The simplest illustration of Bell's error

[Justo]

@Justo Well heck, I think eq. (14) is nonsense and I can further prove it if necessary.
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Well, at least it is a coherent way to reject the inequality.

[FrediFizzx]

@Justo Well heck, I think eq. (14) is nonsense and I can further prove it if necessary.
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Well, at least it is a coherent way to reject the inequality.

Where did I say that I reject the inequality? I don't. I reject Bell's interpretation from the inequality because it was shot down in 2007 by Joy.
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[Justo]

You guys never stop surprising me!. I can't even disagree with you. You just proved wrong the logical principle of excluded the middle.

[minkwe]

The term P(b,c) comes from what Justo calls "elementary and valid mathematical operations on P(a,b) - P(a,c)". This is uncontested. The only remaining disagreement here is that you guys believe, P(b,c) pertains to a measurement on an independent series of particle pairs with the same standing as P(a,b) and P(a,c) while I believe P(b,c), is not independent of P(a,b) and P(a,c), having been generated using "elementary and valid mathematical operations on P(a,b) - P(a,c)".

We just need intellectually honest individuals to accept that this is the core of the disagreement.

minkwe, you accused before, more than once, of not reading and not understanding what you say. I don't like to accuse people when discussing ideas, I only mention it because you use that technic of discussion that I don't like.

I don't like weaselly tactics. So when people play games like that, I call a spade a spade. I asked a perfectly reasonable question, which identifies our disagreement clearly and to it you gave the following masterpiece:

I guess we could say that.
I know that Pf is not P. It is only that Pf does not exist for me. It is your way of understanding Bell's derivation.
Equation 14 is a mathematical expression that has a physical interpretation and you are changing that interpretation.

How am I to interpret this. I don't assume anything. Rather I hope that you will not play silly games with me. You still haven't understood what point we disagree about. I tried to help the discussion along by distilling what we agree about and what we disagree about but getting you to acknowledge the disagreement is like pulling teeth.

The point is what I mentioned in STPEP 1 before. It is about the interpretation of equation (14) in Bell's paper. If you accept that whenever you have that equation it represents P(a,b), for any "a" and "b", then you just have to admit that equation (15) contains the results of three different series of independently performed experiments.

Assume that there was just a single experiment with Alice already measured at setting and Bob already measured using setting . Are you claiming here that equation (14) DOES NOT represent the correlation they would have obtained had Alice measured at instead and Bob had measured at instead? Obviously, it can represent the counterfactual correlation also. So you are playing games when you claim that equation (14) can only refer to actual independent experiments. Both are consistent with equation (14).

What about the situation in which Alice And Bob measured N particle pairs at settings ("a", and "c") and Amy and Bill measure N particle pairs at ("b" and "d"). Are you claiming that equation (14) DOEST NOT represent the correlation they would obtain between Bill's results and Bob's results? This understanding is also consistent with equation (14). In fact, this is the only way of interpreting P(b,c) in equation 15, if you start from two independent experiments P(a,b) and P(a,c). In what universe do you start with two independent experiments and end up with 3 through simple arithmetic operations. What you end up with is a rearrangement of the terms of the two independent experiments, nothing more, nothing less. If you want to claim that such rearrangement will give you the same result as if you started from three independent experiments, then make that claim rather than the unfounded claim that equation 15 must only be three independent experiments.

The only coherent way to claim that (15) does not represent three different series of independent experiments is by rejecting the interpretation of eq. (14) or rejecting the mathematical steps the lead to it, i.e., rejecting the laws of arithmetic.
That is what is called logical inference and is essential in scientific reasoning.

This is absolutely false and I've already explained above. You have no justification for such a claim and it is not based on sound scientific reasoning at the least.

[minkwe]

I guess we could say that.
I know that Pf is not P. It is only that Pf does not exist for me. It is your way of understanding Bell's derivation.
Equation 14 is a mathematical expression that has a physical interpretation and you are changing that interpretation.

Minkwe: I saw an unidentified flying object yesterday afternoon in San Lorenzo.

Justo: I was watching the sky the whole day. I only saw a Paranair plane.

Minkwe: I saw it, and it was not a Paranair plane.

Justo: Yes it was.

Minkwe: Let me clarify our disagreement. I saw something which I call UFO1. You also saw UFO1. Our only disagreement is that I claim that UFO1 =/= Paranair plane. You believe UFO1 = Paranair plane. Is that right?

