## What exactly is Bell's Theorem?

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

### What exactly is Bell's Theorem?

The Wikipedia entry for Bell's Theorem gives,

"If [a hidden-variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local."

Which is from the book "Speakable and Unspeakable in Quantum Mechanics" page 65 which actually says,

"But if his extension is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local. This is what the theorem says."

However, in the paragraph before this statement, Bell gives another description of the so-called theorem.

"$A({\bf a}, \lambda)$
$B({\bf b}, \lambda)$ (3)

With these local forms, it is not possible to find functions A and B and a probability distribution $\rho$ which give the correlation (1). This is the theorem."

Correlation (1) is of course the quantum mechanical prediction of -a.b. Now..., someone with proper definitions could possibly make this into a rigorous mathematical theorem. But there not much point in that since Joy has already found local A and B functions that do give the QM correlation. This should really be the end of the debate.
.
FrediFizzx
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### Re: What exactly is Bell's Theorem?

FrediFizzx wrote:
Correlation (1) is of course the quantum mechanical prediction of -a.b. Now..., someone with proper definitions could possibly make this into a rigorous mathematical theorem. But there is not much point in that since Joy has already found local A and B functions that do give the QM correlation. This should really be the end of the debate.

I of course agree with your conclusion. But just to be the devil's advocate (because the devils seem to have fled the battleground), what the Bell-believers would say is that by considering four EPR-Bohm type experiments and assuming only locality and realism (aka "counterfactual definiteness") they are able to derive the bounds of +/-2 on the CHSH correlator. The theorem, they claim, is then that no local functions of the form $A({\bf a}, \lambda)=\pm1$ and $B({\bf b}, \lambda)=\pm1$ can be used to exceed those bounds.

But their claim is wrong on at least two counts. To begin with, their claim is based on the same mistaken assumption von Neumann's theorem is based on (as I show in my latest paper). And secondly, it is wrong because of what you say above (which is explicitly shown, for example, in my IEEE Access paper).

***
Joy Christian
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### Re: What exactly is Bell's Theorem?

FrediFizzx wrote:The Wikipedia entry for Bell's Theorem gives,

"If [a hidden-variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local."

Which is from the book "Speakable and Unspeakable in Quantum Mechanics" page 65 which actually says,

"But if his extension is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local. This is what the theorem says."

However, in the paragraph before this statement, Bell gives another description of the so-called theorem.

"$A({\bf a}, \lambda)$
$B({\bf b}, \lambda)$ (3)

With these local forms, it is not possible to find functions A and B and a probability distribution $\rho$ which give the correlation (1). This is the theorem."

Correlation (1) is of course the quantum mechanical prediction of -a.b. Now..., someone with proper definitions could possibly make this into a rigorous mathematical theorem. But there not much point in that since Joy has already found local A and B functions that do give the QM correlation. This should really be the end of the debate.
.

When Bell himself talked about his *theorem* he usually meant his *inequality*. As a mathematician, I would rather call the inequality a *lemma*. It’s an easy to prove, elementary mathematical inequality. There are mathematical theorems concerning local realism and quantum mechanics. See for instance Boris Tsirelson’s (RIP) writings on Citizendium, or a not so old paper by Landsman and Cator. In order to qualify as mathematical theorem one has to give careful mathematical definitions of all concepts involved. My *executive summary* of such a theorem is: the formal mathematical structure called “quantum mechanics” is incompatible with mathematical structures possessing the conjunction of three, not two, properties: “locality”, “realism”, and “no-conspiracy”. Of course it is a matter for physicists, or for philosophers of science, to argue about whether or not certain formal mathematical definitions have any “real world” interest.

I have been very busy with Corona virus infection statistics, epidemiology, and hydroxychloroquine treatment. It is a good time for statisticians. Physicists also have very important insights.

Regarding “no-conspiracy: Bell himself of course knew very well that one had to rule out super-determinism or to say it in a positive sense, to assume the existence of something which some people like to call “free will”, but I would say is just a statistical assumption about the possibility to make effectively random choices when setting up experimental parameters of a real world lab experimental.

