User talk:Chjoaygame/archive 2

possible conversation
If you feel able to consider the possibility that you might be mistaken, and could learn why, I am willing to converse quietly here.Chjoaygame (talk) 17:01, 7 October 2015 (UTC)


 * Perhaps. Still in public, but more quietly. Even without asking, whether you feel able to consider the possibility that you might be mistaken, and could learn why. :-) Boris Tsirelson (talk) 17:24, 7 October 2015 (UTC)


 * Let's see how we go.Chjoaygame (talk) 19:19, 7 October 2015 (UTC)


 * Who is supposed to make the first move? Boris Tsirelson (talk) 20:12, 7 October 2015 (UTC)

You.Chjoaygame (talk) 20:41, 7 October 2015 (UTC) Or me, if you wish.Chjoaygame (talk) 00:26, 8 October 2015 (UTC)

conversation
A short preamble and the first question.

By "metaphysics" I mean discussion of large classes of hypothetical physical theories selected by general principles or/and specific predictions (formulated in empirical terms). For example: all theories that respect Einstein locality. Another example: all theories that predict CHSH=sqrt2 in some (at least one) experimental setup.

Let us leave aside the quantum mechanics and concentrate on "metaphysics".

——
Question 1. What is the main distinction between
 * (a) Einstein's "independent existence of spatially separated subsystems" + Einstein locality, and
 * (b) "local realism" in the sense of the theory of Bell(-type) inequalities?

Is it the same class of theories? Is class (a) contained in class (b)? Is class (b) contained in class (a)?

Boris Tsirelson (talk) 05:53, 8 October 2015 (UTC)

Answer 1. Einstein's concepts (a) are about real physical bodies in time and space, not necessarily related to Bell(-type) reasoning. Bell(-type) inequalities refer to concepts based on Bell(-type) reasoning.

Class (a) is defined as not necessarily contained in class (b). Since I hold that Bell(-type) reasoning is logically invalid, I hold that it refers to nothing at all, neither physical realities nor logical objects; in other words, class (b) is the empty set. In particular, class (a) refers to physical realities, and therefore does not contain the empty set. Class (b) is not contained in class (a), and class (a) is not contained in class (b).Chjoaygame (talk) 10:15, 8 October 2015 (UTC)

——
Question 2. How do you interpret Einstein's "independent existence of spatially separated subsystems"? What does it mean (or imply) in terms of probabilities? Boris Tsirelson (talk) 10:55, 8 October 2015 (UTC)

---

Answer 2. Its meaning is primarily in terms of real physical objects, whatever they might be. To calculate probabilities one would need to specify a physical model of those real physical objects.

I think it may save some time if I go a bit further at this point. I think you seem to be looking at probabilistic reasoning about independent events? Please correct me if not.

I think, as I believe you do too, that 'independent' has at least two meanings, and probably more.

Spatial separation entails that neither can, in the relevant circumstances, exert causal efficacy on the other. It does not entail that they do not have a common cause in their common past. One can consider two cases: (1) they have in their common past a causal efficacious event that has significantly affected relevant elements of their present separate states; (2) any past common causal event has negligible effect on relevant elements of their present separate states.

Probabilistic independence, in this context, means that there is no significant causal link between them, neither directly between them nor through a common past cause. (There are, I suppose, other ways of characterizing probabilistic independence; perhaps these may become relevant, but for the present, I am not intending to refer to them.)

It may help now if I cite Kolmogorov, page 3 of the 1956 (second) English edition.


 * § 2. The Relation to Experimental Data4


 * We apply the theory of probability to the actual world of experiments in the following manner:
 * 1) There is assumed a complex of conditions, ₡, which allows of any number of repetitions.
 * 2) We study a definite set of events which could take place as a result of the establishment of the conditions ₡. In individual cases where the conditions are realized, the events occur, gener­ally, in diferent ways. Let E be the set of all possible variants ξ1, ξ2, ••• of the outcome of the given events. Some of these vari­ants might in general not occur. We include in set E all the vari­ants which we regard a priori as possible.
 * 3) If the variant of the events which has actually ocurred upon realization of conditions ₡ belongs to the set A (defined in any way), then we say that the event A has taken place.


 * [example omitted from this quote]


 * 4) Under certain conditions, which we shall not discuss here, we may assume that to an event A which may or may not occur under conditions ₡, is assigned a real number P(A) which has the following characteristics :


 * [details omitted]
 * ____________________________________
 * 4 The reader who is interested in the purely mathematical development of the theory only, need not read this section, since the work following it is based only upon the axioms in § 1 and makes no use of the present discussion. Here we limit ourseves to a simple explanation of how the axioms of the theory of probability arose and disregard the deep philosophical dissertations on the concept of probability in the experimental world. In establishing the premises necessary for the applicability of the theory of probability to the world of actual events, the author has used, in large measure, the work of R. v. Mises, [1] pp. 21-27.

