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 gill1109 » Sun Oct 18, 2020 7:26 am

local wrote:Richard, concentrate on the first page where BT is claimed to be refuted. The glaring error is there. In one of your earlier posts you seem to naively accept the error. Think it through more carefully.

Yes, I will do that. But first of all, I wanted to point out to Gordon that his notation is inconsistent, non-standard (and not explained), and overburdened with unnecessary embellishments, while necessary ones are omitted. Hopefully, if his draft is easier to read, everyone will be able to see the error in his reasoning!
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Re: Bell's theorem refuted via elementary probability theory

Postby local » Sun Oct 18, 2020 8:11 am

gill1109 wrote: Hopefully, if his draft is easier to read, everyone will be able to see the error in his reasoning!

For sure, and thank you for helping Gordon to elucidate his argument. Nevertheless, it is adequate at this point. The problem begins right at his equation (3). If you pull the negative signs out of the terms of the right hand integral in order to change the + between the integrals to a -, then the right hand integral is no longer the same (all the terms have been sign-flipped), and you can't pretend that it is. So it is a simple sign error.

I'm curious to see how Gordon responds to my challenge regarding the functions he supposes will integrate to -a.b.
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Sun Oct 18, 2020 1:52 pm

gill1109 wrote:
Gordon Watson wrote:With my emphasis (to identify bits that confuse me):

gill1109 wrote:Bell argues that functions A, B and rho do not exist such that E(a, b) := int d lambda rho(lambda) A(a, lambda) B(b, lambda) = - a.b for all a and b, where the functions A and B take values in the set {-1, +1}; a and b are arbitrary directions in space; lambda lies in some space of hidden variables Lambda, and rho is a probability density over that space (non-negative, integrates to 1). Naturally, one can write the integral as the sum of two integrals, one over the region where A(a, .)B(b, .) = +1, and one over the complementary region where A(a, .)B(b, .) = -1, and hence find that E(a, b) = P(A_a B_b = +1) - P(A_a B_b = -1), where we now consider Lambda together with the probability density rho as defining a probability space with probability measure P; we can now define A_a and B_b as the random variables (functions of lambda) A(a, .) and B(b, .).

Your notation is confusing since the bar "|" is usually used to mean "conditional on" in probability theory. If you want to integrate over subsets of lambda then you can indicate that by giving the integral sign a subscript (over which set do you integrate); or you can multiply the integrand by an indicator function. In your last line, you have on the right-hand side dropped the variables a and b. You write a lot of stuff which is superfluous. The short form "AB" for a random variable which also depends on a and b is dangerous and misleading. The A^+- is quite superfluous. beta is superfluous. Bell is assuming local realism. The "EPR-Bohm set-up" is irrelevant. You are going to do some mathematics, not physics.

Indeed the bar is also used in "set-builder" notation. But you are not using it to build sets. A(a, lambda)B(b, lambda) is a number. {lambda | A(a, lambda)B(b, lambda) = +1} is a set, and moreover, it is a set which in general might depend on a and b. But I don't see any curly brackets, so I don't see any set-builder sets, and anyway, one integrates *functions* over sets. You are mixing up functions and sets and this kind of mix-up is a recipe for disaster, especially when at the same time you put many symbols into your notation which are superfluous but at the same time omit symbols which are important. You end up writing something which makes no sense - there are a and b on the left-hand side, but not on the right-hand side.


Thanks Richard, very helpful.

Please see if the version just posted meets your needs and allows you to pinpoint my errors.

Note:


Gordon 1. I use β to denote Bohm's experiment, α for Aspect's experiment, γ for GHZ. So when the above 2-page pdf is absorbed into a larger paper on all 3 of them: it all fits.

Richard. OK.

Gordon 2. Further, I do not understand this: "The "EPR-Bohm set-up" is irrelevant. You are going to do some mathematics, not physics." Please elaborate. As an engineer, I seem to find myself doing both.

Richard. Of course you want your mathematics to be relevant to the real world. But I thought your claim is that Bell makes mistakes in logical/mathematical reasoning. "Bell's theorem" is, I would say, a piece of mathematics, with applications to physics.

Gordon 3. Also, from the first underline above: I do not follow the need for A_a B_b, nor do I grasp their meaning and physical significance. Please elaborate.

