Scholarpedia article on Bell's Theorem

[mattt]

Travis, I am having a terrible sinusitis case, but as soon as I recover, I'll try to explain my thoughts about the relation between "The Factorizability Condition (4) in the CHSH-Theorem", a "Fundamental Stochastic Theory that pretends to predict (correctly) the outcomes of that type of experiments", and "the Causal Structure of Special Relativity".

I say again that the CHSH-Theorem is a mathematical statement with a correct mathematical proof.

The only issue for me is that at first I thought that a hypothetical fundamental stochastic theory that does not satisfy "your factorizability condition (4)" would not be necessarily in conflict with the idea of "Causal Structure" of Special Relativity. Then (after reading a very good paper of yours) I changed my mind and agreed with you, but after a second reading of that same paper I kind of started to doubt again but then I got ill, so as soon as I recover the energy I'll try to explain my thoughts about that issue.

 

[ttn]

Travis, I am having a terrible sinusitis case, but as soon as I recover, I'll try to explain my thoughts about the relation between "The Factorizability Condition (4) in the CHSH-Theorem", a "Fundamental Stochastic Theory that pretends to predict (correctly) the outcomes of that type of experiments", and "the Causal Structure of Special Relativity".

I say again that the CHSH-Theorem is a mathematical statement with a correct mathematical proof.

The only issue for me is that at first I thought that a hypothetical fundamental stochastic theory that does not satisfy "your factorizability condition (4)" would not be necessarily in conflict with the idea of "Causal Structure" of Special Relativity. Then (after reading a very good paper of yours) I changed my mind and agreed with you, but after a second reading of that same paper I kind of started to doubt again but then I got ill, so as soon as I recover the energy I'll try to explain my thoughts about that issue.

Sounds good. There's no hurry. Hope you feel better soon, and I'll look forward to discussing it a little more when you do.

 

[billschnieder]

Earlier this thread, when I brought it up you said that CFD is either metaphysical and unimportant, or insofar as it is important it is so essential for all scientific theories that it shouldn't be questioned. Yet I think that quantum mechanics does not possesses it; to wit, if you measure the polarization of a photon at 0 degrees, then in QM the question "What would have been the result if you had instead measured the polarization at 45 degrees" has no definite answer.

I believe ttn is wrong, that there is no CFD involved but I think you are also misunderstanding how CFD comes in. Take for example the CHSH inequality from ttn's article. Forget for a moment about how it is derived and just focus for the moment only on the terms within the inequality and their meanings:

|C(a,b)−C(a,c)|+|C(a′,b)+C(a′,c)|≤ 2

Now, consider that you have measured along (a,b) and you now have C(a,b) as factual. The remaining terms are therefore necessarily counter-factual. This therefore begs the question, is it possible to test such an inequality experimentally when measuring one term necessarily makes measurement of the other terms impossible?

ttn tries to argue that you can still test it experimentally by assuming so-called "no-conspiracy", which is an affirmative defense which effectively says: "the remaining terms are indeed counterfactual but we can substitute factual measured correlations in their place because what is measured can only be different if there is conspiracy". Now this is a strange argument which can be rephrased as follows:

"if local causality is true, factual outcomes and counterfactual predictions can only differ if conspiracy is involved"

Assuming for the moment that this argument is true, it means if you start by using ONLY factual terms you should end up with the same inequality as if you start by assuming counterfactual terms. However, Starting with factual terms ONLY we end up with an inequality:

|C1(a,b)−C2(a,c)|+|C3(a′,b)+C4(a′,c)|≤ 4

Which is different from the one with counterfactual terms. Therefore the only argument which allows ttn to avoid the counterfactual problem, leads to a contradiction and we must reject his defense.

 

[billschnieder]

Another criticism which has not been addressed is the one concerning the locality assumption. Note that "Locality" and "local causality" mean the same thing in the context of this discussion, because locality simply means there is no causal connection between two remote events A and B that can propagate faster than the velocity of light. Therefore the locality requirement is the same as a requirement for "no causality" between the two remote events and non-locality means there IS a causal connection between the two.

ttn says P(AB|X) = P(A|BX)P(B|X) = P(A|X)P(B|X) because according to him, the lack of a causal connection between A and B implies that P(A|BX) = P(A|X). Ttn's argument then is effectively:

~C(L) -> I and ~I -> C(~L). L=Locality, C=causality, I=Independence.

A single counter example of a case in which lack of causality (~C,L) does not imply independence (I), or lack of independence (~I) does not imply causality (C, ~L) is sufficient to demolish the argument. For that purpose, I will repeat the Bernouli's urn example:

X = Our urn contains N balls, M of them are red, the remaining (N-M) white. They are drawn out blindfolded without replacement."
Ri = Red on the i'th draw, i = 1, 2, ..."