Kolmogorov does not pursue this topic, but he does recognize that it is needed for applications of probability theory. Kolmogorov's "complex of conditions, ₡", is made explicit in the notation of Jeffreys who insists, very wisely and rightly in my mind, on writing every, yes every, probability in the form P(A|₡). Quick and clever time-savers like to avoid this by taking it that the |₡ notation is necessary only for what they regard as 'conditional probabilities'. Of course, as usual, the quick and clever time-savers create muddle thereby. In new-fangled terminology that I don't like, Kolmogorov's ₡ is called the "context". The word 'context' is here used more or less metaphorically and perhaps that is partly why it can lead to confusion.

In this word usage, however, "contextuality" is written by Kolmogorov into the very foundations of mainstream orthodox routine conventional probability theory, too often left only implicit (and apparently forgotten) by quick and clever time-savers. To them, who forget Kolmogorov and Jeffreys, contextuality, as they like to call it, seems suprising or exotic. Bell is a perfect example of a quick and clever time-saver, creating an industry for his fellows.

Jeffreys was savagely opposed by the very powerful R.A. Fisher, and many writers in his days therefore did not follow him. I do not recall Jeffreys' quoting Kolmogorov. I don't know if Jeffreys spoke Russian.

Jeffreys includes conditions in the notion ₡. Jaynes is a follower of Jeffreys. I think it important, for every student of science, to read, mark, learn, and inwardly digest Jaynes' Probability Theory: The Logic of Science.

Here what matters is that the separate photons of Bell experiments have a common cause in their common past and so they are not probabilistically independent. Consequently, Bell has misused the formula
 * $$P(\overrightarrow a \overrightarrow b) = \int d\lambda \, \rho (\lambda) A (\overrightarrow a, \lambda) B (\overrightarrow b, \lambda)$$

which is for probabilistically independent events. This error of Bell would make one fail an undergraduate answer that made it. It is gross and unforgivable. One can be forgiven for thinking that perhaps Bell was playing a joke when he wrote that formula.

Some years ago, the error was pointed out by Jaynes. Jaynes pointed out the correct formula:


 * $$P(AB|ab) = \int P (AB|ab\lambda) \, P (\lambda|ab) \, d\lambda.$$

Most regrettably, Jaynes failed to go ahead and use this formula to actually derive an inequality. It was left to Khrennikov to do that. You are far more familiar with the literature than I am, and I guess perhaps you may recall papers that show that Khrennikov is not the first to do this?

I have reluctantly to observe that you have used the pejorative term 'excuse' to refer to this situation. I can only ask you to reconsider your judgement, taking into account a fair reading of Kolmogorov, Jeffreys, and Jaynes. Khrennikov cites Kolmogorov, and uses the same ideas and reasoning as Jeffreys and Jaynes, in his own notation, which in some ways I find not as handy and clear as that of Jeffreys and Jaynes. He fails to refer to Jeffreys, perhaps because of historical events.

Perhaps that is enough for the moment. Since I have here written a fair amount, I would expect you might take your time to respond.Chjoaygame (talk) 13:51, 8 October 2015 (UTC)

——
No, not a long time. Do not think that shared randomness is for me a revelation. I understand it, believe me. That is, I understand (like you) what it can do; but also (unlike you), what it cannot do.

Question 3. How is "independent existence" related to "local elements of reality"? Boris Tsirelson (talk) 15:07, 8 October 2015 (UTC)

Before I come to answer 3, I wish to go further with your response to my Answer 2. I find it hard to see how you can understand "shared randomness" when you have not responded to the error in Bell's reasoning, and you seem to carry on as if there was no such error, and you have not dealt with the principle that a statement of the data on which it is based must accompany every statement of a probability. I have no doubt that you have far greater mental facility than I in understanding these things, but I find a glaring discrepancy when I see you dismiss in a moment what you have previously called "contextualism". You have dismissed it in a cavalier way, not offering any reason for doing so. Your further phrase "shared randomness" is vague and you have not offered to me any support for it. At present it seems like another name for "contextuality", but it is so vague that I don't know if that is so. If this is to be a conversation, I cannot let this pass: your just telling me I don't understand without your offering any reason. You would need to persuade me that Kolmogorov, Jeffreys, and Jaynes are somehow profoundly deceived, and that you know better than they. Since you have, till now at least, seemed to deny validity to what you call "contextuality", which they support, though not using that particular language, I have no reason to take seriously your unsupported dismissal of it. I think I would be unwise to go further in this with you until you give me a good response to this concern of mine. So, for the present, my answer to your Question 3 is on hold.Chjoaygame (talk) 16:24, 8 October 2015 (UTC)


 * No, I did not dismiss anything (yet). I really wish you to see (at last) the other side of the coin. There is a special condition, in addition to the usual probabilistic reasoning. In addition! No one is dismissing any general rule of probability. They all hold. (You'll see it ultimately, if you'll have the patience to go the whole hog.) There is an important, more special, physical input here, not contained in axioms of probability theory (since its origin is "metaphysics", not mathematics). In order to see it, please do answer my questions. They are steps toward it. (Or, alternatively, just dwell further on your one-sided view, infinitely long.) Boris Tsirelson (talk) 17:11, 8 October 2015 (UTC)


 * You ask me to trust you. You ask for my patience. You deny that you have dismissed anything. It seems you wish to take me on a course of Socratic questioning, intending to teach me metaphysics in that way, on the assumption you know it all and that eventually I will come to see that. In your favor, at least you do not dismiss the very word 'metaphysics' as many physicists seem to do. Then they trot out their own homespun amateurish wanna-be metaphysics, of course not labelling it as such.


