Is action at a distance possible as envisaged by the EPR Paradox.

[DevilsAvocado]

the issue is interaction.

(Let’s skip "the moon" for awhile ) If we assume that nothing exist without interaction, and BB is correct.

How can anything start to interact (exist) if it has nothing to interact with??


(I have to go, there’s real thunderstorm outside...)

 

[RUTA]

Okay! Now I’m making a fool of myself... You are taking "the moon" literally??

I don't know of any limit to the size of something that can be "screened off" in principle. In practice? Well, that's another matter

 

[RUTA]

But who cares about macroscopic objects that interact with nothing.

Exactly! What does that even mean in an empirical enterprise such as physics?

The question of Einstein was simple: "Do you think that the moon is not there when you do not look at it?" (quoted from memory). Too many people have deduced from this question and similar ones that Einstein believed in naive realism (e.g., as in the theories of de Broglie and Bohm). IF Einstein believed in HVs, those would not have let one predict values on pairs of conjugate variables in usual coordinates. But if one knows what Einstein was making fun of (see for instance his correspondence with Born (where Born gives to the reader his opinion , but clearly from those letters, Born never understood the completeness issue, in the EPR paper or the way Einstein stated it) and also the books of Jammer and Fine, there is little that I know of about progress that he or Shrödinger would have made in that direction, assuming any progress was made for one of them. Unfortunately, the search for such HVs, or the effort to prove them illusory has been pushed aside by a rear-guard fight on naive HVs of the kind that most physicists had dismissed much before 1964, and even before 1955 when Einstein died (I have posted elsewhere a few positive effects that I know of Bell's work, as well as some other negative comments). How do you guys/girls do: I type slowly (thanks god some will say) and will need to stop as my fingers hurt.

I would love to hear Einstein's thoughts about the situation now, given the vast experimental evidence in support of QM over "Einsteinian reality."

 

[charlylebeaugosse]

I would love to hear Einstein's thoughts about the situation now, given the vast experimental evidence in support of QM over "Einsteinian reality."

What evidence, or rather what do yu mean is proven, especially over "Einsteinian reality". Most of what one reads as "Einsteinian reality" is false attribution (like doing as if the EPR paper had his imprimatur, if not represented the way he thought) or lack of comprehension. As I have developed in previous posts, on the basis of writings of Einstein, Fine, and Jammer, Einstein was not a naive realist, at least after 1927, and in fact provided the first (only so far) proof of non-realism in 1931 with Tolman and Podolsky. He considered that as long as one uses the classical observables, the UP is here to stay (DrChinese would write HUP, but I have advocated using UP before since the first general version was not under the hand pf Heisenberg, as far as I know: in fact Jean-Marc Levy-Leblond would explain to us that we should rather said Indeterminacy principle that one could represent for short as IP). Please attack on more precise statements that I have made, mostly earlier today, if you disagree on the fact that legend and and not history serves as the major way one now believe Einstein thought of all matters that concern QM (the subject he most thought about did he say at least once). The experimentalists such as Aspect who think having proved Einstein wrong have generally miss-represented his thinking about the matters relevant to said experiments. Honesty has not been the trademark of such attacks. I can document, but prefer sticking to physics. I was myself abused and went into this field (one of my youth goals) because I felt that non-locality was too cool, too green would they say in the 5th element (a fun movie filled with serious non-local events). Well, reading the original sources and the history books (be it as sources of originals citations) I realized that dishonesty had taken command of the subject most central to physics: Quantum Mechanics. So please, look into what I clam and negate with basis if needed. I may be wrong thinking I was wrong in the first place (but I read a lot to see the light, starting with Fine's book although for now I only by his analysis and history part: no idea about his prism nor his philosophy, although I saw him being VERY sharp).

 

[RUTA]

What evidence, or rather what do yu mean is proven, especially over "Einsteinian reality". Most of what one reads as "Einsteinian reality" is false attribution (like doing as if the EPR paper had his imprimatur, if not represented the way he thought) or lack of comprehension. As I have developed in previous posts, on the basis of writings of Einstein, Fine, and Jammer, Einstein was not a naive realist, at least after 1927, and in fact provided the first (only so far) proof of non-realism in 1931 with Tolman and Podolsky. He considered that as long as one uses the classical observables, the UP is here to stay (DrChinese would write HUP, but I have advocated using UP before since the first general version was not under the hand pf Heisenberg, as far as I know: in fact Jean-Marc Levy-Leblond would explain to us that we should rather said Indeterminacy principle that one could represent for short as IP). Please attack on more precise statements that I have made, mostly earlier today, if you disagree on the fact that legend and and not history serves as the major way one now believe Einstein thought of all matters that concern QM (the subject he most thought about did he say at least once). The experimentalists such as Aspect who think having proved Einstein wrong have generally miss-represented his thinking about the matters relevant to said experiments. Honesty has not been the trademark of such attacks. I can document, but prefer sticking to physics. I was myself abused and went into this field (one of my youth goals) because I felt that non-locality was too cool, too green would they say in the 5th element (a fun movie filled with serious non-local events). Well, reading the original sources and the history books (be it as sources of originals citations) I realized that dishonesty had taken command of the subject most central to physics: Quantum Mechanics. So please, look into what I clam and negate with basis if needed. I may be wrong thinking I was wrong in the first place (but I read a lot to see the light, starting with Fine's book although for now I only by his analysis and history part: no idea about his prism nor his philosophy, although I saw him being VERY sharp).

I'm using the term "Einsteinian reality" generically to mean "local and separable." I have no idea what he would say, I wouldn't even begin to argue that.

 

[charlylebeaugosse]

(responding to Devils...) I don't know of any limit to the size of something that can be "screened off" in principle. In practice? Well, that's another matter

Ruta, Devilsavocado, DrChinese, etc. (people of good will and not realist): it seems to me that

- 1) Phase one is making sure of non-existence of classical realism at the miscroscopic scale.

- 2) Next, deciding if quantal realism holds true or not (I mean here a form of realism defended by some advocates of CQT (Consistent Quantum Theory))

- 3)Later then seeing the issue about the moon.

- 4) Even later, or any time but this is hard, checking if HVs compatible with QM (a theory that would permit exact predictions, but never on both members of conjugate pairs) can be constructed, if they would help, and how to do it: at some point we will have to negate or support scientifically the belief of Heisenberg and others that "one should not look for other variables". As long as one speaks of the HVs of de Broglie, Bohm, or Bell, no question left in my mind, but for some that respect the UP, I do not know if they can exist nor if they could help in anything if found. Solving that positively would be an immense achievement, but the other questions seems more in present time reach, at least the first one.

The above is a proposal for emergencies ordering (1 to 4, but I may have forgotten steps or independent questions that should belong here), but if anyone can solve the later items before the earlier ones, that is fine: I just would not like to spend time defending vague ideas about the moon while the crucial problem of local realism calls for a solution (thanks to Dr Bell planting a doubt in many minds, but as I said, if the issue can be decided by physics, it was worthwhile the big confusion about locality and the miss - attributions to Einstein starting with Bell).

 

[RUTA]

Ruta, Devilsavocado, DrChinese, etc. (people of good will and not realist): it seems to me that

Phase one is making sure of non-existence of classical realism at the miscroscopic scale.

Next, deciding if quantal realism holds true or not.

Later then seeing the issue about the moon.

