Bell's theorem refuted via elementary probability theory

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Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Sun Oct 11, 2020 3:36 pm

.
Bell's theorem has been described as the most profound discovery of science, one of the few essential discoveries of 20th Century physics, indecipherable to non-mathematicians. Let's see.

https://vixra.org/abs/2010.0068

ALL comments welcome, especially those that are educative and/or critical.

Thanks; Gordon
.
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Sun Oct 11, 2020 9:43 pm

Gordon Watson wrote:.
Bell's theorem has been described as the most profound discovery of science, one of the few essential discoveries of 20th Century physics, indecipherable to non-mathematicians. Let's see.

https://vixra.org/abs/2010.0068

ALL comments welcome, especially those that are educative and/or critical.

Thanks; Gordon
.

Just two pages! Excellent.

Notation. Why do you have “beta” in your notation, standing for “Bell’s experiment”? Doesn’t every P(... | ...) in the whole two pages stand for probability in Bell’s experiment?

What do you mean by A^+, A^- and A^ +- ; similarly for B. Bell calls the actually observed results A and B; their possible values are +1 and -1.
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Sun Oct 11, 2020 11:23 pm

gill1109 wrote:
Gordon Watson wrote:.
Bell's theorem has been described as the most profound discovery of science, one of the few essential discoveries of 20th Century physics, indecipherable to non-mathematicians. Let's see.

https://vixra.org/abs/2010.0068

ALL comments welcome, especially those that are educative and/or critical.

Thanks; Gordon
.

Just two pages! Excellent.

Q1. Notation. Why do you have “beta” in your notation, standing for “Bell’s experiment”?

Q2. Doesn’t every P(... | ...) in the whole two pages stand for probability in Bell’s experiment?

Q3. What do you mean by A^+, A^- and A^ +- ; similarly for B. Bell calls the actually observed results A and B; their possible values are +1 and -1.


A1. β (beta) denotes the experiment IN Bell (1964). That experiment is the EPR-Bohm experiment. Elsewhere I chose β to honour David Bohm. In trying to keep to 2 pages here, I omitted such background.

A2. NO; see the line before B(2): Bell uses P to denote an expectation value. Further: nowhere does Bell use this crucial bit in any probability function -- |...) -- whereas you rightly show it in your notation above.

A3. Please note: Bell is confused here; see the line before B(1)! He uses A and B for both functions and the related results. Since the related results are +1 or -1, it is much clearer to use my terms in probability relations. See my move from eqn (3) -- with its functions -- to eqn (4) with its more conventional [encoded, and therefore more compact] results.

HTH; Gordon
.
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Mon Oct 12, 2020 2:58 am

Gordon Watson wrote:
gill1109 wrote:
Gordon Watson wrote:.
Bell's theorem has been described as the most profound discovery of science, one of the few essential discoveries of 20th Century physics, indecipherable to non-mathematicians. Let's see.
https://vixra.org/abs/2010.0068
ALL comments welcome, especially those that are educative and/or critical.
Thanks; Gordon

Just two pages! Excellent.
Q1. Notation. Why do you have “beta” in your notation, standing for “Bell’s experiment”?
Q2. Doesn’t every P(... | ...) in the whole two pages stand for probability in Bell’s experiment?
Q3. What do you mean by A^+, A^- and A^ +- ; similarly for B. Bell calls the actually observed results A and B; their possible values are +1 and -1.

A1. β (beta) denotes the experiment IN Bell (1964). That experiment is the EPR-Bohm experiment. Elsewhere I chose β to honour David Bohm. In trying to keep to 2 pages here, I omitted such background.
A2. NO; see the line before B(2): Bell uses P to denote an expectation value. Further: nowhere does Bell use this crucial bit in any probability function -- |...) -- whereas you rightly show it in your notation above.
A3. Please note: Bell is confused here; see the line before B(1)! He uses A and B for both functions and the related results. Since the related results are +1 or -1, it is much clearer to use my terms in probability relations. See my move from eqn (3) -- with its functions -- to eqn (4) with its more conventional [encoded, and therefore more compact] results.
HTH; Gordon

Just trying to be educative, but also trying to make the job of reading the two pages more pleasant:

A1. The whole paper assumes the EPR-Bohm experiment as background. You can safely omit beta *everywhere*. Removing superfluous letters on the pages would make them *much* easier to read.

