Both from the points of view of recognition and understanding, the knowledge of fallacies can arm us against many kinds of mistakes and can help us to overcome serious scientific disagreements in a reasonable way. Being able to avoid or to detect fallacies can been viewed as a supplement to criteria of good scientific reasoning. Fallacies can be identified in several different ways. One way is to derive a logical contradiction such as +0 = +1. Still, a unified theory of fallacies giving us a systematic framework for demarcating fallacies and other kinds of mistakes is to be achieved. The present inquiry focuses on the logical and mathematical content of Bell's theorem. As we shall see, Bell's theorem itself is a false but popular belief, a deceptively bad argument. Bell's theorem is the most profound logical fallacy of physics and of science as such.
Both from the points of view of recognition and understanding, the knowledge of fallacies can arm us against many kinds of mistakes and can help us to overcome serious scientific disagreements in a reasonable way. Being able to avoid or to detect fallacies can been viewed as a supplement to criteria of good scientific reasoning. Fallacies can be identified in several different ways. One way is to derive a logical contradiction such as +0 = +1. Still, a unified theory of fallacies giving us a systematic framework for demarcating fallacies and other kinds of mistakes is to be achieved. The present inquiry focuses on the logical and mathematical content of Bell's theorem. As we shall see, Bell's theorem itself is a false but popular belief, a deceptively bad argument. Bell's theorem is the most profound logical fallacy of physics and of science as such.
Bell's theorem has been described as the most profound discovery of science; one of the few essential discoveries of 20th Century physics; indecipherable to non-mathematicians. However, taking elementary analysis to be an adequate logic here, we refute Bell's theorem, correct his inequality and identify his error. Further, we do this under the principle of true local realism, the union of true locality (or relativistic causality: no influence propagates superluminally) and true realism (or non-naive realism: some existents change interactively). We thus lay the foundation for a more complete physical theory: one in line with Einstein's ideas and Bell's hopes. Let's see.
While it is widely agreed that Bell's theorem is an important result in the foundations of quantum physics, there is much disagreement about what exactly Bell's theorem shows. It is agreed that Bell derived a contradiction with experimental facts from some list of assumptions, thus showing that at least one of the assumptions must be wrong; but there is disagreement about what the assumptions were that went into the argument. In this paper, I make a few points in order to help clarify the situation. Introduction Different authors have expressed very different views about what Bell's theorem shows about physics. The disagreement concerns particularly the question of which assumptions go into the argument. Since we have to give up one of the assumptions leading to the empirically violated Bell inequality, knowing what the assumptions were is crucial. For example, if author X believes that the argument requires only one assumption, A 1 , while author Y believes that it requires two, A 1 and A 2 , then X will conclude that A 1 ¬ must be abandoned, while Y will conclude that either A 1 or A 2 must abandoned, so A 1 may well be true in our world if A 2 is false. In this paper, I consider several assumptions that have been mentioned in connection with Bell's theorem, and I look into their roles in the proof of Bell's theorem. While I do not say anything here that has never been said before, I hope that my remarks can nevertheless be helpful to some readers. For the sake of definiteness, the version of the relevant experiment that I will consider involves two spin-1\2 particles, initially in the singlet spin state ψ = 2 −1/2 (|↑↓ ⟩ − |↓↑ ⟩). Two widely separated experimenters, Alice and Bob, have one particle transported to each of them. At spacelike separation, they each choose a direction in space, represented by the unit vectors a and b , and carry out a quantum measurement of the spin observable in this direction on their particle, ( a · σ ) ⊗ I and I ⊗ ( b · σ ), with I the 2 × 2 unit matrix and σ the triple of Paulimatrices. The possible outcomes, A and B , are±1 on each side. Bell's theorem is the statement that under the assumptions we are about to discuss, Bell's inequality holds; quantum theory predicts violations of Bell's inequality, and experiments have confirmed these predictions.
The Bell theorem [103] can be seen in several different ways, as the EPR argument. Initially, Bell invented it as a direct logical continuation of the EPR theorem: the idea is to take the existence of the EPR elements of reality seriously, and to push it further by introducing them explicitly into the mathematics with the notation λ; one then proceeds to study all possible kinds of correlations that can be obtained from the fluctuations of one or several variables λ, making the condition of locality explicit in the mathematics (locality was already useful in the EPR theorem, but not used in equations). The reasoning develops within determinism (considered as proved by the EPR reasoning) and classical probabilities; it studies in a completely general way all kinds of correlations that can be predicted from the fluctuations of some classical common cause in the past – if one prefers, from some random choice concerning the initial state of the system. This leads to the famous inequalities. But subsequent studies have shown that the scope of the Bell theorem is not limited to determinism; for instance, the λ variables may determine the probabilities of the results of future experiments, instead of the results themselves (see Appendix B), without canceling the theorem. We postpone the discussion of the various possible generalizations to §4.2.3. For the moment, we just emphasize that the essential condition for the validity of the Bell theorem is locality: all kinds of fluctuations can be assumed, but their effect must affect physics only locally.
2014.06.03 – the 25th anniversary of my phone-call re Mermin (1988) – 1989.06.03 Abstract: Generalizing Bell 1964:(15) to realizable experiments, CHSH (1969) coined the term “Bell’s theorem”. Since the results of such experiments (eg, see Aspect 2002) contradict Bell’s theorem: at least one step in his supposedly com-monsense analysis must be false. Using undergraduate maths and logic, we find a mathematical error, a false equality, in Bell (1964). Uncorrected, and therefore continuing, this error undermines all of Bell’s EPR-based analysis and many later variants, rendering them false. We can therefore predict with certainty that all loophole-free EPRB-style experiments will also give the lie to Bell’s theorem. “It is a matter of indifference... whether λ denotes a single variable or a set, or even a set of functions, and whether the variables are discrete or continuous. However, [Bell writes] as if λ were a single continuous parameter, ” Bell (1964:195). λ may denote “any number of hypothetical additional complementary variables needed to complete quantum mechanics in the way envisaged by EPR, ” Bell (2004:242); from his final essay. #1. Re Bell (1964) (which, like other key essays here, is available online; see References): Let the unnumbered equations between Bell’s (14)-(15) be (14a)-(14c). Let a,b, c replace his ~a,~b,~c. Given Bell’s (2004:242) λ specification above (from his final essay), let A(a, λi) and B(b, λ′i) denote the respective outcomes (±1) when Alice and Bob test the i-th particle-pair. #2. NB: Primes (′) distinguish λs in Bob’s domain from those in others. With i = 1, 2,.., N (and j similarly), there is no requirement here that any two particle-pairs should be the same. #3. Let expectation 〈A(a)B(b) 〉 replace Bell’s equivalent term P (~a,~b); etc. Let P (. | Z) denote a probability conditioned on Z: where Z is shorthand for EPRB, the experiment based
On the 50th anniversary of Bell's monumental 1964 paper, there is still widespread misunderstanding about exactly what Bell proved. This misunderstanding derives in turn from a failure to appreciate the earlier argument of Einstein, Podolsky and Rosen. I retrace the history and logical structure of these arguments in order to clarify the proper conclusion, namely that any world that displays violations of Bell's inequality for experiments done far from one another must be non-local. Since the world we happen to live in displays such violations, actual physics is non-local. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.