At this moment, Sabine Hossenfelder and Tim Palmer are two high-profile proponents of super-determinism. Earlier, Gerard ‘t Hooft has been an influential but somewhat lonely proponent of that point of view.
gill1109
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### Re: What exactly is Bell's Theorem?

gill1109 wrote:
When Bell himself talked about his *theorem* he usually meant his *inequality*.

Usually, but not always. Fred's observation is correct. For Bell his "inequality" was incidental --- a means to an end. For him, the "theorem" was about the impossibility of reproducing the correlations -a.b by averaging over the product A(a, h)*B(b, h) of the local functions A(a, h) and B(b, h). On page 65 of his book quoted by Fred, Bell explicitly says "This is the theorem."

However, his theorem, whichever form it takes, is simply wrong. But many things are wrong in this world and people still believe in them. That is the characteristic of a belief system.

gill1109 wrote:
Bell himself knew that one had to rule out super-determinism or to say it in a positive sense, to assume the existence of something which some people like to call “free will” ...

Here I agree with you. Super-determinism is a nonstarter. It dictates not only the answers but also questions that we can ask Nature. That is equivalent to giving up on scientific enterprise.

***
Joy Christian
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### Re: What exactly is Bell's Theorem?

Gentlemen, perhaps this video will be of interest to you.

local

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### Re: What exactly is Bell's Theorem?

Joy Christian wrote:
FrediFizzx wrote:
Correlation (1) is of course the quantum mechanical prediction of -a.b. Now..., someone with proper definitions could possibly make this into a rigorous mathematical theorem. But there is not much point in that since Joy has already found local A and B functions that do give the QM correlation. This should really be the end of the debate.

I of course agree with your conclusion. But just to be the devil's advocate (because the devils seem to have fled the battleground), what the Bell-believers would say is that by considering four EPR-Bohm type experiments and assuming only locality and realism (aka "counterfactual definiteness") they are able to derive the bounds of +/-2 on the CHSH correlator. The theorem, they claim, is then that no local functions of the form $A({\bf a}, \lambda)=\pm1$ and $B({\bf b}, \lambda)=\pm1$ can be used to exceed those bounds.
...
***

It's trivial that if you have local A and B functions that produce the QM correlation of -a.b, you will exceed the bounds of CHSH also. So, all it takes is Bell's original "theorem" that I have shown above. With that disproven, all of the rest of it also falls into junk physics theory territory.
.
FrediFizzx
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### Re: What exactly is Bell's Theorem?

FrediFizzx wrote:
Joy Christian wrote:
FrediFizzx wrote:
Correlation (1) is of course the quantum mechanical prediction of -a.b. Now..., someone with proper definitions could possibly make this into a rigorous mathematical theorem. But there is not much point in that since Joy has already found local A and B functions that do give the QM correlation. This should really be the end of the debate.

I of course agree with your conclusion. But just to be the devil's advocate (because the devils seem to have fled the battleground), what the Bell-believers would say is that by considering four EPR-Bohm type experiments and assuming only locality and realism (aka "counterfactual definiteness") they are able to derive the bounds of +/-2 on the CHSH correlator. The theorem, they claim, is then that no local functions of the form $A({\bf a}, \lambda)=\pm1$ and $B({\bf b}, \lambda)=\pm1$ can be used to exceed those bounds.
...
***

It's trivial that if you have local A and B functions that produce the QM correlation of -a.b, you will exceed the bounds of CHSH also. So, all it takes is Bell's original "theorem" that I have shown above. With that disproven, all of the rest of it also falls into junk physics theory territory.

Sure. But to recognize that elementary fact requires honesty, scientific integrity, and competence in physics. That is asking too much from those irrationally committed to Bell orthodoxy.

***
Joy Christian
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### Re: What exactly is Bell's Theorem?

Joy Christian wrote:
FrediFizzx wrote:
Joy Christian wrote:
FrediFizzx wrote:
Correlation (1) is of course the quantum mechanical prediction of -a.b. Now..., someone with proper definitions could possibly make this into a rigorous mathematical theorem. But there is not much point in that since Joy has already found local A and B functions that do give the QM correlation. This should really be the end of the debate.

