Mathematics: Science of the Possible, or the Probable? by Thomas Howard Ray

An enjoyable essay Tom..

It took me a while to work through this paper, but it appears your logical reasoning is solid, even though I found some portions confusing. There is a lot I agree with, though it runs counter to prevailing opinion. I especially like that you wove Euler's equation into the story in such a meaningful way. I'll likely have more to say, but a ratings boost is all for now.

Regards,

Jonathan

Hi Jonathan,

Good to hear your take on this. I do believe that the proof Albert Jan did was all done analytically here on FQXi in the debates of the past. But now there is proof via a geometric algebra computer program that Joy's classical realistic model does in fact produce the prediction of QM, -a.b.

FQXi is going to have some real embarrassment to deal with in the not too far future concerning their so called panel of experts that claimed the model was wrong. The model works as advertised.

Hi everyone

I see there is some discussion here of Albert Jan Wonnink's GAViewer program with which he attempted to verify one line in the first "Bell refutation" paper of J J Christian ... from way back in 2007: quant-ph/0703179

He immediately ran into Christian's trademark algebraic error. This is the (-1)^2 = -1 error which Christian needs in order to make embarassing bivectorial terms cancel out of his "correlation" (the answer has to be scalar, right?).

Of course it is easy to "fix" that mistake locally, by an ad hoc subtraction of what you don't want to have. He shows what you have to subtract to Christian's (17) in order to make the left hand side equal to the right hand side.

What Albert Jan *didn't* do is simulate the whole model. His computer program verifies a patched version of formulas (17) and (19) of quant-ph/0703179. So on the one hand, he shows that Christian's original math was wrong. On the other hand, he still has not checked whether the patch which he introduces to fix the gap between LHS of (17) and RHS of (19) is consistent with the rest of the story. After all, if you change the very definition of geometric product locally in one formula in a complex story, that might have repercussions elsewehere, right?

In fact, Albert Jan hasn't addressed the challenge yet of actually generating the measurement outcomes A_n(mu) of Christian's equation (16). Most readers, on a superficial reading, would suppose that (16) was a definition of the two measurement functions A(a, mu) and B(b, mu). However if you read carefully it is not a definition at all: in order to make it a definition one still has to specify the mapping from bivectorial measurement outcomes to {-1, +1}. Many are possible. None, of course, can deliver the goods ... because of Bell's theorem.

It's at this point that one sees the other major error in Christian's attempts to refute Bell: he computes some kind of bivectorial correlation between the outcomes represented as points in S^2, instead of the correlation between the corresponding values +/- 1. Bell is about experiments with binary outcomes. His "correlation" is just the probability that Alice's outcome equals Bob's, minus the probability that it doesn't. Quantum mechanics predicts probabilities; experimentalists observe relative frequencies. The problem is how to explain the relative frequencies?

In Christian's one page paper, four years later, it is clear that he has seen that there was something missing. He does give a definition of the measurement outcomes and now comes up with his daring bivectorial Pearson correlation instead of the "straight" correlation between the binary outcomes. After all, the labelling of the outcomes +/-1 is just convention.

His definition of the measurement functions is A(a) = -B(b) = +/- 1 (with equal probabilities for the two possibilities). Thus his model predicts that the correlation is -1. Bob's outcome is always opposite to Alice's.

"Out of the frying pan into the fire".

I have written up a postmortem which includes a tutorial section on geometric algebra, so that no one has any excuse any more not to be able to work through the math of these classics, line by line, and check everything for themselves.

http://www.math.leidenuniv.nl/~gill/GA.pdf

Amusingly, I was not allowed to post this on arXiv.org: it was seen as a personal attack. I must say that my earlier drafts had a title which was a little bit over the top. Now the paper is simply entitled "Does Geometric Algebra provide a loophole to Bell's Theorem?". The answer is of course, "no".

Showing that geometric algebra provides an alternative and beautiful way to describe the maths of spin half, including several spin half particles, entanglement and all that, was one of David Hestenes' greatest achievements. Two whole chapters are devoted to this topic in the textbook of Doran and Lasenby. They are worth careful study.

