Constraints on Determinism: Bell Versus Conway–Kochen

Abstract

Bell's Theorem from 1964 and the (Strong) Free Will Theorem of Conway and Kochen from 2009 both exclude deterministic hidden variable theories (or, in modern parlance, 'ontological models') that are compatible with some small fragment of quantum mechanics, admit 'free' settings of the archetypal Alice&Bob experiment, and satisfy a locality condition akin to Parameter Independence. We clarify the relationship between these theorems by giving reformulations of both that exactly pinpoint their resemblance and their differences. Our reformulation imposes determinism in what we see as the only consistent way, in which the 'ontological state' initially determines both the settings and the outcome of the experiment. The usual status of the settings as 'free' parameters is subsequently recovered from independence assumptions on the pertinent (random) variables. Our reformulation also clarifies the role of the settings in Bell's later generalization of his theorem to stochastic hidden variable theories.

On the notion of free will in the Free Will Theorem

“…This is exactly what has been achieved in Cator & Landsman (2014), at least mathematically: we show that rather than "indeterminism in, indeterminism out", the thrust of the fwt is really: "determinism in, constraints on determinism out' '.…”

“…(b) The same three differences persist also in the new versions of both Bell's Theorem and the fwt proposed by Cator & Landsman (2014), in which the experimentalists' "freedom" of choosing settings is defined rigorously (in a probabilistic and a deterministic framework, respectively).…”

“…Part of this interest has been rather critical (cf. Bassi & Ghirardi, 2007;'t Hooft, 2007;Goldstein et al, 2010;Wüthrich, 2011;Hemmick & Shakur, 2012;Cator &Landsman, 2014;Hermens, 2014Hermens, , 2016, mainly on the following grounds:…”

“…The precise relationship between the fwt and Bell's Theorem was analyzed in detail in Cator & Landsman (2014), with the following conclusion:…”

Randomness? What Randomness?

“…2 Randomness as a family resemblance 'The idea that in order to get clear about the meaning of a general term one had to find the common element in all its applications has shackled philosophical investigation; for it has not only led to no result, but also made the philosopher dismiss as irrelevant the concrete cases, which alone could have helped him to understand the usage of the general term.' (Wittgenstein,Blue Book,(19)(20).…”