Bell’s Theorem Refuted: Bell’s 1964:(15) is False

[Gordon Watson]

Harry, did you just shift ground again? For you now use the phrase "similarly acting lambdas".

Harry writes above: "Lambda is a hypothetical entity introduced by Einstein; Bell puts it to the test. If no similarly acting lambda's occur in later experiments then how can you (or Einstein) produce reproducible experiments??"

Of course there are similarly acting lambdas!

But that seems to have meant to you, until very recently, that MANY lambdas HAD TO BE the SAME!

NOT "have the same ACTION" but "BE the SAME".

Do you now understand that, despite NO two lambdas BEING the SAME, experimental outputs of ±1 are reproduced effortlessly: just like aluminium extrusions?
.

[gill1109]

It's probability zero that two lambda's are the same (if they are continuously distributed). And it's totally irrelevant. How about you guys grow up and study a modern proof of Bell's theorem? Easier proof, weaker conditions, stronger conclusion. Modern terminology.

[Xray]

It's probability zero that two lambda's are the same (if they are continuously distributed). And it's totally irrelevant. How about you guys grow up and study a modern proof of Bell's theorem? Easier proof, weaker conditions, stronger conclusion. Modern terminology.

gill1109 - is it your own theorem that would be better to be discussed as Bell's theorem? And, please, how is the data constructed in your theorem? I mean, does each combination of results or line in the table occur equally or is the table related to an experiment?
Xray

[harry]

It's probability zero that two lambda's are the same (if they are continuously distributed). And it's totally irrelevant.

Exactly - that was discussed and done with! And Gordon's refutation attempt doesn't even address that point so it's totally irrelevant here.

How about you guys grow up and study a modern proof of Bell's theorem? Easier proof, weaker conditions, stronger conclusion. Modern terminology.

This thread is not the place to discuss that. I did not see you give a link to the thread in which that is discussed. Maybe I overlooked it?

[gill1109]

It's probability zero that two lambda's are the same (if they are continuously distributed). And it's totally irrelevant. How about you guys grow up and study a modern proof of Bell's theorem? Easier proof, weaker conditions, stronger conclusion. Modern terminology.

gill1109 - is it your own theorem that would be better to be discussed as Bell's theorem? And, please, how is the data constructed in your theorem? I mean, does each combination of results or line in the table occur equally or is the table related to an experiment?
Xray

You had better read my paper to answer this question, Xray. I try to separate maths from physics. And I discuss the bridge between the two. Or read "Bertlman's socks".

Anyway, the mathematical result which Bell called his theorem is a calculus triviality; the mathematical result which is my "theorem" is a triviality of elementary probability theory.

The difficulty is relating these trivial mathematical results to physics, metaphysics, experiment ...

[gill1109]

It's probability zero that two lambda's are the same (if they are continuously distributed). And it's totally irrelevant.

Exactly - that was discussed and done with! And Gordon's refutation attempt doesn't even address that point.

How about you guys grow up and study a modern proof of Bell's theorem? Easier proof, weaker conditions, stronger conclusion. Modern terminology.

This thread is not the place to discuss that. I did not see you give a link to the thread in which that is discussed. Maybe I overlooked it?

Yes Harry you overlooked it. It was started by Gordon who arrogantly claims that a theorem if mine, which he cannot even parse let alone contemplate its meaning, is false. You can find my theorem here: http://arxiv.org/abs/1207.5103

[harry]

Attempted clarifications/corrections of Harry's statements:
[..] Harry says, "Bell assumed a probability distribution from which approximately the same lambda's must reoccur for those lambda's to reproduce experimental outcome."

Response: Alice's detector has just two outputs, +1 and -1. Are there just two lambdas, or 100, or … (say when).

Harry asks, "explain how your lambda's cause reproducible experiments to happen."

Response, presuming you mean "reproducible outputs (±1) from many repetitions of the same experiments": Alice's detector has just two outputs, +1 and -1. Are there just two lambdas, or 100, or ... (say when).

Please consider an aluminium-extruding machine with two dies producing just two profiles: + -. What limitation is there, then, on the orientation of the allowable billets (feed-stock)? Given N factories (N -> infinity), each producing the same outputs (= two profiles) from machines of their own design, what limit is there on the number and orientation of billets?