A2. Yes, I know you are using better notation than Bell, in having E for expectation. That’s not my point at all.

A3. You can use different letters, or different letter-styles (Italic/Roman/Gothic...; bold, double, ...) or different alphabets (Greek/Hebrew/...) or special diacritical marks (hats, bars, ...) for functions , and for outcomes - possible values taken by functions. Your superscript +, -, and +- are superfluous and confusing.

Just delete every symbol which is superfluous, and the whole thing will be *much* easier to read. Less is more!

Then on to a new question:

Q4. Inside an integration, and apparently to be interpreted as a number multiplying another number, you write “(A(a, lambda) = 1)”. But the result of evaluating “A(a, lambda) = 1” is true or false. Are you wanting us to convert that to 1 or 0, accordingly?
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Mon Oct 12, 2020 6:25 am

gill1109 wrote:
Gordon Watson wrote:
gill1109 wrote:
Gordon Watson wrote:.
Bell's theorem has been described as the most profound discovery of science, one of the few essential discoveries of 20th Century physics, indecipherable to non-mathematicians. Let's see.
https://vixra.org/abs/2010.0068
ALL comments welcome, especially those that are educative and/or critical.
Thanks; Gordon

Just two pages! Excellent.
Q1. Notation. Why do you have “beta” in your notation, standing for “Bell’s experiment”?
Q2. Doesn’t every P(... | ...) in the whole two pages stand for probability in Bell’s experiment?
Q3. What do you mean by A^+, A^- and A^ +- ; similarly for B. Bell calls the actually observed results A and B; their possible values are +1 and -1.

A1. β (beta) denotes the experiment IN Bell (1964). That experiment is the EPR-Bohm experiment. Elsewhere I chose β to honour David Bohm. In trying to keep to 2 pages here, I omitted such background.
A2. NO; see the line before B(2): Bell uses P to denote an expectation value. Further: nowhere does Bell use this crucial bit in any probability function -- |...) -- whereas you rightly show it in your notation above.
A3. Please note: Bell is confused here; see the line before B(1)! He uses A and B for both functions and the related results. Since the related results are +1 or -1, it is much clearer to use my terms in probability relations. See my move from eqn (3) -- with its functions -- to eqn (4) with its more conventional [encoded, and therefore more compact] results.
HTH; Gordon

Just trying to be educative, but also trying to make the job of reading the two pages more pleasant:

A1. The whole paper assumes the EPR-Bohm experiment as background. You can safely omit beta *everywhere*. Removing superfluous letters on the pages would make them *much* easier to read.

A2. Yes, I know you are using better notation than Bell, in having E for expectation. That’s not my point at all.

A3. You can use different letters, or different letter-styles (Italic/Roman/Gothic...; bold, double, ...) or different alphabets (Greek/Hebrew/...) or special diacritical marks (hats, bars, ...) for functions , and for outcomes - possible values taken by functions. Your superscript +, -, and +- are superfluous and confusing.

Just delete every symbol which is superfluous, and the whole thing will be *much* easier to read. Less is more!

Then on to a new question:

Q4. Inside an integration, and apparently to be interpreted as a number multiplying another number, you write “(A(a, lambda) = 1)”. But the result of evaluating “A(a, lambda) = 1” is true or false. Are you wanting us to convert that to 1 or 0, accordingly?


Many thanks, Richard

R (Response) 1: Sure: Beta can generally go; I'll define the primary conditioning space, once, in the text.

NB: The 2nd and 3rd betas in eqn (3) which are TYPOS can go now!

R3: Functions A and B each produce 2 results. So what can be simpler than symmetrically representing such results via A and B and ± superscripts?