This is a proposal for emergencies ordering, but if anyone can solve the later items before the earlier ones, that is fine: I just would not like to spend time defending vague ideas about the moon while the crucial problem of local realism calls for a solution (thanks to Dr Bell planting a doubt in many minds, but as I said, if the issue can be decided by physics, it was worthwhile the big confusion about locality and the miss - attributions to Einstein starting with Bell).

Agreed, the issue will only be decided by physics. I think the goal of threads like this isn't to decide the issue, but merely debate the possibilities.

 

[charlylebeaugosse]

I'm using the term "Einsteinian reality" generically to mean "local and separable." I have no idea what he would say, I wouldn't even begin to argue that.

Ruta, you see, "local and separable" is precise. "Einsteinian reality" to the contrary depends on what you know of Einstein's writing, and others, including misinformation propagated by people who want to show themselves better than Einstein (what would be the public value of being better than Podolsky?). Also, in "Einsteinian reality" there is reality, and while I (we?) love locality and separability, reality is for me THE ENEMY, so attributing him to the great guy does not help, except the glory of people who have greatly contributed t the confusion (see the book of Asher Peres, and let me know if he does not hint at QM being non-local, or look at the very public writings of the great Roger Penrose that deal with Bell's Theory, and let me know if he does not propagate the dark forces?). If no-locality was only defended by imbeciles, we would not have to worry. Bell was a crypto-realist who revealed himself later as realist, this pushed hard the fact that QM is non-local.
Now: look at Bell 1964, and check by yourself that Bell claim that QM had been proved to be non-local so that one should try HVs to restore its locality: see the 2 first sentences.
Now the Bell Theorem as state it there is about predictive HVs compatible with QM, but isn't non-locality suggested as a way out? So please be precise here, e.g., if you have students (graduate or else) or read papers that have some success. Lack of truth or words that can help that have to chased actively.

 

[charlylebeaugosse]

Agreed, the issue will only be decided by physics. I think the goal of threads like this isn't to decide the issue, but merely debate the possibilities.

I won't mind being part of a group that solves a question (but perhaps I do not care about having publication with one or few authors). In fact, groups formed over the www might be the best chance of progress in some very hard questions, including precise questions in physics. This being said, when you write "the issue will only be decided by physics", do you mean that you are sure that physics can solve that or that if there is a solution it can only come from physics (sorry for the mad precision, but I have spent a few years in pure math before coming back to physics (but not mathematical physics that is for me a branch of applied math that requires talents that I do not have)). I do believe though that it would be useful to all be as precise as we can, be it only to avoid unnecessary confusions and disputes due only to misunderstanding. I feel some convergence, despite apparent divergences of writing: we might mostly need to adjust vocabulary.

 

[RUTA]

This being said, when you write "the issue will only be decided by physics", do you mean that you are sure that physics can solve that or that if there is a solution it can only come from physics.

Both: I'm confident that physics can solve this and the solution will, ipso facto, come from physics.

 

[unusualname]

I won't mind being part of a group that solves a question (but perhaps I do not care about having publication with one or few authors). In fact, groups formed over the www might be the best chance of progress in some very hard questions, including precise questions in physics. This being said, when you write "the issue will only be decided by physics", do you mean that you are sure that physics can solve that or that if there is a solution it can only come from physics (sorry for the mad precision, but I have spent a few years in pure math before coming back to physics (but not mathematical physics that is for me a branch of applied math that requires talents that I do not have)). I do believe though that it would be useful to all be as precise as we can, be it only to avoid unnecessary confusions and disputes due only to misunderstanding. I feel some convergence, despite apparent divergences of writing: we might mostly need to adjust vocabulary.

Hardly any remaining questions in physics have been or will be solved by people who aren't competent in mathematical physics. At the very least you should be aware of the existing models and their dificiencies before attempting to "solve" any questions.

Bell's result attracts amateurs and crackpots since it can be understood without a huge investment of effort into learning real mathematics and physics. Unfortunately, the resulting discussions are mostly an amusing illustration of mental difficulties rather than anything worthwhile.

 

[charlylebeaugosse]

Hardly any remaining questions in physics have been or will be solved by people who aren't competent in mathematical physics. At the very least you should be aware of the existing models and their dificiencies before attempting to "solve" any questions.

Bell's result attracts amateurs and crackpots since it can be understood without a huge investment of effort into learning real mathematics and physics. Unfortunately, the resulting discussions are mostly an amusing illustration of mental difficulties rather than anything worthwhile.

TYVM. I do have over 120 papers, collaboration with some of leading figures in math, a long past in physics as well, and about 80 patents. Yet I have seen in these pages, besides stupid remarks, posts by a few people who either are smart professionals, or that we miss in the labs. Most of the stupid hings about Bell Theory were written by pros: I have no much patience with those papers, and even less with non-professional writings, except if it of very good quality. The same applies to delayed choice, delayed erasure, interferences in general, but of course Bell and related matter is the main crackpots attractor. Yet I think it worthwhile to see if collective thinking can lead us to otherwise hard or get results. I have collaborated all my life and am curious of the value of large scale collaboration (on a single well defined theory problem). We'll see...

 

[JesseM]

As I have developed in previous posts, on the basis of writings of Einstein, Fine, and Jammer, Einstein was not a naive realist, at least after 1927, and in fact provided the first (only so far) proof of non-realism in 1931 with Tolman and Podolsky.

When you say he was not a "naive realist", is that in contrast with some other form of realism, or do you think he was not a realist of any kind? And you mention Jammer, is that Max Jammer's book "Einstein and Religion" or some other publication? (if it is that book, do you know what pages discuss Einstein's views on realism?) Also, what publications of Einstein and Fine are you referring to?

 

[RUTA]

TYVM. I do have over 120 papers, collaboration with some of leading figures in math, a long past in physics as well, and about 80 patents. Yet I have seen in these pages, besides stupid remarks, posts by a few people who either are smart professionals, or that we miss in the labs. Most of the stupid hings about Bell Theory were written by pros: I have no much patience with those papers, and even less with non-professional writings, except if it of very good quality. The same applies to delayed choice, delayed erasure, interferences in general, but of course Bell and related matter is the main crackpots attractor. Yet I think it worthwhile to see if collective thinking can lead us to otherwise hard or get results. I have collaborated all my life and am curious of the value of large scale collaboration (on a single well defined theory problem). We'll see...

My PhD was in general relativity, but I've been working in the foundations community since 1994. It's just my impression (and I'm a nobody ... ), but I haven't seen any real collaboration, per se. There are some general "groups," the largest seems to be Many Worlds, then the Bohmians, followed by variations on backwards causation, but within any "group" it's pretty much a collection of independent researchers -- nothing unified like research in string theory. I don't know the social dynamics, all I can report is what I perceive. The point is, I wouldn't hold out much hope of generating a large scale unified assault on this problem

Let me ask you, what approach are you looking to advance?

 

[charlylebeaugosse]

My PhD was in general relativity, but I've been working in the foundations community since 1994. It's just my impression (and I'm a nobody ... ), but I haven't seen any real collaboration, per se. There are some general "groups," the largest seems to be Many Worlds, then the Bohmians, followed by variations on backwards causation, but within any "group" it's pretty much a collection of independent researchers -- nothing unified like research in string theory. I don't know the social dynamics, all I can report is what I perceive. The point is, I wouldn't hold out much hope of generating a large scale unified assault on this problem

Let me ask you, what approach are you looking to advance?