Harry, hoping this poor analogy helps, and with best regards; Gordon

PS: Please see my response to Richard above: re the similarity of patients and lambdas under experimentation; no two of them the same!
.

That's just a beginning of an answer. I think that there should be very many lambda's to produce a smooth cosine function for the varying angles. Therefore, Bell assumes that during an experiment lambda's are picked out of a continuous distribution, ever more filling up all points on the curve. For N->infinity in good approximation all points will be covered multiple times, ever more closely approaching a prediction (if the derivation is done correctly) which is based on all lambda's - possibly an infinite number - occurring. Of course a mathematician would be tempted to say that never any experimental point will exactly match an earlier point - and that is both true and irrelevant.

The only aspect of this that is relevant to your paper, is that Bell chose the same lambda everywhere for each integration step - which, as I explained, is necessarily so - in contradiction to your simplified version of his 14a in your paper.

Now I will take a "time-out" from this forum.

[gill1109]

Of course a mathematician would be tempted to say that never any experimental point will exactly match an earlier point - and that is both true and irrelevant.

The only aspect of this that is relevant to your paper, is that Bell chose the same lambda everywhere for each integration step - which, as I explained, is necessarily so - in contradiction to your simplified version of his 14a in your paper.

Now I will take a "time-out" from this forum.

Exactly. True and irrelevant.

Bell assumes that for each of the three correlations, lambda is chosen at random from the same set of possible values according to the same probability distribution. He does not assume the distribution is continuous, but just uses notation corresponding to the continuous case, because his readers don't know 20th century mathematics.

[Gordon Watson]

It's probability zero that two lambda's are the same (if they are continuously distributed). And it's totally irrelevant. How about you guys grow up and study a modern proof of Bell's theorem? Easier proof, weaker conditions, stronger conclusion. Modern terminology.

Richard,

To be clear, I take it that you as underlined above refer to your own theorem: http://arxiv.org/abs/1207.5103 eqn. (3)?

Gordon

[Xray]

It's probability zero that two lambda's are the same (if they are continuously distributed). And it's totally irrelevant. How about you guys grow up and study a modern proof of Bell's theorem? Easier proof, weaker conditions, stronger conclusion. Modern terminology.

Richard,

To be clear, I take it that you as underlined above refer to your own theorem: http://arxiv.org/abs/1207.5103 eqn. (3)?

Gordon

Gordon -- see replies to harry and me above -- the answer is yes. Xray

[Gordon Watson]

How about you guys grow up and study a modern proof of Bell's theorem? Easier proof, weaker conditions, stronger conclusion. Modern terminology.

Yes Harry you overlooked it. It was started by Gordon who arrogantly claims that a theorem if mine, which he cannot even parse let alone contemplate its meaning, is false. You can find my theorem here: http://arxiv.org/abs/1207.5103

Thanks Xray,
I missed the above due to current small-screen problems.

Dear Richard,

Re "Gill's theorem": http://arxiv.org/abs/1207.5103 eqn. (3).

You insist that your "theorem" should be discussed in this thread because it is a better version of the Bell-CHSH inequality ; ie, you offer a better inequality than the original CHSH inequality that was based on Bell 1964:(15) that I (in my essay) claim to refute.

So let me apologise in advance for any error here; some earlier notes of mine were possibly also astray, in that they too were written-up in haste.

For I'm here presenting a preliminary analysis of your theorem, based on rough notes that I put on the back of an envelope when I first heard of it.

In this way you can correct my misconceptions, errors, etc. For, given the authorities that you acknowledge (p.21-22), it seems that there must be serious mistakes below.

Apropos Gill's Theorem: http://arxiv.org/abs/1207.5103 eqn. (3).

Introduction:

Given:

Independent observations: AB, AB', A'B, A'B' = ±1. (1)

W = <AB> + <AB'> + <A'B> - <A'B'>. (2)

η ≥ 0. (3)

N random observations; N ≥ 1. (4)

Then, with P denoting probability, Gill's theorem http://arxiv.org/abs/1207.5103 eqn. (3) asserts:

P(W ≤ 2 + η) ≥ 1- 8e^(-N(η^2)/256). (5)


Analysis 1:

From "Gill's theorem", our (5), for η = 0: (6)

P(W ≤ 2) ≥ -7, for any countable N. (7)

NB: The limit as N -> oo is excluded, since N is defined in the context of observations and a spreadsheet; and (in any case), the product of infinity and zero is undefined.


