Commonsense local realism refutes Bell's theorem

[Gordon Watson]

We are discussing Watson (2014, v.3). So, with that understanding, could you (gill1109) couch you criticism in more precise terms, please? For example, since every one of my equations is numbered, why not be specific and say things like this: In eqn. (99) Watson calculates the sample mean and wrongly equates it to Bell 1964:(101) ... because Bell is explicit in .... Or Watson's analysis of CHSH at eqn. (97) assumes ... contrary to their assumption ... .

Hi Gordon, welcome to the forum!

[X] I have many problems with your paper but the perhaps most important one is your transition around formula (6) from ensemble means, to averages in a particular sequence of runs. You fall into this trap through the device of trying to make things simple and discrete. So there are just n different possible values of lambda, each with the same probability. OK, no problem with that, as a pedagogical device.

But this does not mean that in n runs, one will get to see each of those n possible values exactly once, in turn. No: in each run, a new value of lambda is drawn completely at random (and "with replacement") from the list of n possibilities.

[Y] Do you know "Speakable and unspeakable"? I recommend very highly chapters 13 and 16. In chapter 13 there is even an explicit discussion of the transition from theory to experiment, namely from ensemble means to sample averages. The transition is of course only approximate (sample averages are not equal to population mean values) and it requires a random sampling assumption. This is where the random selection of settings comes in, which you do not mention anywhere, thereby missing one of the crucial components of the whole story.

Thanks Richard; and thanks for the questions. And NB: I value your criticisms! For most Bell supporters in my experience promise such and deliver nothing. So please do not let our differences have you withdraw from a good clean tussle -- me with Einstein in my corner; you with JB? -- but we cannot both be right:

Re [X]: I'm not clear about the TRAP?

(6), and ALL my analyses are exactly as any reasonable statistician would require, and more; please see the sentence that introduces (6). Let me know your response, please. Note that in basic CLR there are 2(wn+i) values of lambda TESTED; w, n, i unlimited.

You write: "But this does not mean that in n runs, one will get to see each of those n possible values exactly once, in turn. No: in each run, a new value of lambda is drawn completely at random (and "with replacement") from the list of n possibilities."

Which is what CLR allows IFF you would change the last phrase to read: ... from the infinity of possibilities.

NB; under CLR: P(replacement|EPRB) = 0; for the set is infinite.

Richard; there is no limit on any setting in CLR, so you are free to select any that you like. SO: In this instance, and (I'm sure) in any other: there is NO "thereby missing one of the crucial components of the whole story."

Re [Y]: I know Bell's "Speakable" -- and just wish his supporters had included a comprehensive Index; as well as his "Speakings"! The latter similarly unimpressive under my analysis. The former a remarkable lapse given the number of his devotees. DO you by chance know of such an Index? (I've often offered to co-operate in the production of such -- to be freely available online -- no takers so far.)

Chapters 13 and 16 are equally flawed under CLR.

Richard; to help us both progress, how about we start with your first specific niggle in the paper and proceed from there.

I appreciate your focus on (6), and trust that your concerns will soon be allayed via that introductory sentence. If not; please, just come back with one specific objection: I'm not beyond missing something.

Thanks again; Gordon

[gill1109]

Gordon, are you aware of the difference between a sample average and a population mean? Do you know the one over root N law, and are you familiar with the concept of error bar, standard error?

What is your favourite textbook on statistics?

Bell's book can be obtained from Amazon (Kindle eBook) for 30 dollars or something like that. You can find the table of contents at the publisher's web pages (Cambridge University Press). Most of the chapters appeared as articles in various journals and many are available free, online, though unfortunately not (as far as I know) Chapter 13.

I am not a "Bell supporter". I have read his work carefully and admire it. Like everyone else, he is not perfect. As in Borge's story about the maps, any summary of Bell's work which does it full justice would be at least as long as the original. So let's go back to the original, please, to the basis. I don't want to argue about the idiot things which so-called Bell supporters have said. About the corners they have cut off, about their own misconceptions. Remember: there is no "Bell's theorem" in Bell's work. There is a description of a thought experiment, and there is a list of four positions which one might like to argue, given the earlier discussion. The list is not exhaustive.

From experience, I am learning that there before getting into details about formula (6) or formula (7) or whatever, we need to establish some common language and common background. If we go into details, I will probably refer frequently to passages, formulas, and illustrations in chapters 13 and 16 of "Speakable and unspeakable" and also to concepts and results from statistics and probability. Sometimes Wikipedia will be adequate for this latter purpose, but it would be useful to have a decent textbook to hand which we both know and appreciate.