Justo: I guess we could say that. I know that UFO1 is not Paranair plane It is only that UFO1 does not exist for me. It is your way of understanding air flight.

and to make things even more interesting you accuse me of not being used to abstract thinking.

[FrediFizzx]

Posts between Justo and I were moved to the Coming Soon Topic.
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[Mikko]

I'm amazed that a whole bunch of learned people can't follow a simple argument. Richard, Justo, Mikko etc please answer one question. And don't go off on a tangent talking about CHSH and Bell's other papers. Please focus on just Bell's 1964 paper and answer this one question:

In Equation 14a, Bell has P(a,b) and P(a,c). There is no P(b,c) anywhere in that expression. Then all of a sudden, P(b,c) appears in equation 15.

Where did P(b,c) come from?

The answer is very easy. Just look at the arithmetic between eq 14 and eq 15. The only thing in doubt is whether any of you are intellectually honest enough to admit it.

As stated on the line before equation 15, from the previous unnumbered equation, using the definition of P as rewritten in equation 14.

Duh?! and where do those terms under the integral come from? I'm sure you will say from the numbered equation just before that line, and we can keep going until we arrive at P(a,b) - P(a,c).

The question about the equation 15 is already answered. Ultimately, P comes from equation 2 and a, b, and c are unspecified directions, i.e., what is said about a or b or c is equally valid about any direction.

[minkwe]

I'm amazed that a whole bunch of learned people can't follow a simple argument. Richard, Justo, Mikko etc please answer one question. And don't go off on a tangent talking about CHSH and Bell's other papers. Please focus on just Bell's 1964 paper and answer this one question:

In Equation 14a, Bell has P(a,b) and P(a,c). There is no P(b,c) anywhere in that expression. Then all of a sudden, P(b,c) appears in equation 15.

Where did P(b,c) come from?

The answer is very easy. Just look at the arithmetic between eq 14 and eq 15. The only thing in doubt is whether any of you are intellectually honest enough to admit it.

As stated on the line before equation 15, from the previous unnumbered equation, using the definition of P as rewritten in equation 14.

Duh?! and where do those terms under the integral come from? I'm sure you will say from the numbered equation just before that line, and we can keep going until we arrive at P(a,b) - P(a,c).

The question about the equation 15 is already answered. Ultimately, P comes from equation 2 and a, b, and c are unspecified directions, i.e., what is said about a or b or c is equally valid about any direction.

At this point, you could as well argue that comes from the alphabet.

[FrediFizzx]

At this point, you could as well argue that comes from the alphabet.

LOL! The Bell fans get pretty ridiculous most of the time.
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[gill1109]

In a recent thread with Justo, I mentioned this problem but only in passing. Here I want to highlight it because, by itself, it shows Bell's faulty logic right from his original paper. I don't know if anyone has described this problem before and I have not come across any attempt in Bell's other papers to address the problem. In any case, the fact that such a clear error exists in Bell's original paper should be a red flag for anyone who takes him seriously. Now unto the argument.

In his paper, Bell is dealing with correlations between pairs of particles formed in the singlet state and moving in opposite directions towards measurement stations and those correlations are specified as the average of the paired product of outcomes observed at those stations .

Michel, I think you mis-read Bell’s 1964 paper, and I believe you also have missed an important later paper by Bell.

P(a, b) is *not* specified as the average of products of outcomes. P(a, b) is defined in equation (2) of Bell(1964) for any a, b, as the *expectation value* of the product of two measurement outcomes under a certain mathematical model, the so-called local hidden variables model. “P” is therefore a mathematical function of two directions, by definition obtained from two other functions “A” and “B” and a probability measure “rho” by the familiar formula: P(a,b) is the integral over some space Lambda of A(a, lambda)B(b, lambda) rho(lambda) d lambda.

“P” is *not* defined as an average of products of actual observations done in any actual experiment.

Your issue was raised long ago, in the early days of Bell’s theorem. Bell wrote a paper (chapter 8 in the book of his collected works) entitled “Locality in quantum mechanics: reply to critics”. It was published in 1975, https://cds.cern.ch/record/980330/files/CM-P00061609.pdf. He treats issues raised in some recent publications. The last issue discussed was raised by three physicists in a paper published in 1972. I think that it is essentially your issue. It has been raised again and again over the years in different forms, especially by physicists who are already happy with quantum mechanics and who use mathematics merely as a language in which to express *their* current physical understanding of the world. The idea of the mathematical study of “stand alone” mathematical structures is alien to many physicists.