R4: Inside the integral over the space of λ, Bell has a PRODUCT of two functions [A(a,λ) and B(b,λ)] and each produces 2 discrete outcomes (+1 or -1).

However, for a given λ, only one paired-outcome is produced! That is: IF A = +1 and B = +1 is produced in the first term (under a given λ), THEN, via the definition of a function, the next three terms are zero.

So the output, as in (4), is the SUM over (the WEIGHT, ie, the probability, of each AB output) times (the value of that output).

Further, as you'll see via eqn (7): the sum of the weights equals unity.

PS: I guess you see that (4) is equally the definition of an expectation over 4 discrete outcome-pairs:

respectively, AB = (+1, +1) = +1, AB = (+1, -1) = -1, AB = (-1, +1) = -1, AB = (-1, -1) = +1.

HTH. Thanks again; Gordon
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Mon Oct 12, 2020 10:36 pm

R4: I think you are confirming that your definition of “A = 1” inside an *arithmetical* expression is “1 if true, otherwise 0”

R3: I think you are saying that by the superscript +, -, +-, you mean the *value* of the function (the outcome of the hypothetical measurement) in that instance. The A or B is a reminder of what function delivered that value. I don’t see the point of keeping track whose +1 it is. If you insist, write the 1 normal size, and write the A or B as an index (a decoration).

I find your works generally lead to head-ache, since as a very original person working in splendid isolation you have developed alternative notations to what the rest of the world is using. Already, maths, physics and computer science each have their own dialects of talking math. And within maths there are as many variations as accents of English in the world. So for me, used to decoding those many dialects, it is confusing to meet a new one in which you do not explain your notation using the words which are used in university classrooms around the world.

My general suggestion is: remove superfluous symbols! Less is more!
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Wed Oct 14, 2020 1:19 am

Thanks Richard, I'll post version-2 here as soon as it's online. Gordon
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Wed Oct 14, 2020 1:03 pm

Richard, etc.,

Seeking to be clearer: here's Version 2, https://vixra.org/pdf/2010.0068v2.pdf

In eqn (3), to identify each integrand, I've this time used (the triple bar) to denote equivalence: and thereby separate the results.

I personally prefer for equivalence.

So maybe this would be better; and even clearer?

and ?

Thanks; Gordon
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Wed Oct 14, 2020 6:14 pm

Gordon Watson wrote:Richard, etc.,

Seeking to be clearer: here's Version 2, https://vixra.org/pdf/2010.0068v2.pdf

In eqn (3), to identify each integrand, I've this time used (the triple bar) to denote equivalence: and thereby separate the results.

I personally prefer for equivalence.

So maybe this would be better; and even clearer?

and ?

Thanks; Gordon

It looks to me this is overkill. You say “something” is in the set of x such that x is equivalent to 1. So you could replace that by just saying that “something” is equivalent to 1. And what does “equivalent” mean? Don’t you mean “equals”?
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Wed Oct 14, 2020 6:47 pm

gill1109 wrote:
Gordon Watson wrote:Richard, etc.,

Seeking to be clearer: here's Version 2, https://vixra.org/pdf/2010.0068v2.pdf

In eqn (3), to identify each integrand, I've this time used (the triple bar) to denote equivalence: and thereby separate the results.

I personally prefer for equivalence.

So maybe this would be better; and even clearer?

and ?

Thanks; Gordon

It looks to me this is overkill. You say “something” is in the set of x such that x is equivalent to 1. So you could replace that by just saying that “something” is equivalent to 1. And what does “equivalent” mean? Don’t you mean “equals”?


Thanks Richard,

Given your earlier comments, I thought my earlier use of "equals" could be improved. (For you seemed to suggest it was some unique form of mathematics.)

Equivalence is a more general term than equals. So we are separating those AB functions (with a λ-pair that delivers 1) from those AB functions (with a λ-pair that delivers -1).

That way you should not be confused into thinking that the integrands can be reduced to ±1 before integrating. And in this way -- under the "overkill" -- the integral in (1) reduces to eqn (4) rigorously.