You might also find it useful, if you want to discuss things with me, to take a look at some of my own work. I refer you especially to:

Accardi contra Bell (cum mundi): The Impossible Coupling
https://arxiv.org/abs/quant-ph/0110137
Appeared in the IMS monograph series (Institute of Mathematical Statistics) though actually the specific volume in question is a collection of contributions by many different authors. Invited, peer reviewed. Pretty tough, careful, thorough review process, spanning about two years.

Time, finite statistics, and Bell's fifth position
http://arxiv.org/abs/quant-ph/0301059
Appeared in a Vaxjo conference proceedings volume. Invited, presented, discussed. Little formal peer review.

Statistics, causality, and Bell's theorem
https://arxiv.org/abs/1207.5103
To appear in a special issue on causality in the journal Statistical Science.
Invited and peer reviewed. Very, very tough review procedure, I can tell you! Highly critical reviewers both from physics and from mathematics. Gone through four revisions already, spanning probably (when the process is complete) four years.

However these works are mainly written with an audience in mathematical statistics and probability theory in mind, and only to a lesser extent physicists or philosophers of science or the general public. So they contain a lot of hidden assumptions as to background knowledge in the mind of the reader.

[Gordon Watson]

Gordon, are you aware of the difference between a sample average and a population mean? Do you know the one over root N law, and are you familiar with the concept of error bar, standard error?

What is your favourite textbook on statistics?

Bell's book can be obtained from Amazon (Kindle eBook) for 30 dollars or something like that. You can find the table of contents at the publisher's web pages (Cambridge University Press). Most of the chapters appeared as articles in various journals and many are available free, online, though unfortunately not (as far as I know) Chapter 13.

I am not a "Bell supporter". I have read his work carefully and admire it. Like everyone else, he is not perfect. As in Borge's story about the maps, any summary of Bell's work which does it full justice would be at least as long as the original. So let's go back to the original, please, to the basis. I don't want to argue about the idiot things which so-called Bell supporters have said. About the corners they have cut off, about their own misconceptions. Remember: there is no "Bell's theorem" in Bell's work. There is a description of a thought experiment, and there is a list of four positions which one might like to argue, given the earlier discussion. The list is not exhaustive.

From experience, I am learning that there before getting into details about formula (6) or formula (7) or whatever, we need to establish some common language and common background. If we go into details, I will probably refer frequently to passages, formulas, and illustrations in chapters 13 and 16 of "Speakable and unspeakable" and also to concepts and results from statistics and probability. Sometimes Wikipedia will be adequate for this latter purpose, but it would be useful to have a decent textbook to hand which we both know and appreciate.

Hi Richard, and thanks for spelling out your position and providing extra information.

I tend not to have favourite books and I'm not at all sure that we need go anywhere near error bars, etc, as I'll explain below. But, if it should somehow become essential that we move beyond Wikipedia: you could give me some favoured-book-details from your collection and I could attempt to get one of them. However, for my part: any difficulties or likely errors in my essay do not require any more statistics than I discuss below.

Re "Speakable": Yes, the book itself has a Table of Contents; but I was suggesting that we would all benefit from a comprehensive Index; of names, phrases, key words ++. Surely someone has done it?

Now, re your specific question: I tend to use the words sample mean and population mean. (Is there a technical basis for your use of "average"?) And I expect the sample mean to approach the population mean as the sample size increases. So, do you see that I have covered this by the line of text that precedes (6)? To be clear: I allow you to choose any n that delivers an accuracy adequate for your purposes; and I will accept that n if it is reasonable. NB: You do see that, if n=10^6, then <AB> would be averaged over 10^6 tests, and <AC> would be averaged over another 10^6 tests: there being no requirement (and certainly no expectation) that "replacement" occurs.

So, since (6) is pure text-book and rigorously based on Bell 1964:(14a), it seems to me that you need now to satisfy yourself re (7).

PS, added: Of all Bell's "positions" I take my own: Bell's theorem is nonsense; a positioned reinforced by his errors -- identified and corrected in my essay -- all my results being experimentally valid (in accord with quantum theory).

[gill1109]

Re "Speakable": Yes, the book itself has a Table of Contents; but I was suggesting that we would all benefit from a comprehensive Index; of names, phrases, key words ++. Surely someone has done it?