I think about some experiments (thought and/or real) that may help establish or help significantly the non-realist point of view (to be co-authored by all people whose contribution is used, more or less, and in anonymous form if people insist-in which case the PF pseudos would be acknowledged as representing contributors. I have several lines of ideas in mind, probably some based on mistakes of mine. I would propose a few from one line to start with. OR I would start with a less ambitious project such as the analysis of Wheeler Delay type experiments (and then would hope to have Cthugha for instance on board - I am relatively new to QM (6 years) where I hope to bring my experience in qualitative methods acquired in non-linear dynamics (mostly, both math and physics): I am bad at what most pros are good at and better in arcane methods and view points. In fact, what I am most interested in is try this idea that the www can help create big collective brains. This is more important that the first question(s) that would be solved as then, many other could follow. The main reason to have soon a few teams on a few subjects would be to explore what rules work best. We could perhaps even begin with two threads on the same basic subject (two questions about said subject), one with full freedom, one where a subgroup would soon form a sort of police on what is relevant or not and taking care of re-launching the life when needed. That might be more fun and contribution than solving one physics question (of course, not great for people still looking for a job, or a Ph. D.). Maybe I am turning into a relatively young crackpot after all.

 

[DevilsAvocado]

Ruta, Devilsavocado, DrChinese, etc. (people of good will and not realist):

...

I won't mind being part of a group that solves a question

I think RUTA and unusualname has some really good points here. Yes, it would be marvelous to put together a group and solve some real mysteries in science, but I think that is to underestimate the problem, to say at least... Right now, in this very thread, we are experiencing "one" who has wander off into "The Hazy Swamp of Crackpots of No Return", believing he has solved "everything" alone, using nothing else but probability. While the real probability for doing just that is not good, not good at all, at least if you are alone...

Fundamentally, EPR-Bell is not a question (or fight) between locality/realism/FTL/LHVT, etc – it’s much bigger than that (and I think RUTA agrees?). The genius(es) that solves this question are going to present the next paradigm in physics, where QM + GR + Gravity = True, and perhaps even TOE.

To me EPR-Bell is a parallel to the Michelson–Morley experiment, and what followed after that, but even more complicated (to solve at least).

I don’t think this is something one solves in a discussion over internet. It’s just too big.

Furthermore, I think it’s a big mistake to make any hasty conclusions on what is right or wrong, if you plan to solve this... no offense, but talking about "dark forces" and "THE ENEMY" and so on, can’t be fruitful before we know for sure, can it?

Also if we look back, it all becomes a little 'amusing'. For many nonlocality is repulsive, unnatural, etc, but it was not that long ago that one of the brightest minds in history, Isaac Newton, found his own law of gravity and the notion of "action at a distance" deeply uncomfortable, so uncomfortable that he made a reservation in 1692:

That one body may act upon another at a distance through a vacuum without the mediation of anything else, by and through which their action and force may be conveyed from one another, is to me so great an absurdity that, I believe, no man who has in philosophic matters a competent faculty of thinking could ever fall into it.

This is funny! And future generations will of course laugh at us and our current 'limitations'!

And it also fun to discuss this and learn more, so let’s continue!

 

[charlylebeaugosse]

When you say he was not a "naive realist", is that in contrast with some other form of realism, or do you think he was not a realist of any kind? And you mention Jammer, is that Max Jammer's book "Einstein and Religion" or some other publication? (if it is that book, do you know what pages discuss Einstein's views on realism?) Also, what publications of Einstein and Fine are you referring to?

Max Jammer indeed, but the book is (in Amazon):

The Philosophy of Quantum Mechanics: The Interpretations of Quantum Mechanics in Historical Perspective by Max Jammer (Hardcover - June 1974)
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Fines book is:
The Shaky Game (Science and Its Conceptual Foundations series) by Arthur Fine (Paperback - Dec. 15, 1996)
Buy new: $25.00 $22.28

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Get it by Friday, Aug. 13 if you order in the next 22 hours and choose one-day shipping.
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There is from there an easy way to get to original writings by Einstein.

As for Einstein's realism, he did believe that the Moon did not even need apes I would bet.
But the real issue, I think, is realism at the microscopic level. But I would not like to be considered as an unconditional supporter of A.E., since I am not. I am only bothered a lot by all lies and false info that have lead us to a situation where more physicist (in or close to QM) would relinquish locality and not realism at the miscroscopic level.
For me, it is like having Algebra under the control of bandits, or biology controlled by the creationists. I have posted a lot on my views of Einstein's realism so I would rather stop on that for a while. Also, I only add my physicist's sensitivity to real work done by Jammer and Fine (see also the conference where Fine (?), Jammer, Peirls and Rosen contributed for the 50th anniversary of EPR and other paper here and there, mostly the correspondence of Einstein (mainly with Born, but there are other gems), the Schlipp book, and one pocket book on AE's views on the world where there is more politics than physics but some good pieces anyway) and as much reading of Einstein as I could put my hands on. But as I do not read German, I loose lots of first hand material.

 

[DevilsAvocado]

I don't know of any limit to the size of something that can be "screened off" in principle. In practice? Well, that's another matter

Okay, thanks. I think I have a question regarding "screened off" + BB + CMB... but I must think it over. Hope to see you tomorrow!

 

[charlylebeaugosse]

Furthermore, I think it’s a big mistake to make any hasty conclusions on what is right or wrong, if you plan to solve this... no offense, but talking about "dark forces" and "THE ENEMY" and so on, can’t be fruitful before we know for sure, can it?

Also if we look back, it all becomes a little 'amusing'. For many nonlocality is repulsive, unnatural, etc, but it was not that long ago that one of the brightest minds in history, Isaac Newton, found his own law of gravity and the notion of "action at a distance" deeply uncomfortable, so uncomfortable that he made a reservation in 1692:



This is funny! And future generations will of course laugh at us and our current 'limitations'!

And it also fun to discuss this and learn more, so let’s continue!

Action at a distance was very odd indeed, like the lack of realism (even if only microscopic) is now. That is what I have read too. I do believe that the reason why locality is more abandoned than realism by professionals of quantum physics is as follows:
Realism is coded in our brain for millions of years, or at least 100,00 years or about, while
the discovery of finite speed of light is very new.
Modern physics has been marked by the destruction of credos (simultaneity, continuity, parity, etc...). Destroying realism by physicist argument, for good (and not as a new credo) would be great. Perhaps more modest goals should be tried first.

BTW: Someone wrote about the need of mathematical physicists in order to solve any big problem. What have they brought to physics that is acknowledged by the rest of the physicists? I have great respect for them, some of the best ones are my friends, but their contributions are more considered as math. There is a funny story about Simon and Feynman where RF asked BS "who are you young man" to which "BS" answered "I am BS", to which it was replied: and "what is your field?". adn BS comments: can you imagine that F did not know about my work? i.e., for me: BS did not even understand that RF couldn't care less about the type of things he was doing.

I hope that mathematical Physicist will have some recognition as physicists some day. Some of them have deep physical intuition beside tremendous technical power, but so far, ...