 * Your Question 3 reads "How is "independent existence" related to "local elements of reality"?" I do not find in the EPR paper the phrase "local elements of reality". There I find the phrase "element of (the) physical reality". I think you have thereby tried to put words into EPR's mouth. Or have you invented a new concept, without offering a definition of it? Moreover, I do not find EPR referring to "independent existence". Where did this term come from? Is it an invention of yours? How do you define it? I have no strong reason to try to answer a question constructed in that way.


 * You are right to be interested in the word 'independent', but I am disappointed that you make no attempt to respond to my above-offered attempt to deal with it. You leave me with the impression, perhaps mistaken, that you are not interested in the distinction between probabilistic and serial causal (in)dependence, nor the difference between serial causal (in)dependence and common causal (in)dependence. Perhaps you are in fact so interested, but to persuade me so, you would need to do more than continue to question me by your present technique, apparently ignoring my responses as above.


 * As an act of charity, I will have a try at answering your Question 3, setting aside my reservations about it. Taking a guess at your concepts, I would say that local elements of reality are in general related to oneChjoaygame (talk) 03:41, 9 October 2015 (UTC) another by common causal antecedents and by common causal consequents, and in general are therefore not independent existents. It may be possible in particular cases to ignore such dependence, but good reason needs to be advanced for doing so in each case. They may or may not be related by serial causal (in)dependence; spatially separated local elements of reality are not related by serial causal dependence, and are related by serial causal independence. Broadly speaking, I think this view is pretty much common sense, recognized by most or all well-trained scientists. For explicitness, I will repeat my above comment, that photons that are Bell pairs are linked by common cause and are therefore very definitely not mutually independent existents; indeed they exhibit an exceptionally strong form of mutual common cause dependence.Chjoaygame (talk) 18:43, 8 October 2015 (UTC)


 * Well, it seems, the discussion approaches its end. You do not understand my terminology, and I do not understand yours. I have no idea of "serial causal (in)dependence" etc. Probably we are educated in too different cultures. Also, textology is not my hobby, and moreover, the history of science is not my hobby. When I feel that I understand the idea(s), I ignore details of the language. I do not want to bother, whether Einstein wrote "local elements of reality", or "element of (the) physical reality" etc; I guess, he also used such phrases rather interchangeably; but maybe not. Now, you ask: "independent existence", where did this term come from? Sorry, this is too much. I gave you exact Einstein's text in German, and its partial English translation, with this phrase; moreover, you found yourself another translation; did you forget it all so quickly?? It is not a small detail; for Einstein, it was the central idea!
 * Some days ago, I wrote you "Even without asking, whether you feel able to consider the possibility that you might be mistaken, and could learn why"; indeed I felt that you are not. Now I see that really, you are not. As for me, you are just finding miserable formal excuses to escape seeing the Sun. Probably you feel it different. Sorry, I did my best. Boris Tsirelson (talk) 19:12, 8 October 2015 (UTC)


 * Ah, I see you refer not to EPR 1935, which is written in English, and which I checked for the terms, and explicitly referred to. You refer to the text of Quantenmechanik und Wirklichkeit 1948, which you did indeed quote on the Hidden variable theory talk page, and indeed in German. Your quote was


 * "Wenn ich die mir bekannten physikalischen Phänomene betrachte, auch speziell diejenigen, welche durch die Quanten-Mechanik so erfolgreich erfasst werden, so finde ich doch nirgends eine Tatsache, die es mir als wahrscheinlich erscheinen lasst, dass man die Forderung II aufzugeben habe." p.323 "QUANTEN-MECHANIK UND WIRKLICHKEIT", A. Einstein 1948 "Dialectica".
 * translated thus by Irene Born


 * "when I consider the physical phenomena known to me, and especially those which are being so successfully encompassed by quantum mechanics, I still cannot find any fact anywhere which would make it appear likely that requirement II will have to be abandoned."


 * I don't see 'independent existence' there in your quote of Einstein's German. It is, however, present in what I added of Irene Born's English: "principle II, i.e. the independent existence of the real state of affairs existing in two separate parts of space Ri and Rj." Looking again at Irene Born's translation indeed I find further examples, such as "An essential aspect of this arrangement of things in physics is that they lay claim, at a certain time, to an existence independent of one another, provided these objects 'are situated in different parts of space'. Unless one makes this kind of assumption about the independence of the existence (the 'being-thus') of objects which are far apart from one another in space—which stems in the first place from everyday thinking—physical thinking in the familiar sense would not be possible." In this latter quote, Einstein has "the things in physics ... lay[ing] claim, at a certain time, to an existence independent of one another, provided these objects 'are situated in different parts of space'." As I read this, "at a certain time" rules out serial causal dependence, but allows, or even in general assumes, common causal dependence. So yes, I ought to have looked beyond the EPR 1935 paper, which I cited, for the term "independent existence". But the important thing for Bell-type reasoning is the difference between independence at a certain time in Einstein's sense, one might say ontic or bare spatio-temporal independence, and independence in the sense that Bell needs, one might say epistemic or probabilistic independence, crucially not implied in Einstein's sense, of independent existence at a certain time. Though I don't feel he is very bad at it, I find Einstein not very good at formal expression of metaphysics, not nearly as professional as Whitehead, whom I am sorry Einstein seems to have ignored. As I have previously said, I think much of the confusion was generated by the peculiar character of Niels Bohr.


