## Gill's theorem refuted (GTR)

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

### Gill's theorem refuted (GTR)

IF Richard Gill's theorem supports Bell's theorem (BT) in some way, please provide a link and commentary to the latest version of that theorem here.

It will then be refuted, as time permits.

Until then, just regard this as prophecy.
Gordon Watson

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### Re: Gill's theorem refuted (GTR)

minkwe wrote:The strongly objective view is that P(H) is the probability of repeatedly tossing the same coin many times, while the "weakly objective" view is that P(H) is the probability of tossing many different "similar" coins each just one time. Once you pick an interpretation, you must consistently use that interpretation, otherwise you shoot yourself in the foot and drown in paradoxes.
...
The short version of the two questions are: What interpretation does Gill use for the terms in Bell's inequality strongly objective or weakly objective? What interpretation does Gill use for the terms from QM strongly objective or weakly objective?

gill1109 wrote:I take the weakly objective interpretation of E(a, b), both with respect to QM and with respect to a possible LHV theory "behind" QM.

minkwe wrote:Are you sure that is your choice, because you just opened Pandoras box.

gill1109 wrote:Yes.

See

https://en.wikipedia.org/wiki/Interpret ... rpretation

It's just the usual frequentist interpretation of probability. It works pretty well in science. See "Introduction to mathematical statistics and data analysis" by John A Rice. Excellent text book with practical introduction to probability theory and then to statistics.

minkwe wrote:Secondly, since you have now committed that you are relying on the "weakly objective" interpretation of those expectation values, could you please explain how "counterfactual definiteness" is relevant to Bell's inequalities. Surely you must understand that there are no counterfactual terms in the "weakly objective" interpretation as opposed to the "strongly objective" one. So since your recent paper spends a lot of ink talking about counterfactual definiteness, please explain how it arises in the "weakly objective view".
minkwe

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### Re: Gill's theorem refuted (GTR)

Gordon: Gill has a theorem. You should find out what it is before trying to refute it.

Bell does not have a theorem.

Clauser, Horne, Shimony, Holt (1970) coined the name "Bell's theorem" for a statement which they seemed to believe in, but I am pretty sure that Bell would not have agreed with it, though maybe complied with a useful "soundbite" or "lies for children" one sentence take-home message.

The Bertlmann's socks paper (Chapter 16 of "Speakable and unspeakable") gave a list of four possible conclusions to draw from Bell's analysis and the list was not supposed to be exhaustive. The list is also in the lesser known but also very significant Chapter 13.

Later Bell admitted one particular further possibility, which I later called "Bell's fifth position". Emilio Santos has been an eloquent proponent for decades.
Last edited by gill1109 on Thu Jun 05, 2014 8:15 pm, edited 1 time in total.
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### Re: Gill's theorem refuted (GTR)

minkwe wrote:
minkwe wrote:The strongly objective view is that P(H) is the probability of repeatedly tossing the same coin many times, while the "weakly objective" view is that P(H) is the probability of tossing many different "similar" coins each just one time. Once you pick an interpretation, you must consistently use that interpretation, otherwise you shoot yourself in the foot and drown in paradoxes.
...
The short version of the two questions are: What interpretation does Gill use for the terms in Bell's inequality strongly objective or weakly objective? What interpretation does Gill use for the terms from QM strongly objective or weakly objective?

gill1109 wrote:I take the weakly objective interpretation of E(a, b), both with respect to QM and with respect to a possible LHV theory "behind" QM.

minkwe wrote:Are you sure that is your choice, because you just opened Pandoras box.

gill1109 wrote:Yes.

See

https://en.wikipedia.org/wiki/Interpret ... rpretation

It's just the usual frequentist interpretation of probability. It works pretty well in science. See "Introduction to mathematical statistics and data analysis" by John A Rice. Excellent text book with practical introduction to probability theory and then to statistics.

minkwe wrote:Secondly, since you have now committed that you are relying on the "weakly objective" interpretation of those expectation values, could you please explain how "counterfactual definiteness" is relevant to Bell's inequalities. Surely you must understand that there are no counterfactual terms in the "weakly objective" interpretation as opposed to the "strongly objective" one. So since your recent paper spends a lot of ink talking about counterfactual definiteness, please explain how it arises in the "weakly objective view".