The power of collective thinking is worthwhile trying if the crackpots are kept away de-facto by ignoring them and if we get some of the people I have seen in my short experience with Phys-Forum. 99% of chance of failure, perhaps 99.99%. But solving one problem would be great, and perhaps those interested in the experience should find 1 or 2 problems that are "a bit" less ambitious than the issue of microscopic realism. Anyway I have spent some of my life trying to solve questions with ow odd for solutions, which helped me solving lesser questions. I would not advise a grad student to spend (much) time on that, but think of the reward if we even only begin to understand how to solve hard question as an open group we can get in other threads to advertise those with questions being attacked, can't we? Now for the "enemy" and the "dark forces", it is not defined by the position but by the use (knowing the truth) or not of false information. I was a realist most of my life and a supporter of non-locality when I heard about the subject and talked to some of the lead authors: this is what brought me into the field. I would not consider badly people with provably wrong positions if I am convinced that they do that by ignorance. Anyway, I do not expect everyone to be exited by trying, and I am prepared to failure, as when I tackle "hard problems" (or very hard ones). I do not expect people to spend much energy before some hope of success becomes a bit more reasonable.

 

[billschnieder]

I gave you an abstract list. No mention of anything such as trial. No mention of anything such a physical process. I asked you to give me the probability of one of the entries from the list, and you told me it was impossible despite the fact that this is what is done everyday in your favorite frequentist approach to probability.

Not if we are excluding "finite frequentism", which I already told you I was doing.

So you you are saying if you were not exclusing "finite frequentism" you will be able to give an answer? So, you are effectively picking and triming your definition of probability for argumentative purposes as more of your statements will show below. You are not being serious.

Does your list of four give us enough information to know the frequency of ++ in the limit as the sample size goes to infinity?

Bah! This list is the entire context of the question! The list is the population. True probability of the (++) in the list, is the relative frequency of (++) in the list. This is the frequentist approach, which you now want to abandon in order to stay afloat.

When ever you say the probability of Heads and Tails is 0.5 you are doing it, whenever you say the probability of one face of a die is 1/6, you are doing the exact same thing you now claim is impossible. Go figure.

[bu]No, in those cases I am just using the physical symmetry of the object being flipped/rolled to make a theoretical prediction about what the limit frequency would be[/u], perhaps along with the knowledge that empirical tests do show each option occurs with about equal frequency in large samples

Hehe! Do you know of anybody who has ever performed an infinite number of coin or die tosses? I think not. So you can not know what the limit will be as the number of tosses tends toward infinity. And since you have continued to insist on your ridiculous idea that the "true probability" must be defined as the limit as the number of trials tends towards infinity, the above response is very telling.

Furthermore, did you really think I will not notice the fact that you have now abandoned your favorite frequentist approach and now you are using the bayesian approach (see underlined text above) to decide that the P(Heads) = 0.5. If you can use symmetry of the coin to decide that P(Heads) = 0.5, why couldn't you also use symmetry of my abstract list to decide that P(++) is 1/4?? I'm sure if I looked, I will not need to look hard to find a post in which you wrote a list not very different from mine and also wrote P(++) to be 1/4 or similar, without having performed an infinite number of damned "trials". So as I mentioned earlier, you are not being serious, just finding anything you can hang-on to, even if it means contradicting yourself.

I already gave you the answer which is 1/4.

Yes, and that answer is incorrect if we are talking about the "limit frequentist probability", as I already made clear I was doing.

You say it is impossible to calculate an answer, then when I give you the answer, you then say the answer is wrong. How do you know it is wrong, if you are unable to calculate the correct one? You are way off base, and the answer is correct in ANY probabilistic approach.

The relative frequency of (++) in my list is 1/4. Since my list is the population, P(++)=1/4 you do not need any trials to determine this.

Note that the wikipedia article says "close to the expected value", not "exactly equal to the expected value".

An "expectation value" like E(a,b) would be interpreted in frequentist terms as the expected average result in the limit as the number of trials (on a run with detector settings a,b) goes to infinity

Um, how do you figure? The two statements of mine are entirely compatible, obviously you are misunderstanding something here

It is quite clear from the two statements that if average from the law of large numbers is close to but not equal to the true expectation value, it can not be the definition of the expectation value! Which one is it? The definition of the expectation value can not at the same time be only approximately equal to it!?

Yes, and with that context there isn't enough information to estimate the limit frequentist probability which is the only notion of probability I want to use
...

You can visualize it by thinking that if you would randomly pick an entry from the the list I gave you

Well, that's an entirely separate question, because then you are dealing with a process that can repeatedly pick entries "randomly" from the list for an arbitrarily large number of trials. But you didn't say anything about picking randomly from the list, you just presented a list of results and asked what P(++) was.

It is not an entirely separate question. I did not mention any trials in my question. But you have stated that the only notion of probability you want to use is the "limit frequentist probability", even though initially you just said "frequentist", but if you want to stick to that limited approach, which is only interested in "trials", you could still have provided an answer to the question by imagining what the limit will be if you actually randomly picked items from my list. Is it your claim that this is also impossible?

Secondly, despite my repeated correction of your false statements that I presented "results" or "trials", you keep saying it. You quickly jumped to claim I never mentioned trials, yet in the next sentence, you say I presented "results", even though I never characterized the list as such, and corrected your attempts to characterize it as such multiple times! You are not being honest.

 

[billschnieder]

Failing to do the specific thing I said it should do, yes.

According to you, the wikipedia article is wrong. Why don't you correct it. It is obvious you are the one who is way off base and you know it. All the grandstanding is just a way to stay afloat, not a serious argument agains the well accepted meaning of expectation value.

Wikipedia: http://en.wikipedia.org/wiki/Mean

In statistics, mean has two related meanings:
* the arithmetic mean (and is distinguished from the geometric mean or harmonic mean).
* the expected value of a random variable, which is also called the population mean.

Wikipedia: http://en.wikipedia.org/wiki/Expected_value

In probability theory and statistics, the expected value (or expectation value, or mathematical expectation, or mean, or first moment) of a random variable is the integral of the random variable with respect to its probability measure.

That you continue to pursue this strange objection to standard mathematics is very telling.

The law of large numbers says if you would randomly pick a large number of pairs from our given abstract list, the average value will get close to the true expectation value as the number of pairs you pick tends towards infinity.

Again, you said nothing about "randomly picking" from a list, you just gave a list itself and asked for the probabilities of one entry on that list.

Yes, that is exactly what I did, and you answered that it was impossible to do because you wanted to use ONLY a probability approach that involved "trials". So I said, if you really were serious about using ONLY a probability approach that involved "trials", you would have imagined randomly picking an infinite number of pairs from the given list, and still be able to give an answer very close to the "true expectation" I wanted which is simply the relative frequency of (++) in my list, obtained without any trials. You do the same thing for dice and coins and you have done the same thing in you famous scratch-lotto examples, but when doing it here would have proven fatal to your line of argument, you balked.

Well, excuse me for thinking your question was supposed to have some relation to the topic we were discussing, namely Bell's theorem.

While discussing Bell's INEQUALITIES, Not Bell's theorem which we haven't discussed at all, you claimed, and continue to claim that Bell's equation (2) is not a standard mathematical definition for the expectation value of a paired product. So we went down this rabbit trail in order to force you to admit that you are wrong, or be humiliated in the process of trying to defend the ridiculous claim. And you know very well that once you admit that you were wrong, you have no valid response to the rest of my argment, so you are standing your ground. Even though you know fully well that what you are arguing is wrong and borderline dishonest.

Only if you assume by symmetry that it's a "fair" die or coin, in which case you have a reasonable theoretical basis for believing the "limit frequency" of each result would appear just as often as every other one. If you had an irregularly-shaped coin (say, one that had been partially melted) it wouldn't be very reasonable to just assume the limit frequency of "heads" is 0.5.