I will give some further explanation about what Albert Jan did with GAViewer so that others are not hoodwinked by Gill's misrepresentations.

The geometric algebra program GAViewer is fixed in a right handed bivector basis. So in order to see the left handed bivector basis part of Joy's model, all one has to do is reverse the order of the geometric product AB to BA. IOW, from a right handed perspective, one sees the order reversed for the left handed part. Lambda simply toggles between the two orders to correctly represent the model from the right handed only perspective. Pretty easy to understand and the model works as advertised. And what Albert Jan did is simple and elegant.

Fred,

Thanks for pointing that out, GAViewer sounds like 'graphic arts drawing' tool to me, and would not be built to select between a right or left mathematical handedness. It also points out the importance of thinking through any proposed experimental protocol, and especially with the growing reliance on computerized simulation. In Theory, a process is an interactive dynamic. In physical practice, a process is a something - like the bump at the end of a bone that the muscles connect to. In Theory all you need for a mathematical process to be complete is to specify an initial state, it will stick together just fine. In Experiment, to get two things to physically stick together initially, you need to add a mechanical process and that can introduce an asymmetry into an otherwise mathematically symmetrical dynamic. Measure twice, cut once. jrc

Tom,

Thanks, I admit looking first at the entrance to the rabbit hole. A deceased friend of mine had been a machinist, and we were talking once about the decline of the U.S. pre-eminence in hard production capability. He told of a domestic precision spring manufacturer that had sent a Japanese competitor a sample of their tiniest, finest caliber coil spring, and the competitor sent it back with a hole drilled through the wire and one of theirs threaded through it. :-( jrc

Hi Tom,

Actually it is more simple that what you describe. It is just due to the non-communitivity in the algebra. Which is familiar by just rotating a book two different ways. Anyways, the classical local realistic model does in fact produce the prediction of QM, -a.b.

I just respond to the remark "GAViewer sounds like 'graphic arts drawing' tool to me, and would not be built to select between a right or left mathematical handedness".

GAViewer is a research tool built by the authors of Geometric Algebra for Computer Science: Leo Dorst, Daniel Fontijne, Stephen Mann; see http://www.geometricalgebra.net/

I wouldn't call it a "graphic arts drawing tool".

It is built to select between right or left handedness: if "a b" is a right-handed geometric product then "b a" is left-handed.

The challenge to program Joy Christian's model in GAViewer is still open. Going back to Christian's 2007 paper there would not appear to be much work to do: the model is contained in formulas (16) to (19). Albert Jan Wonnink has therefore got about half way - he has done the second half. Only the first half still to do.

This essay appears indeed highly erudite but I wonder how much the author has understood of the authors he is citing. Let me illustrate this with the citations he gives to papers in an area I know well: Bell inequalities.

Instead of plugging the Joy Christian model, Thomas Ray is now plugging Hess and Philipp, who more than ten years ago published a bunch of papers with the main theme that Bell had forgotten about time. Actually, if you take the care the read Bell's famous Bertlmann's socks paper, you will see that Bell was very aware of the role of time, and gave specific experimental instructions so that one would *not* be able to blame a violation of his inequality on time.

Apart from an erudite verbal discussion of the issue of time in these experiments, Hess and Philipp also professed to have constructed a local hidden variables model which reproduced the singlet correlations but unfortunately -- inevitably, of course, by Bell's theorem -- their elaborate construction concealed a little math error. They forgot one of three subscripts somewhere deep in the computations and failed to normalise a measure to be a probability measure. This was pointed out by myself and others, and the model died a natural death. The time issues they raised *are* interesting. The possibility of a memory loophole was already being studied by several researchers including myself. All this activity led to Jan-Ake Larsson and myself discovering a "new" loophole in Aspect type experiments due to the fact that experimenters do not use a framework of predetermined time intervals for measurements. Instead, the detection times of two photons being close to one another is used to post-select the pair i.e. to discard all detection events which seem not to be paired. Disaster! A very non-local selection of which outcomes to keep, which to throw away. Biased sampling ... It turns out to be a far *worse* loophole than the famous detection loophole.