Now, re your specific question: I tend to use the words sample mean and population mean. (Is there a technical basis for your use of "average"?) And I expect the sample mean to approach the population mean as the sample size increases. So, do you see that I have covered this by the line of text that precedes (6)? To be clear: I allow you to choose any n that delivers an accuracy adequate for your purposes; and I will accept that n if it is reasonable. NB: You do see that, if n=10^6, then <AB> would be averaged over 10^6 tests, and <AC> would be averaged over another 10^6 tests: there being no requirement (and certainly no expectation) that "replacement" occurs.

So, since (6) is pure text-book and rigorously based on Bell 1964:(14a), it seems to me that you need now to satisfy yourself re (7).

I have no idea if anyone has provided the comprehensive index you would so much appreciate.

There is a technical or you could say pedagogical basis for my use of the word average. I wish to powerfully emphasize the difference between an average computed on a sample of numbers, e.g., numbers coming out of an experiment, by the formula x bar equals 1 over n summation over i runs from 1 to n of x subcript i; and a population mean value which is a somewhat more abstract concept though it might be understood in some double thought experiment sense: suppose we repeated a certain thought experiment infinitely often then ...

Very glad you know about the law of large numbers.

I agree that in experimental physics, <AB> would be approximated by an average over 10^6 runs and <AC> by an average over another 10^6 runs. But the theoretical numbers <AB> and <AC> come out of theory and refer to a weighted average over possible values of some physical quantity, weighted by the relative frequency of that quantity. Moreover, what that physical quantity actually is, depends on theory. According to one theory, there is an underlying physical quantity lambda involved. According to another theory, such a quantity "lambda" does not exist at all. In fact, the real experiment which we would like to do, is done precisely in order to compare those two theories.

To give an example. Suppose that there is a theory, that the mean difference in height between married men and women within US Catholic married couples is at least 3 centimeters. Then an experimenter might randomly sample 100 married US Catholic men and compute their average height, randomly sample 100 US Catholic women and compute their average height, look at the difference (say: 4.2 cm), report the standard deviation, and claim that his theory has been resoundingly vindicated. Though none of the men in his sample are married to any of the women in his sample.

[Gordon Watson]

You might also find it useful, if you want to discuss things with me, to take a look at some of my own work. I refer you especially to:

Accardi contra Bell (cum mundi): The Impossible Coupling
https://arxiv.org/abs/quant-ph/0110137
Appeared in the IMS monograph series (Institute of Mathematical Statistics) though actually the specific volume in question is a collection of contributions by many different authors. Invited, peer reviewed. Pretty tough, careful, thorough review process, spanning about two years.

Time, finite statistics, and Bell's fifth position
http://arxiv.org/abs/quant-ph/0301059
Appeared in a Vaxjo conference proceedings volume. Invited, presented, discussed. Little formal peer review.

Statistics, causality, and Bell's theorem
https://arxiv.org/abs/1207.5103
To appear in a special issue on causality in the journal Statistical Science.
Invited and peer reviewed. Very, very tough review procedure, I can tell you! Highly critical reviewers both from physics and from mathematics. Gone through four revisions already, spanning probably (when the process is complete) four years.

However these works are mainly written with an audience in mathematical statistics and probability theory in mind, and only to a lesser extent physicists or philosophers of science or the general public. So they contain a lot of hidden assumptions as to background knowledge in the mind of the reader.

Thanks for these, Richard.

First; because maybe urgent: I wonder if this needs to be corrected in your Abstract https://arxiv.org/abs/1207.5103 : "Violation of Bell's inequality in experiments such as that of Aspect et al. (1982) provides empirical proof of non-locality in the real world."

I wonder because this reads to me as if you are presenting a fact and not just a popular (and erroneous) opinion. My reading appears to be warranted by your own acceptance of locality -- something that we agree on!

Second: I don't wonder anymore that you come to my essay with extreme doubt. ALMOST every issue that you discuss in your diligent articles is totally irrelevant to my position.

WHY? Because I spotted the error in Bell's analysis on my first reading about it in Mermin (1988). So I have never considered (nor employed) loopholes and 101 other extraneous and irrelevant possibilities.

To me it was always clear that Bell's theorem and his analysis is just plain wrong.

The surprise for me came later: when I saw the chain of errors attaching to his EPR-work over his entire lifetime.

The further surprise remains as a search: Who else has found them?

PS: MY essay is wholly based on analysing Bell's EPR-work from "first-principles" -- using little more than undergraduate maths and logic. So its "errors" (IF ANY) will be found at that level of analysis. NOT in any use of loopholes; time-sequences; coincidence errors; fancy/erroneous statistics; etc. In a nutshell: no favourite books required!

