On Conway-Kochen

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. The proof of the theorem relies on three axioms.
  • 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.
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. Note that this is a weaker claim than the anthropomorphic claim, this post only discusses free will.

Cool, what are we looking at?

Spin is not interesting, we have a particle with a certain behavior, it consistently shows a measurable property in two dimensions. Twin is entanglement, I'll come back to that. Fin is the assumption that information cannot travel faster than the speed of light.

All axioms are true, at least, according to quantum mechanics. The only silly thing is that 'entanglement' is nothing more than a mathematical property, and 'a state space collapse' is nothing more than a mathematical action. Its entirely similar to, if I have 'x = 3 + y,' than determining a value for 'x,' or 'y,' immediately determines the other side of the equation. Einstein and Bohr had fervent debates about this, Bohr decided that the equations just work, a pragmatic argument. A lot of philosophers unfortunately decided that 'equations' are related to the world, and build 'free will' arguments on that.

Casually, there is a lot wrong with the free will theorem. Free will is normally associated with nondeterminism, and there's where the argument goes wrong immediately. Nondeterminism cannot be observed. If I give you a coin, it is impossible to state anything probabilistic about the outcome of what you give back to me, except for the fact that it'll be head or tail. This is essential different behavior than a coin flip, which follows stochastic rules. So say we have a nondeterministic system E which forces a system P into a state. It remains impossible to observe whether system P is a deterministic or nondeterministic system since nondeterminism cannot be observed. We will never know if P flips or turns a coin. Thus the implication, free will for experimenters plus axioms implies free will for the particle, cannot be true. At least the anthropomorphic claim cannot hold, even given all the axioms.

Quantum mechanics predicts a 50%/50% distribution of spin up or down, and 100% correlation at two sites. This cannot be nondeterminism -since that cannot be observed- it is stochastic behavior. Stated differently, if particles have free will we should observe traces where particles just decided to be up for no apparent reason. But if we have stochastic behavior then the underlying model can be expected to be fully deterministic by Einstein's 'God doesn't throw dice' argument. Assuming a dice leads to a world where some guy with a beard does something when we observe something and also keeps the distribution in order. Moreover, now the anthropomorphic argument leads to the immediate conclusion we don't have free will, which is irrelevant, since following the above reasoning, it was flawed anyway.

So what do we have? A theorem where the implication cannot hold, the axioms are debatable, and the conclusion contradicts experiment. Great.

I didn't know about Conway-Kochen, this is the condensed result of a discussion over at LtU, for which I should thank the other contributors.