Dear Michel

Thanks for the nice questions. But before answering them, let me mention that this thread was opened by Gordon Watson to discuss Gill's theorem. I have no idea what Gordon thinks is Gill's theorem, but it might be an idea to start the discussion by quoting it. There is just one theorem in my paper which maybe is what he is referring too.

Question one: pandora's box. I usually like to think about probability in the frequentist manner and that seems to be what you call "weakly objective". It doesn't make any difference to me whether we are talking about QM theory or LHV theories. Probability is probability.

(If you are a follower of Jaynes then you should be a Bayesian and have a completely different picture of what probability means ... )

Question two: counterfactual definiteness. Local hidden variables implies counterfactual definiteness. If the outcome of Alice's measurement in direction a, when the particle turns up carrying the hidden variable lambda, is A(a, lambda), then it should not be a crime to talk about what the outcome would have been if the setting had actually been b? It seems to me that it's A(b, lambda).

Is it useful to talk about it? I think so.

Niels Bohr didn't believe in local hidden variables and according to him it was a waste of time to talk about what the outcome would have been if the setting had actually been b. QM is not a LHV theory and doesn't talk about what goes on behind the scenes. It does not explain how the probabilities come to be what they are. It just assumes that they are given by weird formulas involving vectors in Hilbert space and projections onto subsets thereof.

Now, Einstein, in EPR, gave a rather good reason - from quantum mechanics - for believing in a local hidden variables theory! Bell brilliantly developed Einstein's brilliant reasoning further and was able to turn Einstein's conclusion on its head. Pretty cool, I think.

Your simulation models use local hidden variables and by doing mathematics we can derive their performance limits ... but that is another thread, another topic. I am looking forward to your comments. I hope I can no longer be accused of creating a travesty of your beautiful work, and misrepresenting it. That certainly wasn't my intention; on the contrary, I want everyone to know how nice it is.
Last edited by gill1109 on Thu Jun 05, 2014 8:26 pm, edited 3 times in total.
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### Re: Gill's theorem refuted (GTR)

Gordon Watson wrote:IF Richard Gill's theorem supports Bell's theorem (BT) in some way, please provide a link and commentary to the latest version of that theorem here.

It will then be refuted, as time permits.

Until then, just regard this as prophecy.

Gordon unconventionally gives the name "Bell's theorem" to Bell's (1964) three correlation inequality, aka the original Bell inequality.

My paper is not about Bell's original inequality. I do have a lot of CHSH related stuff in there. And there is a theorem in there too. I hope Gordon will study it, sometime.

Physicists don't prove theorems. They uncover the secrets of Nature. Mathematicians do prove theorems. Every true mathematical theorem is just a tautology. No more, no less. Physicists sometimes invent slogans and call them theorems. But Bell never did that.
gill1109
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### Re: Gill's theorem refuted (GTR)

gill1109 wrote:Physicists sometimes invent slogans and call them theorems. But Bell never did that.

False.

Most of what Richard Gill writes about Bell and his so-called theorem are inventions of his own, without any bases in actual facts.

For example, Bell repeatedly called his theorem a "theorem", both in conferences and in his writings (not to mention in private discussions). I know this because I was present in a number of such conferences and closed meetings, and also had private discussions with Bell about his so-called "theorem" on a number of occasions.

But you don't have to believe me. Just read Bell's book, written in his own words, where you will find that he repeatedly calls his theorem a "theorem." See, for example, the first edition of his book, page 65, after equation (3), where he writes: "This is the theorem. The proof will not be repeated here."

The real question then is: Why does Richard Gill repeatedly make false and misleading assertions and claims about almost everything? What does he gain by this?
Joy Christian
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### Re: Gill's theorem refuted (GTR)

Joy Christian wrote:Just read Bell's book, written in his own words, where you will find that he repeatedly calls his theorem a "theorem." See, for example, the first edition of his book, page 65, after equation (3), where he writes: "This is the theorem. The proof will not be repeated here."

In the preface to the first edition John Bell writes "Seeing again what I have written on the locality business, I regret never having written up the version of the locality inequality theorem that I have been mostly using in talks on this subject in recent years. But the reader can easily reconstruct that. It begins by emphasizing the need for the concept ‘local beable’, along the lines of the introduction to 7. (If local causality in some theory is to be examined, then one must decide which of the many mathematical entities that appear are supposed to be real, and really here rather than there). Then the simpler locality condition appended to 21 is formulated (rather than the more elaborate condition of 7). With an argument modelled on that of 7 the factorization of the probability distribution again follows. The Clauser–Holt–Horne–Shimony inequality is then obtained as at the end of 16."