 * You find me "finding miserable formal excuses". I find you dismissive of my sources in a cavalier way; for example, you write above: "When I feel that I understand the idea(s), I ignore details of the language." You do not offer any reason, and deny that you have dismissed them. As well as practically ignoring my cited sources, you seem to ignore my act of charity, answering your Question 3 even though I object to its formulation.


 * Unless more is forthcoming, I think that I am going to come away from this conversation believing that you do not distinguish, in the relevant context, between serial causal dependence and common causal dependence. You say you do not understand my terminology in that, and you seem not to wish to find common understanding and terminology, and indeed not to be interested in considering the distinction. It is indeed grounds for sorrow that we have made so little, if any, progress.Chjoaygame (talk) 22:56, 8 October 2015 (UTC)


 * For you, probably, no progress. For me, the progress is as follows. I felt that you (felt to) understand Bell's position and see his error, while I had only a slight idea of your position. Now I feel that I understand your position at least as much as you (feel to) understand Bell's, and see your error. Thus I do not bother anymore. Boris Tsirelson (talk) 05:50, 9 October 2015 (UTC)

——
It is indeed good news that you have now understood my position.

The best maths teacher I had was Arthur Erdélyi, whose undergraduate course in Schwartz distributions I was fortunate to attend. I recall how he corrected the student exercises that I handed in. He exercised an insight that made me feel he had read my mind to see just how and why I had gone wrong, and to put me right.

Since you too now have read my mind, and can see how and why I have gone astray, perhaps you may very kindly be willing to do an Erdélyi for me, and express for me, in your own words, more clearly than mine, just what is in my mind, and how and why it is mistaken, and put me right?Chjoaygame (talk) 16:40, 9 October 2015 (UTC)


 * Wow! Hardly I can be as good as Erdélyi in this job. But if you are (temporarily) ceased to be furious toward Bell and all his admirers, then I can try.
 * 1. No one says that Bell inequalities follow from probability theory. Surely not. (Otherwise they would be satisfied by the quantum theory as well.)
 * 2. No one says that independent existence of spatially separated subsystems implies their statistical independence. (Otherwise all correlations would be zero in Bell framework; but Bell inequality is CHSH ≤ 2, not CHSH = 0.)
 * 3. Einstein's "independent existence of spatially separated subsystems" is not just hand-waving; it is falsifiable, that is, it does restrict the class of physical theories. (Einstein would not attach great scientific importance то а non-scientific assumption.)
 * 4. The independent existence does not imply statistical independence in repeated experiments. In a single experiment, it does not imply deterministic outcomes of experiments. Here is what it does imply in a single experiment: there exists a, say, local state (in other words, a set of local elements of reality) such that, given the local state, the outcome of any experiment is conditionally independent of any remote data.
 * 4a. Clarification. It is not assumed that a mortal is able to know (to read, to measure) the local state. Maybe he can, maybe he cannot.
 * 4b. Clarification. In repeated experiments, statistical correlations are not "postulated out", since the local state may be statistically correlated with remote data.
 * 4c. Clarification. Why does such local state exists? Since otherwise the situation is quite "holistic" even in a single experiment.
 * 5. Now we calculate the expectation of a linear combination of products (as in CHSH, or anything like that). We first calculate the conditional expectation, given the local state; and then we average it over all local states (according to their probabilities). On the first stage of this calculation we have independence; on the second stage correlations are welcome.
 * 6. Thus, on the first stage of calculation, the expectation of each product factorizes into the product of expectations; and so, in the case of CHSH, we get $$|A_1 B_1 + A_1 B_2 + A_2 B_1 - A_2 B_2| \le |A_1| |B_1+B_2| + |A_2| |B_1-B_2| \le 2.$$
 * 6a. Clarification. But under independence correlations must vanish?? No, wait; "correlations" as physicists use this word in this context are rather expectations of products (as noted in our article on Bell inequalities); above, A1 is the conditional expectation, given the local state; generally, not 0, even if the corresponding unconditional expectation is 0.
 * 7. Now we take the unconditional expectation (of the conditional expectation calculated above). No matter what is the distribution of the local state, and the corresponding distribution of the conditional expectation, we are taking the expectation of a random variable that never leaves the interval [-2,+2]. Therefore the final result also cannot leave this interval!
 * The end. Boris Tsirelson (talk) 18:08, 9 October 2015 (UTC)
 * Oops, sorry, I forgot to say... you surely want E(AB|1,2) instead of E(A1B2), etc. But this is unproblematic. When you understand the idea in one form, you rewrite it easily in several equivalent forms. Boris Tsirelson (talk) 19:47, 9 October 2015 (UTC)
 * And, frankly, I do not think in these terms. I prefer to think in terms of a game. You may see it on my CZ article here. Or here. Then, all the metaphysics turns into such simple idea: if you need the optimal strategy for Alice and Bob, just find it, as a mathematician would find it before invention of quantum mechanics. The metaphysical assumption just says that advanced physical apparata apparatus should be useless when you need only information; either signals are allowed, or they are disallowed. Boris Tsirelson (talk) 20:31, 9 October 2015 (UTC)
 * In this or that form, anyway, the mathematics is: convex combination of factorized (rank 1, that is) correlation matrices (with elements in [-1,1]). Boris Tsirelson (talk) 05:50, 10 October 2015 (UTC)


