious statement, to say the least, given that it uses a term like “free will”.

Here is the full statement of the theorem, from their paper:

SPIN Axiom: Measurements of the squared (components of) spin of a spin 1 particle in three orthogonal directions always give the answers 1, 0, 1 in some order.

The TWIN Axiom: For twinned spin 1 particles, suppose experimenter A performs a triple experiment of measuring the squared spin component of particle a in three orthogonal directions x, y, z, while experimenter B measures the twinned particle b in one direction, w . Then if w happens to be in the same direction as one of x, y, z, experimenter B’s measurement will necessarily yield the same answer as the corresponding measurement by A.

The MIN Axiom: Assume that the experiments performed by A and B are space-like separated. Then experimenter B can freely choose any one of the 33 particular directions w , and a’s response is independent of this choice. Similarly and independently, A can freely choose any one of the 40 triples x, y, z, and b’s response is independent of that choice.

The Free Will Theorem. The axioms SPIN, TWIN and MIN imply that the response of a spin 1 particle to a triple experiment is free—that is to say, is not a function of properties of that part of the universe that is earlier than this response with respect to any given inertial frame.

The definition of “free” used in the MIN axiom is the same as that used in the Free Will Theorem – some event is “free” in this sense just in case multiple versions of it are all compatible with everything before that event in any reference frame. Mathematicians express this notion in terms of functions, and philosophers would say that the event doesn’t supervene on anything outside the future light cone.

When we note this definition of “free”, it seems that the initial summary of the theorem is trivial – if some human action doesn’t supervene on the past in any way, then of course this is also true for some subatomic particle, namely, the first one whose movement would be different under the different choices of action by the human.

However, the theorem points out something stronger than this – nothing in the axioms involved assumes that the experimenter is a physical being made up of subatomic particles. Even if you think it’s a conceptual necessity that the experimenter (or at least, the experimenting apparatus) is made up of subatomic particles, nothing requires that there be a *first* such particle whose motion is different in the choices of how to set up the experiment. So without the theorem, it’s conceptually possible that human movements are free in the sense described, even though the motions of any specific particle are determined by the motions at earlier times, because human actions are at least in part composed of chains of motions of particles with no earliest member. So the theorem really does prove that indeterminacy at the human level requires indeterminacy at the particle level.

However, it seems to me that Conway and Kochen go on to make some bad interpretations of what this theorem says about freedom, determinism, and interpretations of quantum mechanics. They say, “our theorem asserts that if experimenters have a certain freedom, then particles have exactly the same kind of freedom.” This is true for a very specific type of freedom (namely, non-supervenience on the past) but their theorem says nothing else about any other kind of freedom, or whether their freedom has anything to do with the kind of freedom that matters. It may be that this kind of freedom is an important component of free will in the ordinary sense, but it may be that free will essentially requires not just non-supervenience, but also some sort of complex structure that just isn’t possible for the motions of individual particles.

They do make some good points about how the sort of freedom allowed for the particles is merely “semi-freedom” – that is, it is really spacelike separated pairs of particles whose motions are free, because the TWIN axiom says that the motions are in fact correlated in certain ways. They are right to point out that this means the freedom is different from “classically stochastic processes”, which clearly don’t provide any help in explaining free will. However, it really isn’t clear to me that this semi-freedom is any more help – correlations between twinned particles seem exceedingly unlikely to be relevant to the notion of free will.

“Granted our three axioms, the FWT shows that nature itself is non-deterministic. It follows that there can be no correct relativistic deterministic theory of nature. In particular, no relativistic version of a hidden variable theory such as Bohm’s well-known theory can exist.”

I agree that their axioms entail non-determinism. However, I don’t see why this should cause any trouble for the proponent of Bohm’s theory. It seems to me that a proponent of Bohm’s theory would just never grant the MIN axiom. Since the theory is deterministic, it entails that the choices of experimenters (assuming they are part of the physical world) aren’t free in the sense required by the axiom. Presumably, Bohmians are either compatibilists about free will (so that it doesn’t require freedom in the sense of the theorem) or insist that apparent free will is just an illusion. In either case, the seeming freedom of experimenters to set up their apparatus how they like gives us no evidence that this process is non-deterministic.

I suspect that a similar move can be made by the proponent of GRW theory, but I am unfamiliar with the details. They spend the last page or so of this paper engaged in a dialectic with a proponent of GRW theory who responded to some earlier papers of theirs, and give a modified version of the MIN axiom that they claim should be acceptable to the defender of GRW, but I suspect that a lot will depend on the interpretations of the words “independent”, “free”, and “choice” that they use.

In summary, I think the Free Will Theorem does a nice job of showing that a few facts about quantum mechanics (SPIN and TWIN) show that a certain type of macro-scale indeterminacy (MIN) entails a certain type of micro-scale indeterminacy. Additionally, the micro-scale indeterminacy is required not to be like most standard stochastic processes (because of the correlations over distances), so it may well be a place to look for interesting explanations of incompatibilist free will.

However, the theorem tells us nothing about compatibilism itself (which, contra Conway and Kochen, is not “a now unnecessary attempt to allow for human free will in a deterministic world”), because the theorem does nothing to prevent deterministic interpretations of quantum mechanics, whether Bohmian or otherwise. It may do something to constrain the shape that GRW-style theories can take, but this is less clear to me.

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