It's clear from this that at that time, Bell thought of the CHSH inequality as being "the theorem". Something like: "if such and such conditions hold, then this inequality will hold".

This is not what Aspect in the preface to the second edition calls "Bell's theorem". Aspect formulates the theorem as "it is not possible, in general, to understand EPR-type correlations by ‘complementing’ the quantum theory along the lines proposed by Einstein". Well, it is certainly a common opinion, and one of the opinions which Bell considered legitimized by his analysis, but it is not the only possible position which Bell considered it logically possible to hold.

In chapter 7 of "Speakable" Bell does again refer to "a theorem of his own" and it is clear that again he means his inequality, and the logical consequence thereof that it is not possible to find functions A(a, lambda) and B(b, lambda) (with co-domain {-1, +1}) and a probability density rho(lambda) such that ... reproduces the negative cosine correlation function.

I understand that dr J. Christian (Oxford) agrees with this, and has famously said: one does not disprove Bell's theorem, one circumvents it (or words to that effect). Obviously one cannot disprove an elementary and correct calculus derivation. One tries to show that it is not relevant.

The same things happens again in Chapter 8 of "Speakable".Here he also says "Here I must concede at once that the hypothesis becomes quite inadequate when weakened in this way. The theorem no longer follows. I was mistaken. At this point I had in mind the possibility of exploiting the freedom, in conventional physical theories, of initial conditions. I am now embarrassed not only by the inadequacy of this particular phrase in the hypothesis, but also by the necessity of paying attention in such a study to the creation of the world."

What he says there is that his elementary derivation is irrelevant if we take away one of the conditions which he made. He moreover is saying in a very British way that this relaxation of his conditions is pretty stupid. He is ridiculing "super-determinism" aka "conspiracy". OK well Gerard 't Hooft would be forced to put in a good word for super-determinism. It's an opinion.

OK so I have to correct my earlier claim. Bell does think he proves a theorem. What he calls a theorem is an elementary and indisputable true mathematical derivation. Start with some assumptions and do some correct calculus or whatever and correctly deduce a theorem.

What Bell calls Bell's theorem is not what CHSH called Bell's theorem and what the world nowadays understands as Bell's theorem!

But sure: I stand corrected. Some physicists do occasionally prove what they call theorems, and admit to it to. And these "theorems" do have a right to be called theorems by mathematicians to. A theorem is just a tautology. Bell indeed proved an elementary tautology.
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### Re: Gill's theorem refuted (GTR)

Bell did indeed think he proved a mathematical theorem. His mathematical theorem (simplest form) might be the following.

Theorem. Let A and B be two functions from the product of a set of settings (a, b ...) and a set of values (lambda ....) of a so-called hidden variable to the set {-1, +1}. Let P be a probability measure on the space of values of lambda and let E denote expectation with respect to this probability distribution. Denote A(a), B(b) etc fas the random variables defined by the maps lambda -> A(a, lambda) etc etc. Then for any a, b, a', b'

E(A(a)B(b)) - E(A(a)B(b')) + E(A(a')B(b)) + E(A(a')B(b')) <= 2

Remark. We neglect measurability issues here.

Proof: Since all random variables here are bounded there are no issues in exchanging expectation values and summation. By elementary algebra we see that
A(a)B(b) - A(a)B(b') + A(a')B(b) + A(a')B(b') <= 2

Take the expectation left and right, and write expectation of a sum (and difference) of four terms as sum and difference of four expectation values.

QED.

I think that according to Bell that is a prototypical statement and proof of Bell's theorem. (The "remark" is added by me). Is there a problem with it? I mean, an internal problem? You may think it is irrelevant, that is a different issue. Is it indeed (modulo some technical niceties about what is a function, exactly, and what is integration, exactly) a tautology?

Bell like most sensible physicists was not bothered by technical mathematical niceties concerned with what exactly do we mean by a function, what exactly do we mean by integration, and so on. He could integrate anything he came across, no problem. Gerard 't Hooft has a very low opinion of mathematicians because when he did a course on measure theoretic integration he found out that mathematicians forbid him to integrate many things which he knew perfectly well how to integrate. Well who needs mathematicians then.