 * Thank you for this very kind response. I should say, if one didn't previously believe in miracles, one would come to do so when Erdélyi corrected one's homework. I have previously pointed out that my mind does not work quickly. You have said a great deal here. I will need some large amount of time to study it and think it over. Please be patient in the meantime. On a different tack. As far as I know, 'apparatus' is sometimes thought of as a Latin word, and sometimes as an English one. The Latin word is masculine, of the fourth declension, plural 'apparatūs'. In a sense, the Latin word is already a sort of virtual plural, like 'furniture' and 'equipment'. In suitable circumstances, 'apparatus' (Latin singular) already means 'furniture, equipment, instruments, machines, movables' and doesn't need the plural form when many objects are involved. I guess you have been told a million times that 'apparata' would be a plural for a putative second declension neuter noun 'apparatum'. Often one can just write 'apparatus' and no one will ask if one means the singular or the plural. Or is there something else here, that I haven't imagined? Perhaps this kind of thought partly explains why my mind works slowly!  Chjoaygame (talk) 21:28, 9 October 2015 (UTC)


 * I see: Latin "apparatus" is like Russian "аппаратура" (rather than "аппарат"). Bad news... Boris Tsirelson (talk) 05:58, 10 October 2015 (UTC) I am sorry I didn't learn Russian as a schoolboy, and my Latin was halting at best. I am working away on things. I will need time to digest what you have written.Chjoaygame (talk) 08:40, 10 October 2015 (UTC)
 * No problem. Einstein was thinking decades on such matter. :-) Boris Tsirelson (talk) 09:03, 10 October 2015 (UTC)


 * I have been silent here, but I haven't stopped working on this and thinking about it. I have gained by your help, for which I thank you. I haven't written here because I don't want to occupy your time with half-baked chatter.Chjoaygame (talk) 07:55, 19 October 2015 (UTC)


 * In nutshell, the manipulations of "factors" are invalid and irreparable.Chjoaygame (talk) 08:30, 21 October 2015 (UTC)

in more detail
In more detail than a nutshell.

For the validity of the derivations of the Bell and congener inequalities, it is necessary that Bell's $ρ$ be one and the same distribution for all integrations.

Bell explicitly says that he requires this, but further, Fine shows that Bell needs it, that his proof is invalid without it. Careful examination of the physical meanings of the combinations of "factors" shows that they cannot be described by one and the same $ρ$. In ordinary physical terms, only one setting of a polarizer is possible at one time. Different physical settings are described by non-commuting operators. One and the same joint simultaneous $ρ$ does not exist, as explained by Vorob'ev. This non-existence of joint distributions, even when separate marginal distributions exist, is routine physics. These reasons prove that Bell and congener inequalities cannot be validly derived.

I have not seen any serious attempt by a Bell devotee to acknowledge, let alone reply to, this refutation, which was perhaps first offered by de la Peña, Cetto, and Brody (1972). The same thing is said by Jaynes, in different terminology, and by others.

For me to be persuaded that this refutation is not valid, I would need to see a careful and explicitly detailed recognition and acknowledgement of its strength, followed by an adequately careful demonstration of why, nonetheless, it is believed to be invalid. I have seen plenty of handwaving, and cavalier assurances that mother knows best, but no persuasive and sound argument by Bell believers, to deal with this refutation. I have long ago read various papers on this subject by Mermin, including the one about the moon; they do not provide an adequate reply to this refutation. It is true that Mermin has recently made significant steps to improve his understanding of probability theory, well beyond his primitive views of 1985. In my judgement, however, he has resorted to another inadequate homespun doctrine with its attendant Newspeak, and has not yet advanced to the stage of grasping the basic understanding provided by Jeffreys.

Chjoaygame (talk) 13:48, 23 October 2015 (UTC)


 * First of all, you cannot criticize Bell inequality using quantum notions (non-commuting operators etc). Everyone knows that Bell inequalities are violated in quantum mechanics. Local realism means (nearly by definition) that all operators commute. I recall you were furious toward Bohr's style... and now you accept it?! Boris Tsirelson (talk) 15:47, 23 October 2015 (UTC)