Conclusions 1:

(1a) Since no probability is less than zero, all probabilities exceed any negative number (including - 7). So Gill's theorem appears to add nothing to probability theory, statistics, or Bell's theorem.

(1b) Any theorem delivering (7) is absurd.

(1c) In short: It is trivial to claim that a probability is greater than a negative number; and absurd to consider such.



Analysis 2:

From (2), by observation:

W ≤ 4. So P(W ≤ 4) = 1 (ie, certainty). (8)

So, since in this example, η = 2; using (8) and (5):

P(W ≤ 4) = 1 ≥ 1 - 8e^(-N/64), for any countable N. (9)

(9) yields:

For N = 1, P(W ≤ 4) = 1 ≥ - 6.87.

For N = 10, P(W ≤ 4) = 1 ≥ - 5.84.

For N = 100, P(W ≤ 4) = 1 ≥ - 0.676.

For N = a countable infinity, P(W ≤ 4) = 1 > 1 - ε; since ε > 0. (10)

NB: In the limit, "ε -> 0 as N -> oo" is excluded, since N is defined in the context of observations and a spreadsheet.

Conclusions 2:

(2a) Per (10 above, Gill's theorem asserts the trivial: the probability of a certainty is greater than a number less than one.

(2b) It is trivial for any probability to be greater than a negative number; and absurd to consider such.

(2c) It is trivial (indeed absurd) that η be > 2: yet Gill (p.23) discusses a restriction of η to ≤ 2√2.

(2d) As shown at (10), Gill's claim (p.23) that (1) is trivial for η > 2 should read: for η ≥ 2, at least.

(2e) From (10), we have the trivial: 1 ≥ 1 - ε for ε > 0.

(2f) Inequalities (7) and (10) hold trivially for any N, up to a countable infinity of observations.

(2g) Re (2d) above: Cursory analysis of (7) and (10) has Gill's theorem trivial for η ≥ 0.

General conclusion:

Gill's theorem is trivial for all η; ie, for any η ≥ 0. So Gill's claim (p.3) to have established "a new version of the famous Bell-CHSH inequality" is absurd. Further, to the extent that Gill's theorem is in any way associated with the Bell-CHSH inequality: it is refuted by commonsense local realism (CLR); see the Opening Post (OP) in this thread -- viewtopic.php?f=6&t=62#p2612

PS: Richard, I will be pleased to be corrected, but I note: Your theorem is refuted by my commonsense local realism (CLR) -- which, incidentally, appears to be a philosophy very similar to your own: given your rejection of nonlocality.

E and OE: Gordon

[FrediFizzx]

We warned Richard that there was a lot of mistakes and much just plain BS in that paper but he went ahead and had it published in Statistical Science. I guess that journal is not peer reviewed. Or if it is, it doesn't say very much for their peer review process.

[gill1109]

General conclusion:

Gill's theorem is trivial for all η; ie, for any η ≥ 0. So Gill's claim (p.3) to have established "a new version of the famous Bell-CHSH inequality" is absurd. Further, to the extent that Gill's theorem is in any way associated with the Bell-CHSH inequality: it is refuted by commonsense local realism (CLR); see the Opening Post (OP) in this thread -- viewtopic.php?f=6&t=62#p2612

PS: Richard, I will be pleased to be corrected, but I note: Your theorem is refuted by my commonsense local realism (CLR) -- which, incidentally, appears to be a philosophy very similar to your own: given your rejection of nonlocality.

E and OE: Gordon

Gordon: try picking N = 10^6 and eta = 0.1
Perhaps you will discover that my theorem does say some interesting things.

You have shown that for some values of N and eta it merely tells us something trivially true. OK. That's not a problem. Please look at some interesting values of N and eta, now.

