John Conway and the "Free Will Theorem"


John Conway presented a fascinating look at quantum indeterminacy, in outlining his joint work with Simon Kochen on what he terms the "Free Will Theorem".


John Conway presented his "Free Will" theorem on Friday night in Berkeley.

The gist of the argument: when the squared spin of a "spin-1" particle is measured from 3 orthogonal directions, the answers are always 1,1, and 0 in some order. However, it is impossible to label all orthogonal triplets (simplified: all white spheres in the above picture) with 0s and 1s in a way that satisfies this rule. This is the "Kochen-Specker paradox". Conway and Kochen extend this result, via the phenomenon of quantum entanglement and the finite speed limit of information transfer, to show that the particular result obtained from 3 measurements must be "decided" on-the-fly by the particle, just as the experimenter "decides" which set of directions to measure. Conway's conclusion: if the experimenter was "free" in their choice of which directions to measure, so must the particle be "free" to choose what answer to give.

There are a few takes on this, and I'll try to write about each of them.

The first thing that caught me, even before the talk, was - this has a very restricted application to the entire body of thought on free will. Conway only considers the libertarian idea of "free will" and completely ignores compatibilism. His idea of free will was explicitly at odds with compatibilism, as one of the first examples he gave of why we might not have free will took the form of the idea of eternal recurrence. Paraphrasing, 'since this could be simply the second re-playing of the universe, we might not really have free will'. The compatibilist says that this does not bear on the issue of free will: even if this is the 2000th time the universe has gone exactly this way (leaving aside what it would mean for this to be the '2000th time'), as long as there are agents whose actions are influenced in accordance with their desires, those agents have free will. Under compatibilism, complex enough agents in Conway's own cellular automaton, the Game of Life, could have free will, although Conway himself would not agree with this.

Leaving this aside, however, it does seem that Conway and Kochen have found a real paradox, wherein the properties of some object logically cannot have been "set" before they were measured. In other words, let's say an experimenter measures spin magnitude in 3 orthogonal directions x, y, and z, and gets the results 1, 0, and 1 in that order. Now, *if* the experimenter had had the freedom to choose three different directions x', y', and z', then it cannot have been the case that the particle's spin was already set in such a way to have given the answer 1, 0, and 1 for directions x, y, and z. So either the particle's spin was set beforehand, in which case the experimenter could not have done otherwise than they did, or something else is going on. In Conway's opinion, this "something else" is best termed "free will". His "demonstration" of his free will went like this: he held up an object and made a great show of not knowing whether he would drop it or not - in this case, he didn't drop it. There you go, that's free will. He may as well have said, against the idea of determinism, "I refute it thus."

There is another question, whether the observed spin magnitudes really are completely independent of the particle's causal history. If this were true it would be at odds with Leibniz's principle of sufficient reason, stated thus in The Monadology:

"... we can find no true or existent fact, no true assertion, without there being a sufficient reason why it is thus and not otherwise, although most of the time these reasons cannot be known to us."

Posted: Mon - March 21, 2005 at 06:20 PM | | | |

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