P(R1|X) = M/N

Now if we know that red was found on the first draw, then that changes the contents of the urn for the second:

P(R2|R1, X) = (M - 1)/(N - 1) ≠ P(R2|X) = M/N

and this conditional probability expresses the causal influence of the first draw on the second. But suppose we are told only that red was drawn on the second draw; what is now our probability for red on the first draw? If ttn is being consistent he would say:

P(R1|R2, X) = P(R1|X) = M/N

because whatever happens on the second draw cannot exert any physical influence on the condition of the urn at the first draw (ie C -> ~I). But this result is wrong! The correct answer should be

P(R1,R2, X) = P(R2 |R1, X)

To see this consider the case in which we have only 1 red ball (M = 1); if we know that the one red ball was taken in the second draw, then it is certain that it could not have been taken in the first.

Therefore P(R1|X) = 1/N ≠ P(R1|R2,X) = 0.

So we have a case in which there is no causality (~C) and there is no independence (~I) and the proof fails. So far ttn's only response to this argument has been to ask us to provide a better way of representing local causality, which is a tacit admission that the locality causality condition is fatally flawed.

 

[ttn]

I believe ttn is wrong, that there is no CFD involved but I think you are also misunderstanding how CFD comes in. Take for example the CHSH inequality from ttn's article. Forget for a moment about how it is derived and just focus for the moment only on the terms within the inequality and their meanings:

|C(a,b)−C(a,c)|+|C(a′,b)+C(a′,c)|≤ 2

Now, consider that you have measured along (a,b) and you now have C(a,b) as factual.

C(a,b) is a correlation coefficient. It is the product of the outcomes *averaged over many runs*. So it already takes many many runs to measure even just this one individual term.

The remaining terms are therefore necessarily counter-factual.

Each term is an average (of outcome products) over many many runs. All the runs are "factual" in the sense that they all really happened.

This therefore begs the question, is it possible to test such an inequality experimentally when measuring one term necessarily makes measurement of the other terms impossible?

What's true is that each individual "run" (particle pair) contributes data to one and only one of the four terms. That is, you can only measure along one of (a,b), (a',b), etc., at a time. This hardly makes measurement of the other terms impossible. It just means you need to make lots and lots of runs and take averages. But you already have to do that even to measure just one of the terms!

ttn tries to argue that you can still test it experimentally by assuming so-called "no-conspiracy", which is an affirmative defense which effectively says: "the remaining terms are indeed counterfactual but we can substitute factual measured correlations in their place because what is measured can only be different if there is conspiracy". Now this is a strange argument which can be rephrased as follows:

"if local causality is true, factual outcomes and counterfactual predictions can only differ if conspiracy is involved"

Anybody who wants to understand the actual argument should see the scholarpedia article.

 

[ttn]

ttn says P(AB|X) = P(A|BX)P(B|X) = P(A|X)P(B|X) because according to him, the lack of a causal connection between A and B implies that P(A|BX) = P(A|X).

Here I get no credit at all. I'm just following Bell. And this explanation of the factorization condition is inadequate. It makes sense as a necessary condition of locality only if certain assumptions are made about X. Interested people should see my recent AmJPhys paper on "JS Bell's Concept of Local Causality" for a detailed presentation.

Ttn's argument then is effectively:

~C(L) -> I and ~I -> C(~L). L=Locality, C=causality, I=Independence.

A single counter example of a case in which lack of causality (~C,L) does not imply independence (I), or lack of independence (~I) does not imply causality (C, ~L) is sufficient to demolish the argument. For that purpose, I will repeat the Bernouli's urn example:

X = Our urn contains N balls, M of them are red, the remaining (N-M) white. They are drawn out blindfolded without replacement."
Ri = Red on the i'th draw, i = 1, 2, ..."

P(R1|X) = M/N

Now if we know that red was found on the first draw, then that changes the contents of the urn for the second:

P(R2|R1, X) = (M - 1)/(N - 1) ≠ P(R2|X) = M/N

and this conditional probability expresses the causal influence of the first draw on the second. But suppose we are told only that red was drawn on the second draw; what is now our probability for red on the first draw? If ttn is being consistent he would say:

P(R1|R2, X) = P(R1|X) = M/N

because whatever happens on the second draw cannot exert any physical influence on the condition of the urn at the first draw (ie C -> ~I). But this result is wrong! The correct answer should be

P(R1,R2, X) = P(R2 |R1, X)

To see this consider the case in which we have only 1 red ball (M = 1); if we know that the one red ball was taken in the second draw, then it is certain that it could not have been taken in the first.

Therefore P(R1|X) = 1/N ≠ P(R1|R2,X) = 0.

So we have a case in which there is no causality (~C) and there is no independence (~I) and the proof fails. So far ttn's only response to this argument has been to ask us to provide a better way of representing local causality, which is a tacit admission that the locality causality condition is fatally flawed.

No, Bill, my response to this argument is that you need to do some homework, because you simply have not understood Bell's formulation of locality. You *think* that Bell is saying "any time the probability for one thing depends on another thing, that means the other thing causally influences the one thing". And then you think you can refute Bell by this kind of example where there is a probabilistic/statistical dependence of this sort, but no causal dependence.