I of course agree with your conclusion. But just to be the devil's advocate (because the devils seem to have fled the battleground), what the Bell-believers would say is that by considering four EPR-Bohm type experiments and assuming only locality and realism (aka "counterfactual definiteness") they are able to derive the bounds of +/-2 on the CHSH correlator. The theorem, they claim, is then that no local functions of the form $A({\bf a}, \lambda)=\pm1$ and $B({\bf b}, \lambda)=\pm1$ can be used to exceed those bounds.
...
***

It's trivial that if you have local A and B functions that produce the QM correlation of -a.b, you will exceed the bounds of CHSH also. So, all it takes is Bell's original "theorem" that I have shown above. With that disproven, all of the rest of it also falls into junk physics theory territory.

Sure. But to recognize that elementary fact requires honesty, scientific integrity, and competence in physics. That is asking too much from those irrationally committed to Bell orthodoxy.

***

Competence in physics involves having certain broad mathematical skills (and even statistical skills).
gill1109
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### Re: What exactly is Bell's Theorem?

gill1109 wrote:
Joy Christian wrote:
Sure. But to recognize that elementary fact requires honesty, scientific integrity, and competence in physics. That is asking too much from those irrationally committed to Bell orthodoxy.

Competence in physics involves having certain broad mathematical skills (and even statistical skills).

Absolutely. And we must give the benefit of the doubt to those irrationally committed to Bell-orthodoxy that they do possess broad mathematical skills and even statistical skills. Indeed, Bell himself certainly possessed those skills (as evidenced by his excellent work in theoretical particle physics and phenomenological accelerator physics); and so did his followers such as Wigner (a Nobel Laureate) and Shimony (a Lakatos Laureate). And yet, each one of them failed, as did the mighty von Neumann, in applying those skills to the question of hidden variables underlying quantum mechanics, which is of course a profoundly physical question. von Neumann misapplied his superhuman mathematical ability to this physical question, and so did Bell, who ended up making exactly the same mistake as the one von Neumann had made after ridiculing him for making that mistake. Oh ... the irony!

Ref: https://arxiv.org/abs/1704.02876 (2020).

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Joy Christian
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### Re: What exactly is Bell's Theorem?

Sabine Hossenfelder has a recent post about nonlocality and Bell's theorem on her blog:

http://backreaction.blogspot.com/2020/0 ... 3-non.html
Heinera

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### Re: What exactly is Bell's Theorem?

Heinera wrote:
Sabine Hossenfelder has a recent post about nonlocality and Bell's theorem on her blog:

http://backreaction.blogspot.com/2020/0 ... 3-non.html

With all due respect to Sabine, I prefer my own explanation of quantum correlations. My explanation is far more lucid, compelling, and non-mystical: https://arxiv.org/abs/1911.11578.

***
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### Re: What exactly is Bell's Theorem?

FrediFizzx wrote:The Wikipedia entry for Bell's Theorem gives,

"If [a hidden-variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local."

Which is from the book "Speakable and Unspeakable in Quantum Mechanics" page 65 which actually says,

"But if his extension is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local. This is what the theorem says."

However, in the paragraph before this statement, Bell gives another description of the so-called theorem.

"$A({\bf a}, \lambda)$
$B({\bf b}, \lambda)$ (3)

With these local forms, it is not possible to find functions A and B and a probability distribution $\rho$ which give the correlation (1). This is the theorem."

Correlation (1) is of course the quantum mechanical prediction of -a.b. Now..., someone with proper definitions could possibly make this into a rigorous mathematical theorem. But there not much point in that since Joy has already found local A and B functions that do give the QM correlation. This should really be the end of the debate.
.

For some unknown reason, the Bell fans don't want to even accept Bell's own definition of the "theorem". So, now we have Gill's "theorem" which is basically that a local model can't simulate Nature and the experiments. But quantum mechanics can't predict correct individual outcome events for A and B either. So, what is the point? Now what is more, Jay Yablon has successfully demonstrated that quantum mechanics is local for the EBR-Bohm scenario so locality is no longer an issue. QM and... Nature must be local so non-local simulations are completely out.
.
FrediFizzx
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### Re: What exactly is Bell's Theorem?