 * Recall Weierstrass function. It was a shock for many famous mathematician. Now imagine that someone, call him Ch., says: what is the fuss? Weierstrass' argument is erroneous from the beginning. He starts with the idea that a continuous function must me differentiable at most points. But he did not prove this. No one did. There is no reason to claim this. Thus, his function is not a paradox. Indeed, differentiability is a stronger condition... and so on.
 * But this is not at all Weierstrass' error. Probably it is a feature of Ch.'s intuition and that of others. He is not astonished. Well, this is a fact of his biography, not of math. Whether something is astonishing or not, is beyond mathematics, and cannot be proved or disproved. Boris Tsirelson (talk) 15:55, 23 October 2015 (UTC)
 * Likewise, many physicists are astonished by Bell theorem; but some are not. Especially, all those who are used to non-commutative operators, are not. That is OK. Tastes differ. But why some persons claim this is Bell's error? Bell did not, and could not, prove mathematically that this is astonishing. This is up to the reader. Quite subjective. Boris Tsirelson (talk) 15:59, 23 October 2015 (UTC)
 * By the way, Mermin writes explicitly from the beginning that physicists react differently to Bell theorem (and classify them). But it is illogical, to say "an error is found in the proof" when you just disagree with the assumptions of the theorem. No one of the authors cited above found an error in Bell's proof (it is, after all, a very simple mathematics). They all are dissatisfied with its assumption. OK, if someone finds these assumptions of no interest, he should just find another hobby. But not cry "do not be fooled". We are not fooled; we are shocked, while he is not. That's all. Boris Tsirelson (talk) 16:06, 23 October 2015 (UTC)
 * "I have not seen any serious attempt by a Bell devotee to acknowledge, let alone reply to, this refutation" — Oh! I gloat! They do not deserve attention. I am a mathematician; maybe physicists are different, but for a mathematician it would be very embarrassing, to write a paper, and seek credit, just for discussing, whether Weierstrass function is astonishing, or not. It's kind of an intimate experience. I do not ask anyone's opinion, when to be astonished, and when not. You feel no interest in Bell's assumptions, and therefore, in Bell's theorem? OK, no problem. Find another hobby. But why do you call it "refutation"?? It is rather a personal disappointment. Boris Tsirelson (talk) 17:25, 23 October 2015 (UTC)

thank you
Thank you for your frank expression of feeling.

I can agree that one might feel shocked by the Bell argument. I have to confess, if you like, that immediately on seeing it, as I previously mentioned, I felt sure that there was a flaw in the reasoning. Perhaps I ought not to have felt so, but as you say "I do not ask anyone's opinion, when to be astonished, and when not."

From your response, considering that I have an idea that you are more or less, likely more than I, familiar with the literature, I seem to have confirmation (or perhaps not?) that there has been no serious attempt by a Bell devotee to acknowledge, let alone reply to, this "refutation", noting that you do not regard it as a refutation. I read you as saying that it does not deserve a reply. If you do know of such a reply, I would be glad to learn of it.

I am very cautious or reluctant to use Bellspeak, the local dialect of George Orwell's Newspeak. With such a reservation, I think I can agree with your view, as I understand you, that when it comes to considering what quantum theorists call "measurement", 'counterfactual definiteness' is wrong. For my part, I reject the use of the word 'measurement' made by quantum theorists to refer to their observations. I think their usage leads to endless muddle. To be explicit about 'counterfactual definiteness', I think it wrong in general to say that when an observation is made with a momentum analyzer, if it had been made with a position analyzer, the result would have been such and such. For the sake of clarity, may I check that such is your meaning when you reject 'counterfactual definiteness'?

The physical reason here is that, in general, a quantum observation is made by occasioning a collision between an object of interest, known as the quantum system, with another important object, known as the observing apparatus. Here we come to another very problematic word, the 'state'. Loosely speaking, the "states" of the quantum system and of the observing apparatus are, in general, not definitely and exactly known for the occasion. I think that quantum theorists are very loose and inconsistent in their use of the word 'state'. I think they are vaguely aware of this looseness, and that such awareness drives them to be ever more elaborate in offering what they regard as strict mathematical clarification. But I think those efforts move them away from, not towards, physical clarity. Howsoever, I think that in general a quantum observation is not a measurement in the ordinary-language sense of the word. Taking it that it is a measurement is I think a very fertile source of hopeless muddle. Coming back to our topic, with reservations, I think that counterfactual definiteness, in the present context, is wrong and that we agree about that?

Coming back to Bell, I think it is an assumption of Bell that his single $ρ$ exists. He does not actually write 'Let us assume that such a distribution exists'. He just writes, in the course of his derivation, "If $ρ(λ)$ is the probability distribution of $λ$ ..." and "Because $ρ$ is a normalized probability distribution, ..." I think these statements express a most important assumption for Bell.Chjoaygame (talk) 21:52, 23 October 2015 (UTC)


 * Imagine the following.
 * As a result of careless experiments with a time machine you are in 1900. To fix the time machine and go back you desperately need money. A millionaire is ready to give you a million for an optimal strategy in a cooperative game. Just explain him
 * (a) How should conspire Alice and Bob to maximize the probability of winning, and
 * (b) Why he may be sure that no one will be able to play better than this.
 * But what is the game? Well, it is exactly "Example 2" from Lecture 20 of Watrous.
 * Question: what will you say him? Boris Tsirelson (talk) 05:50, 24 October 2015 (UTC)


 * I will not need to say anything to him. He will be able to read my mind directly, by use of Bell's theorem, which he has learnt about by mental telepathy with the future. Also, being a kind-hearted chap, he will just fix the time machine, not fussing about the money. Fortunately, he has a laboratory that is well equipped with photon detectors that have 100% quantum detection efficiency and zero dark current.Chjoaygame (talk) 07:07, 24 October 2015 (UTC)
 * Surely you are joking, Mr. Chjoaygame! Boris Tsirelson (talk) 07:36, 24 October 2015 (UTC)


 * Yes, I am joking.Chjoaygame (talk) 11:21, 24 October 2015 (UTC)


 * You cite Mermin, who quotes Pauli: “One should no more rack one’s brain about the problem of whether something one cannot know anything about exists all the same, than about the ancient question of how many angels are able to sit on the point of a needle. But it seems to me that Einstein’s questions are ultimately always of this kind.”