[FrediFizzx]

@gill1109 Bell's eq. (14) is a piece of junk as I demonstrated a few months ago. All you have to do is make the substitutions from eq. (1) to see it is a piece of junk. You end up with,


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[gill1109]

@gill1109 Bell's eq. (14) is a piece of junk as I demonstrated a few months ago. All you have to do is make the substitutions from eq. (1) to see it is a piece of junk. You end up with,

I'm afraid I am pretty sure, Fred, that this is not true. Possibly you are misreading Bell's formulas? Bell's mathematical language is a bit sloppy, from the point of view of mathematics, though it seems that most people do understand him. Early misunderstandings, which are indeed still common newcomers' misunderstandings, were discussed and cleared up years and years ago. The reason why it is useful to try to correct Joy's mistakes is pedagogical: they are the common mistakes of independent amateurs who instinctively believe that Bell must have been wrong, and who come up with their own refutations of Bell's theorem. Usually, they have simply misunderstood the theorem.

[Joy Christian]

@gill1109 Bell's eq. (14) is a piece of junk as I demonstrated a few months ago. All you have to do is make the substitutions from eq. (1) to see it is a piece of junk. You end up with,

I'm afraid I am pretty sure, Fred, that this is not true. Possibly you are misreading Bell's formulas? Bell's mathematical language is a bit sloppy, from the point of view of mathematics, though it seems that most people do understand him. Early misunderstandings, which are indeed still common newcomers' misunderstandings, were discussed and cleared up years and years ago. The reason why it is useful to try to correct Joy's mistakes is pedagogical: they are the common mistakes of independent amateurs who instinctively believe that Bell must have been wrong, and who come up with their own refutations of Bell's theorem. Usually, they have simply misunderstood the theorem.

There is no mistake in my understanding of Bell's junk argument. Only uninformed fools believe in such a stupid argument as the one by Bell: https://arxiv.org/pdf/1704.02876.pdf.
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[FrediFizzx]

@gill1109 Bell's eq. (14) is pure junk. Joy's prescription is much better.
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[gill1109]

@gill1109 Bell's eq. (14) is a piece of junk as I demonstrated a few months ago. All you have to do is make the substitutions from eq. (1) to see it is a piece of junk. You end up with,

I'm afraid I am pretty sure, Fred, that this is not true. Possibly you are misreading Bell's formulas? Bell's mathematical language is a bit sloppy, from the point of view of mathematics, though it seems that most people do understand him. Early misunderstandings, which are indeed still common newcomers' misunderstandings, were discussed and cleared up years and years ago. The reason why it is useful to try to correct Joy's mistakes is pedagogical: they are the common mistakes of independent amateurs who instinctively believe that Bell must have been wrong, and who come up with their own refutations of Bell's theorem. Usually, they have simply misunderstood the theorem.

There is no mistake in my understanding of Bell's junk argument. Only uninformed fools believe in such a stupid argument as the one by Bell: https://arxiv.org/pdf/1704.02876.pdf.
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Dear Joy
Of course you are welcome to say in public that you think that Bell’s argument is stupid. Bell is dead, anyway. But I think it is rather unscientific (not to mention, foolish) to say in public that only uninformed fools believe in it. You are insulting fellow scientists. We put forward our arguments. Our peers inspect the arguments. If you don’t want to write here what you think is wrong with my mathematics, please just say you disagree with it. You’ll be given an opportunity to write a Reply to my Comment, if the Comment gets accepted by the editors of IEEE Access.
Richard

@gill1109 Bell's eq. (14) is pure junk. Joy's prescription is much better.
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Thanks, Fred. Yes, I know your opinion on this. I don’t follow your arguments.

[minkwe]

P(a, b) is *not* specified as the average of products of outcomes. P(a, b) is defined in equation (2) of Bell(1964) for any a, b, as the *expectation value* of the product of two measurement outcomes under a certain mathematical model, the so-called local hidden variables model.

I think it is you who has misunderstood Bell.

