Bell's theorem refuted via elementary probability theory

Foundations of physics and/or philosophy of physics, and in particular, posts on unresolved or controversial issues

Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Wed Dec 23, 2020 3:00 am

gill1109 wrote:
Gordon Watson wrote:
TOWARD SOME PRELIMINARY CLARIFICATIONS IN RELATION TO BELL AND GILL (2014):
(1) What statement (with source) does Bohr make that agrees with my position (as seen by you)?
(2) What statement did Bell make about Bohr's statement; ie, what did Bell point out?

Read section 5 (“Envoi”) of “Bertlmann’s socks”. Bell starts this section writing “By way of conclusion I will comment on four possible conclusions which might be taken”. The fourth position is what Bell says would have been Bohr’s position: Bell summarises this as “there is no ‘reality’ below some classical macroscopic level”.

I am not a scholar of the works of Niels Bohr, so I will not answer your question (1). Above, in my answer to your question (2), I have told you what Bell made of Bohr’s position.


Thanks. This statement, relating to Bohr's intuition, does NOT agree with my position. I believe that reality extends — ie, that things exist — below "some classical macroscopic level".
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Wed Dec 23, 2020 5:31 am

Gordon Watson wrote:
gill1109 wrote:
Gordon Watson wrote:
TOWARD SOME PRELIMINARY CLARIFICATIONS IN RELATION TO BELL AND GILL (2014):
(1) What statement (with source) does Bohr make that agrees with my position (as seen by you)?
(2) What statement did Bell make about Bohr's statement; ie, what did Bell point out?

Read section 5 (“Envoi”) of “Bertlmann’s socks”. Bell starts this section writing “By way of conclusion I will comment on four possible conclusions which might be taken”. The fourth position is what Bell says would have been Bohr’s position: Bell summarises this as “there is no ‘reality’ below some classical macroscopic level”.

I am not a scholar of the works of Niels Bohr, so I will not answer your question (1). Above, in my answer to your question (2), I have told you what Bell made of Bohr’s position.


Thanks. This statement, relating to Bohr's intuition, does NOT agree with my position. I believe that reality extends — ie, that things exist — below "some classical macroscopic level".

Then you accept non-locality.
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Re: Bell's theorem refuted via elementary probability theory

Postby minkwe » Wed Dec 23, 2020 10:17 am

gill1109 wrote:Then you accept non-locality.

You want to define "non-locality"?
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Wed Dec 23, 2020 11:38 am

minkwe wrote:
gill1109 wrote:Then you accept non-locality.

You want to define "non-locality"?

I want to hear what Gordon thinks about it. I think it it’s difficult to disentangle the concepts of ‘locality’ and of ‘realism’. What you mean by “local” can depend on what you mean by “real”. For instance, the wave function is a hidden variable, but *where* is the wave function? Does it “exist” or is it just a mathematical device for doing calculations? If it exists in real 3D space, then wave-function collapse on measurement seems rather non-local. But if if it is “just a device for doing the calculations” then who cares...” Remember, the wave function of two particles in 3D space is a function of 6 spatial coordinates. It isn’t “located” anywhere. Let alone, in two “locations”.
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Sun Dec 27, 2020 7:16 pm

gill1109 wrote:
Gordon Watson wrote:
gill1109 wrote:
Gordon Watson wrote:
TOWARD SOME PRELIMINARY CLARIFICATIONS IN RELATION TO BELL AND GILL (2014):
(1) What statement (with source) does Bohr make that agrees with my position (as seen by you)?
(2) What statement did Bell make about Bohr's statement; ie, what did Bell point out?

Read section 5 (“Envoi”) of “Bertlmann’s socks”. Bell starts this section writing “By way of conclusion I will comment on four possible conclusions which might be taken”. The fourth position is what Bell says would have been Bohr’s position: Bell summarises this as “there is no ‘reality’ below some classical macroscopic level”.

I am not a scholar of the works of Niels Bohr, so I will not answer your question (1). Above, in my answer to your question (2), I have told you what Bell made of Bohr’s position.


Thanks. This statement, relating to Bohr's intuition, does NOT agree with my position. I believe that reality extends — ie, that things exist — below "some classical macroscopic level".

Then you accept non-locality.


Richard: I REJECT NON-LOCALITY.

