Bell’s theorem (BT) was formulated in 1964 as a continuation of the 1935 Einstein, Podolski, Rosen (EPR) criticism on the completeness of quantum mechanics (QM).
Based on the principle of locality, i.e., instantaneous spooky actions at a distance are not allowed, EPR showed through a thought experiment that QM was an incomplete theory in the sense that it fails to describe the underlying reality of the microscopic world.
The complete theory is supposed to be obtained by supplementing QM with hidden variables (HV) that would allow superseding quantum uncertainties with deterministic predictions.
John Stewart Bell in 1964 continued the EPR reasoning and managed to replace the thought experiment by an inequality that this time can be tested by an experiment. In a nutshell, BT postulates a complete theory complying with two hypothesis: locality and realism and from these hypotheses he derived an inequality known as Bell’s inequality (BI) that must be satisfied by any local realistic theory. Since then many experimental tests have been performed with increasing orders of accuracy and in all of them, BI is found to be violated.
These experimental results are widely interpreted as the impossibility of completing QM in order to restore realism and, what is considered even more astonishing, locality. There is, however, controversy with this interpretation since there are many researchers and commentators that view BI violations differently thus refuting the “orthodox” interpretation.
The intentions to disprove Bell’s conclusions, in a sense, resembles the vain attempts to prove what in mathematics is known as Fermat’s last theorem. This theorem was famous for the futile attempts to prove it until it was finally proved in 1995 by the English mathematician Andrew Wiles.
Although some rebuttals of BT are logically correct they are achieved by attacking some necessary implicit hypothesis which denial are widely regarded as implausible, e.g., the free will of the experimenter to choose his device settings. Other rejections are simply based on misinterpretations of various kinds.
Such refutations of Bell’s result appeared soon after publication and Bell himself reacted to this criticisms by relaxing and generalizing the hypothesis of his theorem. One of these generalizations consisted in the abandonment of determinism as a necessary hypothesis for the derivation, another was the inclusion of the issue of contextuality.
As regards the relinquishment of determinism he considered it unnecessary and in fact, it was later explicitly proven by Arthur Fine (Phys. Rev. Lett.1982,48,291–295) that it was a superfluous generalization. On the other hand, the contextuality issue deals with the inclusion of the effect that the measuring devices might have on the measurement results.
One of the misunderstandings regarding BT goes by the name of contextuality loophole alleging that Bell’s derivation is meaningless because it does not properly include the effect of the measuring devices.
This misunderstanding is explained in the article “On Nieuwenhuizen’s Treatment of Contextuality in Bells Theorem” published in Foundations of Physics by J. P. Lambare from the University of FIUNI and FaCEN of Paraguay.