FrediFizzx wrote:
FrediFizzx wrote:The Wikipedia entry for Bell's Theorem gives,

"If [a hidden-variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local."

Which is from the book "Speakable and Unspeakable in Quantum Mechanics" page 65 which actually says,

"But if his extension is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local. This is what the theorem says."

However, in the paragraph before this statement, Bell gives another description of the so-called theorem.

"$A({\bf a}, \lambda)$
$B({\bf b}, \lambda)$ (3)

With these local forms, it is not possible to find functions A and B and a probability distribution $\rho$ which give the correlation (1). This is the theorem."

Correlation (1) is of course the quantum mechanical prediction of -a.b. Now..., someone with proper definitions could possibly make this into a rigorous mathematical theorem. But there not much point in that since Joy has already found local A and B functions that do give the QM correlation. This should really be the end of the debate.
.

For some unknown reason, the Bell fans don't want to even accept Bell's own definition of the "theorem". So, now we have Gill's "theorem" which is basically that a local model can't simulate Nature and the experiments. But quantum mechanics can't predict correct individual outcome events for A and B either. So, what is the point? Now what is more, Jay Yablon has successfully demonstrated that quantum mechanics is local for the EBR-Bohm scenario so locality is no longer an issue. QM and... Nature must be local so non-local simulations are completely out.
.

The easiest proof of the CHSH inequality shows it is a mathematical triviality, following directly from the assumptions of local realism and a basic probability theory inequality called Boole’s inequality. Bell’s earlier three correlations inequality is even more so, a triviality. It’s a special case of CHSH. Boole has it as an exercise in his book from 18-whatever. “Bell’s theorem” is the equally trivial logical corollary that local realism and quantum mechanics are incompatible. A local realist model does generate in a local way individual outcomes of measurements whose correlations are the same as the correlations predicted by QM. Now, superdeterminism, retrocausality, and postselection (detection loophole) are all ways to get out of Bell’s theorem. They are collectively thought of as conspiratorial, or violations of the freedom of the experimenters to choose whatever settings they like and then get to see a pair of outcomes.
gill1109
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### Re: What exactly is Bell's Theorem?

gill1109 wrote:
The easiest proof of the CHSH inequality shows it is a mathematical triviality, following directly from the assumptions of local realism and a basic probability theory inequality called Boole’s inequality. Bell’s earlier three correlations inequality is even more so, a triviality. It’s a special case of CHSH. Boole has it as an exercise in his book from 18-whatever. “Bell’s theorem” is the equally trivial logical corollary that local realism and quantum mechanics are incompatible. A local realist model does generate in a local way individual outcomes of measurements whose correlations are the same as the correlations predicted by QM. Now, superdeterminism, retrocausality, and postselection (detection loophole) are all ways to get out of Bell’s theorem. They are collectively thought of as conspiratorial, or violations of the freedom of the experimenters to choose whatever settings they like and then get to see a pair of outcomes.

This is complete and utter nonsense. None of the Bell inequalities follow from local realism and basic probability theory. For example, if you assume local realism and basic probability theory for the CHSH correlator, you end up with the bounds of +/-4. A further assumption of the additivity of expectation values is required to reduce the bounds of +/-4 to the bounds of +/-2. But the additivity of expectation values is not a valid assumption for any hidden variable theory, regardless of the assumption of local realism. Thus Bell's theorem is a non-starter.

To put this differently, what is ruled out by the Bell-test experiments is not local realism. What is ruled out by the Bell-test experiments is the additivity of expectation values.

***
Joy Christian
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### Re: What exactly is Bell's Theorem?

Joy Christian wrote:
gill1109 wrote:
The easiest proof of the CHSH inequality shows it is a mathematical triviality, following directly from the assumptions of local realism and a basic probability theory inequality called Boole’s inequality. Bell’s earlier three correlations inequality is even more so, a triviality. It’s a special case of CHSH. Boole has it as an exercise in his book from 18-whatever. “Bell’s theorem” is the equally trivial logical corollary that local realism and quantum mechanics are incompatible. A local realist model does generate in a local way individual outcomes of measurements whose correlations are the same as the correlations predicted by QM. Now, superdeterminism, retrocausality, and postselection (detection loophole) are all ways to get out of Bell’s theorem. They are collectively thought of as conspiratorial, or violations of the freedom of the experimenters to choose whatever settings they like and then get to see a pair of outcomes.