[Gordon Watson]

General conclusion:

Gill's theorem is trivial for all η; ie, for any η ≥ 0. So Gill's claim (p.3) to have established "a new version of the famous Bell-CHSH inequality" is absurd. Further, to the extent that Gill's theorem is in any way associated with the Bell-CHSH inequality: it is refuted by commonsense local realism (CLR); see the Opening Post (OP) in this thread -- viewtopic.php?f=6&t=62#p2612

PS: Richard, I will be pleased to be corrected, but I note: Your theorem is refuted by my commonsense local realism (CLR) -- which, incidentally, appears to be a philosophy very similar to your own: given your rejection of nonlocality.

E and OE: Gordon

Gordon: try picking N = 10^6 and eta = 0.1
Perhaps you will discover that my theorem does say some interesting things.

You have shown that for some values of N and eta it merely tells us something trivially true. OK. That's not a problem. Please look at some interesting values of N and eta, now.

Richard, using your data (N = 10^6 and η = 0.1) and not being a wasteful academic; ie, I'm bound by DIL (Dill's Inefficiency Lemma):

I'd rather pick N < 10^3.

Would that be OK?

Gordon

[FrediFizzx]

This is a physics forum. Does eta have any physical meaning? Gill seems to have not defined it.

[gill1109]

This is a physics forum. Does eta have any physical meaning? Gill seems to have not defined it.

No I have not defined it, the theorem is a mathematics theorem and it says, "given a number N and an N x 4 spreadsheet of numbers +/-1, then for any eta > 0 and any N, a certain probability is at least as large as a certain number". The certain number depending on N and eta.

You can choose whatever values of N and eta you like, the statement is true. For instance, let's choose N = 10^6 and eta = 0.1. Here's a little piece of R output:

Code: Select all

> N <- 10^6
> eta <- 0.1
> 8 * exp(- N * (eta/16)^2)
[1] 8.678842e-17
> 8 * exp(- N * ((eta/16)^2))
[1] 8.678842e-17


We see that if our spreadsheet has one million rows, and if for each row we toss two coins to choose to observe A or A', and B or B', and calculate the CHSH quantity S in the usual way, the probability that it exceeds 2.1 is smaller than 8.7 times 10 to the minus 17.

For another example, pick eta = 0.4. So we are asking about the probability that S could exceed 2.4. Remember, QM predicts 2.8.... Put N = 10 000

Code: Select all

> N <- 10000
> eta <- 0.4
> 8 * exp(- N * ((eta/16)^2))
[1] 0.01544363


This tells me that if we performed Christian's experiment, with random choice of settings for each of 10 000 runs, then the chance is not more than 1 in 100 that Christian would win his bet with me.

Is this a theorem about physics? Well, it is a theorem about mathematical-physical models, and also of computer simulations of mathematical-physical models. Whether that means it is about physics or not, everyone can decide for themselves. Physicists use mathematical models to describe the real world. Mathematicians prove theorems about those models.

[gill1109]

General conclusion:

Gill's theorem is trivial for all η; ie, for any η ≥ 0. So Gill's claim (p.3) to have established "a new version of the famous Bell-CHSH inequality" is absurd. Further, to the extent that Gill's theorem is in any way associated with the Bell-CHSH inequality: it is refuted by commonsense local realism (CLR); see the Opening Post (OP) in this thread -- viewtopic.php?f=6&t=62#p2612

PS: Richard, I will be pleased to be corrected, but I note: Your theorem is refuted by my commonsense local realism (CLR) -- which, incidentally, appears to be a philosophy very similar to your own: given your rejection of nonlocality.

E and OE: Gordon

Gordon: try picking N = 10^6 and eta = 0.1
Perhaps you will discover that my theorem does say some interesting things.

You have shown that for some values of N and eta it merely tells us something trivially true. OK. That's not a problem. Please look at some interesting values of N and eta, now.

Richard, using your data (N = 10^6 and η = 0.1) and not being a wasteful academic; ie, I'm bound by DIL (Dill's Inefficiency Lemma):

I'd rather pick N < 10^3.

Would that be OK?