Via my true local realism (TLR), as I've written: (1) Under true locality (AKA relativistic causality): no influence propagates superluminally. (2) Under true (non-naive) realism: some existents change interactively.

So please: how does my rejection of Bohr's unrealistic position lead you to draw such a false conclusion?
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Sun Dec 27, 2020 10:02 pm

gill1109 wrote:I am not a scholar of the works of Niels Bohr, so I will not answer your question (1). Above, in my answer to your question (2), I have told you what Bell made of Bohr’s position.


Gordon Watson wrote:
Thanks. This statement, relating to Bohr's intuition, does NOT agree with my position. I believe that reality extends — ie, that things exist — below "some classical macroscopic level".


gill1109 wrote:
Then you accept non-locality.


Gordon Watson wrote:
Richard: I REJECT NON-LOCALITY.

Via my true local realism (TLR), as I've written: (1) Under true locality (AKA relativistic causality): no influence propagates superluminally. (2) Under true (non-naive) realism: some existents change interactively.

So please: how does my rejection of Bohr's unrealistic position lead you to draw such a false conclusion?
.

You wrote “existents change interactively”. Both measurement outcomes come to exist through both settings being made. That is Bohr’s point of view. You don’t separate the processes which generate them. They can’t be separated, that is proved by Bell. You don’t disprove Bell.

You may assert that no influence propagates superluminally, but you don’t prove that.
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Tue Dec 29, 2020 4:23 am

gill1109 wrote:
gill1109 wrote:
[EDIT via re-insertion by GW] Read section 5 (“Envoi”) of “Bertlmann’s socks”. Bell starts this section writing “By way of conclusion I will comment on four possible conclusions which might be taken”. The fourth position is what Bell says would have been Bohr’s position: Bell summarises this as “there is no ‘reality’ below some classical macroscopic level”.[ End GW EDIT}
Above, in my answer to your question (2), I have told you what Bell made of Bohr’s position.


Gordon Watson wrote:
Thanks. This statement, relating to Bohr's intuition, does NOT agree with my position. I believe that reality extends — ie, that things exist — below "some classical macroscopic level".


gill1109 wrote:
Then you accept non-locality.


Gordon Watson wrote:
Richard: I REJECT NON-LOCALITY.

Via my true local realism (TLR), as I've written: (1) Under true locality (AKA relativistic causality): no influence propagates superluminally. (2) Under true (non-naive) realism: some existents change interactively.

So please: how does my rejection of Bohr's unrealistic position lead you to draw such a false conclusion?
.

(A) You wrote “existents change interactively”. (B) Both measurement outcomes come to exist through both settings being made. (C) That is Bohr’s point of view. (D) You don’t separate the processes which generate them. (E) They can’t be separated, that is proved by Bell. (F) You don’t disprove Bell.

(G) You may assert that no influence propagates superluminally, but you don’t prove that. [With (A), (B), etc., added by GW.]


Richard, please: it would help if you did not cut crucial sentences and ignore questions that relate to statements by you, etc., from my inputs.

So I'll try again: Richard, please. How does my rejection of Bohr's unrealistic position lead YOU to draw this false conclusion: that Gordon Watson accepts non-locality?

Then, re your new matters (A)-(G) above:

A: Thus, in my maintaining that “existents change interactively”, my ideas are in full accord with Bohr's insight:
the result of a 'measurement' does not in general reveal some preexisting property of the 'system', but is a product of both 'system' and 'apparatus.' Indeed, It seems to BELL that full appreciation of this would have aborted most of the 'impossibility proofs', and most of 'quantum logic'. See Bell's 1987 book, pp. viii-ix; my emphasis.


Note that Bell was still on the horns of an unresolved dilemma three years later (1990). See the Introduction in my draft.

B, C: In my terms (and Bohr's): each result comes into existence via the transformational interaction of a particle with a polariser-analyser.

D, E. But I do work with separations, since EPRB interactions are 2-step processes.

F. But the inequalities by Bell (1964) and CHSH-Bell are made in the context of quantum experiments. AND my results agree with those experiments while Bell (1964) and CHSH-Bell do not.

Further:

Note, from Bell's 1964 Introduction: "The paradox of Einstein, Podolsky and Rosen was advanced as an argument that quantum mechanics could not be a complete theory but should be supplemented by additional variables. These additional variables were to restore to the theory causality and locality. In this note that idea will be formulated mathematically and shown to be incompatible with the statistical predictions of quantum mechanics.