But the truth is that you are just *wrong* about what Bell says. Bell is light years ahead of you. Read what he writes about Lille and Lyon, about the cooking of the egg and the ringing of the alarm, and many other examples like this where he goes into great, explicit, careful detail about the need to *distinguish* causality from mere correlation. Read what he writes about how he actually formulates locality and try to appreciate how it is a response to this need, i.e., how the locality conditions is violated *only* when there is genuine nonlocal causation and *not* when there is "mere statistical dependence". Sound impossible? Sound too good to be true? Go and do your homework and find out for sure, and I'll be happy to discuss it after you make it clear somehow that you actually understand what Bell said. So far all of your objections are of the straw man variety.

 

[lugita15]

I don't think I said that latter about CFD. Or at least that's not exactly what I meant.

OK, you had said this earlier, but maybe I misinterpreted it:

"We just have to remember that we are talking about *theories* -- and a theory, by definition, is something that tells you what will happen *if you do such-and-such*. *All* of the predictions of a theory are in that sense hypothetical / counterfactual. Put it this way: the theory doesn't know and certainly doesn't care about what experiment you do in fact actually perform. It just tells you what will happen if you do such-and-such.

So back to your #2 above, of course it makes sense to ask what would have happened if you had turned the polarizers some other way. It makes just as much sense (after the fact, after you actually turned them one way) as it did before you did any experiment at all. How could the theory possibly care whether you've already done the experiment or not, and if so, which one you did? It doesn't care. It just tells you what happens in a given situation. QM works this way, and so does every other theory. So there really is no such thing as option #2."

I thought you meant that insofar as counterfactual definiteness is a necessary assumption for Bell's theorem, it is a trivial feature of all scientific theories.

What I think is more like this: you will realize that this whole issue of CFD simply melts away into nothingness (I mean, it becomes clear that there is no issue here at all) if you think of Bell's theorem as a constraint on *what theories say* -- as opposed to trying to think of every last character in the math as somehow referring directly to some real experimental outcome.

I'm not sure what you're talking about here. I certainly agree that there are some parts of most if not all theories that do not directly relate to experiments. Quantum mechanics contains plenty of that: Hilbert space theory and spectral theory and representation theory, oh my! But what does that have to do with counterfactual definiteness?

But this question does have a definite answer: "If you had instead measured at 45 degrees, what would the possible results have been, and what are their probabilities?" That is, the reason QM gives no definite answer to your question is only that QM is not deterministic.

I agree that in the case of quantum mechanics, counterfactual definiteness is closely related to "future definiteness" AKA determinism. But I think that these two notions should still be logically distinguished from each other.

But that certainly doesn't matter. You can derive the Bell inequality just fine, from locality, without invoking determinism.

I agree that there are local probabilistic theories for which you can derive a Bell inequality. But it is not so clear to me that you can derive a Bell inequality from a local theory, deterministic or not, which does not have counterfactual definiteness.

Of course, you might (as many people have) look at some derivation of the Bell inequality in some textbook and see that it seems to *start with* -- to *presume* -- pre-existing (deterministic) answers/outcomes to all these different possible questions/measurements. But that's just because many commentators and textbook authors confuse (what we call in the article) "Bell's inequality theorem" for the full "Bell's theorem". The full "Bell's theorem" starts just with the assumption of locality and *derives* the pre-existing (deterministic) answers/outcomes, using basically the EPR argument. So really the whole thing leading to this red herring about CFD is simply missing this, failing to realize that "Bell's inequality theorem" and "Bell's theorem" are not the same thing.

I think I do recognize two-step nature of Bell's proof:

1. EPR, in which hidden variables is a conclusion, not an assumption of the argument
2. "Bell's inequality theorem" in which the hidden variables conclusion of EPR is used as an assumption in order to derive the Bell inequality

I think that the basic structure of the argument is valid (although I am curious about Demystifier's contention that the hidden variables conclusion of EPR cannot be quite the same as the hidden variables assumption of the inequality theorem). The only place where I think we differ on this is that you don't think counterfactual definiteness needs to be an assumption of EPR.

Put it this way: it's true that QM is not the type of theory that is assumed in standard derivations of "Bell's inequality theorem". But this is of no real relevance whatsoever. Actually what is shows is just this: QM is not a local theory! (Because, if it were, it would have to explain the perfect correlations with pre-existing values, the way the EPR argument proves any local theory must.)

I pretty much agree with you the QM is nonlocal, only because of wavefunction collapse (although I think DrChinese has some arguments to the effect that QM has "quantum nonlocality" but not "regular" nonlocality). And I also think that it would be pretty hard to come up with an explanation of perfect correlations that did not invoke either nonlocality or conspiracy. But I think it may not quite be logically impossible.

Your "several axes" version of EPR seems to avoid counterfactual definiteness, but I'm not completely convinced that there isn't a leap of logic somewhere, even though I haven't come up with a definitive counterargument yet. You're basically saying that anyone who denies the following statement (and rejects nonlocality) must be a superdeterminist: "If you WOULD have been able to predict with certainty the result of the 0-degree polarization measurement of the distant photon if you HAD performed a 0-degree polarization measurement of your photon, then there IS a pre-existing element of reality corresponding to the 0-degree polarization, even if you do not actually carry out a 0-degree polarization measurement." I think that someone could reject this statement and also reject superdeterminism, but I'm still trying to show how this could be possible.