I gave you the list [(++), (+-), (-+), (--)] and you claimed it was impossible to calculate the probability of (++) in the list. So had I given you [(++), (++), (+-), (-+), (--)] and asked the same question, you will still have claimed it was impossible. But anyone who has ever heard anything about probability will immediately realize that each item in the list occurs once and since there are 4 items, P(++) must be 1/4 in the list, for the second one, P(++) will be 2/5. I haven't done anything here other than use the symmetry which is present in the given list to calculate the probability. But you already said those values are wrong, which is very telling.

In statistics, if you are given the population, you can calculate the true probabilities without any trials. It is done every day in the frequentist approach, which you claim to understand!

Not in the "limit frequentist" approach where we are talking about frequencies in the limit as number of times the population is sampled approaches infinity (unless we make some auxiliary assumptions about how the population is being sampled, like the assumption we're using a process which has an equal probability of picking any member of the population)

Oh so now you are saying if given a population from which you can easiliy calculate relative frequencies, you will still not be able to use your favorite "limit frequentist" approach to obtain estimates of true probabilities because the process used to sample the population might not be fair. Wow! You have really outdone yourself. If the "limit frequentist" approach is this useless, how come you stick to it, if not just for argumentation purposes?

There is also such a thing as "finite frequentism" which just says if you have a finite set of N trials, and a given result occurred on m of those trials, then the "probability" is automatically defined as m/N

I have already explained to you multiple times that the list I gave you in the question is an abstract list, not a "result" of "trials". So what you say above is a straw-man argument. And you agreed that I never said anything about trials. So for the last time, be honest about what I asked.

 

[billschnieder]

Bell's theorem, and your odd criticisms of it which seem to presuppose a notion of probability different from the limit frequentist notion

We have been discussing Bell's inequality, NOT Bell's theorem. There is a difference.

So can you please just answer the question: are you using (or are you willing to use for the sake of this discussion) the limit frequentist notion of probability, where "probability" is just the frequency in the limit as the number of trials goes to infinity?

No! I am not willing to pick and choose the definition of probability for argumentation purposes. First you said it was ONLY the "frequentist" view you wanted. Now it is ONLY a particular variant of frequentism that you want, except when it involves coins and dice, you really use the "bayesian" view. I'm not interested in that type of pointless excercise.

No, the "standard mathematical definition" of an expectation value involves only the variable whose value you want to find the expectation value for, in this case the product of the two measurement results.
...
In the standard definition would give us:
$$\sum{i=1}^{N}R_i P(R_i)$$

Wikipedia: http://en.wikipedia.org/wiki/Expected_value

In general if X is a random variable defined on a probability space (Ω, Σ, P), then the expected value of X, denoted by E(X), <X>, $$\overline{x}$$, or E(X), is defined as

$$E(X) = \int_{\Omega}XdP$$

...

The expected value of an arbitrary function of X, g(X), with respect to the probability density function f(x) is given by the inner product of f and g:
$$E(g(X))\int_{-\infty}^{\infty} g(X)f(X)$$

You are way off base. Bell's equation two is the standard mathematical definition.The only difference between Bell's equation (2) and the last equation above, is that the symbols:
X = λ
g(X) = g(λ) = A(a,λ)*B(b,λ)
f(X) = ρ(λ)

Bell is not trying to redefine anything. He is simply using the standard mathematical definition of expectation value for the paired product. Word games will not save you here.

For the last time, you are the one misrepresenting Bell:

$$E(a,b) = \int \rho (\lambda ) A(a,\lambda )B(b,\lambda ) d\lambda$$

Note the dλ, at the end of the expression! There is no expression in Bell's paper as the following:

E(a,b) = (+1)*P(detector with setting a gets result +1, detector with setting b gets result +1) + (-1)*P(detector with setting a gets result +1, detector with setting b gets result -1) + (-1)*P(detector with setting a gets result -1, detector with setting b gets result +1) + (+1)*P(detector with setting a gets result -1, detector with setting b gets result -1)

Your claim that such an expression is missing, because Bell was simplifying for physicists is a cop-out. Furthermore, there is no mention of "limit frequentist", let alone "frequentist" in Bell's paper. You are invoking those terms now only to escape humiliation.

Hopefully you at least agree that in the limit as the number of trials becomes large, the expression for the empirical average below should approach my definition

Your so called definition is not a definition, but an approximation of a certain expectation value, which is different from the one used by Bell. Bell's expectation value is obtained by integrating over all λ as equation (2) of his paper clearly shows. Yours is discretely added over outcomes without regard for λ. It is a non-starter deliberately designed to avoid the pitfall of the uniform ρ(λ) requirement which you know is fatal to your argument.

In that case, does your whole argument hinge on the fact that you think Bell's equation (2) was giving an alternate definition of "expectation value", one which would actually differ from the one I give?

You wish. But NO! I have explained my argument very clearly in point-by-point form and in detail. It is your argument which hinges on the hope that Bell's equation (2) is something other than the application of the standard mathematical definition of the expectation value of a paired product to his situation of interest.

 

[billschnieder]

True! Bell was writing for an audience of physicists, who would understand that he didn't mean for (2) to indicate he was totally rewriting the standard meaning of "expectation value"

You have provided no proof that equation (2) from Bell's paper is not simply an application of the standard definition of expectation value of a paired product of functions of λ as the wikipedia article clearly shows:

The expected value of an arbitrary function of X, g(X), with respect to the probability density function f(x) is given by the inner product of f and g:
$$E(g(X))\int_{-\infty}^{\infty} g(X)f(X)$$

Please answer my question about whether you are willing to just use the "limit frequentist" notion of probability in this discussion--and if you are, do you see why with this understanding it doesn't make sense to say "ρ(λ) in the sample is not significantly different from ρ(λ) in the population" when you are really just talking about the frequencies of different values of λi in the finite sample, not the frequencies that would be found if we took an infinite sample under the same conditions?

Translation: My argument doesn't make sense in the alternate universe in which your limited frequentist view is what I'm using to make my argument?! Is that the best you can do? I'm done with this rubbish!

Your answer only seems to address the part that your ability to do this "resorting" doesn't guarantee the value of λ was really the same for all three (and you basically seemed to agree but say it doesn't matter), but you didn't address the point that even the "hidden triples" may be different than the imaginary triples you created via resorting. For example, suppose after resorting we find the 10th iteration of the first run is a=+1,b=-1, the 10th iteration of the second run is b=-1,c=-1 and the 10th iteration of the third is a=+1,c=-1. Then we are free to self-consistently imagine that each run had the same triple for iteration #10, namely a=+1,b=-1,c=-1. However, in reality this might not be the case--for example, the 10th iteration of the first run might actually have been generated from the triple a=+1,b=-1,c=+1. So, the statistics of the imaginary triples you come up with after resorting might not match the statistics of actual triples on each run, or on all three runs combined.

You missed this part

If ρ(λ) in the sample is not significantly different from ρ(λ) in the population, then the distribution of the outcomes will not be significantly different. However, just because the distribution of the outcomes is not significantly different is not proof that ρ(λ) is the same. It is a necessary but not a sufficient condition as you still must be able to resort the data.

 

[zonde]

I would love to hear Einstein's thoughts about the situation now, given the vast experimental evidence in support of QM over "Einsteinian reality."