A_a is a function of lambda, the hidden variable, and it is the outcome which would have been observed by Alice if the experimenter (Alice) had chosen to fix the setting to the direction a. Your notation is clumsy /misleading /wrong. But this is in actual fact what you and Bell are talking about. In a CHSH experiment we look at A_a for two specific values of a, and B_b for two specific values of b. In one trial of the experiment, one gets to observe one of each. Their "existence" is guaranteed by "local realism".

Gordon 4. Re the second underline above, please identify "the many superfluous symbols" in the latest version.

Richard. Sorry, I am not going to do that *yet again*.

Gordon 5. All of which still leaves you needing to identify an error in my refutation of BT. Also my rebuttal of Bell's inequality: for it is not changed by the new notation.

Richard. Indeed, I'm ready to start reading the two pages, now.


Thanks again Richard, I look forward to the results of your reading. In the meantime, please clarify these matters:

gill1109 wrote:"Gordon 3. Also, from the first [u]underline above: I do not follow the need for A_a B_b, nor do I grasp their meaning and physical significance. Please elaborate.[/u]

Richard 3: A_a is a function of lambda, the hidden variable, and it is the outcome which would have been observed by Alice if the experimenter (Alice) had chosen to fix the setting to the direction a. Your notation is clumsy /misleading /wrong. But this is in actual fact what you and Bell are talking about. In a CHSH experiment we look at A_a for two specific values of a, and B_b for two specific values of b. In one trial of the experiment, one gets to observe one of each. Their "existence" is guaranteed by "local realism".


1. I thought you criticised me for suppressing λ in an expression. Aren't you doing the same with A_a? And how does A_a differ from A(a,λ)?

2. Please, most importantly for me, give examples of my "clumsy /misleading /wrong" notation. Believe me: when I find them I'll fix them.

3. Since, in my experience, the devil is in the detail: How do you define "local realism" in terms of "local = ..." and "realism = ..." ??

All the best; Gordon
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Re: Bell's theorem refuted via elementary probability theory

Postby local » Sun Oct 18, 2020 2:14 pm

Your equation (3) is false. Take a look.
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Sun Oct 18, 2020 3:00 pm

local wrote:Your equation (3) is false. Take a look.


Thanks for the pointer. But please be more specific. To be sure you are reading the latest version posted above, here is its Eqn (3):

E(a,b|β)

= ∫ dλ ρ(λ)A(a,λ)B(b,λ) − ∫ dλ ρ(λ)A(a,λ)B(b,λ) with the first integral over Λ_+ and the second over Λ_-.

If you think this is wrong, please let me know ASAP; with reasons.

Thanks again, Gordon
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Sun Oct 18, 2020 3:14 pm

local wrote:
All of which still leaves you needing to identify an error in my refutation of BT.

The error is very obvious but before pointing it out I would like to give you a chance to find it yourself by offering you a little challenge.

You claim to have shown that Bell's integral in your equation (1) is in fact equal to -a.b. If this is so you will easily be able to show a set of functions rho(alpha), A(a,alpha), and B(b,alpha) that generate -a.b upon integration. Of course A() and B() must have the same form. Depending on what functions you propose, we will be able to verify them either by direct integration, or if that is intractible, by means of a simple event-by-event simulation generating two outcome streams that can then be correlated. Your case will be greatly strengthened by showing such a simulation. I will be happy to assist you with programming the simulation if you are not a programmer. The problem for you is that you will fail at this task. When you think about why that is the case you may find the glaring error in your proof.

People have been trying unsuccessfully for decades to use Malus-law-like detection to implement local models that generate the -a.b correlation. Yes, you can use variable detection (data discarding à la Pearle), but your proof is lacking anything like that.


Thanks local! I missed this (it must have been in editing) and I appreciate your interest.

I am not a programmer: I'm a mechanical engineer and I welcome your offer. I suspect you will need to convince your opponents that you must be allowed to work with the laws of nature in your simulation. Let's see how we go!

I agree that A and B must be the same function with different arguments: so I use A = A(a,λ) and B = A(b,-λ) where, pair-wise [and thus in the same instance], λ + (-λ) = 0 via the pairwise conservation of total angular momentum.

IF YOU disagree with this, please let me know WHY and HOW now and I will spell it out more clearly.

I will write the functions up -- I think they are already on viXra -- and will post another 2-page PDF. I do not rely on any tricks.

