A Mathematician's Refutation of Bell's Theorem

[gill1109]

He did not. Here is what he did. Critics argued that he was using a single 3xN spreadsheet instead of 3 disjoint 2xN spreadsheets. He replied that he was using 3 disjoint 2xN spreadsheets. But based on his equations, that is false. He is using 2 2xN spreadsheets and from each of them he separates one column, recombines them to generate a third 2xN spreadsheet that is not independent of the other two in the same way the first two are independent from each other. While it appears to diffuse the criticism, it does not eliminate it. Perhaps he did not understand the root of the problem.

Michel, Bell does not use the idea of a spreadsheet. His first Bell inequality paper does not talk much about data. If you did do the experiment you would have data sets of various sizes for each of the setting pairs, as well as a data set testing the anti-correlation at equal settings.

The criticism is invalid. It has been adequately answered. But people will continue to misunderstand Bell’s arguments, that we can be sure of!

Why must the terms be cyclical? Why has nobody been able to "prove" Bell's theorem without relying on a cyclicity? What manner of "non-realism" or "non-locality" is this that only shows up when terms are cyclically related? I would have expected more curiosity from you on such topics.

There are numerous other ways to prove Bell’s theorem. Yes, I am very curious about this question. See Gull’s proof. https://arxiv.org/abs/2012.00719. No cyclicity. Fourier analysis and time series. Very elegant.

Oh so now you agree with me that it is very important to know "What" statistics converge to 2? You admit that there are local realistic statistics that converge to 2 and non-local statistics that do not converge to 2?

There are simulation experiments which give all kinds of answers. You wrote some yourself. I recall Joy Christian showing how you could get 2, 2 sqrt 2, or 4, by changing the value of a parameter in a simulation model which, he I think he said, was your simulation of his model, as improved by myself and others, done in R.

P(b,c) in the inequality is more restricted than P(a,b) and P(a,c) due to the cyclic nature of the terms. This is the core ingredient of all variants of Bell's theorem. They include a hidden reduction of independence that is simply ignored when comparing the results with experimental data where no such reduction in independence is present.

Proofs all exploit the *manifest* dependence of the various correlations, since they all depend on the same functions A, B and rho. In an experiment, each observed correlation could be anything from -1 to +1. But if Bell’s assumption is correct they will tend to converge to those integrals. And they will be related by Bell’s inequality.

You can test that yourself by writing your own simulation experiments! Dream up and A, B and rho satisfying the usual conditions. Take a number of independent, big, samples of lambda. Dream up a, b, c…

[minkwe]

He did not. Here is what he did. Critics argued that he was using a single 3xN spreadsheet instead of 3 disjoint 2xN spreadsheets. He replied that he was using 3 disjoint 2xN spreadsheets. But based on his equations, that is false. He is using 2 2xN spreadsheets and from each of them he separates one column, recombines them to generate a third 2xN spreadsheet that is not independent of the other two in the same way the first two are independent from each other. While it appears to diffuse the criticism, it does not eliminate it. Perhaps he did not understand the root of the problem.

Michel, Bell does not use the idea of a spreadsheet. His first Bell inequality paper does not talk much about data. If you did do the experiment you would have data sets of various sizes for each of the setting pairs, as well as a data set testing the anti-correlation at equal settings.

Thanks for pointing that out, I didn't know that

I expressed in terms of spreadsheets so that perhaps you could understand it finally but it appears to confuse you more.

Proofs all exploit the *manifest* dependence of the various correlations, since they all depend on the same functions A, B and rho. In an experiment, each observed correlation could be anything from -1 to +1. But if Bell’s assumption is correct they will tend to converge to those integrals. And they will be related by Bell’s inequality.

Again it appears I confuse you some more when I talk about proofs of Bell's theorem. When I talk of proofs of Bell's theorem, I mean proofs of Bell's theorem, not proofs of Bell's inequalities. Do you know the difference? If so, it seems you don't believe me when I tell you that the proof of Bell's inequality is true and uncontested, at least by me, because you keep trying to convince me that the proof is true. Do you not understand that the validity of Bell's inequalities as a relationship in mathematics is separate from the validity of Bell's theorem in physics?

