The Strong Free Will Theorem

6 02 2009

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.




10 responses

8 03 2009
Dave Baker

Hi Kenny,

Nice to see that you’re looking into this stuff. I agree with you about the Bohmian response; K&C seem to be entirely off-base in their criticisms of that position. One of their starting assumptions is indeterminism about measurement decisions — obviously if that’s your premise, you’re going to conclude determinism is false.

In fact, they seem to presume that a relativistic version of Bohm’s theory will have to be deterministic, like the non-relativistic version, which isn’t true. Some of the newer forms of the theory are stochastic (chancy) rather than deterministic.

There also has to be something fishy about their appendix replying to Tumulka. Tumulka has papers with existence proofs of the “flash function” they’re trying to say can’t exist, and while I haven’t read these in detail I can’t imagine they’re wrong.

It surprises me that Kochen and Conway are still pushing this “free will” line, given the amount of cogent criticism they’ve received from philosophers and fellow mathematicians. I imagine the physical upshot of their result is being misunderstood by them or their opponents or both, so I’d advise agnosticism about almost everything they say.

Hope all’s well!


8 03 2009

Hi Dave,

Thanks for the comment! I haven’t really seen what other philosophers have said about this, just some idle comments from mathematicians. There’s actually a summer math program for high school students (the Canada/USA Mathcamp) where I’ve taught for several times, and Conway comes to visit every summer for a week. I don’t think we’ve overlapped there at the same time since I heard about this theorem, but I’ve definitely learned some useful tricks from him, both mathematical and non (like the trick I use to know what day of the week various dates will be). So I’m a bit surprised at how much he’s stuck with this.

14 04 2009
R. T. Mercuri

I’ve been attending Conway & Kochen’s lecture series in Princeton on this subject. There, they’ve asserted that the Free Will Theorem is “IF humans have free will, then elementary particles do too.” What you have stated as the Free Will Theorem in your blog posting (above) is actually the Strong FWT, as implied from the paradoxes of SPIN, TWIN and MIN. Conway was pretty emphatic about not knowing whether free will or determinism is actually the case, although he admits to a personal bias toward the concept of free will (as defined within the constraints of their quantum mechanics discussion). Baker (above) is incorrect about their assumption of indeterminism of measurement decisions — indeed they say that IF free will does NOT exist, then the measurement decisions are deterministic. So it seems that they are being consistent on these points.

Incidentally, videos of the lectures (there will be 6 in total) are viewable at:

14 04 2009

That’s useful to hear the clarification about their views. However, I still think that they seem to be conflating free will with a certain type of indeterminism. It seems to me that a better statement of the content of their result is that “If humans are non-deterministic, then elementary particles are too”. They have a technical definition of free will on which your statement is correct, but this technical definition of free will certainly isn’t one that we have any natural intuitions about, or any evidence that humans actually have.

Anyway, I’ll have to check out the lectures some time – I know that Conway tends to be a great lecturer, so it’ll be really good to see.

14 04 2009
R. T. Mercuri

I think they’ve tried to be very clear about the terminology and seem to have put a lot of thought into the words they’ve selected to use.

Personally, I wouldn’t use the word “non-deterministic,” because that term has a rather specific connotation, especially in computer science as related to finite automata (Turing machines, etc.), that I believe is not the same as what Conway and Kochen intend to convey by “free will.” They’ve been careful to explain that other indeterministic aspects, such as “randomness” or “probabilities” are also excluded.

So one might consider “free will” as a constrained type of non-determinism. Conway did say, though, in (I think) lecture 2, that if you’re uncomfortable with the phrase “free will” it could be substituted with “free whim.”

As for the lack of natural intuitions or evidence, many said the same about quantum mechanics decades ago, but it’s generally accepted nowadays. Conway points out that SPIN and TWIN are measurable (although perhaps not yet with sufficient accuracy), and results to date do seem to confirm the theory (as well as the paradoxes).

14 04 2009
Dave Baker

Posting again to correct some misconceptions above:

Baker (above) is incorrect about their assumption of indeterminism of measurement decisions

In their argument against Bohm, which is what I was talking about, they do assume “free will” (via the MIN/FIN axioms) as a premise.

indeed they say that IF free will does NOT exist, then the measurement decisions are deterministic.

They definitely don’t think that, since they believe that “free will” is incompatible with the GRW model, which is indeterministic.

14 04 2009
R. T. Mercuri

True. They do say that if the observer has free will “then the particle’s response is not determined by the entire previous history of the universe” and they say that “there’s no longer any evidence to support determinism,” but they’ve also said that they haven’t disproven the possibility of determinism either.

I think the confusion occurs when we try to consider free will as the dual of determinism which I’m not sure they are saying it necessarily is or is not. I’ll try to get a clarification on this during next Monday’s Q&A if I have a chance to ask.

20 04 2009
Charlie Stromeyer

I emailed Conway and Kochen because their Strong Free Will Theorem appears to me to be a computational skill within the context of G. Castagnoli’s work on the fundamentally non-deterministic nature of quantum computation. For example, see this newly revised paper by Castagnoli et al. which argues that quantum entanglement is essential for quantum computation and that the latter transcends conventional physical notions of both causality and dynamics:

2 05 2009
Charlie Stromeyer

Let us suppose that quantum mechanics (QM) is truly more fundamental than classical mechanics. Then, the “free will” in the above theorem only has three possible origins: it somehow evolved into existence, for example, with something like the Big Bang, or it is from a physical reality that is more fundamental than QM or it is from a metaphysical source.

Well, we can eliminate the first possibility because this brief paper shows that the wavefunction of QM is both non-local and non-sequential, meaning that the wavefunction of QM transcends conventional notions of causality, space and time:

14 07 2009
Stephen Harris

I’m very suspicious of using logical results such as Penrose using Goedel’s Incompleteness Theorem (GIT) to limit the possibility of strong AI machines. I’m not even comfortable with going the other direction, using the Heisenberg Uncertainty Principle used as a physical basis supporting/establishing GIT. (Also causality does not exactly mean determinism.) by David Wolpert
Abstract: “In this paper strong limits on the accuracy of real-world physical
computation are established. To derive these results a non-Turing Machine (TM)
formulation of physical computation is used. First it is proven that there
cannot be a physical computer C to which one can pose any and all computational
tasks concerning the physical universe. Next it is proven that no physical
computer C can correctly carry out every computational task in the subset of
such tasks that could potentially be posed to C. This means in particular that
there cannot be a physical computer that can be assured of correctly “processing
information faster than the universe does”. … Note that since the universe is
microscopically deterministic, (be it classical or quantum-mechanical, if we
adopt the many-worlds interpretation for the latter case)…”

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