In the meantime, you might like to consider the nature of my functions AND comment thereon: for they are required to covert Bell's continuous variable λ into discrete outcomes.

Thanks again; Gordon
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Re: Bell's theorem refuted via elementary probability theory

Postby local » Sun Oct 18, 2020 3:54 pm

Pay attention. I explained why your (3) is false. It's an ingenious little swindle. Congratulations!

Let us know when you have the functions ready for integration/simulation, and we will get right on it. Best wishes going forward!

Gordon Watson wrote: I will write the functions up -- I think they are already on viXra -- and will post another 2-page PDF.

Should be three lines, one for A, one for B, and one for rho.
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Sun Oct 18, 2020 8:57 pm

Gordon Watson wrote:In the meantime, you might like to consider the nature of my functions AND comment thereon: for they are required to covert Bell's continuous variable λ into discrete outcomes.

Bell wrote “integral rho(lambda) d lambda ... “ [formula for expectation value, in absolutely continuous case]

He could just as well have written “sum p(lambda) ...” [formula for expectation value, in discrete case]

He could have written “integral P(d lambda) ...” [formula for expectation value, *all cases*, using notation from measure theory, following Borel, Kolmogorov, ...].

He did it his way because he was talking to physicists in 1964. He mentioned that of course his proof did not need the assumption of a probability density and lambda in R^something. He says he writes it that way just for convenience, and says of course it goes through in the discrete case.

In CHSH we have four binary hidden variables - the counterfactual outcomes of Alice and Bob to each of two settings. So we need a probability space with 16 outcomes only. This has been known for half a century...
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Mon Oct 19, 2020 1:02 am

local wrote:
gill1109 wrote: Hopefully, if his draft is easier to read, everyone will be able to see the error in his reasoning!

For sure, and thank you for helping Gordon to elucidate his argument. Nevertheless, it is adequate at this point. The problem begins right at his equation (3). If you pull the negative signs out of the terms of the right hand integral in order to change the + between the integrals to a -, then the right hand integral is no longer the same (all the terms have been sign-flipped), and you can't pretend that it is. So it is a simple sign error.

I'm curious to see how Gordon responds to my challenge regarding the functions he supposes will integrate to -a.b.

.
RE EQN (3): Thanks local and Richard, that's a TYPO. That MINUS should be a PLUS, as seen by the next line.

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

Postby gill1109 » Mon Oct 19, 2020 1:18 am

local wrote:Pay attention. I explained why your (3) is false. It's an ingenious little swindle. Congratulations!

Let us know when you have the functions ready for integration/simulation, and we will get right on it. Best wishes going forward!

Gordon Watson wrote: I will write the functions up -- I think they are already on viXra -- and will post another 2-page PDF.

Should be three lines, one for A, one for B, and one for rho.

Yes, I'm looking forward to seeing those three lines.

In the mean-time, the notation of https://vixra.org/pdf/2010.0068v4.pdf is still abominable.

It is still not adequately explained in the document itself, but at least we have now heard the explanation on this forum.

The sets Lambda^+ and Lambda^- depend on a and b. Why do you leave them out of your notations???? It could be a good way to start cheating further down (deliberately or accidentally, doesn't matter).

Your random latent variable has apparently a probability distribution depending on a and b. You do stuff between (6) and (7) without showing how you can choose A, B and lambda to make that work. Sorry Gordon, let me tell you in advance: you can't.
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Mon Oct 19, 2020 2:05 am

gill1109 wrote:
local wrote:Pay attention. I explained why your (3) is false. It's an ingenious little swindle. Congratulations!

Let us know when you have the functions ready for integration/simulation, and we will get right on it. Best wishes going forward!

Gordon Watson wrote: I will write the functions up -- I think they are already on viXra -- and will post another 2-page PDF.

Should be three lines, one for A, one for B, and one for rho.

Yes, I'm looking forward to seeing those three lines.

In the mean-time, the notation of https://vixra.org/pdf/2010.0068v4.pdf is still abominable.

It is still not adequately explained in the document itself, but at least we have now heard the explanation on this forum.

The sets Lambda^+ and Lambda^- depend on a and b. Why do you leave them out of your notations???? It could be a good way to start cheating further down (deliberately or accidentally, doesn't matter)


1. To help me improve things for the benefit of all, please identify the abominable for me.

2. We want E(a,b|β). To that end, Λ^+ and Λ^-, defined via a and b in the line above eqn (3), are self-selecting subsets of Λ that sum to Λ. They complete Bell's integral, independent of his functions.