[minkwe]

I gave you three it's very easy to pick one of them unless you just want to argue.

Two of them invoke the coincidence "loophole". That has been uninteresting for many years as the recent experimental protocols eliminate it.

So what? Perhaps you should review the "claim" that you were responding to.

BTW, I do not agree with other claims in your comment that recent protocols eliminate the coincidence loophole.

[gill1109]

He did not. Here is what he did. Critics argued that he was using a single 3xN spreadsheet instead of 3 disjoint 2xN spreadsheets. He replied that he was using 3 disjoint 2xN spreadsheets. But based on his equations, that is false. He is using 2 2xN spreadsheets and from each of them he separates one column, recombines them to generate a third 2xN spreadsheet that is not independent of the other two in the same way the first two are independent from each other. While it appears to diffuse the criticism, it does not eliminate it. Perhaps he did not understand the root of the problem.

Michel, Bell does not use the idea of a spreadsheet. His first Bell inequality paper does not talk much about data. If you did do the experiment you would have data sets of various sizes for each of the setting pairs, as well as a data set testing the anti-correlation at equal settings.

Thanks for pointing that out, I didn't know that

I expressed in terms of spreadsheets so that perhaps you could understand it finally but it appears to confuse you more.

Proofs all exploit the *manifest* dependence of the various correlations, since they all depend on the same functions A, B and rho. In an experiment, each observed correlation could be anything from -1 to +1. But if Bell’s assumption is correct they will tend to converge to those integrals. And they will be related by Bell’s inequality.

Again it appears I confuse you some more when I talk about proofs of Bell's theorem. When I talk of proofs of Bell's theorem, I mean proofs of Bell's theorem, not proofs of Bell's inequalities. Do you know the difference? If so, it seems you don't believe me when I tell you that the proof of Bell's inequality is true and uncontested, at least by me, because you keep trying to convince me that the proof is true. Do you not understand that the validity of Bell's inequalities as a relationship in mathematics is separate from the validity of Bell's theorem in physics?

Yes, I do understand the difference between (1) a theorem about the incompatibility of local realism and quantum mechanics, (2) a trivial inequality used as a lemma in some proofs of the theorem, and (3) the use of Bell's reasoning in order to argue that some experimental data is incompatible with local realism.

BTW here's a new way to do a Bell experiment. Of course, it should be "loophole free" in the sense people understand that nowadays. No experimentally avoidable loopholes. Experiment: estimate correlations by taking lots of measurements at lots and lots of pairs of settings. Draw the correlation curve. You hope to see a negative cosine, right? It must have amplitude at least 1/sqrt 2. https://studenttheses.universiteitleiden.nl/handle/1887/3196304

[Joy Christian]

He did not. Here is what he did. Critics argued that he was using a single 3xN spreadsheet instead of 3 disjoint 2xN spreadsheets. He replied that he was using 3 disjoint 2xN spreadsheets. But based on his equations, that is false. He is using 2 2xN spreadsheets and from each of them he separates one column, recombines them to generate a third 2xN spreadsheet that is not independent of the other two in the same way the first two are independent from each other. While it appears to diffuse the criticism, it does not eliminate it. Perhaps he did not understand the root of the problem.

Michel, Bell does not use the idea of a spreadsheet. His first Bell inequality paper does not talk much about data. If you did do the experiment you would have data sets of various sizes for each of the setting pairs, as well as a data set testing the anti-correlation at equal settings.

Thanks for pointing that out, I didn't know that

I expressed in terms of spreadsheets so that perhaps you could understand it finally but it appears to confuse you more.

Proofs all exploit the *manifest* dependence of the various correlations, since they all depend on the same functions A, B and rho. In an experiment, each observed correlation could be anything from -1 to +1. But if Bell’s assumption is correct they will tend to converge to those integrals. And they will be related by Bell’s inequality.

Again it appears I confuse you some more when I talk about proofs of Bell's theorem. When I talk of proofs of Bell's theorem, I mean proofs of Bell's theorem, not proofs of Bell's inequalities. Do you know the difference? If so, it seems you don't believe me when I tell you that the proof of Bell's inequality is true and uncontested, at least by me, because you keep trying to convince me that the proof is true. Do you not understand that the validity of Bell's inequalities as a relationship in mathematics is separate from the validity of Bell's theorem in physics?