[Gordon Watson]

<SNIP>
I agree that in experimental physics, <AB> would be approximated by an average over 10^6 runs and <AC> by an average over another 10^6 runs. But the theoretical numbers <AB> and <AC> come out of theory and refer to a weighted average over possible values of some physical quantity, weighted by the relative frequency of that quantity. Moreover, what that physical quantity actually is, depends on theory. According to one theory, there is an underlying physical quantity lambda involved. According to another theory, such a quantity "lambda" does not exist at all. In fact, the real experiment which we would like to do, is done precisely in order to compare those two theories.
<SNIP>

Richard; I understand your expertise in statistics, but must again insist: they are irrelevant to any possibility of error in my theory.

I must again stress the extreme simplicity of my work.

(1) We study EPRB-correlations and seek causal agents, beables, just like Bell.

(2) And just like Bell, we use the good old Greek λ as a surrogate -- let it represent "latent = hidden".

(3) We find λ, A NATURAL PHYSICAL VARIABLE associated with spin. NB: We even see spin 1/2 and spin 1 in our results.

(4) Our results are at one with the views of Einstein and EPR.

(5) Moreover, we can predict with certainty the results of experiments. ...

(6) NO knowledge beyond undergrad know-how is required.

[gill1109]

First; because maybe urgent: I wonder if this needs to be corrected in your Abstract https://arxiv.org/abs/1207.5103 : "Violation of Bell's inequality in experiments such as that of Aspect et al. (1982) provides empirical proof of non-locality in the real world."
I wonder because this reads to me as if you are presenting a fact and not just a popular (and erroneous) opinion. My reading appears to be warranted by your own acceptance of locality -- something that we agree on!

I am presenting a popularly accepted opinion. Moreover this is the abstract of a long paper where the many subtleties involved in making any sense out of that popularly accepted opinion are laid out on the table.

[gill1109]

I must again stress the extreme simplicity of my work.

(1) We study EPRB-correlations and seek causal agents, beables, just like Bell.

(2) And just like Bell, we use the good old Greek λ as a surrogate -- let it represent "latent = hidden".

(3) We find λ, A NATURAL PHYSICAL VARIABLE associated with spin. NB: We even see spin 1/2 and spin 1 in our results.

(4) Our results are at one with the views of Einstein and EPR.

(5) Moreover, we can predict with certainty the results of experiments. ...

(6) NO knowledge beyond undergrad know-how is required.

I stress the essential simplicity of mine. I've pointed out a major conceptual error in yours, very near the beginning. Good luck trying to explain the whole thing to other people. Or alternatively, just win the QRC or one of my many bets and challenges. Actions speak louder than words.

[minkwe]

I agree that in experimental physics, <AB> would be approximated by an average over 10^6 runs and <AC> by an average over another 10^6 runs. But the theoretical numbers <AB> and <AC> come out of theory and refer to a weighted average over possible values of some physical quantity, weighted by the relative frequency of that quantity. Moreover, what that physical quantity actually is, depends on theory. According to one theory, there is an underlying physical quantity lambda involved. According to another theory, such a quantity "lambda" does not exist at all. In fact, the real experiment which we would like to do, is done precisely in order to compare those two theories.

Gordon,
The problem is Richard does not understand that lambda varies from particle to particle. He needs lambda constant so that he can apply his statistics to calculate a population mean. He fails to understand that no two particles are alike in your description.

[gill1109]

I agree that in experimental physics, <AB> would be approximated by an average over 10^6 runs and <AC> by an average over another 10^6 runs. But the theoretical numbers <AB> and <AC> come out of theory and refer to a weighted average over possible values of some physical quantity, weighted by the relative frequency of that quantity. Moreover, what that physical quantity actually is, depends on theory. According to one theory, there is an underlying physical quantity lambda involved. According to another theory, such a quantity "lambda" does not exist at all. In fact, the real experiment which we would like to do, is done precisely in order to compare those two theories.

Gordon,
The problem is Richard does not understand that lambda varies from particle to particle. He needs lambda constant so that he can apply his statistics to calculate a population mean. He fails to understand that no two particles are alike in your description.

No two samples from the same standard normal distribution are alike. Not any single observation in the first sample is the same as one in the second sample. Not ever. Yet both of their averages are close to the common population mean value (zero) with large probability, if the two samples are both large. Michel Fodje not only believes that there is no use whatsoever in thinking about experiments which have not yet actually been done (as far as we know) in any terrestrial laboratory in the past thousand years, he also has a lot of quite remarkable home-grown wisdom concerning statistics. Altogether quite an unusual view on science. He really ought to study Jaynes' book. And after that, a book about applied statistics.