... Let this more complete specification be effected by means of parameters .


Thus, under TLR, my added variables provide a more complete specification:*** a specification that satisfactorily restores to the theory causality and locality. And I do this on Bell's terms:

And it is via that I identify latent elements of physical reality with equivalence classes which locally mediate each local result.

G. You say: I may assert that no influence propagates superluminally, but I don’t prove that.

But I do prove my point: under special relativity and relativistic-causality, see the flow charts (7)-(9) in my draft. There you'll see two pairwise-correlated particles proceeding from the Source to interact with their respective polariser-analysers. Thus: no influence propagates superluminally in my theorising!

*** NB: not a wholly complete specification; just the additional variables required to restore causality and locality to the theory.
.
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Tue Dec 29, 2020 6:57 am

Dear Gordon, someone calling themselves “Mikko” has carefully explained the errors in your paper in the Disqus discussion section of your viXra paper. He is the first to comment on the paper. I completely agree with everything he says. I refer you to his comments at
https://vixra.org/abs/2011.0073
Your paper is incoherent, incomplete, illogical. You think Bell was wrong but you do not establish any errors in Bell’s reasoning. You seem to think that you have a counterexample to Bell’s theorem but you nowhere exhibit functions A and B taking values +/-1, and a probability distribution rho of a hidden variable lambda, which together reproduce the singlet correlations through the usual formula

.

By the way, it can’t be done, so I suggest you don’t waste your time trying to find A, B and rho which do the trick!

I discovered that you publish on viXra under two names (with and without a second given name).
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Re: Bell's theorem refuted via elementary probability theory

Postby Esail » Tue Dec 29, 2020 8:49 am

Gordon Watson wrote:.
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. Let's see.

https://vixra.org/abs/2010.0068

ALL comments welcome, especially those that are educative and/or critical.

Thanks; Gordon
.


Gordon:
Bell assumed that an LHV model is not contextual. This means that the functions A (a, lambda) and B (b, lambda) exist. Under this assumption, Bell's inequality is correct and the transition in his 1964 paper from Eq. 14a to Eq. 14b is also correct. What you've done with your equations 15 and 16 is only to show that the quantum mechanical expectation values do not satisfy Bell's inequality. This is known and not a proof for the invalidity of Bell's theorem.

To refute Bell's theorem, you would have to present a contextual model that correctly predicts the quantum mechanical expectation values. I have already done that, read under
https://www.researchgate.net/deref/http ... qJdKwYZAJQ

In a contextual LHV model unambiguous functions A(a, lambda) do not exist. Then Bell's Eq. (14b) does not follow from Eq. (14a) and Eq. (15) is wrong.
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Tue Dec 29, 2020 11:53 am

Esail wrote:
Gordon Watson wrote:.
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. Let's see.

https://vixra.org/abs/2010.0068

ALL comments welcome, especially those that are educative and/or critical.

Thanks; Gordon
.


Gordon:
Bell assumed that an LHV model is not contextual. This means that the functions A (a, lambda) and B (b, lambda) exist. Under this assumption, Bell's inequality is correct and the transition in his 1964 paper from Eq. 14a to Eq. 14b is also correct. What you've done with your equations 15 and 16 is only to show that the quantum mechanical expectation values do not satisfy Bell's inequality. This is known and not a proof for the invalidity of Bell's theorem.

To refute Bell's theorem, you would have to present a contextual model that correctly predicts the quantum mechanical expectation values. I have already done that, read under
https://www.researchgate.net/deref/http ... qJdKwYZAJQ

In a contextual LHV model unambiguous functions A(a, lambda) do not exist. Then Bell's Eq. (14b) does not follow from Eq. (14a) and Eq. (15) is wrong.

This is not true. Bell's model is as contextual as you could possibly wish. See Bell's own "answer to critics" in which he explicitly refutes this criticism. "Esail" has not bothered to have his model programmed and won't do it himself, so his claims are merely his claims. They haven't even passed peer review in a decent journal (and God knows that is not much of a barrier these days). They contradict well-known and much-studied and long established purely mathematical theorems. No reason to believe remarkable claims without very, very strong evidence.
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=441&start=20#p12423
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Re: Bell's theorem refuted via elementary probability theory

Postby Esail » Wed Dec 30, 2020 2:18 am

gill1109 wrote:This is not true.