Richard, in that you seemed to recently think that Bell's use of P denoted probabilities, why not begin with the refutation of Bell's inequality that I now include in the 2-page format that you favoured? For if that simple analysis is false ... I'm out of here!

That way, in the interim, those who understand set-notation can assist with the following.

I think this is the best more-expansive way to go, thus far, in my eqn (3):

and ?
.
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Wed Oct 14, 2020 10:18 pm

Gordon Watson wrote:
gill1109 wrote:
Gordon Watson wrote:Richard, etc.,

Seeking to be clearer: here's Version 2, https://vixra.org/pdf/2010.0068v2.pdf

In eqn (3), to identify each integrand, I've this time used (the triple bar) to denote equivalence: and thereby separate the results.

I personally prefer for equivalence.

So maybe this would be better; and even clearer?

and ?

Thanks; Gordon

It looks to me this is overkill. You say “something” is in the set of x such that x is equivalent to 1. So you could replace that by just saying that “something” is equivalent to 1. And what does “equivalent” mean? Don’t you mean “equals”?


Thanks Richard,

Given your earlier comments, I thought my earlier use of "equals" could be improved. (For you seemed to suggest it was some unique form of mathematics.)

Equivalence is a more general term than equals. So we are separating those AB functions (with a λ-pair that delivers 1) from those AB functions (with a λ-pair that delivers -1).

That way you should not be confused into thinking that the integrands can be reduced to ±1 before integrating. And in this way -- under the "overkill" -- the integral in (1) reduces to eqn (4) rigorously.

Richard, in that you seemed to recently think that Bell's use of P denoted probabilities, why not begin with the refutation of Bell's inequality that I now include in the 2-page format that you favoured? For if that simple analysis is false ... I'm out of here!

That way, in the interim, those who understand set-notation can assist with the following.

I think this is the best more-expansive way to go, thus far, in my eqn (3):

and ?
.

I did not think Bell’s use of P denoted probability.

A times B is one function. You can’t “separate those functions which deliver this, or that”. It makes no sense at all. You can separate their arguments. Or you can create new functions by writing e’g. f = f 1_A + f 1_Ac. Write f as a sum of two parts, each zero on complementary parts of their domain; I use 1_A to denote the indicator function of the set A. It takes the values 1 and 0 depending on whether x in A or not.
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Thu Oct 15, 2020 12:44 am

gill1109 wrote:
Gordon Watson wrote:
gill1109 wrote:
Gordon Watson wrote:Richard, etc.,

Seeking to be clearer: here's Version 2, https://vixra.org/pdf/2010.0068v2.pdf

In eqn (3), to identify each integrand, I've this time used (the triple bar) to denote equivalence: and thereby separate the results.

I personally prefer for equivalence.

So maybe this would be better; and even clearer?

and ?

Thanks; Gordon

It looks to me this is overkill. You say “something” is in the set of x such that x is equivalent to 1. So you could replace that by just saying that “something” is equivalent to 1. And what does “equivalent” mean? Don’t you mean “equals”?


Thanks Richard,

Given your earlier comments, I thought my earlier use of "equals" could be improved. (For you seemed to suggest it was some unique form of mathematics.)

Equivalence is a more general term than equals. So we are separating those AB functions (with a λ-pair that delivers 1) from those AB functions (with a λ-pair that delivers -1).

That way you should not be confused into thinking that the integrands can be reduced to ±1 before integrating. And in this way -- under the "overkill" -- the integral in (1) reduces to eqn (4) rigorously.

Richard, in that you seemed to recently think that Bell's use of P denoted probabilities, why not begin with the refutation of Bell's inequality that I now include in the 2-page format that you favoured? For if that simple analysis is false ... I'm out of here!

That way, in the interim, those who understand set-notation can assist with the following.

I think this is the best more-expansive way to go, thus far, in my eqn (3):

and ?
.

I did not think Bell’s use of P denoted probability.