Einstein was die hard empiricist. Take for example this remark from his http://www.marxists.org/reference/subject/philosophy/works/ge/einstein.htm" :
"To his [Margenau] Sec. I: "Einstein's position . . . contains features of rationalism and extreme empiricism..." This remark is entirely correct."

You should not expect that Einstein as extreme empiricist would have left matters of interpretation of experiments into hands of Aspect and Zeilinger and would not have formulated his own viewpoint.

 

[zonde]

I love Einstein, he’s my hero. The question is – do you think that he would have rejected Bell's Theorem and EPR-Bell experiments?

Let me give longer quote from Einstein http://www.marxists.org/reference/subject/philosophy/works/ge/einstein.htm" :
"One arrives at very implausible theoretical conceptions, if one attempts to maintain the thesis that the statistical quantum theory is in principle capable of producing a complete description of an individual physical system. On the other hand, those difficulties of theoretical interpretation disappear, if one views the quantum-mechanical description as the description of ensembles of systems.

I reached this conclusion as the result of quite different types of considerations. I am convinced that everyone who will take the trouble to carry through such reflections conscientiously will find himself finally driven to this interpretation of quantum-theoretical description (the Psi-function is to be understood as the description not of a single system but of an ensemble of systems).

Roughly stated the conclusion is this: Within the framework of statistical quantum theory there is no such thing as a complete description of the individual system. More cautiously it might be put as follows: The attempt to conceive the quantum-theoretical description as the complete description of the individual systems leads to unnatural theoretical interpretations, which become immediately unnecessary if one accepts the interpretation that the description refers to ensembles of systems and not to individual systems. In that case the whole "egg-walking" performed in order to avoid the "physically real" becomes superfluous. There exists, however, a simple psychological reason for the fact that this most nearly obvious interpretation is being shunned. For if the statistical quantum theory does not pretend to describe the individual system (and its development in time) completely, it appears unavoidable to look elsewhere for a complete description of the individual system; in doing so it would be clear from the very beginning that the elements of such a description are not contained within the conceptual scheme of the statistical quantum theory. With this one would admit that, in principle, this scheme could not serve as the basis of theoretical physics. Assuming the success of efforts to accomplish a complete physical description, the statistical quantum theory would, within the framework of future physics, take an approximately analogous position to the statistical mechanics within the framework of classical mechanics. I am rather firmly convinced that the development of theoretical physics will be of this type; but the path will be lengthy and difficult."


As you can see direction taken by EPR-Bell experiments is exactly the one Einstein was talking (dreaming) about. They try to investigate behavior of individual particles.
However their interpretation is mainly based on assumption that QM is valid description of individual systems contrary to what Einstein believed.

So I think that Einstein would have discarded without regret any restrictions placed by orthodox QM on local realistic interpretation.

Restriction I am talking about is that the same measurement settings at both sites should give the same outcome with probability of 1.

And how does the Ensemble Interpretation explain if we decide to have very long intervals between every entangled pair in EPR-Bell experiments, let’s say weeks or months? Where is the "Global RAM" situated in a case like this? That fixes the experimentally proved QM statistics, for the whole 'spread out' ensemble??

If we view Ensemble Interpretation as physically realistic interpretation and not as some other metaphysical interpretation we of course can not talk about some "Global RAM".
We can talk only about some "local RAM" that is justifiable by physical dynamics inside equipment used in experiments.

If we decide to have very long intervals between every entangled pair we should expect complete decoherence of entanglement.

 

[ThomasT]

... trying to talk reasonable to Bill is a waste of time. He lives in his own little bubble; firmly convinced he represents the "universe", when the fact is that he’s totally lost and totally alone in his "reasoning".

DA, I don't know if you know this, but billschnieder is a working scientist. I don't think that either DrC or JesseM are.

I admire your honest efforts to understand the conundra surrounding Bell's theorem. I think that billschnieder, and JesseM, and DrC, and all of us are interested in understanding this stuff. And, honestly, I don't think that any of us have a definitive way of expressing anything about the nature of reality.

billschnieder's expertise and knowledge exceeds yours, and I think you should take that into account, just as you apparently do wrt JesseM and DrC.

These are not easy considerations. If they were, then notable physicists and mathematicians wouldn't still be arguing about them. And, while I appreciate your input and your apparent interest, I think you should focus on the precise arguments being made. I'm not sure they're good arguments. Maybe you can sort it out, and clarify it, for all of us. But, please, focus on the the arguments. They're there to be refuted. So, refute them, or agree with them, or just say that you don't understand them -- and ask some questions. But, please, you and nismaratwork, stop with the 'fanboy' stuff.

 

[ThomasT]

I'm with DrC, I also don't believe "the Moon is there when nobody looks." By "when nobody looks" I mean "when not interacting with anything."

And when is it ever the case that something is not interacting with anything?

In Relational Blockworld, if the entity "isn't there," i.e., is "screened off," it doesn't exist at all. So, the answer to your question is that there is no Moon to wonder.

Come on RUTA, are you saying that your Relational Blockworld is a description of the physical reality?

 

[RUTA]

And when is it ever the case that something is not interacting with anything?

When it exhibits wave-like behavior. Once it interacts with its environment, it acquires definite position (particle-like behavior) per decoherence.

RBW is not the only interpretation in which "non-interacting" means "non-existent." I got that idea from Bohr, Ulfbeck and Mottelson. Zeilinger has also been credited with that claim regarding photons.

Come on RUTA, are you saying that your Relational Blockworld is a description of the physical reality?

Absolutely, RBW is an ontological interpretation of QM. What in particular strikes you as unreasonable about this ontology? The non-existence of non-interacting entities (manifested as nonseparability of the experimental equipment)? Or, blockworld?

 

[DrChinese]

I'm done with this rubbish!

I can only hope...

An interesting note for those still reading: GHZ theorem, another no-go theorem for local realism, shows that EVERY trial will have QM and local realism giving opposite predictions. I.e.

LR=+1, +1, +1, ...
QM=-1, -1, -1, ...

Guess which is experimentally demonstrated? No statistics required! No ensemble interpretation required!

 

[zonde]

An interesting note for those still reading: GHZ theorem, another no-go theorem for local realism, shows that EVERY trial will have QM and local realism giving opposite predictions. I.e.

LR=+1, +1, +1, ...
QM=-1, -1, -1, ...

Guess which is experimentally demonstrated? No statistics required! No ensemble interpretation required!

Are you familiar with GHZ experiments at all?

Anyways
from this paper - http://prl.aps.org/abstract/PRL/v91/i18/e180401"

"In conclusion, we have demonstrated the statistical and nonstatistical conflicts between QM and LR in fourphoton GHZ entanglement. However, it is worth noting that, as for all existing photonic tests of LR, we also had to invoke the fair sampling hypothesis due to the very low detection efficiency in our experiment."

Guess what? Fair sampling hypothesis does not quite hang together with ensemble interpretation.

 

[charlylebeaugosse]

I can only hope...

An interesting note for those still reading: GHZ theorem, another no-go theorem for local realism, shows that EVERY trial will have QM and local realism giving opposite predictions. I.e.

LR=+1, +1, +1, ...
QM=-1, -1, -1, ...

Guess which is experimentally demonstrated? No statistics required! No ensemble interpretation required!