Didn't you admit that you didn't read my paper? Why do you keep objecting against it without any rationale? But this is not about my paper but about what Gordon wrote.
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Wed Dec 30, 2020 4:10 am

Esail wrote:
gill1109 wrote:This is not true.


Didn't you admit that you didn't read my paper? Why do you keep objecting against it without any rationale? But this is not about my paper but about what Gordon wrote.

Dear Eugene

I have read your paper, several times. Your model is not local. “delta” depends on the hidden variable “phi”and Alice’s detector setting “alpha”, it then needs to be known by Bob’s particle so that when it encounters detector setting “beta” it can figure out what to do, it also needs to know what Alice’s particle did. Your model comes down to writing p(x, y|a, b) = p(x |a, b)p(y |x, a , b) = p(x | a)p(y | x, a, b). Unfortunately, p(y | x, a, b) depends on x and b - a.

Please find someone who would like to program it for you. Create an app which can be downloaded, or create an interactive web site. I will test whether or not it satisfies the specifications I wrote out at http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=460#p12307, correction note at http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=460#p12316.
Please let me know if you think your program would satisfy the side conditions which I wrote down. Or do you have some reason to object to them? That would interest me very much.

I would also be very interested indeed if you could tell me what you think is wrong with the two proofs of Bell’s theorem which I gave in https://arxiv.org/abs/2012.00719. I formulated Bell’s theorem as a no-go theorem in computer science (distributed computing). I find it useful to de-couple the maths and logic on the one hand, from the physics and meta-physics (philosophy of science) on the other. Everyone ought to be able to agree on the maths and logic. They may disagree on the metaphysics, depending on their conception of “reality”. However, I believe that active forum participants Joy Christian, Fred Diether, Gordon Watson and Michel Fodje all disagree with me on the maths and logic. Michel and Fred are both good at programming. Maybe they can help you.

In your paper you first describe how photon 1 makes its decision. This depends of course on the setting “alpha” which it encounters at Alice’s detector. Then you describe the behaviour of photon 2. It depends on photon 1’s choice and on the difference between the beta and alpha. You call that “contextual”, I call that “non-local”.

You refer to two predecessors: De Zela, and Khrennikov. The latter writes a lot of words, but says nothing. The former makes grand claims, but proves nothing. Two very, very weak papers.
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Wed Dec 30, 2020 10:36 pm

Esail wrote:
Gordon Watson wrote:.
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. Let's see.

https://vixra.org/abs/2010.0068

ALL comments welcome, especially those that are educative and/or critical.

Thanks; Gordon
.


Gordon:
Bell assumed that an LHV model is not contextual. This means that the functions A (a, lambda) and B (b, lambda) exist. Under this assumption, Bell's inequality is correct and the transition in his 1964 paper from Eq. 14a to Eq. 14b is also correct. What you've done with your equations 15 and 16 is only to show that the quantum mechanical expectation values do not satisfy Bell's inequality. This is known and not a proof for the invalidity of Bell's theorem.

To refute Bell's theorem, you would have to present a contextual model that correctly predicts the quantum mechanical expectation values. I have already done that, read under
https://www.researchgate.net/deref/http ... qJdKwYZAJQ

In a contextual LHV model unambiguous functions A(a, lambda) do not exist. Then Bell's Eq. (14b) does not follow from Eq. (14a) and Eq. (15) is wrong.


Dear Esail,

Esail wrote: "What you've done with your equations 15 and 16 is only to show that the quantum mechanical expectation values do not satisfy Bell's inequality. This is known and not a proof for the invalidity of Bell's theorem."


1. You appear to have missed the foundation on which my irrefutable counter-inequality to Bell (1964) is built.

FOR: IF you are referring to https://vixra.org/pdf/2010.0068v5.pdf,

THEN eqn (15) is derived via high-school math, without any reference to QM; nor any restriction to non-contextual or contextual models.***

Imho, my eqn (15) shows that Bell’s 1964:(15) inequality is mathematically and physically false.

Then, in my eqn (16), since Bell is supposedly deriving his inequality under EPRB, I rightly use the related QM expectations to show the disparity (under EPRB) between Bell’s inequality and my own irrefutable one.

Via my (19), I show that disparity graphically.

2. The CHSH-Bell inequality falls similarly.

3. I therefore question the second paragraph on p.2 of your essay: "On a contextual model refuting Bell's Theorem, 20 Nov 2020."