 

[billschnieder]

C(a,b) is a correlation coefficient. It is the product of the outcomes *averaged over many runs*. So it already takes many many runs to measure even just this one individual term.

You mean averaged over many photon pairs. In my vocabulary, angle pair(a,b) is one run, in which many photon pairs are measured, I understand that in your vocabulary one "run" corresponds to one photon pair. This is a non-issue as far as my point is concerned.

All the runs are "factual" in the sense that they all really happened.

They are factual in the experiment but not in the inequalities. Nothing in the inequality happened. The series of particles you measure to be able to average and obtain C(a,b) can not be restored to measure C(b,c) therefore C(b,c) is counterfactual as soon as C(a,b) is measured. Your argument is that it doesn't matter because a different series of particles can be used to measure C(b,c). This is what I debunked in post #125. With a different series of particles, you have many more degrees of freedom (64 vs the original 16) and the resulting inequality is different.

What's true is that each individual "run" (particle pair) contributes data to one and only one of the four terms. That is, you can only measure along one of (a,b), (a',b), etc., at a time. This hardly makes measurement of the other terms impossible. It just means you need to make lots and lots of runs and take averages. But you already have to do that even to measure just one of the terms!

Obviously you have a different meaning what a "run" is which leads you to misunderstand my point.

 

[billschnieder]

Here I get no credit at all. I'm just following Bell. And this explanation of the factorization condition is inadequate. It makes sense as a necessary condition of locality only if certain assumptions are made about X. Interested people should see my recent AmJPhys paper on "JS Bell's Concept of Local Causality" for a detailed presentation. No, Bill, my response to this argument is that you need to do some homework, because you simply have not understood Bell's formulation of locality. You *think* that Bell is saying "any time the probability for one thing depends on another thing, that means the other thing causally influences the one thing". And then you think you can refute Bell by this kind of example where there is a probabilistic/statistical dependence of this sort, but no causal dependence.

But the truth is that you are just *wrong* about what Bell says. Bell is light years ahead of you. Read what he writes about Lille and Lyon, about the cooking of the egg and the ringing of the alarm, and many other examples like this where he goes into great, explicit, careful detail about the need to *distinguish* causality from mere correlation. Read what he writes about how he actually formulates locality and try to appreciate how it is a response to this need, i.e., how the locality conditions is violated *only* when there is genuine nonlocal causation and *not* when there is "mere statistical dependence". Sound impossible? Sound too good to be true? Go and do your homework and find out for sure, and I'll be happy to discuss it after you make it clear somehow that you actually understand what Bell said. So far all of your objections are of the straw man variety.

I'm responding to your argument in your article which is flawed. You keep repeating that I'm wrong and yet you do not refute the argument in any way.

 

[lugita15]

I'm responding to your argument in your article which is flawed. You keep repeating that I'm wrong and yet you do not refute the argument in any way.

I think he does claim to have refuted it, in his paper "JS Bell's Concept of Local Causality".

 

[billschnieder]

I think he does claim to have refuted it, in his paper "JS Bell's Concept of Local Causality".

For example let us look at his equation (1) in the article "J.S. Bell's Concept of Local Causality" which according to him lays out mathematically what Bell's means by "local causality"

P(b_1 |B_3 , b_2 ) = P (b_1 |B_3 )

What then does

P(b_1 |B_3 , b_2 ) \neq P (b_1 |B_3 )

imply? Non-local causality?

 

[billschnieder]

Oh and another misunderstanding: "stochastic" and "complete specification" are incompatible despite ttn's claims. He admits on page 10 that:

Of course, if one insists that any stochastic theory is ipso facto a stand-in for some (perhaps unknown) under-lying deterministic theory (with the probabilities in the stochastic theory thus resulting not from indeterminism in nature, but from our ignorance), Bell’s locality concept would cease to work.

Even if we were to accept that it is possible to have a complete specification and still only have a stochastic theory, he would be admitting that Bell's locality concept is invalid for deterministic hidden variable theories.

 

[billschnieder]

Another falsehood in the "JS Bell's Concept of Local Causality" paper.

ttn says:

Bell is not asking us to accept that any particular theory (stochastic or otherwise) is true; he’s just asking us to accept his definition of what it would mean for a stochastic theory to respect relativity’s prohibition on superluminal causation. And this requires us to accept, at least in principle, that there could be such a thing as a genuinely,irreducibly stochastic theory, and that the way “causality” appears in such a theory is that certain beables do,and others do not, influence the probabilities for specific events.

This is clearly a misrepresentation of Bell

Since we can predict in advance the result of measuring any chosen component of σ2, by previously measuring the same component of σ1, it follows that the result of any such measurement must actually be predetermined. Since the initial quantum mechanical wave function does not determine the result of an individual measurement, this predetermination implies the possibility of a more complete specification of the state.

