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

My attachment of 2 April -- which demonstrates reversibility of the counting function by the natural properties of recursion and parity -- got me thinking about the Monty Hall problem and why reversibility of the time metric is equivalent to experimenter free will in a Bell-Aspect type experiment. Switching choices implies physical time reversibility, and here's why:

Mathematicians will always agree -- that given n contestants choosing 1 of 3 doors, two of which hide a goat, and the third a new car -- one can predict that over many iterations, or even if many contestants simultaneously choose from sets of doors, that by the law of large numbers 1/3 of the contestants will win cars. This how Richard Gill describes the independent "counts" of four 2 X 2 tables of results in a Bell-Aspect type experiment, with 4 instead of 3 "doors.".

The singular case in which the host (Monty) opens one door of the two that a contestant has not chosen -- and reveals a goat, then asks the contestant if she would like to switch choices -- raises the question of whether the contestant has a winning advantage by switching the choice, or staying with the first.

Naively, one thinks that -- because Monty has shown one of two doors that the car is *not* behind, that the odds of choosing the winning door have been increased some 16% (from 1/3 to 1/2) by choosing to switch. In fact, though, the odds are still 1 in 3 whether the contestant switches the choice or not. The question is whether one has a better choice of winning the car by switching choice, or not.

Even though the contestant knows in advance that Monty will never open the door with a car behind it, this information adds nothing to her knowledge of what door the car is behind. In other words, a potential choice (the door identified but not yet opened), does not change the energy state of the system. It does, however, add to the information of the energy state -- one now has a 66.6% chance of winning the car if one switches choices of door -- and this is equivalent to the Hess-Philipp result ( 3) for their Bell-Aspect type inequality that I cited in the attachment; i.e., there is a 3 to 1 advantage (my paper explains) for the result not observed, over the P(1/2) probability for the result that is observed. That difference of initial condition vs. measurement outcome is a hidden variable.

To see why, compare this scenario to the Schrodinger Cat experiment. The decay rate of the substance that emits a particle and triggers the hammer that breaks the vial that releases the poison that kills the cat -- is precisely known. The energy potential of the hammer is identical to the pre-choice of door in the MH problem -- If Monty lifts the lid on the box and declares "the cat is alive," or "the cat is dead," it has no effect on the decay rate of the material or the energy potential of the hammer.

Monty, however, *cannot choose* to say "the cat is dead," because we *know* that the conditions under which the cat dies are fully determined, even though hidden in a black box. There is absolutely no point in Monty communicating to us that the cat is dead, because:

If the cat were dead, the experiment is ended -- just as if Monty opened the door with the car behind it while the contestant still has a choice pending. It doesn't happen, because Monty knows which door the car is behind. He isn't an observer making a binary choice; he's the guiding principle *behind* the measurement choice. This is the same principle by which Joy Christian successfully argues for the choice that Nature makes independently of conscious observers, and which guarantees real binary measurement in a locally real and objective way.

Ultimately, the free will hypothesis prevails, because -- and I made this point repeatedly in the great "debate" over Christian's result -- *unless* Nature has a choice, human observers have no free will. The energy cost to remove the middle value is equal to the observer's choice to change the state of the system.

So in support of Tegmark's hypothesis and Christian's measurement framework (which was published by the International Journal of Theoretical Physics recently as "Macroscopic Observability of Sign Changes under 2(pi) Rotations"-- nature is not fundamentally random, even though conscious observers switch their choices.

As my attachment shows, observer choices that change the measurement outcome deterministically, also change the initial condition randomly -- consonant with my claim that free will exists IFF nature is not fundamentally random. The question of whether the initial condition is positive or negative obviates the independence of tables that Richard Gill describes. The measurement is observer entangled -- and that entanglement is equivalent to classical orientation entanglement (spinor property).

More to come.

Hello Tom

I read your essay twice and am still not very clear about what you are saying. The fault is entirely mine - for example a lack of background in the type of mathematical/logical arguments you used. Another reason is that at my age (73) I was reading not to explore new ideas, but to confirm my own half-cooked ones in a view to develop them further! For example, based on my Beautiful Universe Theory concepts I think there is no inherent probability in Nature, nor is there particle-wave duality, no wave function collapse (which you agree with) and that Bell's Theorem only confuses the issue. At least about the latter I can point to Edwin Klingman's essay here for a more substantive analysis than my one-liners. In my essay here I argue that in a causal local absolute discrete Universe mathematics and physics reduce to the same thing - not themselves, but the micro structure of Nature itself. As always I value your feedback.

With best wishes

Vladimir

Hi Tom,

I have printed out your essay and I am going to study it on my next lie down rest my back break. :-)

Looks like you have been busy here but thought I would inform you if you don't already know that the core part of Joy Christian's model has been proven by Albert Jan Wonnink via the computer program GAViewer. For those interested look here.

Looks like FQXi's panel of experts were very wrong. But you already knew that.

I would not hold my breath Fred..

Let each out breath inexorably lead to an in breath, then vice versa, and so on. That each inward path leads to an outward one, and outward to inward, is a suitable imitation of the connectedness of S3. I too enjoyed your summary, Tom, and I have to wonder what it would take - because it appears Gill's criteria are a moving target, just as he claims for Joy.

It is ironic that RG wants to use Joy's first paper as the bellwether, given that it is only a sketch of the full proof, and that he has ignored that Bell's first paper contained an error that was later glossed over or corrected - as clearly pointed out by M. Goodband. I think perhaps Joy's use of the Kronecker delta is partially at fault; it is a convenient but lossy abbreviation that leaves too much room for interpretation.

More later,

Jonathan

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