It is definitely not the intention of this paper to question the derivation of Bell's inequality. Bell's proof is mathematically correct. The point is simply that the assumptions of Bell's inequality—namely, the restriction to non-contextual models—do not comprehensively describe conceivable physical reality.


How then do you move, in a mathematically correct way, from Bell's 1964:(14a) to his (14b)?

4. Also, is this next point made in your essay?

Esail wrote: Then Bell's Eq. (14b) does not follow from Eq. (14a) and Eq. (15) is wrong.


*** 5. I’ll happily leave you and Richard to deal with the issues surrounding non-contextual and contextual models.

Thanks.
.
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Thu Dec 31, 2020 3:29 am

Gordon Watson wrote:... since Bell is supposedly deriving his inequality under EPRB, I rightly use the related QM expectations to show the disparity (under EPRB) between Bell’s inequality and my own irrefutable one.

Bell is *not* assuming the usual QM description of EPR-B. He doesn’t assume any quantum mechanics at all.

The whole point of Bell’s work is that there is an unbridgeable disparity between certain results derived in QM (namely, the singlet correlations) and anything which could hold under a classical physical description of the world.

It seems that Gordon agrees with Bell. Esail (Eugen) disagrees.

Gordon: you had better find out from Esail how he uses contextuality to escape from Bell’s conclusion. And maybe Esail can explain to Gordon where he (Gordon) is going astray. Esail agrees with Bell’s proof that functions A, B, rho - with the usual properties - do not exist. He thinks that Bell’s formulation of local realism is too narrow (I disagree).
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Fri Jan 01, 2021 3:38 am

Gordon Watson wrote:
gill1109 wrote:
minkwe wrote:
Gordon Watson wrote:2. Let the three mathematical relations between Bell's equations (14) and (15) be, respectively: (14a), (14b), (14c).

3. Then Bell's error is his move from (14a) to (14b).

The contradiction is present even on the left-hand side of 14a.

P(a,b) - P(a,c) = "The correlation obtained if Alice and Bob measure at settings (a,b)" - "The correlation obtained if Alice and Bob measure at settings (a,c)"

The two terms contain contradictory premises. If Alice and Bob measured at (a,b) then they did not measure at (a,c). P(a,c) is counterfactual to P(a,b). The antecedents are contradictory therefore the combination of terms does not make physical sense since there is no universe in which True is False.

Michel and Gordon:just read Bell’s own answer to this criticism. It’s as old the hills, often been repeated, and it’s wrong. Chapter 8 of “Speakable and Unspeakable” is a two page paper, and I wrote it out for you here: http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=441&start=20#p12423


Richard,

1: I've pointed out that minkwe's claim re contradictions in LHS (14a) is mistaken; see above.

2: Then, re your point, my principal objection to Chapter 8 of “Speakable and Unspeakable” relates to this:

"With these local forms, it is not possible to find functions and and a probability distribution which give the correlation (1). This is the theorem. The proof will not be repeated here." [GW emphasis.]


For Bell's proof relies on a false inequality: derived under EPRB, which is a quantum experiment. That is: Bell (1964) derives an inequality THAT CANNOT HANDLE quantum mechanical expectation values.

So, under EPRB, Bell made a mistake. And that mistake is in his move from (14a) to (14b): ie, the EPRB quantum observables are pairwise-bound and Bell breaches that bond.

This breach — Bell's erroneous move from Bell 1964:(14a) to Bell 1964:(14b) — is defined via this next (or similar, for there are other false routes to Bell's inequality):



(Similar errors produce the equally false CHSH-Bell inequality.)

Yet, via nothing more than high-school math: I derive irrefutable inequalities that refute those by Bell and CHSH-Bell. And I test them with quantum mechanical expectation values. I find that my inequalities survive all possible combinations of such expectations: the Bell and CHSH-Bell inequalities do not.

So, Richard, please explain the "realism" or the CFD that allows Bell and CHSH-Bell to proceed so incorrectly; ie, please explain the "false" realism that Bell CHSH-Bell employ but you reject.

Further. While I spot Bell's errors, and can explain and correct them: your 2014 reads to me like an attempt to save Bell via a vaguely-diluted Bohrian realism.