[Heinera]

Since this is related to the issue in this thread, I quote from two posts in another thread:

Just calculate the correlations separately and sample without replacement. Enough is enough! What are you afraid of? That is the starting point anything else is just garbage. Until then see ya.

Later in the same thread Richard replied:

And here are some new results, with many thanks to Zen for new ideas and new result: running [Joy's] program *four* times, to make four separate files, each analysed separately, the four correlations were:

-0.75156, 0.24978, -0.75114, -0.24862

whose absolute values add up to 2.0011. Not exactly 2. But not far off (about 1 / sqrt N off target?).

Eagerly awaiting Michel's response, that somehow never turned up in the original thread.

[Joy Christian]

And here are some new results, with many thanks to Zen for new ideas and new result: running [Joy's] program *four* times, to make four separate files, each analysed separately, the four correlations were:

-0.75156, 0.24978, -0.75114, -0.24862

whose absolute values add up to 2.0011. Not exactly 2. But not far off (about 1 / sqrt N off target?).

These are wrong results, invented by Richard Gill. The correlations calculated by Richard Gill contradict what is evident form my simulation.

The correct calculations consistent with what is manifest in the simulation are given here: http://rpubs.com/jjc/16531. They look like this:

[minkwe]

Since this is related to the issue in this thread, I quote from two posts in another thread:

Just calculate the correlations separately and sample without replacement. Enough is enough! What are you afraid of? That is the starting point anything else is just garbage. Until then see ya.

Later in the same thread Richard replied:

And here are some new results, with many thanks to Zen for new ideas and new result: running [Joy's] program *four* times, to make four separate files, each analysed separately, the four correlations were:

-0.75156, 0.24978, -0.75114, -0.24862

whose absolute values add up to 2.0011. Not exactly 2. But not far off (about 1 / sqrt N off target?).

Eagerly awaiting Michel's response, that somehow never turned up in the original thread.

Heinera, What has that got to do with this thread, which is discussing Watson's paper. If you want to taunt me, do it in the appropriate thread. Have you learnt Richard's bad habbit of forum thread stalking?

[Heinera]

That thread is closed, for reasons completely unrelated to the two posts I quoted above. So this was pretty much the only way to do it. And I do not want to taunt you; I am genuinely interested in your response.

[minkwe]

That thread is closed, for reasons completely unrelated to the two posts I quoted above. So this was pretty much the only way to do it. And I do not want to taunt you; I am genuinely interested in your response.

There are other more appropriate open threads about Richard's various bets. This is not the "only way to do it".

[Heinera]

That thread is closed, for reasons completely unrelated to the two posts I quoted above. So this was pretty much the only way to do it. And I do not want to taunt you; I am genuinely interested in your response.

There are other more appropriate open threads about Richard's various bets. This is not the "only way to do it".

Ok. So to make this quick, what thread do you suggest?

[minkwe]

he also has a lot of quite remarkable home-grown wisdom concerning statistics. Altogether quite an unusual view on science

Unable to argue the merits of arguments infront of him, Richard Gill often resorts to delusional ad-hominem, attacks about the background of people he has no information on. He has made it his prime mission to assassinate the character of anyone who dares disagree with him. Shameful indeed.

Now we were talking about Watson's paper, an why there is no confusion between sample mean and population mean in Watson's paper. Watson is analyzing the EPR experiment, the same one Bell purports to be analyzing. Watson makes sure that his analysis respects the requirements of performable experiments. Bell's analysis failed in that respect. Watson's argument is to point out the disconnect between Bell's analysis, and performable experiment. Richard Gill instead is criticizing Watson for failing to preserve the impossibility inherent in Bell's analysis.

He says silly things like:

Different sets of infinitely many runs. But the same values of lambda will turn up in both series, and they'll both turn up equally often.

Even though Watson clearly says in his paper that each lambda is unique, ie P(lambda_i) = 1/N.

I'm beginning to suspect the LG paper was mostly the work of Larsson:

http://arxiv.org/pdf/quant-ph/0312035v2.pdf
The problem here is that the ensemble on which the correlations are evaluated changes with the settings, while the original Bell inequality requires that they stay the same. In effect, the Bell inequality only holds on the common part of the four different ensembles ΛAC′ , ΛAD′ ,ΛBC′ , and ΛBD′

Otherwise, how can somebody who writes and publishes the above take any issue with Watson's analysis in equations (6-15)?