“P” is therefore a mathematical function of two directions, by definition obtained from two other functions “A” and “B” and a probability measure “rho” by the familiar formula: P(a,b) is the integral over some space Lambda of A(a, lambda)B(b, lambda) rho(lambda) d lambda.

P(a,b) for a local hidden variable model according to Bell is equation (2) of the paper. But P(a,b) in general is the expectation value of the product of outcomes irrespective of the model.

“P” is *not* defined as an average of products of actual observations done in any actual experiment.

You are wrong. For an experiment, "P" is the average of products of actual outcomes in the limit of large N. For QM "P" is the expectation value of the product of outcomes. For local hidden variable models, "P" is the expectation value of the product of outcomes which is also the average of the product of outcomes in the limit of large N. You've badly misread Bell. "P" is always the expectation value of the product of outcomes irrespective of the model. That is why in actual experiments, it is estimated as the average of the product of outcomes for large N. No need to play word games. You know what I mean.

Your issue was raised long ago, in the early days of Bell’s theorem. Bell wrote a paper (chapter 8 in the book of his collected works) entitled “Locality in quantum mechanics: reply to critics”. It was published in 1975, https://cds.cern.ch/record/980330/files/CM-P00061609.pdf. He treats issues raised in some recent publications. The last issue discussed was raised by three physicists in a paper published in 1972. I think that it is essentially your issue. It has been raised again and again over the years in different forms, especially by physicists who are already happy with quantum mechanics and who use mathematics merely as a language in which to express *their* current physical understanding of the world. The idea of the mathematical study of “stand alone” mathematical structures is alien to many physicists.

Handwaving. The issue is not addressed by Bell in the paper you cite or in any other paper of his. Perhaps you don't understand the issue yourself. Bell had something to say about the types of silly mistakes mathematicians make about physics.

[gill1109]

P(a, b) is *not* specified as the average of products of outcomes. P(a, b) is defined in equation (2) of Bell(1964) for any a, b, as the *expectation value* of the product of two measurement outcomes under a certain mathematical model, the so-called local hidden variables model.

I think it is you who has misunderstood Bell.

“P” is therefore a mathematical function of two directions, by definition obtained from two other functions “A” and “B” and a probability measure “rho” by the familiar formula: P(a,b) is the integral over some space Lambda of A(a, lambda)B(b, lambda) rho(lambda) d lambda.

P(a,b) for a local hidden variable model according to Bell is equation (2) of the paper. But P(a,b) in general is the expectation value of the product of outcomes irrespective of the model.

“P” is *not* defined as an average of products of actual observations done in any actual experiment.

You are wrong. For an experiment, "P" is the average of products of actual outcomes in the limit of large N. For QM "P" is the expectation value of the product of outcomes. For local hidden variable models, "P" is the expectation value of the product of outcomes which is also the average of the product of outcomes in the limit of large N. You've badly misread Bell. "P" is always the expectation value of the product of outcomes irrespective of the model. That is why in actual experiments, it is estimated as the average of the product of outcomes for large N. No need to play word games. You know what I mean.

Your issue was raised long ago, in the early days of Bell’s theorem. Bell wrote a paper (chapter 8 in the book of his collected works) entitled “Locality in quantum mechanics: reply to critics”. It was published in 1975, https://cds.cern.ch/record/980330/files/CM-P00061609.pdf. He treats issues raised in some recent publications. The last issue discussed was raised by three physicists in a paper published in 1972. I think that it is essentially your issue. It has been raised again and again over the years in different forms, especially by physicists who are already happy with quantum mechanics and who use mathematics merely as a language in which to express *their* current physical understanding of the world. The idea of the mathematical study of “stand alone” mathematical structures is alien to many physicists.

Handwaving. The issue is not addressed by Bell in the paper you cite or in any other paper of his. Perhaps you don't understand the issue yourself. Bell had something to say about the types of silly mistakes mathematicians make about physics.

Bell writes clear English. Bell does address exactly the issue we are talking about. I admit, he is talking about the usual functions A, B… but what he says applies to the function P defined as a functional of A, B and rho. In different contexts, “P” can mean something different. Of course the different contexts are related. You could overburden the notation by using a different font for each differently defined “P”. Or you write informally, mixing them up.

Mathematicians make silly mistakes about physics. Physicists and chemists make silly mistakes about mathematics. Oh well… probably you and I will never understand one another. We need an intermediary to translate. It’s great that Fred has given us this forum.