Let this more complete specification be effected by means of parameters λ.

...

So in this simple case there is no difficulty in the view that the result of every measurement is determined by the value of an extra variable and the statistical features of quantum mechanics arise because the value of this variable is unknown in individual instances.

Remember that this is directly relevant to EPR because they were discussing "completeness" and "prediction with certainty". It is impossible to predict with certainty the outcome of a measurement in a stochastic theory. Bell clearly understands that "incomplete" and "statistical"/stochastic/probabilistic are synonymous. Einstein understood that too.

 

[ttn]

I thought you meant that insofar as counterfactual definiteness is a necessary assumption for Bell's theorem, it is a trivial feature of all scientific theories.

The point is that it's not a necessary assumption at all. The "assumption" that you actually need is instead completely trivial: a theory predicts what will happen in some situation.

I agree that there are local probabilistic theories for which you can derive a Bell inequality. But it is not so clear to me that you can derive a Bell inequality from a local theory, deterministic or not, which does not have counterfactual definiteness.

I don't even think it makes sense to talk about being able to derive a Bell inequality for/from some particular theory. Bell's theorem is the derivation of the inequality from certain (mathematically formulated) physical principles (e.g. "locality" and "no conspiracies"). And the point here is that "counterfactual definiteness" is not among the principles needed to derive the inequality.

I think I do recognize two-step nature of Bell's proof:

1. EPR, in which hidden variables is a conclusion, not an assumption of the argument
2. "Bell's inequality theorem" in which the hidden variables conclusion of EPR is used as an assumption in order to derive the Bell inequality

I think that the basic structure of the argument is valid (although I am curious about Demystifier's contention that the hidden variables conclusion of EPR cannot be quite the same as the hidden variables assumption of the inequality theorem). The only place where I think we differ on this is that you don't think counterfactual definiteness needs to be an assumption of EPR.

OK.

I pretty much agree with you the QM is nonlocal, only because of wavefunction collapse (although I think DrChinese has some arguments to the effect that QM has "quantum nonlocality" but not "regular" nonlocality).

...where "quantum nonlocality" is defined as "the kind of nonlocality that QM has and which we don't have to worry about because we decided in advance that we weren't going to worry about QM"?

And I also think that it would be pretty hard to come up with an explanation of perfect correlations that did not invoke either nonlocality or conspiracy. But I think it may not quite be logically impossible.

OK, I mean, that's the whole thing right there then. There's a derivation of it, so tell me where you think it's wrong if you think it's wrong. Or maybe it can be an official challenge. I'll gladly give $20 and a kiss on the cheek to anybody who can come up with a way to explain perfect correlations that is both local and non-conspiratorial and which doesn't involve "pre-existing values" or the equivalent.

Your "several axes" version of EPR seems to avoid counterfactual definiteness, but I'm not completely convinced that there isn't a leap of logic somewhere, even though I haven't come up with a definitive counterargument yet. You're basically saying that anyone who denies the following statement (and rejects nonlocality) must be a superdeterminist: "If you WOULD have been able to predict with certainty the result of the 0-degree polarization measurement of the distant photon if you HAD performed a 0-degree polarization measurement of your photon, then there IS a pre-existing element of reality corresponding to the 0-degree polarization, even if you do not actually carry out a 0-degree polarization measurement." I think that someone could reject this statement and also reject superdeterminism, but I'm still trying to show how this could be possible.

What can I say? I will not refuse to listen if/when you show how it's possible. But basically here it sounds like you are conceding that, after scrutinizing the argument, you can't see any flaw and thus think the argument is good... but your feelings haven't quite caught up with your conscious judgment yet. OK, that's cool, sometimes it takes some time to get yourself fully lined up behind a new and surprising conclusion that you realize the evidence compels you to embrace.

 

[ttn]

I think he does claim to have refuted it, in his paper "JS Bell's Concept of Local Causality".

Well, sure, but there's a (shorter) version of this in the scholarpedia article too. But people who have followed this whole (admittedly long) thread know that I've gone far above and beyond in giving Bill the benefit of the doubt, responding to his criticisms and questions, patiently refuting his arguments (over and over again one might say), etc. So forgive me if I don't get into it with him yet again here.

 

[ttn]

For example let us look at his equation (1) in the article "J.S. Bell's Concept of Local Causality" which according to him lays out mathematically what Bell's means by "local causality"

P(b_1 |B_3 , b_2 ) = P (b_1 |B_3 )

What then does

P(b_1 |B_3 , b_2 ) \neq P (b_1 |B_3 )

imply? Non-local causality?

Try reading the nearby words that say clearly what the various symbols *mean*.

 

[ttn]

Bell clearly understands that "incomplete" and "statistical"/stochastic/probabilistic are synonymous. Einstein understood that too.

