minkwe wrote:Gordon Watson wrote:Eqn (9) in my essay is
Bell's inequality for EPRB: its upper-bound is 1.
Eqn (10) is my inequality for EPRB: its upper-bound is 3/2.
Now (10) provides -- precisely -- a one-to-one correspondence of "my variables to the variables of the original relationship".
Gordon Watson wrote:
It should be obvious from the contradiction between (9) and (10) in your paper that there is no one-to-one correspondence. E(a,b) is incompletely specified. What exactly is E(a,b) in equation (9)?
E(a,b) is defined in ¶2.1 of my essay. It is Bell's P(a,b) and is intended by him to be under EPRB.
minkwe wrote:What you have done is to claim that there is one-to-one correspondence, and therefore Bell must have made an error. In other words, you are assuming one-to-one correspondence and then concluding that there is an error in Bell's mathematics. That may very well be the case, but if that is granted, the effect of the error is to break the one-to-one correspondence.
I am comparing apples with apples. The 3 terms in my (9) are the 3 terms in Bell's (15). The grub in Bell's apple does not change the one-to-one correspondence; it simply means that no one -- who understands apples --- buys it!
minkwe wrote:What I see in fact is that Bell's mathematics is correct, but his error is in failing to preserve the one-to-one correspondence between his mathematics and the induction made by comparing the mathematics with QM.
Bell is offering his math under the auspices of EPRB. He fails under EPRB without any reference to QM; the more so since the correct EPRB result can be derived without reference to QM.
minkwe wrote:Although the results are the same, I believe this analysis is more accurate than yours because Bell's assumptions in Bell 1964 implies an assumption of counterfactual definiteness that will not be meaningful were he following the logical steps you impute to him.
Since when is counterfactual definiteness allowable under EPRB? You are making my point here: Bell's erroneous assumption takes him to settings less-correlated than EPRB. So his critique of Einstein, etc., under EPRB is based on an experiment that is NOT EPRB.
minkwe wrote: Thus, Bell starts out with paired relationships drawn from a set of triples and ends up making comparisons with paired relationships drawn from three disjoint sets of pairs. Whereas his mathematics and resulting inequalities are valid for paired relationships drawn from a set of triples, the variables from paired relationships drawn from three disjoint sets of pairs (like from QM and experiments) do not have one-to-one correspondence to those representing paired relationships drawn from a set of triples.
Whatever Bell starts with, he finishes up BEYOND the EPRB BOUNDARY-CONDITIONS. You are describing the EPRB experiment -- disjoint pairs -- which Bell claims to critique.
minkwe wrote:Although the results are the same, I believe this analysis is more accurate than yours because Bell's assumptions in Bell 1964
(i): So what do you mean 'we need further “coordinates” to complete the specification'?
This means you need to label each E(a,b) with coordinates fully specifying all relevant facts of how they are obtained. For example your equation (9) will become.
And your equation (10) will become
See
https://iopscience.iop.org/article/10.1 ... 60007/meta
E(a,b) is perfectly clear under EPRB; the experimenters know exactly what to measure. NB: Bell's nominated context is EPRB; his erroneous analysis is NOT relevant to EPRB. That is the point of my essay. Whose analysis under EPRB is valid: mine (agreeing with elementary algebra and experiment) or Bell's which fails both tests?
Well Aspect (2004) -- see the link in my essay -- is a real experiment with real measurements. And my generalized inequality -- see fn.5 and my eqn (10a) there that covers Aspect (2004) as well-- again has a one-to-one correspondence of "my variables to the variables of the original relationship".
minkwe wrote:Precisely, equation (10) is meaningful for actual measurements, whereas, equation (9) is meaningless as far as measurements are concerned. The variables in (9) can never be measured.
Of course (9) can be measured under EPRB-basics; eg, when adjusted and tested via Aspect's experiments. You seem to be saying that the experimenters are saying that they did not test (9) or its equivalent. But they routinely take the EPRB expectations, they show that Bell erred, AND they confirm my results!
minkwe wrote:PS: WRT your hemisphere analogy, I suggest Bell's claim is akin to this: all spherical triangles have an upper-bound of angles summing to 180º. And his first [and inappropriate] assumption confines his inequality to 2D planes. Which, as with his EPRB/Aspect inequalities, means his spherical claim is sometimes valid. BUT false in general!...
It means simply that there is no violation, just lack of correspondence of the terms.
So Bell's analysis has terms that lack correspondence with the underlying EPRB physical reality. You again appear to be making my point; which will soon be reinforced via Bell's critique of von Neumann.
minkwe wrote:The most devastating criticism of Bell was delivered by Bell himself before Bell 1964 was ever published.
John S. Bell wrote:Thus the formal proof of von Neumann does not justify his informal conclusion. ....... It was not the objective measurable predictions of quantum mechanics which ruled out hidden variables. It was the arbitrary assumption of a particular (and impossible) relation between the results of incompatible measurements either of which might be made on a given occasion but only one of which can in fact be made.
This is well known. But you are giving it a new twist -- it seems to me -- and one that helps make my case.
Bell knew of this defect before he published his 1964 essay. So you are confirming that he published his EPRB-based essay with the same defect.
So we agree: He did repeat von Neumann's mistake,
but only because he did not recognize his own errors under EPRB-- as my essay is intended to make clear.
.