# Free will theorem

The free will theorem of John H. Conway and Simon B. Kochen states that, if we have a free will in the sense that our choices are not a function of the past, then, subject to certain assumptions, so must some elementary particles. Conway and Kochen's paper was published in Foundations of Physics in 2006.[1] They published a stronger version of the theorem in 2009.[2]

## Axioms

The proof of the theorem as originally formulated relies on three axioms, which Conway and Kochen call "fin", "spin", and "twin". The spin and twin axioms can be verified experimentally.

1. Fin: There is a maximum speed for propagation of information (not necessarily the speed of light). This assumption rests upon causality.
2. 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).
3. 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 quantum entanglement, but full entanglement is not necessary for the twin axiom to hold (entanglement is sufficient but not necessary).

In their later paper, "The Strong Free Will Theorem",[2] Conway and Kochen replace the Fin axiom by a weaker one called Min, thereby strengthening the theorem. Min asserts only that two experimenters separated in a space-like way can make choices of measurements independently of each other. In particular it is not postulated that the speed of transfer of all information is subject to a maximum limit, but only of the particular information about choices of measurements.

## The theorem

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. As stated by Cator and Landsman regarding hidden variable theories:[3] "There has been a similar tension between the idea that the hidden variables (in the pertinent causal past) should on the one hand include all ontological information relevant to the experiment, but on the other hand should leave the experimenters free to choose any settings they like."

## Reception

According to Cator and Landsman, Conway and Kochen prove that "determinism is incompatible with a number of a priori desirable assumptions." They demonstrate that in certain experiments, "the choice an experimenter makes [about what to observe] is not a function of the past."[2] The philosopher David Hodgson supports this theorem as showing determinism is unscientific, that quantum mechanics allows observers (at least in some instances) the freedom to make observations of their choosing, and so leaves the door open for free will.[4] Some critics argue that the theorem applies only to deterministic models.[5] Others have argued that the indeterminism that Conway and Kochen claim to have established was already assumed in the premises of their proof.[6]

## Notes

1. Conway, John; Simon Kochen (2006). "The Free Will Theorem". Foundations of Physics. 36 (10): 1441. arXiv:quant-ph/0604079. Bibcode:2006FoPh...36.1441C. doi:10.1007/s10701-006-9068-6.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
2. Conway, John H., and Simon Kochen (2009). "The strong free will theorem" (PDF). Notices of the AMS. 56 (2): 226–232.CS1 maint: multiple names: authors list (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
3. Cator, Eric, and Klaas Landsman (2014). "Constraints on determinism: Bell versus Conway–Kochen". Foundations of Physics. 44 (7): 781–791. arXiv:1402.1972. Bibcode:2014FoPh...44..781C. doi:10.1007/s10701-014-9815-z.CS1 maint: multiple names: authors list (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
4. David Hodgson (2012). "Chapter 7: Science and determinism". Rationality + Consciousness = Free Will. Oxford University Press. ISBN 9780199845309.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
5. Sheldon Goldstein, Daniel V. Tausk, Roderich Tumulka, and Nino Zanghì (2010). What Does the Free Will Theorem Actually Prove?. Notices of the AMS, December, 1451-1453, http://www.ams.org/notices/201011/rtx101101451p.pdf
6. Wüthrich, Christian (September 2011). "Can the world be shown to be indeterministic after all?". In Beisbart, Claus; Hartmann, Stephan (eds.). Probabilities in Physics. Oxford University Press. pp. 365–389. doi:10.1093/acprof:oso/9780199577439.003.0014. ISBN 978-0199577439.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> (preprint available at PhilSci-Archive)