Hogwash. Bell went out of his way to *avoid* any assumption of determinism, i.e, to formulate everything (in particular the concept of "locality") from the very beginning in a way that embraced the idea of irreducibly stochastic theories (deterministic theories being, in his words, just a special case where the probabilities are delta functions). He did this precisely because early commentators on his theorem already -- erroneously in his view -- said it only applies to deterministic theories. (One still hears, in textbooks and such, statements like "Bell refuted the idea of local determinism.") I explain all of this in my papers (quoting extensively from Bell). Or you could just read Bell. But Bill, contra Bell, simply doesn't know what he's talking about and seems unable to understand what he reads. But I am happy to see he's doing his homework!

 

[billschnieder]

Try reading the nearby words that say clearly what the various symbols *mean*.

This is a very simple question. Why don't you answer what P(b_1 |B_3 , b_2 ) \neq P (b_1 |B_3 ) implies. I'm not asking you to define the symbols. What does it mean for the LHS to be different from the RHS in the above "definition" of local causality.

You say local causality implies P(b_1 |B_3 , b_2 ) = P (b_1 |B_3 ). So I'm simply asking you what P(b_1 |B_3 , b_2 ) \neq P (b_1 |B_3 ) imples? Very simple question.

 

[Gordon Watson]

... I explain all of this in my papers (quoting extensively from Bell). Or you could just read Bell. But Bill, contra Bell, simply doesn't know what he's talking about and seems unable to understand what he reads. But I am happy to see he's doing his homework!

ttn, about a week ago I PMed for a copy of an AmJPhys article (of yours) that you cited. No reply so far received. Any chance? Also, can you link to papers of yours that you cite?

PS: I annotated the Scholarpedia article and will happily do the same on others.

Regards, GW.

 

[billschnieder]

Hogwash. Bell went out of his way to *avoid* any assumption of determinism, i.e, to formulate everything (in particular the concept of "locality") from the very beginning in a way that embraced the idea of irreducibly stochastic theories (deterministic theories being, in his words, just a special case where the probabilities are delta functions). He did this precisely because early commentators on his theorem already -- erroneously in his view -- said it only applies to deterministic theories. (One still hears, in textbooks and such, statements like "Bell refuted the idea of local determinism.") I explain all of this in my papers (quoting extensively from Bell). Or you could just read Bell. But Bill, contra Bell, simply doesn't know what he's talking about and seems unable to understand what he reads. But I am happy to see he's doing his homework!

I just quoted to you Bell's own words which refute what you claim here. Besides in your article you state clearly that Bell's definition of local causality does not work in a local deterministic theory.

 

[ttn]

ttn, about a week ago I PMed for a copy of an AmJPhys article (of yours) that you cited. No reply so far received. Any chance? Also, can you link to papers of yours that you cite?

I never got any PM from you. Maybe you forgot to put a stamp on it? A pre-print of the paper is online here:

http://arxiv.org/abs/0707.0401

 

[ttn]

I just quoted to you Bell's own words which refute what you claim here.

No, they don't. I don't think you understood either what I wrote, or what Bell wrote.

Besides in your article you state clearly that Bell's definition of local causality does not work in a local deterministic theory.

No, I don't say that at all. (See what I meant just above...) I say that if somebody refuses to accept the possibility of an irreducibly stochastic theory -- i.e., if they assume that determinism is true, such that stochasticity already implies incompleteness of the descriptions -- then they will think (erroneously) there is some kind of problem with the formulation. But that's their problem (indeed, your problem, since this seems to be your view!) not Bell's.

As to your other question, about what it means for two probabilities to be different... what kind of answer are you looking for?

 

[ttn]

This is a very simple question. Why don't you answer what P(b_1 |B_3 , b_2 ) \neq P (b_1 |B_3 ) implies. I'm not asking you to define the symbols. What does it mean for the LHS to be different from the RHS in the above "definition" of local causality.

You say local causality implies P(b_1 |B_3 , b_2 ) = P (b_1 |B_3 ). So I'm simply asking you what P(b_1 |B_3 , b_2 ) \neq P (b_1 |B_3 ) imples? Very simple question.

Oh, now I get the question. I thought you were asking what it *meant*, but your just trying to get me to say that P(b_1 |B_3 , b_2 ) \neq P (b_1 |B_3 ) implies non-locality.

Yes, it does. P(b_1 |B_3 , b_2 ) \neq P (b_1 |B_3 ) implies nonlocality.

But you have to read and understand and remember the words -- in particular that B_3 denotes a complete description of the physical state of a certain spacetime region, that b_2 has to live in a certain spacetime region (and can't be just any old extra thing you want to throw in), and that the P's are the fundamental dynamical probabilities assigned by some physical theory (as opposed to the kinds of probabilities that are based on ignorance of various things, etc.). If you actually hold all this in mind, it's trivial to see why all your examples, with the balls in the urns and whatnot, don't show what you think they show. Seriously, you have to actually slow down and read and process Bell's formulation. Let it marinade. (Sorry, I'm watching American Idol in the background.) Understand and appreciate what he's doing. Bell is not a dummy and he didn't formulate "locality" in a way that would diagnose, as nonlocal, trivial cases of correlation-without-causation like the ones you bring up. If you think it's so easy to refute -- if you think Bell is a dummy -- it only shows that you haven't taken the time to understand and appreciate what he accomplished.