A times B is one function. You can’t “separate those functions which deliver this, or that”. It makes no sense at all. You can separate their arguments. Or you can create new functions by writing e’g. f = f 1_A + f 1_Ac. Write f as a sum of two parts, each zero on complementary parts of their domain; I use 1_A to denote the indicator function of the set A. It takes the values 1 and 0 depending on whether x in A or not.


Thanks Richard,

1. Apologies re my P statement and what seemed to be! I see now that I misinterpreted the related reference to Bell's experiment.

2. Re the math, here's what I legitimately separate:

(i). Bell's integral is over the AB product across all λ-pairs.

(ii). I split the integral into 2 parts: the first part over all the λ-pairs that deliver AB = +1; the second part over all the λ-pairs that deliver AB = -1.

(iii). Since all λ-pairs are covered -- with no exceptions -- the integral over the AB product can surely be evaluated in this way? Yes?

(iv). So this is where I need help: I need a compact way of expressing this "separation of variables" -- one that is well-understood.

(v). So which of these do you prefer, or is there yet a better way. Thanks

X: and

Y: and

Z: ??

PS: Also, if you now understand what my math intends, you can now follow it to the end.

Which means to the end of BT and to the end of BI -- in very elementary ways; ways that lead to agreement with QM and an irrefutable inequality whose LHS can be expressed identically with Bell's but whose RHS is extremely different and never false.

HTH; thanks again; Gordon
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Thu Oct 15, 2020 9:20 pm

I would use

Z: AB (AB == 1) ,

having explained that in an arithmetical expression, logical expressions are converted from T/F to 1/0. I would use == when asking are LHS and RHS equal, forcing AB == 1 to be read as logic. I would use := for defining LHS by RHS, and plain = for a chain of expressions asserting equality.

In Z, three functions are multiplied together: A, B, and (AB == 1). The third function is a function of a, b and lambda and it takes values in {0, 1}.

A(a, lambda) is a number
A is a function
You did not define “x is equivalent to 1”, and the set of numbers x equal to 1 contains only one number.

So X and Y are awful. Say simple things in a simple way. Mathematical notation is designed in order to say difficult things as clearly and simply as possible. Mathematics is built up logically and sequentially. Every time you introduce new notation, or you change notation, you should say so. Usually you are writing in a context which means that you know the notation which your readers are familiar with, and you use it too, if possible, or at least, to start with. You build on foundations which are there and which are known by your readers.

I realise how difficult it must be! I have had these discussions with students for 45 years and once I was a student myself. I remember a student asking me (after the lecture) “does the less than or equals symbol” mean less than or equals, or less than and equals? I wish he had dared to ask it during the lecture.
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Thu Oct 15, 2020 9:48 pm

gill1109 wrote:I would use

Z: AB (AB == 1) ,

having explained that in an arithmetical expression, logical expressions are converted from T/F to 1/0. I would use == when asking are LHS and RHS equal, forcing AB == 1 to be read as logic. I would use := for defining LHS by RHS, and plain = for a chain of expressions asserting equality.

In Z, three functions are multiplied together: A, B, and (AB == 1). The third function is a function of a, b and lambda and it takes values in {0, 1}.

A(a, lambda) is a number
A is a function
You did not define “x is equivalent to 1”, and the set of numbers x equal to 1 contains only one number.

So X and Y are awful. Say simple things in a simple way. Mathematical notation is designed in order to say difficult things as clearly and simply as possible. Mathematics is built up logically and sequentially. Every time you introduce new notation, or you change notation, you should say so. Usually you are writing in a context which means that you know the notation which your readers are familiar with, and you use it too, if possible, or at least, to start with. You build on foundations which are there and which are known by your readers.

I realise how difficult it must be! I have had these discussions with students for 45 years and once I was a student myself. I remember a student asking me (after the lecture) “does the less than or equals symbol” mean less than or equals, or less than and equals? I wish he had dared to ask it during the lecture.


Thanks Richard,

I need to see your proposal in proper math notation.