PS see http://arxiv.org/abs/1402.1972 for another mathematician's version of the theorem.
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### Re: Gill's theorem refuted (GTR)

gill1109 wrote:Theorem. Let A and B be two functions...

I would stop right here.

What do you mean by "functions"?

As I understand the term, a function cannot be meaningfully defined without specifying its domain as well as codomain.

I would define Bell's functions properly (and physically meaningfully), as

$\pm\,1\,=\,{\cal A}({\bf a},\,\lambda):{\rm I\!R}^3\!\times\Lambda\longrightarrow S^3 \sim {\rm SU(2)}$.

It is then very easy to show that

$|\,{\cal E}({\bf a},\,{\bf b})\,+\,{\cal E}({\bf a},\,{\bf b'}) \,+\,{\cal E}({\bf a'},\,{\bf b})\,-\,{\cal E}({\bf a'},\,{\bf b'})\,|\,\leqslant\,2\,\sqrt{2}\,$.

So much for Bell's theorem.

For details see my derivations in this paper, or its simplified explanation on my blog.
Joy Christian
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### Re: Gill's theorem refuted (GTR)

Joy Christian wrote:
gill1109 wrote:Theorem. Let A and B be two functions...

I would stop right here.

What do you mean by "functions"?

If you read to the end of my sentence you'll see exactly what John Bell meant with "functions" (as far as I can tell). I wrote exactly the range which Bell had assumed. I was trying to phrase Bell's theorem (derivation of CHSH), not anyone else's.

I wasn't trying to phrase, for instance, Tsirelson's theorem. Or Christian's theorem.

The range of A and B (what you call their co-domain) was assumed by Bell to be a point set, {-1, +1}, (two points in R). Later he showed his proof could be adapted so as to allow him to replace {-1, +1} by [-1, +1] (a particular line segment in R) and he had reason to do this in order to accommodate also hidden variables located in the measurement stations, but still the actual measurement outcomes were thought of as being elements of the set {-1, +1}, itself a subset of R. Thus allowing us, within R to multiply, add and divide (except by zero).
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### Re: Gill's theorem refuted (GTR)

gill1109 wrote:
minkwe wrote:Secondly, since you have now committed that you are relying on the "weakly objective" interpretation of those expectation values, could you please explain how "counterfactual definiteness" is relevant to Bell's inequalities. Surely you must understand that there are no counterfactual terms in the "weakly objective" interpretation as opposed to the "strongly objective" one. So since your recent paper spends a lot of ink talking about counterfactual definiteness, please explain how it arises in the "weakly objective view".

Question two: counterfactual definiteness. Local hidden variables implies counterfactual definiteness. If the outcome of Alice's measurement in direction a, when the particle turns up carrying the hidden variable lambda, is A(a, lambda), then it should not be a crime to talk about what the outcome would have been if the setting had actually been b? It seems to me that it's A(b, lambda).

If you are using the weakly objective interpretation, which of the correlations in Bell's or your inequality corresponds to what Alice and Bob got and which one corresponds to what they could have gotten but didn't?
I know you like talking about CFD. There is no crime to talk about it. But it is confusion to introduce the concept into your maths when none of your terms are counterfactual. If you are using the weakly objective view, then none of your terms are counterfactual and you can't reasonably claim at the end that CFD must be false due to violation.
In the weakly objective view, each correlation is measured on a different set of particles so it makes absolutely no sense to talk of what might have been measured but wasn't (CFD). It is only in the Strongly objective view that CFD is present, since all the correlations are measured on the same set of particles, and only one of them can practically be measured, then the others represent what they could have measured but didn't (CFD).
Last edited by minkwe on Fri Jun 06, 2014 4:34 am, edited 1 time in total.
minkwe

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### Re: Gill's theorem refuted (GTR)

gill1109 wrote:The range of A and B (what you call their co-domain) was assumed to be a point set, {-1, +1}, (two points in R). Later he showed his proof could be adapted so as to allow him to replace {-1, +1} by [-1, +1] (a particular line segment in R) and he had reason to do this in order to accommodate also hidden variables located in the measurement stations.