 * I think Pauli goes too far. One never knows exactly for sure that a particle is in a certain place. One has only a finite number of occasions of observation that lead one to believe so, thus with finite precision. Consequently there is room for one to know something (commensurately small in amount with one's residual imprecision on its position) about its momentum; small but not nothing. I believe that in English folklore, the usual puzzle is about how many angels can dance on the head of a pin. I believe that this puzzle did not come from the mediaeval scholastics,  but was a satire, invented in the eighteenth century Enlightenment. Do I detect a resonance here?Chjoaygame (talk) 00:44, 25 October 2015 (UTC)


 * In the article you cite, Mermin writes:


 * ... He was sure that Einstein would have been very bothered by Bell's theorem. Then he added,
 * Anybody who's not bothered by Bell's theorem has to have rocks in his head.
 * To this moderate point of view I would only add the observation that contemporary physicists come in two varieties. Type 1 physicists are bothered by EPR and Bell's theorem.


 * I am bothered by Bell's theorem, but not shocked by it. What bothers me is that people seem to take it too seriously, when I think it is logically faulty.Chjoaygame (talk) 01:25, 25 October 2015 (UTC)


 * But you did not answer my question. It was not a joke (on my part). Boris Tsirelson (talk) 04:50, 25 October 2015 (UTC)


 * I think your question "What will you say to him?" is out of order. It does not follow on the line of discussion. It arises from an arbitrary new line that you invented apparently from some outside source: "Imagine the following. ...."


 * Coming back to the line of discussion. You say "First of all, you cannot criticize Bell inequality using quantum notions (non-commuting operators etc)." That is a red herring. The point I made was that the existence of a universal $ρ$ is affected by the fact that one cannot make two different settings of a polarizer at the same time. That is not a peculiarity of quantum mechanics. It is an ordinary physical fact that happens also to be recognized by quantum mechanics. That the settings are represented in quantum mechanics by non-commuting operators is a side comment; I do not need it and I ought not to have made it, since you have used it as a red herring, and it is not strictly to the point.


 * Then you claim that it is almost by definition of local realism that all operators commute. In general, rotations do not commute for geometrical reasons, not restricted to quantum mechanics. Classical theory does not live only on commutative operators. It is far from by definition. And a red herring, as I just pointed out.


 * The material issue is whether there exists such a universal $ρ$ as Bell assumes. You dismiss this question saying that it does not deserve attention.


 * I will remain unpersuaded until I see it given the close and thorough attention that I think it deserves. The Bell arguments rely on more settings than can be combined for single occasion of existence of a singlet pair. It is worthy of close attention as to how they can be combined in one equation. That is why the question of the existence of a universal $ρ$ needs attention, and is the main concern of the present discussion, because its non-existence is how Bell's reasoning is fatally flawed.Chjoaygame (talk) 12:28, 25 October 2015 (UTC)


 * Then, I feel, the discussion is finished. You insist on discussing "refutation of Bell theorem". For me it is as nonsensical as "refutation of Weierstrass function" or "refutation of the equality 2+2=4". A mathematical truth cannot be refuted.
 * Yes, someone may say: "Do not overestimate the relevance of the equality 2+2=4. In reality, the number of objects is often ill-defined. Look at clouds in the sky; are you sure about the number of them?" OK, but this is not a refutation of the equality 2+2=4. It is a (re)assessment of its importance, relevance to something, or whatever; but clearly not a refutation.
 * Yes, someone may say: "Why the fuss around Weierstrass function? Some (even famous) mathematicians believed that this is impossible, without any reason to think so. Their problem." OK, but this is not a refutation; it is a (re)assessment.
 * Yes, someone may say: "Why the fuss around Bell theorem? Some (even famous) physicists believed that this should not happen, without any reason to think so. Their problem." OK, but this is not a refutation; it is a (re)assessment.
 * By my question I tried to let you understand, in which sense Bell's assumptions are properly chosen. But I feel that this understanding is highly undesirable for you; you do your best in order not to understand it. OK, I could not force you.
 * Boris Tsirelson (talk) 16:24, 25 October 2015 (UTC)
 * 'In the last decade of the 20th century I was usually able to convince a skeptic. Frustratingly, in the first decade of the 21th century I faced a small serried cohort of “Bell-deniers”. My gut feeling was telling me that most (but not all) of them realize in the depth of their hearts that they are wrong, but cannot afford to acknowledge defeat and exchange a high status in the opposition for a low status in the coalition.' Quoted from here. Boris Tsirelson (talk) 16:55, 25 October 2015 (UTC)


 * Perhaps you are right that the discussion is finished. I don't know.