3. What is your problem with #2?

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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Mon Oct 19, 2020 5:21 am

Gordon Watson wrote:
gill1109 wrote:
local wrote:Pay attention. I explained why your (3) is false. It's an ingenious little swindle. Congratulations!

Let us know when you have the functions ready for integration/simulation, and we will get right on it. Best wishes going forward!

Gordon Watson wrote: I will write the functions up -- I think they are already on viXra -- and will post another 2-page PDF.

Should be three lines, one for A, one for B, and one for rho.

Yes, I'm looking forward to seeing those three lines.

In the mean-time, the notation of https://vixra.org/pdf/2010.0068v4.pdf is still abominable.

It is still not adequately explained in the document itself, but at least we have now heard the explanation on this forum.

The sets Lambda^+ and Lambda^- depend on a and b. Why do you leave them out of your notations???? It could be a good way to start cheating further down (deliberately or accidentally, doesn't matter)


1. To help me improve things for the benefit of all, please identify the abominable for me.

2. We want E(a,b|β). To that end, Λ^+ and Λ^-, defined via a and b in the line above eqn (3), are self-selecting subsets of Λ that sum to Λ. They complete Bell's integral, independent of his functions.

3. What is your problem with #2?

Gordon

My problem is that they depend on a and on b. Why don't you give them a subscript "ab"? Are you trying to hide something??? You put in lots of decorations on symbols which are unnecessary, eg A+ is a notation for "+1". It's the number +1. No need to attach A or B or whatever to it. You omit decorations which are essential in order to keep track of what depends on what. Lambda+ and Lambda- is *misleading*. They are not "independent" of Bell's functions at all!!!!!

What do you mean by "self-selecting"? They are subsets of Lambda. They are not in any way autonomous, they depend on the functions A and B, and they depend on the values of the settings a and b which you are talking about at that moment.
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Re: Bell's theorem refuted via elementary probability theory

Postby local » Mon Oct 19, 2020 5:25 am

Gordon Watson wrote:RE EQN (3): Thanks local and Richard, that's a TYPO. That MINUS should be a PLUS, as seen by the next line.

Of course it should be a plus sign but it's not a typo. The next line (4) is:

= P(AB = 1|a,b,Λ+) − P(AB = −1|a,b,Λ−)

The incorrect minus sign is still there, and it is propagated right through to the wrong conclusion. Your silly proof is busted!

Still waiting for your function set. Where is it? Surely you must be able to present one.
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Mon Oct 19, 2020 11:27 am

local wrote:
Gordon Watson wrote:RE EQN (3): Thanks local and Richard, that's a TYPO. That MINUS should be a PLUS, as seen by the next line.

Of course it should be a plus sign but it's not a typo. The next line (4) is:

= P(AB = 1|a,b,Λ+) − P(AB = −1|a,b,Λ−)

The incorrect minus sign is still there, and it is propagated right through to the wrong conclusion. Your silly proof is busted!

Still waiting for your function set. Where is it? Surely you must be able to present one.

He can’t, because none exist. That’s Bell’s theorem! (Or the corollary of Bell’s theorem, depending on your terminology).
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Re: Bell's theorem refuted via elementary probability theory

Postby local » Mon Oct 19, 2020 11:45 am

That is not Bell's theorem. Bell's theorem asserts that LHV systems can't duplicate the predictions of QM. But QM does not and cannot predict -a.b for separated measurement (unless one appeals to unphysical Lorenz-invariance-violating Luders projection, or one duplicitously appeals to the QM joint prediction, which is inapplicable to EPRB). There is no hurdle for LHV theories. Bell's theorem is false for space-like separated EPRB.

I do agree that Watson can't produce the functions, but as I said, that is not Bell's theorem. That he cannot produce the functions is confirmation that his proof is busted. We have to feel bad for a guy whose life work has crumbled into nothing. His response so far: crickets!