[-1, +1] is not the only codomain one could assume to describe the measurement results in any EPRB type experiment. It is a completely ad hoc and physically incorrect assumption of the codomain. The error in Bell's argument thus lies in his very first equation. It contradicts the fact that one is trying to chart the spectrum of the Pauli matrices. An obvious fact which quantum mechanical description correctly takes into account. Bell blundered in his very first equation, and misdirected the course of physics for 50 years. But it is not Bell who should be blamed for this colossal travesty, but the followers of Bell, who should have known better.
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### Re: Gill's theorem refuted (GTR)

minkwe wrote:
gill1109 wrote:
minkwe wrote:Secondly, since you have now committed that you are relying on the "weakly objective" interpretation of those expectation values, could you please explain how "counterfactual definiteness" is relevant to Bell's inequalities. Surely you must understand that there are no counterfactual terms in the "weakly objective" interpretation as opposed to the "strongly objective" one. So since your recent paper spends a lot of ink talking about counterfactual definiteness, please explain how it arises in the "weakly objective view".

Question two: counterfactual definiteness. Local hidden variables implies counterfactual definiteness. If the outcome of Alice's measurement in direction a, when the particle turns up carrying the hidden variable lambda, is A(a, lambda), then it should not be a crime to talk about what the outcome would have been if the setting had actually been b? It seems to me that it's A(b, lambda).

If you are using the weakly objective interpretation, which of the correlations in Bell's or your inequality corresponds to what Alice and Bob got and which one corresponds to what they could have gotten but didn't?
I know you like talking about CFD. There is no crime to talk about it. But it is confusion to introduce the concept into your maths when none of your terms are counterfactual. If you are using the weakly objective view, then none of your terms are counterfactual and you can't reasonably claim at the end that CFD must be false due to violation.
In the weakly objective view, each correlation is measured on a different set of particles so it makes absolutely no sense to talk of what might have been measured but wasn't (CFD). It is only in the Strongly objective view that CFD is present, since all the correlations are measured on the same set of particles, and only one of them can practically be measured, then the others represent what they could have measured but didn't (CFD).

Let x be the number of apples and y be the number of oranges in my fruit bowl. Suppose x=5 and y=5. Then x=y. If I say x=y, I am not sayng that apples are oranges.
The word correlation can mean three different things. The value x we get out of an experiment. The value y which we imagine we would get, if we measured infinitely many pairs of particles. The value we get when we take three functions A B and rho and calculate a certain integral. If the experiment could be well described by a lcal hidden variables theory with those particular A, B and rho, then x will probably be close to y and y and z should be equal. The three numbers x, y, z will be (almost) the same (with large probability). They stand for three different things. The values of the things are equal, but the things (concepts) are not.

I would like to see your comments about my translations to R of epr-simple and epr-clocked, I started a new topic for this, hopefully no misrepresentation, hopefully I got the parameters and the model formulas right:
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=61
I can explain the meaning of CFD in the context of a simple specific example. I would like to take my simulation version / travesty of epr-simple as example. Call it epr-gill-stupid if you prefer the name. The code is up there on internet. Even if it's not your model, it is "a" model. Nice violation of CHSH. Event-based. LHV.

There are some lines of code in that program where the "actual" outcomes are computed. We could add next to those lines, two lines computing the "other" outcomes. Give them different names. Do nothing with them. The results woukd be identical. CFD is valid for this program, it seems to me. Is it useful? I'll come back to that later.

Suppose N was 10 000 and one of the correlations has the value x.

In inagination, I could have run the program with bigger and bigger N. The correlations would vary less and less. I can imagine their limit. Call the limiting value y.

I can also write down a formula with functions A, B, rho. The functions implemented in the code. I can calculate some integrals. I get a number. Let its value be z. It will be the same, again.

There are three numbers (values) x, y, z. The first is probably quite close to the second, and the second is exactly equal to the third Three correlations. Equal y, z), or almost equal with large probability (x).

It can be confusing just to say "the correlation E(a, b)". Are we talking about a number or a concept, and if so which? The number x, the number y, the number z; or the ways I explained each of those three numbers is found? That's actually 6 different meanings to the words "the correlation". Three values, each obtained from a different sequence of operations. Three sequences of operations. Three definitions.