Later Hess used the Larsson-Gill approach to build simulation models for past experiments with these defect, together with Hans de Raedt. In his recent book, Hess claims that Larsson and I had *stolen* the idea from him. Well we were certainly inspired by his work, and as he pointed out, other people had noticed that there was an issue, before him.

The hidden title of Hess' book (published as "Einstein was Right!") is "Karl Hess was always right, at least, in retrospect".

Then a second authority whom Ray quotes is my friend Han Geurdes from the Netherlands who recently got a paper published in the journal RIP (results in physics) which is Elseviers' answer to predatory journals where, I am sorry to say, the author gets to pay an exhorbitant fee to have just about anything published. Geurdes' amazing insight is that an experimentally observed correlation might differ by some amount, in either direction, from the "true" theoretical correlation, hence that a local realist simulation model of a loophole free Bell-CHSH type experiment can easily produce a result larger than 2. The paper is discussed on PubPeer: https://pubpeer.com/publications/DBFF182E87F04FB92102CAC7E33046

Yes! The measured value of some physical quantity might be larger than the true value! Disaster? The end of experimental physics?

Smart people have already known that about half the time, the "observed" value of CHSH would be bigger than 2, half the time it would be smaller. That's why physicists who actually do experiments make sure that the sample size is rather larger and they compute a standard error and do a statistical significance test in order to show that the deviation they have observed above 2 could not be ascribed to merely chance variation around a "true" value equal to 2.

Regarding Bell, EPR and all that I note that Ray does *not* refer to Christian's work so I wonder if this means that he has now abandoned support of that direction?

Before spending just a few words on that, it should be mentioned that Doran and Lasenby's book on geometric algebra contains two whole chapters "doing" spin half, the singlet state, and all that, with geometric algebra. It is very elegant, and very interesting, and just the tip of the iceberg in this area. The only thing they don't do is provide a local realist model for the singlet correlations. (For obvious reasons ... Bell's theorem).

However it seems that geometric algebra is not so popular in this area any more. I suppose it did a good job at describing all the facts - all the facts which are also described in the conventional Hilbert space approach. It links them nicely to geometry. But it didn't take us any further. And just like the conventional Hilbert space approach, it does not tell us what is going on "under the hood" event by event. It is "just" another (mathematically isomorphic) way to derive the probabilities of what happens.

My recent analysis of Christian's early works including a tutorial on geometric algebra is on viXra: http://vixra.org/abs/1504.0102 "Does Geometric Algebra Provide a Loophole to Bell's Theorem?"

and Ray has given a link to a pdf discussing my paper here: http://fqxi.org/data/forum-attachments/Continuing_misguided_attacks_by.pdf

The point I want to make is that time and time again in this essay, I am sorry to say, the essayist does show that he does not know what he is talking about. IMHO. Sorry.

Tom,

I thought your approach to the essay topic via the mathematics of probabilities was rather challenging in the first place, and it is provocative of further questioning of what we *do* with math. On the surface its quite a simple thing to correlate probabilities to a space frame, like throwing darts. But having come to understand some of your theoretical thinking, you are reaching well beyond that concept.

I have known a number of people for years whom, once I dealt with it enough working with them, I came to recognize that they don't see things in a geometric sense though they are quite adept at shooting pool, operating cranes, or racing automobiles. They might rough out a sketch on a scrap of paper of how they want a site laid out for a small footer and spacing pilasters, but there will be no proportion at all in the sketch and it will as likely show the short dimension of a rectangle as the longer length. It is the numerical relationship that they see and the actual spatial relationship only in a moment to moment instinctive reaction. The final result, here to there.

There have been recent advances in brain mapping and studies of mathematic abilities that make me wonder. Even though algebra comes from geometry, do we as a species have an inherent disjunct between our spatial perception with its temporal dimension, and the perception of mathematical relationships in an abstract dimension? Brain scans of 'math whizzes' at work show areas larger than common, consuming high levels of oxygen. But is that only in the abstract, does it correspond to a sense of spatial environment?

That correspondence at a foundational level is what you seem to be driving at in your essay. And a perceptual lack of such a correspondence almost guarantees a perception of non-locality as the reality. Good luck getting that across, jrc