minkwe wrote:Justo wrote:There are 16 different values of hidden variables , the numbers are the indices numbering them.

Hidden variables are not measured, let alone numbered in experimental data. You are assuming the availability of information that is not available in experiments.

Of course, hidden variables are not measured. What the reasoing does is to show that if they existed the result obtained with the experimental values would have to be bounded by 2. Since the experimental result exceeds 2 the conclusion is that they cannot exist. That is why the experiments falsify the existence of the hidden variables.

The inequality is evaluated only with the measured data. The results falsify your assumptions.

minkwe wrote:Justo wrote:What I have shown is that "each correlation function pertains to a different set of particle pairs." therefore they are weakly objective, and yes, of course, you can arrange them in a single 4xN (N=16) spreadsheet. The essential point is the four columns pertain to four different series of actual experiments.

You have not. You claim to but then you make an implicit assumption that the weakly objective case is the same as the strongly objective case (eq 24), and then proceed with the strongly objective case.Justo wrote:You keep saying that but you just cannot say where, in what step, my reasoning fails or is wrong

I've mentioned several times and referenced the transition from equation (23) to (24). How can you claim that I have not shown where? You may disagree with me but to suggest that I haven't explicitly said where and how is just wrong.

I am sorry but I just can't make sense of that. Accepting that (23) is built up with actually measured experimental results (weakly objective), why is it incorrect to apply mathematical properties to put it in an equivalent form?. Of course, you do not measure (24), it is a mathematically equivalent form obtained with measured results. That is how mathematics is used to derive results all the time: you transform a given expression to an equivalent form where is it easy to evaluate and draw useful conclusions that were not evident in their original form.

minkwe wrote:Conceptually, your equation (23) is equivalent to 4 independent 2xN spreadsheets of outcomes. Your equation (24) is equivalent to a single 4xN spreadsheet of outcomes. And your equation (29) is equivalent to a single row from the single 4xN spreadsheet in equation (24). You can't derive the inequality without putting those numbers into a single 4xN spreadsheet as you did in equation (24). In fact, a spreadsheet that is not a 4xN spreadsheet can't be expected to obey the inequality. When you obtain experimental data you can't just compare it with the inequality without first converting the data to a 4xN spreadsheet. If this was not a necessary step, you would not have needed to do it before deriving the inequality.Justo wrote:Yes, of course, that is what the derivation is all about. To prove that (23) is equivalent to 4 independent 2xN spreadsheets that reduce to one 4xN spreadsheet with actual data, i.e., "weakly objective".

So let me ask you a simple question. If I take 4 2xN spreadsheets from a weakly objective experiment, will you expect it to obey the inequality you derive without first being converted into a single 4xN spreadsheet? This is the key question.

Yes, I agree this is the key question. The answer is yes because they are mathematically equivalent unless you reject the laws of arithmetic. That is why the Bell theorem is a very elementary result. You just need arithmetic to understand it. You do not need to use delta functions, Fourier transforms, complex variables, etc. just arithmetic.

What happens is that if you do not transform the values contained in the 4 2xN spreadsheets into a mathematically equivalent arrange, the result is not evident. For instance, if I give you 100 numbers from 1 to 100 in a random configuration you cannot guess they are sequential numbers unless you order them. If you want to know the sum, you just order them and apply the formula for an arithmetic progression. That does not change the original value of the sum. It is just that simple end evident.

To evaluate the bound of the Bell inequality you transform the data contained in four 2xN spreadsheets into a single "ordered" 4xN spreadsheet so that you can obtain the bound of the inequality. You do not change the original experimental values, just apply to them valid arithmetic rules, namely, commutativity, associativity, and distributivity. There are no further intricacies involved.

What Adenier objected to is not that you cannot use arithmetic to evaluate the Bell inequality. He objected to the fact that the single 4xN spreadsheet does not contain real experimental values because the four columns are obtained from a single pair of particles (strongly objective interpretation).

What I proved is that, although Adenier is correct when saying that such interpretation is meaningless, the 4xN spreadsheet can be understood weakly objectively, i.e., containing results from 4 2xN spreadsheets of real experimental values, so it does make sense.