CFD is clearly true for epr-simple and epr-clocked. And I found it nice to see that the results fit beautifully to the predictions of the Larsson and Larsson-Gill modified CHSH inequalities, up to statistical variation. In other words: theory based on CFD gave true predictions about my real computer experiments. So CFD is a valid and a useful concept in this context. I used it to derive the performance limits or operating characteristics of epr-gill-stupid. This is computer science applied to epr simulation models. What can you get them to do? What can't you get them to do?
Last edited by gill1109 on Fri Jun 06, 2014 8:36 pm, edited 4 times in total.
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### Re: Gill's theorem refuted (GTR)

Joy Christian wrote:
gill1109 wrote:The range of A and B (what you call their co-domain) was assumed to be a point set, {-1, +1}, (two points in R). Later he showed his proof could be adapted so as to allow him to replace {-1, +1} by [-1, +1] (a particular line segment in R) and he had reason to do this in order to accommodate also hidden variables located in the measurement stations.

[-1, +1] is not the only codomain one could assume to describe the measurement results in any EPRB type experiment. It is a completely ad hoc and physically incorrect assumption of the codomain. The error in Bell's argument thus lies in his very first equation. It contradicts the fact that one is trying to chart the spectrum of the Pauli matrices. An obvious fact which quantum mechanical description correctly takes into account. Bell blundered in his very first equation, and misdirected the course of physics for 50 years. But it is not Bell who should be blamed for this colossal travesty, but the followers of Bell, who should have known better.

You can change the codomain.

Christian's one page paper http://arxiv.org/abs/1103.1879 did more than just change the co-domain, it also changed the definition of correlation. The average of the product of the outcomes was also divided left and right by two bivectorial standard errors.

But this is off-topic. This thread is on Gill's theorem, not Christian's.
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### Re: Gill's theorem refuted (GTR)

gill1109 wrote:Let x be the number of apples and y be the number of oranges ...

There are no counterfactual oranges in your bowl. I've asked you a very specific question based on what you chose as your interpretation of the correlations in Bell's inequalities, and what you discuss in your paper. The question again is: Where exactly in your equations does CFD come in, if you say you are using the "weakly objective interpretation. We don't need another example, we have one already, the exact one Bell was using which you also use. Simply tell us which of the three correlations in Bell's inequalities are counterfactual so that we may evaluate if you are being consistent in using your claimed "weakly objective interpretaion".

The central conclusion of your paper is the claim that since realism = CFD, violation of your theorem and Bell's by QM and experiments should force us to abandon realism (CFD). Clearly this implies that you believe without CFD/realism, you would not have obtained your proof. We don't need to go off discussing oranges or apples, simply point to an equation or paragraph in your paper, or provide a mathematical argument which demonstrates how CFD is essential for you proof or Bell's?

The word correlation can mean three different things...

Right, it can. So then, whatever your definition of correlation, which of the correlations in Bell's inequalities or your theorem are counterfactual. What role does CFD play in your derivations that permits you to make conclusions about it when a violation is observed?

I can explain the meaning of CFD in the context of a simple specific example.

I would rather you explain it in the context of your paper which is the topic of this thread since you make claims about CFD in your paper. None of my simulations make use of any concepts of CFD so those are not appropriate examples. The EPRB experiment you discuss in your paper is a clear enough example so use that. What role does CFD play in your derivation in your paper, that permits you to conclude at the end that CFD is untenable?
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### Re: Gill's theorem refuted (GTR)

It can be confusing just to say "the correlation E(a, b)".

What I'm asking you is very very clear and not confusing at all. If we have a single set of particle pairs, we can measure them at angles (a,b) to get E(a,b). Once they have been measured at those angles, they are destroyed and can no longer be measured at angles (a',b'). However we can imagine what result they might have produced, had they been measured at those angles, eg. E(a,b'), E(a',b'). These terms are counterfactual. Because we could have measured them but didn't. However, if we measure one set of particles at (a,b), we can without any problem also measure a different set of particles at (a',b') and yet a different set at (a,b'). None of these results are counterfactual. But you have claimed that you rely on the "weakly objective view" in which each correlation is measured on a different set of particles. Therefore there are no counterfactual terms. How then do you justify your conclusion that CFD is untenable when you don't even have any counterfactual terms in your inequality? This is the simple question I'm asking you.

CFD is clearly true for epr-simple and epr-clocked.