FOR Gill (2014) includes: "So “realism” actually refers to models of reality, not to reality itself (p.2). In view of the experimental support for violation of Bell’s inequality, the present writer prefers to imagine a world in which “realism” is not a fundamental principle of physics but only an emergent property in the familiar realm of daily life. In this way, we can keep quantum mechanics, locality and freedom (p.3). It seems to me that we are pretty much forced into rejecting realism, which, please remember, is actually an idealistic concept: outcomes “exist” of measurements which were not performed (p.8)." [GW emphasis.]


YET, in Bell's important and extensive 1990 recapitulation of his dilemma and difficulties, I find that Bell makes NO use of these words: REAL, REALITY, REALISM.

Finally, from Bell (1964): "Let this more complete specification*** be effected by means of parameters ."

*** NB: NOT a wholly complete specification!

Thus my use of provides a more complete specification. That is: I match "elements of physical reality" to equivalence classes and thereby deliver quantum mechanical expectation values.

.


I've copied this post, and will soon copy another (by Richard Gill), from the thread : "Some people do not understand Counterfactual Definiteness."

Reason: My reply to Richard will have more to do with mathematics and less to do Counterfactual Definiteness.
.
Last edited by Gordon Watson on Fri Jan 01, 2021 3:55 am, edited 1 time in total.
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Fri Jan 01, 2021 3:51 am

Copied to this thread — from "Some people do not understand Counterfactual Definiteness" (CFD) — by Gordon Watson.

Reason: My response will have more to with mathematics than CFD

gill1109 wrote:Gordon, Bell summarises the physical and metaphysical assumptions and reasoning which would ensure that functions A, B and rho exist *with certain properties* and which would reproduce the singlet correlations. According to that point of view, A(a, lambda) = +/-1 is the outcome which Alice would observe if she chose setting a, while the state of detectors, source and transmissions lines between them was described by lambda. Bell does not say that he agrees with those assumptions. It is however clear, I think, that Einstein, and all physicists of his generation and before, would have agreed with them. Many later physicists do too. Many people active on this forum seem to believe in “local realism”, and they cite approvingly others (e.g. Hess, de Raedt, Michielsen), who do too.

I’m not a physicist nor a philosopher, I try to stick to maths, though of course I do have opinions and a (Buddhist) world view.

Bell’s purely mathematical proof that A, B and rho (*with the desired properties*) do not exist is waterproof, in my opinion, though e.g. Joy Christian thinks it is flawed.

It seems you agree with Bell, on essentials: there are no A, B and rho *with the usual properties* which reproduce the singlet correlations. You just don’t understand his maths and his logic.

I have no idea what you mean by true local realism. I suspect you are an adherent of the Copenhagen school. You have some mystical belief in the reality of noncommuting Hilbert space operators. I think that that is a religious belief, not a scientific viewpoint. I have looked for, but failed to discover, any convincing stand-alone original mathematics in your works, sorry. Seems you are happy with QM as it is; you follow Bohr. I don’t know what you take as hidden variable “lambda”, I haven’t seen any “complete description” of an EPR-B experiment in your works.

You keep mentioning that Bell nowhere talks about realism. He was a physicist, not a philosopher. He had no need to use the word. I do often use the phrase “local realism” because it has become very common. When I use it, I say what I mean by it.
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Re: Bell's theorem refuted via elementary probability theory

Postby Esail » Fri Jan 01, 2021 12:21 pm

Gordon Watson wrote:


How then do you move, in a mathematically correct way, from Bell's 1964:(14a) to his (14b)?



.

Gordon:



With A(b,lambda) *A(b,lambda)=1 see Bell's eq.(1) you will easily arrive from Bell's 14a at 14b.
With A(a,lambda)*A(b,lambda) <1 you will arrive from 14b at 14c provided A(a,lambda) exists.
But A(a,lambda) exists unambiguously only with noncontextual models. Thus, with a noncontextual model Bell's equation is correct but does not apply to QM.

Eugen
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Fri Jan 01, 2021 6:54 pm

Esail wrote:
Gordon Watson wrote:How then do you move, in a mathematically correct way, from Bell's 1964:(14a) to his (14b)?

Gordon:

With A(b,lambda) *A(b,lambda)=1 see Bell's eq.(1) you will easily arrive from Bell's 14a at 14b.
With A(a,lambda)*A(b,lambda) <1 you will arrive from 14b at 14c provided A(a,lambda) exists.
But A(a,lambda) exists unambiguously only with noncontextual models. Thus, with a noncontextual model Bell's equation is correct but does not apply to QM.