Note that I was using a formal set-builder notation (text-book stuff). I now think it clearer if I make conditional statements.

SO: This next is very clear and rigorous to me. How about you?

.......................

This is Bell’s theorem, BT: E(a,b|β) = ∫dλ ρ(λ)A(a,λ)B(b,λ) ≠ −a·b [sic]; (1)

with A(a,λ) = ±1 ≡ A^±, B(b,λ) = ±1 ≡ B^±, A(a,λ)B(b,λ) = ±1 ≡ AB in short form. (2)

REFUTATION: Via RHS (2), and independent of the functions A and B, we distribute (1)’s integrand

(identified in short form as AB = ±1) over two subsidiary integrands: AB = 1 and AB = −1.

Thus:

E(a,b|β) = ∫dλ ρ(λ)[(A(a,λ)B(b,λ)|AB = 1)−(A(a,λ)B(b,λ)|AB = −1)] (3)

= P (AB = 1) − P (AB = −1) , the weighted-sum of the binary AB results ±1. (4)

etc. etc. Gordon
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Fri Oct 16, 2020 5:48 am

Bell argues that functions A, B and rho do not exist such that E(a, b) := int d lambda rho(lambda) A(a, lambda) B(b, lambda) = - a.b for all a and b, where the functions A and B take values in the set {-1, +1}; a and b are arbitrary directions in space; lambda lies in some space of hidden variables Lambda, and rho is a probability density over that space (non-negative, integrates to 1). Naturally, one can write the integral as the sum of two integrals, one over the region where A(a, .)B(b, .) = +1, and one over the complementary region where A(a, .)B(b, .) = -1, and hence find that E(a, b) = P(A_a B_b = +1) - P(A_a B_b = -1), where we now consider Lambda together with the probability density rho as defining a probability space with probability measure P; we can now define A_a and B_b as the random variables (functions of lambda) A(a, .) and B(b, .).

Your notation is confusing since the bar "|" is usually used to mean "conditional on" in probability theory. If you want to integrate over subsets of lambda then you can indicate that by giving the integral sign a subscript (over which set do you integrate); or you can multiply the integrand by an indicator function. In your last line, you have on the right-hand side dropped the variables a and b. You write a lot of stuff which is superfluous. The short form "AB" for a random variable which also depends on a and b is dangerous and misleading. The A^+- is quite superfluous. beta is superfluous. Bell is assuming local realism. The "EPR-Bohm set-up" is irrelevant. You are going to do some mathematics, not physics.

Indeed the bar is also used in "set-builder" notation. But you are not using it to build sets. A(a, lambda)B(b, lambda) is a number. {lambda | A(a, lambda)B(b, lambda) = +1} is a set, and moreover, it is a set which in general might depend on a and b. But I don't see any curly brackets, so I don't see any set-builder sets, and anyway, one integrates *functions* over sets. You are mixing up functions and sets and this kind of mix-up is a recipe for disaster, especially when at the same time you put many symbols into your notation which are superfluous but at the same time omit symbols which are important. You end up writing something which makes no sense - there are a and b on the left-hand side, but not on the right-hand side.
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Sat Oct 17, 2020 2:37 pm

.
NEAR-FINAL VERSION -- https://vixra.org/pdf/2010.0068v4.pdf -- for the Introduction should start with :"Let β denote Bohm’s experiment in Bell (1964). Let B(.) denote Bell’s equations (.)."

All comments, from helpful to CRITICAL, etc., are welcome!

I learn much from such.

Thanks; Gordon
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Re: Bell's theorem refuted via elementary probability theory

Postby Gordon Watson » Sat Oct 17, 2020 3:04 pm

With my emphasis (to identify bits that confuse me):

gill1109 wrote:Bell argues that functions A, B and rho do not exist such that E(a, b) := int d lambda rho(lambda) A(a, lambda) B(b, lambda) = - a.b for all a and b, where the functions A and B take values in the set {-1, +1}; a and b are arbitrary directions in space; lambda lies in some space of hidden variables Lambda, and rho is a probability density over that space (non-negative, integrates to 1). Naturally, one can write the integral as the sum of two integrals, one over the region where A(a, .)B(b, .) = +1, and one over the complementary region where A(a, .)B(b, .) = -1, and hence find that E(a, b) = P(A_a B_b = +1) - P(A_a B_b = -1), where we now consider Lambda together with the probability density rho as defining a probability space with probability measure P; we can now define A_a and B_b as the random variables (functions of lambda) A(a, .) and B(b, .).