[minkwe]

Bell writes clear English. Bell does address exactly the issue we are talking about. I admit, he is talking about the usual functions A, B… but what he says applies to the function P defined as a functional of A, B and rho. In different contexts, “P” can mean something different. Of course the different contexts are related. You could overburden the notation by using a different font for each differently defined “P”. Or you write informally, mixing them up.

Mathematicians make silly mistakes about physics. Physicists and chemists make silly mistakes about mathematics. Oh well… probably you and I will never understand one another. We need an intermediary to translate. It’s great that Fred has given us this forum.

Have you actually read the paper you keep citing?

It is clear from the paper that he is talking about an experiment, despite to your repeated claims to the contrary.

We are interested in correlations between the counts 1 and 2, and define a correlation function

which is the average of the product of A and B over many repetitions of the experiment

Bell doesn't really address any of the criticizms. His paper amounts to statements that he disagrees with various people:
* On de Broglie's refutation: Bell says "As regards ordinary quantum mechanics de Broglie disagrees here with most of the students of the subject, and I am unable to follow his reasons for doing so." That's not really a response.
* On Lochak: Bell simply proclaims without argument that "It may be that Lochak has in mind some other extension of de Broglie's theory to the more-than-one-particle system, than the straightforward generalization from 3 to 3N dimentions that I considered. 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."
* On Pena, Cetto & Brody: Bell resplies to their claim that Bell's 4 functions refer to measurements on the same particle pair, by proclaiming that he is dealing with 4 separate particle pairs with each particle measured only once. But he misses the point of the criticism. It does not matter what Bell claims. When he factorizes random variables he makes an implicit assumption which is equivalent to all the measurements being performed on the same particle pair.

All his replies in that paper are just handwaving.

[gill1109]

On Pena, Cetto & Brody: Bell replies to their claim that Bell's 4 functions refer to measurements on the same particle pair, by proclaiming that he is dealing with 4 separate particle pairs with each particle measured only once. But he misses the point of the criticism. It does not matter what Bell claims. When he factorizes random variables he makes an implicit assumption which is equivalent to all the measurements being performed on the same particle pair.

Bell agrees that in the actual experiment each particle pair is measured just once.

When he does maths, he does maths. If some function P is such that P(a, b) satisfies equation (2) for all a, b, for certain functions A and B taking values in {-1, +1} and for some fixed probability measure rho, then the CHSH inequality is true. If moreover P(a, a) = -1 for all a, then the three-correlation Bell (1964) inequality follows.

Do you agree with those purely mathematical claims?

[minkwe]

On Pena, Cetto & Brody: Bell replies to their claim that Bell's 4 functions refer to measurements on the same particle pair, by proclaiming that he is dealing with 4 separate particle pairs with each particle measured only once. But he misses the point of the criticism. It does not matter what Bell claims. When he factorizes random variables he makes an implicit assumption which is equivalent to all the measurements being performed on the same particle pair.

Bell agrees that in the actual experiment each particle pair is measured just once.

When he does maths, he does maths. If some function P is such that P(a, b) satisfies equation (2) for all a, b, for certain functions A and B taking values in {-1, +1} and for some fixed probability measure rho, then the CHSH inequality is true. If moreover P(a, a) = -1 for all a, then the three-correlation Bell (1964) inequality follows.

Do you agree with those purely mathematical claims?

From day 1, I've always stated that I disagree with the comparison of Bell's mathematical results with QM or experiments. You know this. I have no problem with Bell's mathematics as a purely mathematical excercise. But Bell blundered badly when it comes to comparing his mathematical results with QM or experiments. Mathematicians struggle with this. Bell himself criticized mathematicians for this type of silly mistakes:

The physicists didn't want to be bothered with the idea that maybe quantum theory is only provisional. A horn of plenty
had been spilled before them, and every physicist could find something to apply quantum mechanics to. They were pleased to think that this great mathematician had shown it was so. Yet the Von Neumann if you actually come to grips with it, in your hands! There is nothing not fust flawed, it's silly. If you look assumptions made, it does not hold a moment. It's the work of a mathematician, and he makes assumptions that have a mathematical symmetry to them. When you translate them into terms of physical disposition, they're nonsense. You may quote me on that: The proof of Von Neumann is not merely false but foolish

What irony!