Not many places where the use of locality in the usual derivation of GHZ is explicitly explained: do you know any, some, many?
I am talking about Mermin's version using 3 particles, not the original 4 particles configuration.
The use of realism is obvious, of course.
(I will not tell who (among the great expert) thought that locality was not used in GHZ.)

By the way, GHZ is often called: Bell Theorem without Inequality (I mentioned that before as one reason why one should not equate Bell Inequalities
(a form of Boole's inequalities, as pointed out long ago by Itamar Pitowsky in several papers getting deeper and deeper into that matter - this is related to earlier work by Fine) and Bell Theorem as was claimed in a link related to a dispute around billschnieder.

 

[DrChinese]

Are you familiar with GHZ experiments at all?

Anyways
from this paper - http://prl.aps.org/abstract/PRL/v91/i18/e180401"

"In conclusion, we have demonstrated the statistical and nonstatistical conflicts between QM and LR in fourphoton GHZ entanglement. However, it is worth noting that, as for all existing photonic tests of LR, we also had to invoke the fair sampling hypothesis due to the very low detection efficiency in our experiment."

Guess what? Fair sampling hypothesis does not quite hang together with ensemble interpretation.

What does Fair Sampling have to do with my comment? If I predict a -1 every time, and you predict +1 every time, and it always comes up -1... Then it doesn't really much matter how often that occurs.

As I have said a million times all science involves the fair sampling assumption. There is nothing special about GHZ or Bell tests in that regard.

And as I have also said too many times to count: if the GHZ result is due to some unknown weird bias... what is the dataset we are sampling that produces such a result? I would truly LOVE to see you present that one! Let's see:

LR=+1, +1, +1, ...
QM=-1, -1, -1, ...
Actual sample=Oops!

 

[DrChinese]

Actually, I had the predictions of LR and QM reversed in my little sample. It should be more like:

QM=+1, +1, +1, ...
LR=-1, -1, -1, ...

See this article from Zeilinger and Pan:

Multi-Photon Entanglement and Quantum Non-Locality (2002)

"Comparing the results in Fig. 16.7, we therefore conclude that our experimental results verify the quantum prediction while they contradict the local-realism prediction by over 8standard deviations; there is no local hidden-variable model which is capable of describing our experimental results."

 

[JesseM]

DA, I don't know if you know this, but billschnieder is a working scientist. I don't think that either DrC or JesseM are.

In what field? Search arxiv.org for author "Bill Schnieder" turns up no results. Likewise, a general google search for "Bill Schnieder" and "abstract" doesn't seem to turn up any papers with abstracts (and most papers these days have at least the abstracts online). Can you link to any work by him?

 

[JesseM]

So you you are saying if you were not exclusing "finite frequentism" you will be able to give an answer?

Only if your list was understood as a statistical sample (either a sample drawn from a larger population, or the results of a series of trials), or if you add some conditions like that an experimenter is picking a sample from the "population" represented by the list. In the first case we could use finite frequentism to give probabilities, in the second case we could even use "limit frequentism" if we added the condition that the experimenter was picking randomly using a method that was equally probable (in the limit frequentist sense) to give any entry on the list.

On the other hand, I've never heard of any authority on statistics talking about determining a "probability" from an "abstract list" which is not interpreted as either a sample or a population. If you want to continue with the "abstract list" argument, please find a source where some authority does something like this.

So, you are effectively picking and triming your definition of probability for argumentative purposes as more of your statements will show below.

I had already stated clearly that I was only interested in talking about probabilities defined in the "frequency in limit as number of trials/sample size goes to infinity", since I think these are the only types of probabilities relevant to Bell's derivation. Again, are you willing to at least consider whether Bell's derivation might make sense (and not have the problems of limited applicability you argue for) when his probabilities are interpreted in these terms, or are you basically refusing to consider the possibility of an interpretation of the paper different from your own, suggesting you are not really interested in trying to understand Bell's argument in its own terms but just in making a lawyer-like rhetorical case against him?

Does your list of four give us enough information to know the frequency of ++ in the limit as the sample size goes to infinity?

Bah! This list is the entire context of the question! The list is the population.

You didn't specify that when you first posted the list, and given that all your previous examples of lists of + and -'s involved a series of trials from a run of a given experiment, there was no reason for me to think the list was intended to be something totally different.

True probability of the (++) in the list, is the relative frequency of (++) in the list. This is the frequentist approach, which you now want to abandon in order to stay afloat.

This is just a ridiculous criticism, Bill. I have always been using what I now call the "limit frequentist approach" to avoid your quibbling about finite frequentism (most scientists nowadays also just talk about 'frequentism' when they mean limit frequentism), you can see that in every post where I talked about frequentism I explained I was talking about the limit as the number of trials approached infinity. Go on, find a single post of mine where my own use of probability involved anything other than limit frequentist probabilities; you won't be able to, showing that your "you now want to abandon" comment is either based on totally misreading what I've been saying all along, or knowingly misrepresenting it.

Hehe! Do you know of anybody who has ever performed an infinite number of coin or die tosses? I think not. So you can not know what the limit will be as the number of tosses tends toward infinity.

No, you can never know with absolute certainty what the "limit frequentist" probabilities are, but you can have a high degree of confidence that they are close to some value based on both theoretical arguments (like the symmetry of fair coins) and empirical averages with large numbers of trials. In any case, Bell's derivation does not require us to actually know what the limit frequentist probabilities of anything are, it just assumes they have some objective values (encapsulated in a function like ρ(λ)) and that these objective values have certain properties (like ρ(λ) being independent of the detector settings), and derives inequalities for the expectation values (themselves just weighted sums of objective probabilities for different combinations of results) based on that. If all of Bell's theoretical assumptions about the objective probabilities were correct, then given the law of large numbers it should be very unlikely that the empirical averages for an experiment with a great many trials would violate the inequality if the "true" expectation values (determined by limit frequentist probabilities) obey it.

Furthermore, did you really think I will not notice the fact that you have now abandoned your favorite frequentist approach and now you are using the bayesian approach (see underlined text above) to decide that the P(Heads) = 0.5.

No. First of all, I'm not saying that the P(heads) is actually guaranteed to equal 0.5, just that it's physically plausible that it would be--that's my hypothesis about the true probability, as distinguished from the true probability itself. A theorist who uses limit frequentist definitions when making theoretical arguments about probabilities (like Bell's) is free to use Bayesian methods when trying to come up with an empirical estimate about what the objective probabilities are. But a Bayesian would say the "probability" is just your best estimate (a more 'subjective' definition of the meaning of probability), while a limit frequentist would distinguish between the estimate and the "true" probability.

Second of all, I'm not using the symmetry of the sample space as a basis for my estimate that P(heads)=0.5, I'm using the actual physical symmetry of the coin itself. If I had an irregular coin with more weight on one side than the other I wouldn't make this estimate, despite the fact that the sample space still contains only two possible outcomes so a Bayesian (or Jaynesian) might say the principle of indifference demands our prior distribution assign each outcome an equal probability.

In any case Bell's derivation does not require any estimates of the true limit frequentist probabilities given by ρ(λ). Only once we have derived the inequality do we have to worry about empirical measurements, and here a limit frequentist can just argue that by the law of large numbers, our sample averages are unlikely to differ significantly from the "true" expectation values (determined by the 'true' limit frequentist probabilities) if the number of trials is large enough.

I'm sure if I looked, I will not need to look hard to find a post in which you wrote a list not very different from mine and also wrote P(++) to be 1/4 or similar, without having performed an infinite number of damned "trials".