Your notation is confusing since the bar "|" is usually used to mean "conditional on" in probability theory. If you want to integrate over subsets of lambda then you can indicate that by giving the integral sign a subscript (over which set do you integrate); or you can multiply the integrand by an indicator function. In your last line, you have on the right-hand side dropped the variables a and b. You write a lot of stuff which is superfluous. The short form "AB" for a random variable which also depends on a and b is dangerous and misleading. The A^+- is quite superfluous. beta is superfluous. Bell is assuming local realism. The "EPR-Bohm set-up" is irrelevant. You are going to do some mathematics, not physics.

Indeed the bar is also used in "set-builder" notation. But you are not using it to build sets. A(a, lambda)B(b, lambda) is a number. {lambda | A(a, lambda)B(b, lambda) = +1} is a set, and moreover, it is a set which in general might depend on a and b. But I don't see any curly brackets, so I don't see any set-builder sets, and anyway, one integrates *functions* over sets. You are mixing up functions and sets and this kind of mix-up is a recipe for disaster, especially when at the same time you put many symbols into your notation which are superfluous but at the same time omit symbols which are important. You end up writing something which makes no sense - there are a and b on the left-hand side, but not on the right-hand side.


Thanks Richard, very helpful.

Please see if the version just posted meets your needs and allows you to pinpoint my errors.

Note:

1. I use β to denote Bohm's experiment, α for Aspect's experiment, γ for GHZ. So when the above 2-page pdf is absorbed into a larger paper on all 3 of them: it all fits.

2. Further, I do not understand this: "The "EPR-Bohm set-up" is irrelevant. You are going to do some mathematics, not physics." Please elaborate. As an engineer, I seem to find myself doing both.

3. Also, from the first underline above: I do not follow the need for A_a B_b, nor do I grasp their meaning and physical significance. Please elaborate.

4. Re the second underline above, please identify "the many superfluous symbols" in the latest version.

5. All of which still leaves you needing to identify an error in my refutation of BT. Also my rebuttal of Bell's inequality: for it is not changed by the new notation.

Thanks again; Gordon
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Re: Bell's theorem refuted via elementary probability theory

Postby local » Sun Oct 18, 2020 5:31 am

All of which still leaves you needing to identify an error in my refutation of BT.

The error is very obvious but before pointing it out I would like to give you a chance to find it yourself by offering you a little challenge.

You claim to have shown that Bell's integral in your equation (1) is in fact equal to -a.b. If this is so you will easily be able to show a set of functions rho(alpha), A(a,alpha), and B(b,alpha) that generate -a.b upon integration. Of course A() and B() must have the same form. Depending on what functions you propose, we will be able to verify them either by direct integration, or if that is intractible, by means of a simple event-by-event simulation generating two outcome streams that can then be correlated. Your case will be greatly strengthened by showing such a simulation. I will be happy to assist you with programming the simulation if you are not a programmer. The problem for you is that you will fail at this task. When you think about why that is the case you may find the glaring error in your proof.