This is complete and utter nonsense. None of the Bell inequalities follow from local realism and basic probability theory. For example, if you assume local realism and basic probability theory for the CHSH correlator, you end up with the bounds of +/-4. A further assumption of the additivity of expectation values is required to reduce the bounds of +/-4 to the bounds of +/-2. But the additivity of expectation values is not a valid assumption for any hidden variable theory, regardless of the assumption of local realism. Thus Bell's theorem is a non-starter.

To put this differently, what is ruled out by the Bell-test experiments is not local realism. What is ruled out by the Bell-test experiments is the additivity of expectation values.

***

Additivity of expectation values is, in my book, part of basic probability theory. But of course everything depends here also on your definition of “local realism”. If your definition is different from that of everyone else, then (a) nobody will understand you, (b) Bell’s theorem can switch between true and false, and similar, Bell’s inequality too. Perhaps everyone who participates in this thread should start by giving some precise definitions of what they understand by “Bell’s theorem”. They should also point out if they agree or disagree with various definitions given by what usually might be considered “authoritative sources”. Standard text books and the like.
gill1109
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### Re: What exactly is Bell's Theorem?

gill1109 wrote:
Joy Christian wrote:
gill1109 wrote:
The easiest proof of the CHSH inequality shows it is a mathematical triviality, following directly from the assumptions of local realism and a basic probability theory inequality called Boole’s inequality. Bell’s earlier three correlations inequality is even more so, a triviality. It’s a special case of CHSH. Boole has it as an exercise in his book from 18-whatever. “Bell’s theorem” is the equally trivial logical corollary that local realism and quantum mechanics are incompatible. A local realist model does generate in a local way individual outcomes of measurements whose correlations are the same as the correlations predicted by QM. Now, superdeterminism, retrocausality, and postselection (detection loophole) are all ways to get out of Bell’s theorem. They are collectively thought of as conspiratorial, or violations of the freedom of the experimenters to choose whatever settings they like and then get to see a pair of outcomes.

This is complete and utter nonsense. None of the Bell inequalities follow from local realism and basic probability theory. For example, if you assume local realism and basic probability theory for the CHSH correlator, you end up with the bounds of +/-4. A further assumption of the additivity of expectation values is required to reduce the bounds of +/-4 to the bounds of +/-2. But the additivity of expectation values is not a valid assumption for any hidden variable theory, regardless of the assumption of local realism. Thus Bell's theorem is a non-starter.

To put this differently, what is ruled out by the Bell-test experiments is not local realism. What is ruled out by the Bell-test experiments is the additivity of expectation values.

***

Additivity of expectation values is, in my book, part of basic probability theory. But of course everything depends here also on your definition of “local realism”. If your definition is different from that of everyone else, then (a) nobody will understand you, (b) Bell’s theorem can switch between true and false, and similar, Bell’s inequality too. Perhaps everyone who participates in this thread should start by giving some precise definitions of what they understand by “Bell’s theorem”. They should also point out if they agree or disagree with various definitions given by what usually might be considered “authoritative sources”. Standard text books and the like.

Whichever book additivity of expectation values belongs to, it has been ruled out by the Bell-test experiments. No one has ruled out local realism. The moon is there when I am not looking.

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### Re: What exactly is Bell's Theorem?