Eugen


Thanks Eugen, I should have been more specific. We accept the truth of Bell (14a) but differ over the truth of (14b).

So please show me the (14aa), (14ab), (14ac) — say, if you need 3 steps — that YOU use to arrive at (14b).
.
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Fri Jan 01, 2021 10:25 pm

Gordon Watson wrote:
Esail wrote:
Gordon Watson wrote:How then do you move, in a mathematically correct way, from Bell's 1964:(14a) to his (14b)?


Gordon

With A(b,lambda) *A(b,lambda)=1 see Bell's eq.(1) you will easily arrive from Bell's 14a at 14b.
With A(a,lambda)*A(b,lambda) <1 you will arrive from 14b at 14c provided A(a,lambda) exists.
But A(a,lambda) exists unambiguously only with noncontextual models. Thus, with a noncontextual model Bell's equation is correct but does not apply to QM.

Eugen


Thanks Eugen, I should have been more specific. We accept the truth of Bell (14a) but differ over the truth of (14b).

So please show me the (14aa), (14ab), (14ac) — say, if you need 3 steps — that YOU use to arrive at (14b).
.

Gordon:

Bell defines A(a, lambda) to be the result “A” of measuring the observable “sigma_1 cdot a” when, just before measurement, the state of particles and measurement devices is lambda. That result is either +1 or -1. It’s a number.

“sigma_1 cdot a” is a self-adjoint operator on a Hilbert space, represented mathematically by a 2x2 complex matrix.

The derivation from (14) to (15) is irrefutable, elementary. After some preparatory explanation, Bell assumes given: a set Lambda of points lambda endowed with a probability measure rho, a function A taking the values +/-1, and he defines P(a, b) = - int d lambda rho(lambda) A(a, lambda) A(b, lambda).

In the language of probability theory, he has assumed existence of a collection of random variables A_a on a single probability space, taking values in {-1, +1}, and studies the expectation values P(a, b) = E( A_a A_b). He considers only three values of “a” so we are talking about three binary variables. There are 2x2x2 = 8 elementary events in play, by which I mean subsets of Lambda on which A_a, A_b and A_c take on each if the 8 possible combinations of values.

His inequality is an exercise to the reader in Boole (1853). Boole shows that the 6 one-sided Bell inequalities (I.e., single inequalities without an absolute value sign) are necessary and sufficient for existence of 8 non-negative probabilities adding up to 1 which reproduce the three expectations of products P(a, b), P(a, c), P (b, c).

P(a, b) - P(a, c) - P(b, c) is not greater than +1
P(a, b) - P(a, c) - P(b, c) is not less than -1

P(a, c) - P(b, c) - P(a, b) is not greater than +1
P(a, c) - P(b, c) - P(a, b) is not less than -1

P(b, c) - P(a, b) - P(a, c) is not greater than +1
P(b, c) - P(a, b) - P(a, c) is not less than -1

Eugen:

Bell defines A(a, lambda) to be the result “A” of measuring the observable “sigma_1 cdot a” when, just before measurement, the state of particles and measurement devices is lambda. That result is either +1 or -1. It’s a number. Bell’s model is as contextual as you could possibly desire. It is “local”. Bell assumes A(a, lambda), not A(a, lambda; b). The latter would be contextual, taking the whole system of both measurement devices to be the context for one joint measurement depending on a joint setting (a, b).
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Sat Jan 02, 2021 12:31 am

X1 = X2 and X2 = X3 => X1 = X3
X1 != X3 => X1 != X2 or X2 != X3
Pr(X1 != X3) </= Pr(X1 != X2) + Pr(X2 != X3)
2 * Pr(X1 != X3) </= 2 * Pr(X1 != X2) + 2 * Pr(X2 != X3)
1 + 1 - 2 * Pr(X1 != X3) >/= 1 - 2 * Pr(X1 != X2) + 1 - 2 * Pr(X2 != X3)
1 + E(X1 * X3) >/= E(X1 * X2) + E(X2 * X3)

[the three variables take only the values +/-1, for such r.v.s, E(XY) = Pr(X == Y) - Pr(X != Y) while Pr(X == Y) + Pr(X != Y) = 1]
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