Gordon

Gordon, whatever values of N and eta you pick, my theorem makes a true assertion. It may or may not be interesting. The theorem is only interesting for values of eta, say, smaller than about 0.8 (remember: QM tells us that Bell-CHSH quantity "S" can equal 2.828...). For small values of N, the theorem is not interesting. But the large N becomes, the more interesting it becomes. Because it tells us that a certain probability is exponentially small as N increases.

By the way, I chose to prove with simple means a quite simple assertion. Easy to prove, easy to remember. The result is not "best possible". With a lot more work one could get a much sharper bound (hence an even more useful bound).

But already, my simple (very suboptimal) bound helps me design bets against LHV simulation models which I will win with large probability. It tells me how large I should insist N must be, for a given risk of losing. (I would of course let my chance of losing a bet depend on the amount of money being staked).

Present day Bell-CHSH experiments have N of the order of size of 10s of millions, or more. Michel Fodje rapidly simulates experiments with N = 10^6. Nobody is much interested in an experiment with N smaller than one thousand. (At least: not in physics - in psychology one has to make do with N = 60 psychology students, but they accept ridiculously large error rates).

My bound also allows me to study the limit as N to infinity of S. Which is what usual statements of Bell's theorem are all about. My theorem is about the finite N case, which is of course the only case relevant to experiment.

[Joy Christian]

This tells me that if we performed Christian's experiment, with random choice of settings for each of 10 000 runs, then the chance is not more than 1 in 100 that Christian would win his bet with me.

I have already defeated all the so-called bets and challenges by Richard Gill concerning my proposed experiment. In fact he owes me 10,000 Euros, but I do not expect him to ever pay up. Last month I made the following statement about his ridiculous bets and challenges on my blog:

I have recently won the 10,000 Euros offered by Richard Gill for theoretically producing the 2n angular momentum vectors, and , appearing in the equation (16) of my proposed experiment. He had foolishly claimed that it was mathematically impossible to construct such 2n vectors and had challenged me to produce them as a "proof of concept" for my proposed experiment. I defeated his challenge in May and June of 2014 by explicitly producing the 2n vectors in these four simulations: (1), (2), (3), and (4) (the reason there are four simulations instead of just one is because he kept moving the goalpost each time I, or someone else, defeated his challenge). The correlations in these simulations are calculated as



together with



where is the number of experiments performed. These results strongly suggest that my proposed experiment will be a spectacular success. It will reproduce the manifestly local-realistic and yet strong correlations exactly as I have predicted in my papers, confirming my hypothesis that physical space we live in respects the geometry and topology of , not .

A theoretical description of my 3-sphere model can be found in my latest paper on the subject (see, especially, figure 1): http://arxiv.org/abs/1405.2355.

[gill1109]

This is a physics forum. Does eta have any physical meaning? Gill seems to have not defined it.

Fred, This is a good question. What does the theorem mean? Fix N and a spreadsheet. Toss 2N fair coins, calculate S, the CHSH quantity based on four correlations each based on a different random subset of rows of the table. The value of S could in principle lie almost anywhere between -4 and +4. The theorem says, however, that with large probability it won't be larger than +2 by more than a few multiples of 1 / sqrt N.

[gill1109]

This is a physics forum. Does eta have any physical meaning? Gill seems to have not defined it.

Fred, This is a good question. What does the theorem mean? Fix N and a spreadsheet. Toss 2N fair coins, calculate S, the CHSH quantity based on four correlations each based on a different random subset of rows of the table. The value of S could in principle lie almost anywhere between -4 and +4. The theorem says, however, that with large probability it won't be larger than +2 by more than a few multiples of 1 / sqrt N.

For instance, let's consider eta equals quite a lot of multiples of 1 / sqrt N. For instance eta = 160 / sqrt N. We find that the chance S exceeds +2 by more than 160 / sqrt N is smaller than 8 exp(- 100). You can calculate that number on your pocket calculator. It is astronomically small.

If N = 256 million then eta = 0.01. So that's an interesting value of eta.

It is quite a large value of N. But such a value of N is commonplace in today's experiments. My theorem says something interesting about such experiments when, in the future, they are performed in rigorous experimental conditions: event ready detectors, no detection loophole, no locality loophole, fast random switching of settings. It says that under local realism it would be almost impossible to observe a value of S larger than 2.01.