There are no counterfactual terms in any of my simulations because my simulations use the weakly objective interpretation -- None whatsoever.Part of the quarrel I've had with you about my simulations has been on this very point. There are no counterfactual terms in any of my simulations. Besides you do not need any simulation to answer the question. You didn't need any simulation to write and publish your article so you should be able to answer the question clearly without one, unless you already know that you can not answer the question.
minkwe

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### Re: Gill's theorem refuted (GTR)

minkwe wrote:There are no counterfactual terms in any of my simulations because my simulations use the weakly objective interpretation -- None whatsoever.Part of the quarrel I've had with you about my simulations has been on this very point. There are no counterfactual terms in any of my simulations. Besides you do not need any simulation to answer the question. You didn't need any simulation to write and publish your article so you should be able to answer the question clearly without one, unless you already know that you can not answer the question.

There are no counterfactual terms in your simulation because you didn't put them in. But you could have put them in.

Here is a simulation. I have commented out a block of lines. The block of lines which are commented out are "counterfactual". Imaginary. You can imagine "uncommenting them". What would happen if I un-out-commented the commented-out lines?

Code: Select all
N <- 10^4coincWindow <- 0.0004ts <- pi * 0.03asym <- 0.98spin <- 0.5n <- 2 * spinphase <- pi * nalpha <- c(0, 90) * pi / 180 # Alice's possible two settings (degrees)beta <- c(45, 135) * pi / 180 # Bob's possible two settings (degrees)a <- sample(c(1, 2), N, replace = TRUE) # Alice setting labels (1, 2)b <- sample(c(1, 2), N, replace = TRUE) # Bob setting labels (1, 2)el <- runif(N, 0, 2 * pi)er <- el + phasep <- 0.5 * sin(runif(N, 0, pi / 6))^2ml <- runif(N, asym, 1)mr <- runif(N, asym, 1)Cl <- (0.5/pi) * (-1)^n * cos(n * (el - alpha[a]))Cr <- (0.5/pi) * (-1)^n * cos(n * (er - beta[b]))tdl <- ts * pmax(ml * p - abs(Cl), 0)tdr <- ts * pmax(mr * p - abs(Cr), 0)A <- sign(Cl)B <- sign(Cr)# a_CF <- 3 - a# b_BF <- 3 - b# Cl_CF <- (0.5/pi) * (-1)^n * cos(n * (el - alpha[a_CF]))# Cr_CF <- (0.5/pi) * (-1)^n * cos(n * (er - beta[b_CF]))# tdl_CF <- ts * pmax(ml * p - abs(Cl_CF), 0)# tdr_CF <- ts * pmax(mr * p - abs(Cr_CF), 0)# A_CF <- sign(Cl_CF)# B_CF <- sign(Cr_CF)AB <- A * BE11 <- mean(AB[a == 1 & b == 1 & abs(tdl-tdr) < coincWindow])E12 <- mean(AB[a == 1 & b == 2 & abs(tdl-tdr) < coincWindow])E21 <- mean(AB[a == 2 & b == 1 & abs(tdl-tdr) < coincWindow])E22 <- mean(AB[a == 2 & b == 2 & abs(tdl-tdr) < coincWindow])S <- -(E11 - E12 + E21 + E22)

The block of lines which is "commented out" define some new variables which are related to old ones by adding the _CF to their names, meaning "counterfactual". For instance: a is the list of setting labels which Alice is actually using, a sequence of 1's and 2's. a_CF is the complementary sequence of 2's and 1's. I then go on to define the time delays left and right and the measurement outcomes left and right which Alice and Bob would have seen if their settings had been a_CF and b_CF instead of a and b.

The point is that whether I leave those lines commented out, or "uncomment them", the actual outcomes and actually correlations and actual value of S don't change. So: I can safely "imagine" what would have been the outcomes if Alice or Bob had chosen any sequence of measurement settings at all, and those outcomes don't depend on what setting the other party used. The model is a local hidden variables model so CFD is true. Within the model, the outcomes of the not-performed measurements are defined (or "can be defined") in a completely local way alongside of those actually performed. It's harmless. You can put them in or out just as you like.

OK completely harmless completely irrelevant? What's the big deal then?

The big deal is that now I can define the Nx4 matrix of the outcomes of both possible measurements for both parties, for all N runs. What Alice and Bob actually get to see is obtained from the big matrix by random selection, for each row, of one of the two settings of Alice, and one of the two settings of Bob. Gill's Theorem 1 applies.