Nope, you won't be able to, I have been quite consistent about understanding probabilities in terms of the limit frequentist approach, since some of my earliest discussions with you--for example see post #91 from the 'Understanding Bell's Logic' thread, posted back in June, where I said:

It's still not clear what you mean by "the marginal probability of successful treatment". Do you agree that ideally "probability" can be defined by picking some experimental conditions you're repeating for each subject, and then allowing the number of subjects/trials to go to infinity (this is the frequentist interpretation of probability, its major rival being the Bayesian interpretation--see the article Frequentists and Bayesians) ... Do you think there are situations where even hypothetically it doesn't make sense to talk about repetition under the same experimental conditions (so even a hypothetical 'God' would not be able to define 'probability' in this way?) If so, perhaps you'd better give me your own definition of what you even mean by the word "probability", if you're not using the frequentist interpretation that I use.

Even earlier than my discussions with you, in January of 2009 I explained to a different poster that I understood derivations of Bell inequalities to involve frequentist probabilities defined in the limit as the number of trials goes to infinity, see this post:

I didn't say anything about you knowing the objective facts. Again, the frequentist idea is to imagine a God's-eye perspective of all the facts, and knowing the causal relations between the facts, figure out what the statistics would look like for a very large number of trials.

...

If you believe there are objective facts in each trial, even if you don't know them, then it should be possible to map any statement about subjective probabilities into a statement about what this imaginary godlike observer would see in the statistics over many trials--do you disagree? For example, suppose there is an urn with two red balls and one white ball, and the experiment on each trial is to pick two balls in succession (without replacing the first one before picking the second), and noting the color of each one. If I open my hand and see that the first one I picked was red, and then I look at the closed fist containing the other and guess if it'll be red or white, do you agree that I should conclude P(second will be white | first was red) = 1/2? If you agree, then it shouldn't be too hard to understand how this can be mapped directly to a statement about the statistics as seen by the imaginary godlike observer. On each trial, this imaginary observer already knows the color of the ball in my fist before I open it, of course. However, if this observer looks at a near-infinite number of trials of this kind, and then looks at the subset of all these trials where I saw that the first ball was red, do you agree that within this subset, on about half these trials it'll be true that the ball in my other hand was white? (and that by the law of large numbers, as the number of trials goes to infinity the ratio should approach precisely 1/2?)

If you agree with both these statements, then it shouldn't be hard to see how any statement about subjective probabilities in an objective universe should be mappable to a statement about the statistics seen by a hypothetical godlike observer in a large number of trials. If you think there could be any exceptions--objectively true statements of probability which cannot be mapped in this way--then please give an example. It would be pretty earth-shattering if you could, because the frequentist interpretation of probabilities is very mainstream, I'm sure you could find explanations of probability in terms of the statistics over many trials in virtually any introductory statistics textbook.

So I think it's safe to say I have been quite consistent in my understanding of what "probability" means in the context of Bell's derivation, and any notion of yours that I've been waffling is just another example of your leaping to an uncharitable conclusion when you see any ambiguity in the way I have expressed myself.

You say it is impossible to calculate an answer, then when I give you the answer, you then say the answer is wrong. How do you know it is wrong, if you are unable to calculate the correct one?

What I said in my original response (post #1249) was "No, you can't calculate the probability just from the information provided, not if we are talking about objective frequentist probabilities rather than subjective estimates." If we have some additional information about what the list represents, like that it is a population and we have an experimenter picking a random sample from the population (using a method that we are told has an equal probability of picking any of the four entries, with 'probability' understood in the limit frequentist sense), then we can certainly calculate the probability. I already made this point in post #1277:

Again, you said nothing about "randomly picking" from a list, you just gave a list itself and asked for the probabilities of one entry on that list. If you want to add a new condition about "randomly picking", with "randomly" meaning that you have an equal limit frequentist probability of picking any of the four entries on the list, then in that case of course I agree that P(++)=1/4...well duuuuh! But that wasn't the question you asked.

Now, can we get back to discussing Bell's theorem, and not some silly irrelevant example you came up with to prove I "don't understand probability"?

Note that the wikipedia article says "close to the expected value", not "exactly equal to the expected value".

An "expectation value" like E(a,b) would be interpreted in frequentist terms as the expected average result in the limit as the number of trials (on a run with detector settings a,b) goes to infinity

Um, how do you figure? The two statements of mine are entirely compatible, obviously you are misunderstanding something here

It is quite clear from the two statements that if average from the law of large numbers is close to but not equal to the true expectation value, it can not be the definition of the expectation value! Which one is it? The definition of the expectation value can not at the same time be only approximately equal to it!?

I don't understand the phrase "average from the law of large numbers". The average from any finite number of trials N can be different from the true expectation value, no matter how large of a finite number N we pick. However, the law of large numbers says that in the limit as N approaches infinity, the average approaches the expectation value with probability 1. Another way of putting this is that if we pick some specific real number epsilon between 0 and 1, then no matter how small of an epsilon we pick, the probability that the empirical average (the 'sample mean') differs from the expectation value by an amount greater than or equal to epsilon should become smaller and smaller with greater values of N, approaching 0 in the limit as N approaches infinity. If you're familiar with the official calculus definition of a "limit" in terms of the "epsilon-delta" definition (see here), this should look pretty familiar.

You can visualize it by thinking that if you would randomly pick an entry from the the list I gave you

Well, that's an entirely separate question, because then you are dealing with a process that can repeatedly pick entries "randomly" from the list for an arbitrarily large number of trials. But you didn't say anything about picking randomly from the list, you just presented a list of results and asked what P(++) was.

It is not an entirely separate question.

It is because my original objection was that your problem didn't give enough "information" for any definite answer, and here you are providing more information (the idea that we are randomly picking entries from the list and want to know the probability of picking a given entry).

I did not mention any trials in my question. But you have stated that the only notion of probability you want to use is the "limit frequentist probability", even though initially you just said "frequentist", but if you want to stick to that limited approach, which is only interested in "trials", you could still have provided an answer to the question by imagining what the limit will be if you actually randomly picked items from my list. Is it your claim that this is also impossible?

No, I already told you at the end of post #1277 that this was fine, although you would have to specify that you were picking in a way that gave an equal probability (in limit frequentist terms) of selecting any of the four items on the list, since it's perfectly possible to conceive a method of selection that would make some entries on the list more probable than others (for example, start at the top of the list, if it's 'heads' pick the top entry and if it's 'tails' move to the next entry and repeat this procedure until you either get a heads or get to the last entry on the list...this method gives a probability of 1/2 for picking the first entry, 1/4 for picking the second, 1/8 for picking the third and 1/8 for picking the fourth).

Secondly, despite my repeated correction of your false statements that I presented "results" or "trials", you keep saying it. You quickly jumped to claim I never mentioned trials, yet in the next sentence, you say I presented "results", even though I never characterized the list as such, and corrected your attempts to characterize it as such multiple times! You are not being honest.

Yes bill, every time I colloquially use a word like "result" in a way that could possibly be interpreted as a mischaracterization of something you have said, it proves I am a devious snake who is "not being honest" rather than just that I am an ordinary human who sometimes speaks a bit sloppily. Here I did not intend "result" to explicitly mean the results of a series of trials, it could be any list of data (including a list representing a population of possible 'results' that an experimenter might get when picking randomly from the population)