People have been trying unsuccessfully for decades to use Malus-law-like detection to implement local models that generate the -a.b correlation. Yes, you can use variable detection (data discarding à la Pearle), but your proof is lacking anything like that.
Last edited by local on Sun Oct 18, 2020 6:08 am, edited 8 times in total.
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Re: Bell's theorem refuted via elementary probability theory

Postby gill1109 » Sun Oct 18, 2020 5:32 am

Gordon Watson wrote:With my emphasis (to identify bits that confuse me):

gill1109 wrote:Bell argues that functions A, B and rho do not exist such that E(a, b) := int d lambda rho(lambda) A(a, lambda) B(b, lambda) = - a.b for all a and b, where the functions A and B take values in the set {-1, +1}; a and b are arbitrary directions in space; lambda lies in some space of hidden variables Lambda, and rho is a probability density over that space (non-negative, integrates to 1). Naturally, one can write the integral as the sum of two integrals, one over the region where A(a, .)B(b, .) = +1, and one over the complementary region where A(a, .)B(b, .) = -1, and hence find that E(a, b) = P(A_a B_b = +1) - P(A_a B_b = -1), where we now consider Lambda together with the probability density rho as defining a probability space with probability measure P; we can now define A_a and B_b as the random variables (functions of lambda) A(a, .) and B(b, .).

Your notation is confusing since the bar "|" is usually used to mean "conditional on" in probability theory. If you want to integrate over subsets of lambda then you can indicate that by giving the integral sign a subscript (over which set do you integrate); or you can multiply the integrand by an indicator function. In your last line, you have on the right-hand side dropped the variables a and b. You write a lot of stuff which is superfluous. The short form "AB" for a random variable which also depends on a and b is dangerous and misleading. The A^+- is quite superfluous. beta is superfluous. Bell is assuming local realism. The "EPR-Bohm set-up" is irrelevant. You are going to do some mathematics, not physics.

Indeed the bar is also used in "set-builder" notation. But you are not using it to build sets. A(a, lambda)B(b, lambda) is a number. {lambda | A(a, lambda)B(b, lambda) = +1} is a set, and moreover, it is a set which in general might depend on a and b. But I don't see any curly brackets, so I don't see any set-builder sets, and anyway, one integrates *functions* over sets. You are mixing up functions and sets and this kind of mix-up is a recipe for disaster, especially when at the same time you put many symbols into your notation which are superfluous but at the same time omit symbols which are important. You end up writing something which makes no sense - there are a and b on the left-hand side, but not on the right-hand side.


Thanks Richard, very helpful.

Please see if the version just posted meets your needs and allows you to pinpoint my errors.

Note:


Gordon 1. I use β to denote Bohm's experiment, α for Aspect's experiment, γ for GHZ. So when the above 2-page pdf is absorbed into a larger paper on all 3 of them: it all fits.

Richard. OK.

Gordon 2. Further, I do not understand this: "The "EPR-Bohm set-up" is irrelevant. You are going to do some mathematics, not physics." Please elaborate. As an engineer, I seem to find myself doing both.

Richard. Of course you want your mathematics to be relevant to the real world. But I thought your claim is that Bell makes mistakes in logical/mathematical reasoning. "Bell's theorem" is, I would say, a piece of mathematics, with applications to physics.

Gordon 3. Also, from the first underline above: I do not follow the need for A_a B_b, nor do I grasp their meaning and physical significance. Please elaborate.

A_a is a function of lambda, the hidden variable, and it is the outcome which would have been observed by Alice if the experimenter (Alice) had chosen to fix the setting to the direction a. Your notation is clumsy /misleading /wrong. But this is in actual fact what you and Bell are talking about. In a CHSH experiment we look at A_a for two specific values of a, and B_b for two specific values of b. In one trial of the experiment, one gets to observe one of each. Their "existence" is guaranteed by "local realism".

Gordon 4. Re the second underline above, please identify "the many superfluous symbols" in the latest version.

Richard. Sorry, I am not going to do that *yet again*.

Gordon 5. All of which still leaves you needing to identify an error in my refutation of BT. Also my rebuttal of Bell's inequality: for it is not changed by the new notation.

Richard. Indeed, I'm ready to start reading the two pages, now.
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Re: Bell's theorem refuted via elementary probability theory

Postby local » Sun Oct 18, 2020 5:47 am

Richard, concentrate on the first page where BT is claimed to be refuted. The glaring error is there. In one of your earlier posts you seem to naively accept the error. Think it through more carefully.
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