(Well this was the core of the code of an EPR-B experiment without event-ready-detectors and with some non-detections and with coincidence window post-selection so the story is a bit bigger and more complicated, but the same principle applies).

It's a trick. A conjuring trick. A mathematical device. One wants to study something e.g. the real numbers, one embeds it in something bigger and imaginary, e.g. the complex numbers, and heh presto suddenly we have all kinds of new tools in order to get new results ... about ... the real numbers!
gill1109
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### Re: Gill's theorem refuted (GTR)

Richard, are you going to answer the question or not?

How is CFD important for your derivation, if you are using the weakly objective interpretation?
minkwe

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### Re: Gill's theorem refuted (GTR)

minkwe wrote:Richard, are you going to answer the question or not?

How is CFD important for your derivation, if you are using the weakly objective interpretation?

The weakly objective interpretation tells me what I mean by "probability". I imagine something being repeated many many times and then I look at relative frequencies and then I imagine the number of repetitions going to infinity and the relative frequencies converging. Also known as the frequentist interpretation. But notice the word *imagine*. It's an imaginary sequence of repetitions.

Where does CFD come in? Well forget CFD for just a moment, suppose that we have a LHV model. That means that Nature chooses lambda, Alice chooses a and Bob chooses b. Alice and Bob then get to see A(a, lambda) and B(b, lambda). But since A and B are just a couple of functions I can also think of A(a', lambda) for all possible values of a' and B(b', lambda) for all possible values of b', at the same time. Within my mathematical model they are defined, too, even though they don't correspond to anything in the experiment ... well, what they correspond to, is what the outcome would have been, had Alice chosen a' instead of a, while Nature had still made the same choice lambda.

In your simulation models too, the same thing happens. You pick a lambda, also a and b get picked, there is a function AIa, lambda) which tells us Alice's outcome and so on... but I could add a line to the code where A(a', lambda) is also computed, but nothing else would change.

It is possible to go the other way round and show that CFD + locality => LHV. It's a simple little mathematical theorem. Well ... simple if you are at home with modern probability theory.

Remember what I said about the difference between models and reality?

The frequentist interpretation of probability is a bridge between part of reality and probability theory (which is part of mathematics).
LHV and CFD are both part of mathematics. They are terms which can be used within the domain of mathematical models of reality.

BTW I really appreciate that you keep putting these questions. It means that you feel that there is something going on here which needs to be sorted out and you keep on trying. This is the mark of a true scientist. A true scientist keeps on feeling that itch and keeps on trying to do something about it.

BTW there are also other interpretations of probability, for instance, the subjective (Bayesian) view, cf. Jaynes. I don't like it so much. In practice it tends to come down to the same thing, at least, in physics it does. Fortunately the "calculus" of subjective probability is identical to that of "objective" probability so the mathematical theories are identical. Every theorem of subjective probability theory is also a theorem of objective probability theory, and vice versa. The difference between the two theories is not inside the theories themselves, but in the bridges to the "real world".

We now get into self-reference paradoxes, till we realise that "the real world" is also some kind of idealisation. Hopefully it is inter-subjective. In other words, at a higher level, both Bayesian and frequentist probability concepts are themselves only "models".
gill1109
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### Re: Gill's theorem refuted (GTR)

gill1109 wrote:Where does CFD come in? Well forget CFD for just a moment, suppose that we have a LHV model. That means that Nature chooses lambda, Alice chooses a and Bob chooses b. Alice and Bob then get to see A(a, lambda) and B(b, lambda). But since A and B are just a couple of functions I can also think of A(a', lambda) for all possible values of a' and B(b', lambda) for all possible values of b', at the same time. Within my mathematical model they are defined, too, even though they don't correspond to anything in the experiment ... well, what they correspond to, is what the outcome would have been, had Alice chosen a' instead of a, while Nature had still made the same choice lambda.

Good, you have just clearly and concisely explained how CFD arises in the STRONGLY objective interpretation. This is not what I asked you. You said you were using the WEAKLY objective interpretation please provide a clear and concise explanation of how it arises in the WEAKLY objective interpretation please, without rambling about simulations. My question again in case you still did not understand it is :
How is CFD important for your derivation, if you are using the weakly objective interpretation?
minkwe

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