 * For my part, I think your tactics have been what would be described by the mediaeval scholastics, brought in by Pauli's reference to the oft-quoted satire about angels on the head or point of a pin, in many parts as argumentum ad verecundiam.


 * I have no doubt in my mind that on ninety-nine matters out of a hundred, your intellect and erudition run rings around mine. On this present question, however, I am unpersuaded of that. On reading what you have written here, I think you have not read carefully the references that I gave and have not applied your undoubtedly brilliant mind to the arguments they contain. You content yourself with labels such as "Bell-deniers" and "not a refutation", without actually understanding or confronting the opposing view. Repeating what I wrote above: 'From your response, considering that I have an idea that you are more or less, likely more than I, familiar with the literature, I seem to have confirmation (or perhaps not?) that there has been no serious attempt by a Bell devotee to acknowledge, let alone reply to, this "refutation", noting that you do not regard it as a refutation. I read you as saying that it does not deserve a reply. If you do know of such a reply, I would be glad to learn of it.'


 * I can assure you that at first I could not identify a fault in the Bell reasoning, and in that sense I suppose you might say I "understood" it. I remained bothered by it nevertheless. I spent time examining Bell's paper, trying to find the fault. I wrote to Jaynes with a suggested fault in the reasoning, and he wrote back telling me I was wrong for physical reasons; as shown by his publication on the subject, he thinks the fault is in logic, not in physics. If this does not indicate that I "understood" the Bell argument, I cannot go further. I took me years of unsettled doubt to be fully convinced that the reasoning of de la Peña, Cetto, and Brody (1972) is the right way to tackle the matter. I think it has been reproduced in various forms and notations and conceptual frames many times since. I have not cited all the literature that I think agrees with it, more or less. Besides that, of course there is plenty of Bell-denying literature that I find more or less irrelevant or unhelpful or muddled or beyond me to understand.


 * Coming back to mediaeval rules of debate, one of them is that, at each step of the debate, each speaker should state the opposition's latest argument. I think by quoting Bell I have done that, near enough. But I don't think you have done it.


 * The point that concerns me is that Bell seems to think that the existence of his universal $ρ$ does not need or deserve explicit discussion as an assumption, but may be assumed, as I think you assume, as an obvious and nearly definitional consequence of the Bell-speak notion of "local realism". I think that Bell, like you, has not considered the point with nearly the attention it deserves. I would say that Bell presupposed what it was his task to prove.Chjoaygame (talk) 18:37, 25 October 2015 (UTC)


 * OK, the last word is yours. After all, this is yours talk page. Happy editing. Boris Tsirelson (talk) 19:10, 25 October 2015 (UTC)

Some literature

 * Kolmogorov, A.N.(1933/1956). Foundations of the Theory of Probability, translated from Russian to English by N. Morrison, second English edition, Chelsea, New York.
 * Jeffreys, H. (1939). Theory of Probability, Oxford University Press, Oxford UK.
 * Jaynes, E. T. (2003) Probability Theory: The Logic of Science, Cambridge University Press, ISBN 978-0521592710.
 * Vorob'ev, N.N. (1962). Consistent families of measures and their extensions, Theory of Probability and its Applications, 7(2): 147–163.
 * de la Peña, L., Cetto, A.M., Brody, T.A. (1972). On hidden-variable theories and Bell's inequality, Lettere al Nuovo Cimento, 5(2): 177–181.
 * Fine, A (1982). Hidden variables, joint probability, and Bell inequalities, Phys. Rev. Lett., 48(5): 291–295.
 * Fine, A (1982). Joint distributions, quantum correlations, and commuting observables, J. Math. Phys., 23(7): 1306–1310.
 * Ballentine. L.E. (1986). Probability theory in quantum mechanics, Am. J. Phys., 54(10): 883–889.
 * Jaynes, E.T. (1989). 'Clearing up mysteries – the original goal, pp. 1–27 in Maximum Entropy and Bayesian Methods, edited by J. Skilling, Kluwer Academic Publishers, Dordrecht, ISBN 0-7923-0224-9, (a talk given by Jaynes at the eighth MaxEnt Workshop, held at St John's College, Cambridge UK, August 1–5, 1988).
 * Canals-Frau, D. (1991). La “théorie” de Bell, est-elle la plus grande méprise de l’histoire de la physique? Is Bell’s theory the biggest misunderstanding in the history of Physics?, Annales de la Fondation Louis de Broglie, 16(2): 231–240.
 * Kracklauer, A.F. (2000). "La 'théorie' de Bell, est-elle la plus grande méprise de l’histoire de la physique?", Annales de la Fondation Louis de Broglie, 25(2): 193–205.
 * Khrennikov, A. (2009). Contextual Approach to Quantum Formalism, Springer, ISBN 978-1-4020-9592-4.
 * Vervoort, L. (2012). The instrumentalist aspects of quantum mechanics stem from probability theory, AIP Conf. Proc., 1424: 348–353. arXiv version.
 * Nieuwenhuizen, T.M., (2009). Where Bell went wrong, AIP Conf. Proc., 1101: 127-133. arXiv version.


 * Mermin, David (1985) Is the moon there when nobody looks? Physics Today.