Yes, I do understand the difference between (1) a theorem about the incompatibility of local realism and quantum mechanics, (2) a trivial inequality used as a lemma in some proofs of the theorem, and (3) the use of Bell's reasoning in order to argue that some experimental data is incompatible with local realism.

Michel, Gill does understand the difference between Boole's inequality (incorrectly known as Bell's inequality) and Bell's theorem. At least Gill appears to be intelligent enough to know the difference, considering that we (i.e., at least you, me, and Fred) have time and again stressed that distinction in this forum over the years. The real problem here clearly is that Richard D. Gill is a disingenuous and dishonest debater. He debates like a politician, not a scientist. He will claim that I am making an ad hominem attack on him by saying this, so let me back up my claim by giving concrete proof of what I am saying:

In my official response to his critique of one of my papers published in IEEE Access, namely, this one, I have written the following, which is similar to what I have colored above in red:



But what Gill knowingly and disingenuously keeps doing is quoting my non-yellow-highlighted words from the above image to paint a dishonest picture of what I have actually said. He does that even in his papers he has submitted to RSOS and IEEE Access. Fortunately, at least RSOS does not seem to have accepted his dishonest paper, which he submitted nearly a year ago (in October 2020, to be precise). But his papers might eventually be accepted by the journals, thereby contaminating the scientific record with his dishonesty and misrepresentations.
.

[FrediFizzx]

All this debating about something that is so mathematically simple it is mind boggling that the Bell fanatics don't get it. NOTHING can exceed the bounds on the inequalities!!!!
.

[gill1109]

Boole proved a necessary and sufficient version of Bell's original (three correlation) inequality. Actually, Boole presented it as an exercise to the reader; there is no "solutions manual" to his book, so you have to do some work to figure out the proof. It is not trivial.

Vorob'ev and Fine independently (and not knowing Boole's work either, and Vorob'ev not knowing Bell's work) proved necessary and sufficient versions of the Bell-CHSH four correlation inequality.

Something else is Boole's famous inequality P(A or B or C) is less than or equal to P(A) + P(B) + P(C). One can use it to prove the Bell inequalities (both original and Bell-CHSH).

The inequalities in themselves are very elementary, indeed. The necessary and sufficient versions are not really deep but cost a lot more hard work to prove rigorously, and to state carefully.

My 2014 Statistical Science paper https://arxiv.org/abs/1207.5103 (Statistics, Causality and Bell's Theorem) has a section "Better Bell Inequalities" (it's section 8) about all the possible Bell inequalities and the connection to convex polytopes (the kind of maths on which Frank Lad is an expert).

See also the picture on slide 10 of this talk: https://www.slideshare.net/gill1109/epidemiology-meets-quantum-statistics-causality-and-bells-theorem

Here is a simplified version,

a kind of cartoon. In a Bell type experiment there are 16 conditional probabilities p(x, y|a, b). They add up to 1 in sets of 4, and they are nonnegative. Moreover, there is no-signalling: the marginal probabilities of outcomes x, y given settings a, b (all four variables taking only two possible values) do not depend on the settings on the other side. The vector of 16 conditional probabilities therefore actually lies on the intersection of eight hyperplanes with the positive orthant. It's an 8-dimensional gadget. The cartoon tells us how that would look when you restrict attention to that "affine subspace" where all the action is. Sorry for the fancy word. I don't know a better one. It means the translation of a subspace of R^16 so that it does not contain the origin.

Why 8 dimensions? because once you know the 4 marginal probabilities of +1 given each setting on each side, and the 4 correlations, you can reconstruct all 16 probabilities.

[local]

Another trite lecture from Gill (with superfluous bragging) on stuff we already know, and totally unresponsive to the issues raised. This is called deflection and goes hand-in-hand with projection for narcs. Feigned erudition at its best.

[minkwe]

Boole proved a necessary and sufficient version of Bell's original (three correlation) inequality. Actually, Boole presented it as an exercise to the reader; there is no "solutions manual" to his book, so you have to do some work to figure out the proof. It is not trivial.