[gill1109]

That thread is closed, for reasons completely unrelated to the two posts I quoted above. So this was pretty much the only way to do it. And I do not want to taunt you; I am genuinely interested in your response.

There are other more appropriate open threads about Richard's various bets. This is not the "only way to do it".

Ok. So to make this quick, what thread do you suggest?

The bet between Gill and Christian on the outcome of Christian's experiment is cancelled.

Though actually, if someone ever does do the experiment, and if the two data files would win the bet, I would pay up anyway - I am a gentleman. So if I'm proved wrong I'll accept it and accept the moral consequences, too. Is that clear?

What is still very much on "go" is a computer simulation challenge.

The precise task to be achieved is described here:

http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=52&p=1896&view=show#p1898

This is not a bet. It's a challenge, of the Randi Foundation type. It's open, anyone can take up the challenge. First come first served. It's one-sided. If you submit a claim to the prize but your claim is judged invalid, you do not have to pay anything. The jury's verdict is final. No appeal, no correspondence with the jury is allowed.

Christian however has a prior claim, since today he submitted the data, so now it is basically my move. If I determine that his submission was unsuccessful, then it goes to arbitration.

As I said, first come first served. So if Joy's attempt fails, other people can try later ... what are you waiting for?

I am really interested in Michel Fodje's view of the challenge. He already said that Joy should not accept the bet as then formulated, because as it was then formulated, Christian was certain to lose (I agreed with him, by the way). The challenge has the same formulation as the bet, regarding the analysis of the data! The only difference is that the challenge-data may be simulated, just thought up, borrowed, or stolen, or anything ... how the data is processed is unchanged.

[gill1109]

Now we were talking about Watson's paper, and why there is no confusion between sample mean and population mean in Watson's paper. Watson is analyzing the EPR experiment, the same one Bell purports to be analyzing. Watson makes sure that his analysis respects the requirements of performable experiments. Bell's analysis failed in that respect. Watson's argument is to point out the disconnect between Bell's analysis, and performable experiment. Richard Gill instead is criticizing Watson for failing to preserve the impossibility inherent in Bell's analysis.

Unfortunately, the so-called "disconnect" in Bell's analysis in his early paper is nicely "connect" in his later more full explanations. Chapters 13 and 16 of "Speakable and unspeakable ...". I think I already even quoted the relevant paragraph so I won't do it again.

You can find Chapter 16 on internet (CERN preprint). Unfortunately, for Chapter 13 you have to pay a modest fee to Cambridge University Press some of which hopefully goes to the heirs of the late John S. Bell's estate. Probably Allain Aspect gets a cut too since he wrote the preface to the latest edition.

It's pretty clear to me that neither Watson nor Fodje are aware of this material. Their mistake is very easy to make; indeed, many earlier writers made it earlier, too. The later chapters of Speakable and Unspeakable were specifically written to answer various issues which had arisen on publication of the first key paper. A lot of which was of course just "noise" caused by simple misunderstandings. Sad that these misunderstandings are so prevalent to this day.

[harry]

[..]
To give an example. Suppose that there is a theory, that the mean difference in height between married men and women within US Catholic married couples is at least 3 centimeters. Then an experimenter might randomly sample 100 married US Catholic men and compute their average height, randomly sample 100 US Catholic women and compute their average height, look at the difference (say: 4.2 cm), report the standard deviation, and claim that his theory has been resoundingly vindicated. Though none of the men in his sample are married to any of the women in his sample.

I have now read parts of that paper; and much of it I find hard to follow. However, section 4 was not too hard, and looks very convincing. Thus, in the context of this topic, I find the above example by Gill as argument against the paper surprising as it - especially the last sentence- is not far from a pertinent point of Gordon's paper.

Borrowing from your example, making it more fitting:

Suppose that there is a theory that the mean difference in height between married men and women within US Catholic married couples is at most 3 centimeters. Say that Alice is married to Bob, Ann is married to Ben, and so on. Then a theorist might propose to measure married US Catholic couples and compute their average heights. He asserts that the use of commonsense assumptions leads to a prediction of more than 4 cm difference. However in order to reach that conclusion he compares Alice to Ben, Ann to Bob (and so on); none of the men in his set are married to any of the women in his set. Next an experimentalist samples 1000 couples and finds an average difference of say 2.8 cm, reports the standard deviation, and claims that commonsense has been resoundingly disproved.

Do you agree with me that in such a case the mathematics of the theorist is erroneous?