Mysterians always go on about "passion at a distance" and other nonsense. What is your rigorous theory for "quantum correlation" in separated measurement scenarios? Yeah, we know, no answer because you're just a mathematician. No theory and fake experiments. At least it's marginally better than Hossenfelder's "superdeterminism", which she admits lacks both theory and experiment. Intelligent people call that metaphysical mumbo-jumbo. Could passion be involved? Have a look at Dr Christian's work; you might find a viable theory there in higher topologies. The mysterians have totally failed in normal 3D spacetime, and that's why Christian's work is gaining traction among the open-minded.
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Mon Oct 19, 2020 6:52 pm

gill1109 wrote:
Gordon Watson wrote:
gill1109 wrote:
local wrote:Pay attention. I explained why your (3) is false. It's an ingenious little swindle. Congratulations!

Let us know when you have the functions ready for integration/simulation, and we will get right on it. Best wishes going forward!
...
Should be three lines, one for A, one for B, and one for rho.

Yes, I'm looking forward to seeing those three lines.

In the mean-time, the notation of https://vixra.org/pdf/2010.0068v4.pdf is still abominable.

It is still not adequately explained in the document itself, but at least we have now heard the explanation on this forum.

The sets Lambda^+ and Lambda^- depend on a and b. Why do you leave them out of your notations???? It could be a good way to start cheating further down (deliberately or accidentally, doesn't matter)


1. To help me improve things for the benefit of all, please identify the abominable for me.

2. We want E(a,b|β). To that end, Λ^+ and Λ^-, defined via a and b in the line above eqn (3), are self-selecting subsets of Λ that sum to Λ. They complete Bell's integral, independent of his functions.

3. What is your problem with #2?

Gordon

My problem is that they depend on a and on b. Why don't you give them a subscript "ab"? Are you trying to hide something??? You put in lots of decorations on symbols which are unnecessary, eg A+ is a notation for "+1". It's the number +1. No need to attach A or B or whatever to it. You omit decorations which are essential in order to keep track of what depends on what. Lambda+ and Lambda- is *misleading*. They are not "independent" of Bell's functions at all!!!!!

What do you mean by "self-selecting"? They are subsets of Lambda. They are not in any way autonomous, they depend on the functions A and B, and they depend on the values of the settings a and b which you are talking about at that moment.


Richard,

1. My P(A^-B^-) versus P(A = -1),P(B = -1))? Decorations or rigorous and convenient short-cut identifiers?

2. No matter what you want to talk about, say (a, b) or (c, d): it seems to me that Λ_+ and Λ_- are self-determining and clear from the context.

3. Please provide the other abominables that you have in mind. I can then get on with improving things. Thanks

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

Postby local » Mon Oct 19, 2020 7:10 pm

Start by correcting the false equation (3) and you'll see the conclusion doesn't follow. And where are the functions we asked for?
Last edited by local on Mon Oct 19, 2020 7:54 pm, edited 2 times in total.
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Mon Oct 19, 2020 7:33 pm

local wrote:
Gordon Watson wrote:RE EQN (3): Thanks local and Richard, that's a TYPO. That MINUS should be a PLUS, as seen by the next line.

Of course it should be a plus sign but it's not a typo. The next line (4) is:

= P(AB = 1|a,b,Λ+) − P(AB = −1|a,b,Λ−)

The incorrect minus sign is still there, and it is propagated right through to the wrong conclusion. Your silly proof is busted!

Still waiting for your function set. Where is it? Surely you must be able to present one.

[Identifiers 1-3 added by GW.]


Thanks local,

For the record:

1. It was a TYPO in (3); see the next line.

2. The next line, (4), is unchanged and the analysis flows correctly to its conclusion.

3. I'm waiting for your critique of my refutation of Bell's inequality (p.2), and Richard's definition of the abominables, etc. Best to make ALL the corrections and improvements together.

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

Postby Gordon Watson » Mon Oct 19, 2020 7:43 pm

local wrote:Really? Just going to play dumb? Denial is not a river in Egypt. Start by correcting the false equation (3) and see if the conclusion still follows. And where are the functions? Best thing for you now is to just say "Oops, I was wrong."


local,

1. I've corrected (3) in the next draft. And since it was a TYPO, the balance of the analysis flows to its previous conclusion.

2. Thus, to maintain your position: you still need to find something like a matching error in my math that refutes Bell's inequality (BI).

3. Also: to be clear re your view about Bell's impossible functions: Do you accept that Joy delivers such functions?

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

Postby local » Mon Oct 19, 2020 8:07 pm

Where is Bullwinkle when we need him?
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