Vorob'ev and Fine independently (and not knowing Boole's work either, and Vorob'ev not knowing Bell's work) proved necessary and sufficient versions of the Bell-CHSH four correlation inequality.

Something else is Boole's famous inequality P(A or B or C) is less than or equal to P(A) + P(B) + P(C). One can use it to prove the Bell inequalities (both original and Bell-CHSH).

The inequalities in themselves are very elementary, indeed. The necessary and sufficient versions are not really deep but cost a lot more hard work to prove rigorously, and to state carefully.

My 2014 Statistical Science paper https://arxiv.org/abs/1207.5103 (Statistics, Causality and Bell's Theorem) has a section "Better Bell Inequalities" (it's section 8) about all the possible Bell inequalities and the connection to convex polytopes (the kind of maths on which Frank Lad is an expert).

See also the picture on slide 10 of this talk: https://www.slideshare.net/gill1109/epidemiology-meets-quantum-statistics-causality-and-bells-theorem

Here is a simplified version,

a kind of cartoon. In a Bell type experiment there are 16 conditional probabilities p(x, y|a, b). They add up to 1 in sets of 4, and they are nonnegative. Moreover, there is no-signalling: the marginal probabilities of outcomes x, y given settings a, b (all four variables taking only two possible values) do not depend on the settings on the other side. The vector of 16 conditional probabilities therefore actually lies on the intersection of eight hyperplanes with the positive orthant. It's an 8-dimensional gadget. The cartoon tells us how that would look when you restrict attention to that "affine subspace" where all the action is. Sorry for the fancy word. I don't know a better one. It means the translation of a subspace of R^16 so that it does not contain the origin.

Why 8 dimensions? because once you know the 4 marginal probabilities of +1 given each setting on each side, and the 4 correlations, you can reconstruct all 16 probabilities.

This is like a broken record. GIll is still trying to convince us that Bell's inequalities are true.

[local]

This is like a broken record. GIll is still trying to convince us that Bell's inequalities are true.

No doubt, but I think you meant to say "Bell's theorem", as you earlier posted "I tell you that the proof of Bell's inequality is true and uncontested, at least by me"

Just thought I'd get that in before our forum broken record jumps on you.

[Joy Christian]

This is like a broken record. GIll is still trying to convince us that Bell's inequalities are true.

No doubt, but I think you meant to say "Bell's theorem", as you earlier posted "I tell you that the proof of Bell's inequality is true and uncontested, at least by me"

Just thought I'd get that in before our forumm broken record jumps on you.

No. minkwe said it correctly. He meant to say "Bell's inequalities" and said it correctly. Everyone agrees that Bell's inequalities are trivially true. We don't need a lecture on that from Gill.
.

[local]

Ah yes, I see. Thank you for clarifying that. So he's even worse than I thought. Deflection, obfuscation, and condescension all rolled into one, with a side dose of braggadocio.

[minkwe]

This is like a broken record. GIll is still trying to convince us that Bell's inequalities are true.

No doubt, but I think you meant to say "Bell's theorem", as you earlier posted "I tell you that the proof of Bell's inequality is true and uncontested, at least by me"

Just thought I'd get that in before our forumm broken record jumps on you.

No. minkwe said it correctly. He meant to say "Bell's inequalities" and said it correctly. Everyone agrees that Bell's inequalities are trivially true. We don't need a lecture on that from Gill.
.

Exactly!

[FrediFizzx]

This is like a broken record. GIll is still trying to convince us that Bell's inequalities are true.

No doubt, but I think you meant to say "Bell's theorem", as you earlier posted "I tell you that the proof of Bell's inequality is true and uncontested, at least by me"

Just thought I'd get that in before our forumm broken record jumps on you.

No. minkwe said it correctly. He meant to say "Bell's inequalities" and said it correctly. Everyone agrees that Bell's inequalities are trivially true. We don't need a lecture on that from Gill.
.

Exactly!

And... it is also trivially true that NOTHING!!! can exceed the bounds on the inequalities!!!
.

[Gordon Watson]

This is like a broken record. GIll is still trying to convince us that Bell's inequalities are true.

No doubt, but I think you meant to say "Bell's theorem", as you earlier posted "I tell you that the proof of Bell's inequality is true and uncontested, at least by me"

Just thought I'd get that in before our forumm broken record jumps on you.