Here, I'll put it as a challenge. State clearly, for your balls and urns or whatever example you want, what b_1, b_2, and B_3 are. Convince yourself and me that these satisfy all the conditions Bell laid down. (So, for example, oh, i dunno, B_3 better not turn out to be something like "what somebody who doesn't know the color of the first ball pulled knows about the state of the urn", and b_2 better not turn out to be in the past of b_1 rather than at spacelike separation and also outside the future light cone of region 3.) Then see if you still think there is some counter-example to Bell's formulation here.

 

[billschnieder]

No, they don't. I don't think you understood either what I wrote, or what Bell wrote.

Bell's words are clear as to what he meant, I'm not even interpreting his words, I quote them to you verbatim. You haven't provided any quote to support your claim just a pronouncement without evidence that I'm wrong.

No, I don't say that at all. (See what I meant just above...) I say that if somebody refuses to accept the possibility of an irreducibly stochastic theory -- i.e., if they assume that determinism is true, such that stochasticity already implies incompleteness of the descriptions -- then they will think (erroneously) there is some kind of problem with the formulation. But that's their problem (indeed, your problem, since this seems to be your view!) not Bell's.

Now these are your words which you are now trying to undo:

Of course, if one insists that any stochastic theory is ipso facto a stand-in for some (perhaps unknown) underlying deterministic theory (with the probabilities in the stochastic theory thus resulting not from indeterminism in nature, but from our ignorance), Bell’s locality concept would cease to work.

(1) A *deterministic local hidden variable theory* which attempts to complete QM, is in fact making the *assumption* that the stochastic properties of QM simply arise due to incompleteness of QM, and such incompleteness can be completed by a "more complete specification". Now read Bell's original paper, excepts of which I posted above which clearly state this.
(2) It makes no sense for a *deterministic local hidden variable theory* to allow for the possibility of an irreducable stochastic theory, which is completely contrary to the concept of a *deterministic local hidden variable theory*.

THEREFORE, if you *assume a deterministic local hidden variable theory*, your statement implies that Bell's locality concept would cease to work in the narrow confines of your assumption. Maybe you misspoke in the article but this is clearly the meaning conveyed by the text.

 

[billschnieder]

Yes, it does. P(b_1 |B_3 , b_2 ) \neq P (b_1 |B_3 ) implies nonlocality.

To be more precise then you are saying the above implies non-local causality. What is causing what in the above? Is it your claim that b_1 and b_2 are simultaneous? My next question would be for you to define what you understand by "cause".

But you have to read and understand and remember the words -- in particular that B_3 denotes a complete description of the physical state of a certain spacetime region, that b_2 has to live in a certain spacetime region (and can't be just any old extra thing you want to throw in), and that the P's are the fundamental dynamical probabilities assigned by some physical theory (as opposed to the kinds of probabilities that are based on ignorance of various things, etc.).

Please define what you mean by fundamental dynamical probabilities.

If you actually hold all this in mind, it's trivial to see why all your examples, with the balls in the urns and whatnot, don't show what you think they show.

I do not believe that anyone who understands probability theory can hold all of those things in their mind while being intellectually honest as will soon be evident.

Seriously, you have to actually slow down and read and process Bell's formulation.

I hope you will be patient enough to go through the process with me and we'll see in the end who is right and who has no clue what they are saying. This is my challenge, answer the questions I have given above.

Here, I'll put it as a challenge. ... better not turn out to be something like "what somebody who doesn't know the color of the first ball pulled knows about the state of the urn".

I'm happy you are posting this challenge because now it turns out the issue is about the meaning of probability. So let us start there. I will provide defintion of what probability means, and you will provide yours. then I will provide my definition of "cause" and you will provide yours and then we can discuss who is being consistent and who is not. I'm also happy that you like the urn example because we can use it to illustrate our meanings of probability. Feel free to do so.

So here is my definition of "probability":

a probability is a theoretical construct, which is assigned to represent a state of knowledge, or calculated from other probabilities according to the rules of probability theory. A frequency is a property of the real world, which is measured or estimated.​

And my definition of "cause":

To say "C" (a cause) causes "E" (an effect) means that whenever C occurs, then E follows. Therefore we can not say "C" causes "E" if the two events are simultaneous. Similarly if "E" occurs before "C", then "C" can not be the cause of "E"​

I'll wait for your definitions.

 

[billschnieder]

While waiting for your definitions I thought I should also point out the following mathematical contradictions. NOTE, the following is simply a mathematics exercise, no physics whatever, but it clearly shows the problem Bell proponents are still unable to see:

Consider the CHSH inequality:

|E(a)E(b) - E(a)E(c)| + |E(d)E(b) + E(d)E(c)| ≤ 2, where E(a), E(b), E(c), E(d) ∈ [−1,1]

This inequality is violated IFF

(1) |E(a)E(b) - E(a)E(c)| + |E(d)E(b) + E(d)E(c)| > 2

We are interested to understand the mathematical properties of the 4 terms E(a), E(b), E(c), E(d) when this violation happens

From (1) we have via factorization

(2) |E(a)||E(b) - E(c)| + |E(d)||E(b) + E(c)| > 2

However, since E(a), E(b), E(c), E(d) ∈ [−1,1], it follows that
|E(b) - E(c)| ≤ 2 and |E(b) + E(c)| ≤ 2

Let us consider the different possible extremes of the values for E(b) and E(c).