I agree that the moon is there when I’m not looking. Linearity of expectation values follows immediately from the definition of expectation values. See https://en.m.wikipedia.org/wiki/Expected_value and the many links there. But one has to realise than in present context, “realism” is not a practical, obvious concept. It’s idealistic. Tsirelson and others suggest we use the term “counterfactual definiteness”. Even if, contrary to fact, we did not measure the momentum of a particle, but only measure its position, it still can be thought to have a position. This is not a claim about reality. It’s a claim about mathematical descriptions of reality. They can be augmented in a consistent (and not to forget, local) way with descriptions of alternative realities. A deterministic description allows such augmentation. We can put the planets in a different configuration from where they are today and the equations of motion still tell us where they would then be tomorrow. Also, stochastic descriptions allow this, if the random disturbances can be imagined to be independent of the deterministic parts of the model. From Bohr’s point of view, which you could say found its logical fulfilment with David Bohm’s, the whole debate with Einstein was a waste of time, since everything is interconnected. David Bohm went on to show that the mathematical structure of QM could be augmented so as to make the universe interconnected and deterministic. Bell showed any augmentation *has* to have nonlocal features. I think physicists have to get used to the facts of life. I think actually, most do.
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### Re: What exactly is Bell's Theorem?

gill1109 wrote:
I agree that the moon is there when I’m not looking. Linearity of expectation values follows immediately from the definition of expectation values. See https://en.m.wikipedia.org/wiki/Expected_value and the many links there. But one has to realise than in present context, “realism” is not a practical, obvious concept. It’s idealistic. Tsirelson and others suggest we use the term “counterfactual definiteness”. Even if, contrary to fact, we did not measure the momentum of a particle, but only measure its position, it still can be thought to have a position. This is not a claim about reality. It’s a claim about mathematical descriptions of reality. They can be augmented in a consistent (and not to forget, local) way with descriptions of alternative realities. A deterministic description allows such augmentation. We can put the planets in a different configuration from where they are today and the equations of motion still tell us where they would then be tomorrow. Also, stochastic descriptions allow this, if the random disturbances can be imagined to be independent of the deterministic parts of the model. From Bohr’s point of view, which you could say found its logical fulfilment with David Bohm’s, the whole debate with Einstein was a waste of time, since everything is interconnected. David Bohm went on to show that the mathematical structure of QM could be augmented so as to make the universe interconnected and deterministic. Bell showed any augmentation *has* to have nonlocal features. I think physicists have to get used to the facts of life. I think actually, most do.

Your waffle describes a religion, not science. The additivity of expectation values is not a valid or acceptable assumption for any hidden variable theory, regardless of locality or reality. On the other hand, the only way to derive the bounds of +/-2 on the CHSH correlator is by assuming the additivity of expectation values. If the additivity of expectation values is not assumed, then the bounds on the CHSH correlator are +/-4, not +/-2. In the experiments, the bounds of +/-2 are exceeded. Therefore the assumption of the additivity of expectation values is ruled out by the experiments. Locality and realism remain untouched and unscathed, contrary to what Bell and his followers believe.

***
Joy Christian
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### Re: What exactly is Bell's Theorem?

Joy Christian wrote:Your waffle describes a religion, not science.

That is not acceptable language to use in scientific discourse. It seems to me to be a value judgement, an opinion. You are welcome to your opinion. I have my own. I think your argument is faulty. It seems you don’t want to address arguments. Perhaps someone else on the forum is interested to actually enter into the arguments.
gill1109
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### Re: What exactly is Bell's Theorem?

gill1109 wrote:
Joy Christian wrote:Your waffle describes a religion, not science.

That is not acceptable language to use in scientific discourse. It seems to me to be a value judgement, an opinion. You are welcome to your opinion. I have my own. I think your argument is faulty. It seems you don’t want to address arguments. Perhaps someone else on the forum is interested to actually enter into the arguments.

Ok, forget my first sentence. Here is the rest of what I wrote, and it is an irrefutable scientific argument:

Joy Christian wrote:
"The additivity of expectation values is not a valid or acceptable assumption for any hidden variable theory, regardless of locality or reality. On the other hand, the only way to derive the bounds of +/-2 on the CHSH correlator is by assuming the additivity of expectation values. If the additivity of expectation values is not assumed, then the bounds on the CHSH correlator are +/-4, not +/-2. In the experiments, the bounds of +/-2 are exceeded. Therefore the assumption of the additivity of expectation values is ruled out by the experiments. Locality and realism remain untouched and unscathed, contrary to what Bell and his followers believe."

***
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