No. minkwe said it correctly. He meant to say "Bell's inequalities" and said it correctly. Everyone agrees that Bell's inequalities are trivially true. We don't need a lecture on that from Gill.
.

Exactly!

Everyone DOES NOT agree that Bell's inequalities are trivially true. I'd welcome a lecture on that score.
With no excuses, please. Triviality can surely be demonstrated in a few lines of math: so let's start with Bell (1964), eq.(15).
.

[minkwe]

Everyone DOES NOT agree that Bell's inequalities are trivially true. I'd welcome a lecture on that score.
With no excuses, please. Triviality can surely be demonstrated in a few lines of math: so let's start with Bell (1964), eq.(15).
.

See the proof in

viewtopic.php?f=6&t=482&start=40#p13703

It's a mathematical tautology. I think I know what you mean when you say you disagree.

[Gordon Watson]

Everyone DOES NOT agree that Bell's inequalities are trivially true. I'd welcome a lecture on that score.
With no excuses, please. Triviality can surely be demonstrated in a few lines of math: so let's start with Bell (1964), eq.(15).
.

See the proof in

viewtopic.php?f=6&t=482&start=40#p13703

It's a mathematical tautology. I think I know what you mean when you say you disagree.

Thanks minkwe,

Please, can you address Bell 1964:(15)? Also, please, send PM or email or put here: what do you think I mean when I say I disagree? For that might relate to the crux of the matter.

Thanks again, and best regards; Gordon
.

[Justo]

The third therm appearing there can't be independent of the other two and therefore can't be weakly objective as he later claims, or as Justo claims now.

I show explicitly, for the CHSHS case, why Bell's derivation can be understood as weakly objective in this paper https://doi.org/10.1007/s10701-021-00488-z
Perhaps somebody would want to tell me where I went wrong. I can very well be mistaken. The story that is correct because it was peer reviewed is not valid for me. I'll be happy to recognize any mistake if the argument convinces me.
Marian Kupczynski told me that he was going to issue a comment on my mistakes. I told him that would welcome his criticisms explaining by mistakes. I am still waiting.

I already explained it to you in the other thread but you were too distracted to follow carefully. Perhaps you want to review the thread and the discussion therein: viewtopic.php?f=6&t=482

@minkwe, I went through the thread but could not find any direct indication of what step of my derivation is incorrect. In the other thread, we discussed your arguments sustaining that Bell's derivation is strongly objective. My claim is that it is weakly objective. I also noticed that Richard Gill said that Bell did not mention spreadsheets but I think that is not important because the use of spreadsheets is good for discussing the relevant points.
Of course, you do not have to read my paper and explain to me anything, I'm only saying that you did not tell me where I specifically went wrong in the arguments of my paper.

[gill1109]

But what Gill knowingly and disingenuously keeps doing is quoting my non-yellow-highlighted words from the above image to paint a dishonest picture of what I have actually said. He does that even in his papers he has submitted to RSOS and IEEE Access. Fortunately, at least RSOS does not seem to have accepted his dishonest paper, which he submitted nearly a year ago (in October 2020, to be precise). But his papers might eventually be accepted by the journals, thereby contaminating the scientific record with his dishonesty and misrepresentations.

Sorry, both papers have been accepted. The acceptance emails got labelled as spam, but fortunately, I found them before it was too late.

I have some very minor editing to do.

Then those who reviewed your papers must be idiots.

Many of them are definitely the same as those who reviewed yours.

[Joy Christian]

But what Gill knowingly and disingenuously keeps doing is quoting my non-yellow-highlighted words from the above image to paint a dishonest picture of what I have actually said. He does that even in his papers he has submitted to RSOS and IEEE Access. Fortunately, at least RSOS does not seem to have accepted his dishonest paper, which he submitted nearly a year ago (in October 2020, to be precise). But his papers might eventually be accepted by the journals, thereby contaminating the scientific record with his dishonesty and misrepresentations.

Sorry, both papers have been accepted. The acceptance emails got labelled as spam, but fortunately, I found them before it was too late.

I have some very minor editing to do.

Then those who reviewed your papers must be idiots.
.