If E(b) = E(c) then |E(a)||E(b) - E(c)| = 0 and |E(d)||E(b) + E(c)| must be greater than 2 for equation (1) to hold. But we know that |E(b) + E(c)| ≤ 2 which means |E(d)| must be greater than 2 which is impossible given that E(d) ∈ [−1,1].

If E(b) = -E(c) then |E(a)||E(b) + E(c)| = 0 and |E(a)||E(b) - E(c)| must be greater than 2 for equation (1) to hold. But we know that |E(b) - E(c)| ≤ 2 which means |E(a)| must be greater than 2 which is impossible given that E(a) ∈ [−1,1].

Therefore (1) is mathematically impossible. It is not possible mathematically to violate the CHSH inequality even before we start talking about any physics and what the terms might mean in any physical situation. This is the simple fact that Bell proponents are blind to. I challenge anyone to find values for E(a), E(b), E(c), E(d) ∈ [−1,1] that violate the above inequality from any source whatsover using any means whatsoever. You can even assume that E(a) are averages over many runs or whatever you like.

 

[ttn]

To be more precise then you are saying the above implies non-local causality. What is causing what in the above?

I say explicitly in my papers that you can't say, merely from the failure of this condition, what is causing what. You just know that there is some nonlocality somewhere.

Is it your claim that b_1 and b_2 are simultaneous?

No.

My next question would be for you to define what you understand by "cause".

It's increasingly clear with every question that you haven't read or processed what Bell wrote, or what I've written about what he wrote. I'm not going to play your games if you won't do your homework first.

Please define what you mean by fundamental dynamical probabilities.

That's explained in my papers. It is really simple (though I'm sure this won't satisfy you): it means the probabilities that some candidate fundamental theory attributes to an event.

I do not believe that anyone who understands probability theory can hold all of those things in their mind while being intellectually honest as will soon be evident.


I hope you will be patient enough to go through the process with me and we'll see in the end who is right and who has no clue what they are saying. This is my challenge, answer the questions I have given above.

Right, so accuse me of being intellectually dishonest, and then literally in the next sentence ask me to please be patient enough to answer all your questions (the ones you have because you won't read or can't understand things that you've been referred to). No thanks.

(Your def'n of "probability" is inapppropriate in this context, as I've explained. And your definition of "cause" smuggles in the presupposition of determinism, which is a problem for the reasons I've explained.)

Now I give up.

 

[billschnieder]

That's explained in my papers. It is really simple (though I'm sure this won't satisfy you): it means the probabilities that some candidate fundamental theory attributes to an event.

That is an incomplete definition. What does probability mean in that phrase, that was my question. Define probability.

Right, so accuse me of being intellectually dishonest,

No I'm saying *I* will have to be intellectually dishonest to believe all the things you want me to believe at the same time, in other words, that you do not understand probability theory. Prove me wrong by defining the terms I asked.

(Your def'n of "probability" is inapppropriate in this context, as I've explained. And your definition of "cause" smuggles in the presupposition of determinism, which is a problem for the reasons I've explained.)

You don't have to agree with my definitions but I've clearly explained to you what *I* mean when *I* say "cause", and "probability". You haven't provided any alternate definitions of your own which you think are more appropriate.

After your article "Against Realism" in explained that many people arguing about Bell do not know what "realism" means, I would have thought you would understand the importance of clear definitions of terms. Once, you provide your definitions it would become evident that you do not know what you are talking about. All your claims about having explained things clearly in your articles, when you don't even have consistent definitions of terms will become evident.

I'm still waiting for your definitions for "probabilities" and "cause".

 

[ttn]

I'm still waiting for your definitions for "probabilities" and "cause".

Sorry, I'm really done. You'll have to get the answers you seek from my papers, or better, Bell's. ("La Nouvelle Cuisine" is particularly strongly recommended.) It's just frankly no fun talking with you.

 

[billschnieder]

Sorry, I'm really done. You'll have to get the answers you seek from my papers, or better, Bell's. ("La Nouvelle Cuisine" is particularly strongly recommended.) It's just frankly no fun talking with you.

So you are unable to define here what you mean by "probability" and "cause". Now hopefully you can point exactly to somewhere else where they are defined the way you like. Please provide a reference to a book, or article and specify a page number and paragraph where those terms are defined in a way you approve. This is a simple request. Simply saying, "read all my papers" or "read La Nouvelle Cuisine" would not cut it. Provide a specific location where the definition can be found.

Getting to the truth is not always fun if you are on the wrong side. This is not an entertainment exercise.