Banner
    The Secret Ancient History Of The Higgs Search
    By Amir D. Aczel | August 5th 2012 11:42 PM | 20 comments | Print | E-mail | Track Comments
    About Amir D.

    Amir D. Aczel, Ph.D., studied mathematics and physics at the University of California at Berkeley and also holds a doctorate in statistics. He is...

    View Amir D.'s Profile
    When CERN, the European Organization for Nuclear Research, near Geneva, Switzerland, announced the discovery of the elusive Higgs boson on July 4th this year--the result of years of running of the Large Hadron Collider, the biggest, most expensive, most powerful, and coldest (liquid helium cooled, colder than outer space) machine ever built--it marked a tremendous triumph of experimental physics. A boson (an integer-spin particle, usually associated with conveying a force of nature), and the last undiscovered one, needed to complete the extremely successful Standard Model of particle physics, had finally been found!

    Not many people know, however, that the Higgs boson had been predicted to exist on strictly mathematical grounds--based purely on abstract symmetries usually studied by "pure" mathematicians. As such, the story of the Higgs is even more amazing than you might think--it encompasses a major search that began with the Babylonians and Egyptians and continued to the ancient Greeks, the Arabs, medieval Europe, and on through the nineteenth century to our own time.


    The purpose of this note is to make this part of the history of the monumental discovery of the Higgs boson known to a wider audience. 

    Higgs candidate event from the CMS group at CERN

    I was first drawn to this story in 2001, not long after my book on pure mathematics, Fermat's Last Theorem, was published and I heard the news that the giant machine, to be called the Large Hadron Collider, was being built under the Swiss-French borderlands near Geneva. Having interviewed a number of Nobel Prize-winning physicists about theoretical physics, I knew that there was a direct relationship between the pure-math concepts I explained in that book and the raison d'etre of the $10-billion-investment being made at CERN: the Large Hadron Collider was being built because of a theoretical hunch based on abstract mathematics derived from the most ancient quest in human history - the quest to solve equations! So let me get to the story.

    The quest to solve equations

    Both the ancient Babylonians and the ancient Egyptians could solve linear equations, as well as some quadratic equations. We know this from cuneiform tablets found in Mesopotamia (present-day Iraq) dating from the second millennium BC and from Egyptian papyri, such as the famous Ahmes papyrus, 2000-1800 BC. The ancient Greeks, too, could solve linear and quadratic equations, A more general solution method for quadratic equations came with the Arab-Persian mathematician Al-Khowarizmi (c. 780-850) who worked in the Caliph's "House of Wisdom" in Baghdad, and from whose name we get the word algorithm, and from the name of his book, Al Gabr wa al Muqabbala, we get the word algebra.

    This book laid out methods for systematically solving the quadratic equation. In the sixteenth century, the Italian mathematicians Cardano, del Ferro, Fior, and Tartaglia developed methods for solving the cubic and quartic equations. But for 300 years afterwards no one knew a general method for solving a quintic--or fifth-order--equation, meaning an equation with leading term being x5--no matter how hard they tried! Then in 1829-1830, a brilliant young French mathematician by the name of Evariste Galois applied himself to this age-old mystery. Galois was just a teenager at the time and smarter than all his teachers. In admission tests to the Ecole Polyechnique in Paris, he was asked to explain logarithms and found the question so offensive that he threw the blackboard eraser at the examiner...needless to say, the genius failed. He became depressed, joined the French Artillery of the National Guard and entered political activity as a Republican, meaning a fighter against Louis Philippe d'Orleans - the King of the French. He was arrested and after being jailed became involved with "an infamous coquette" as he referred to her in a desperate letter to a friend - Stephanie Du Motel - who may have been used by the Royalists to entrap him (we don't know the details) - in any case, he died in a bloody duel outside Paris in 1831, perhaps for Stephanie's "honor," at age 20!

    Before he died, he wrote down in letters (and in manuscripts that by then had been rejected by Poisson and Lacroix, famous French mathematicians, as well as not even read by the even more famous French mathematician Cauchy--none of them could or cared or understood it) an entire theory that explained why the fifth-order equation cannot be solved by radicals (meaning in terms  of the coefficients of the equation and any fractions or roots of such coefficients, as we do when solving the quadratic using the famous formula). Galois' amazing answer had to do with symmetry.

    In this case, the symmetry of the set of solutions to an equation. In developing his incredibly insightful theory--which was only finally understood by mathematicians some years after his tragic death--Galois invented the mathematical concept of a group. Groups are the mathematician's way of explaining and handling symmetry.

    Basically, in terms of the application to the original problem: the quintic equation would have solutions with the wrong kind of symmetry, and hence it cannot be solved in radicals (we solve it, today, using numerical methods by computer). But group theory--the young Galois' brainchild--has applications to a huge variety of problems in both pure and applied mathematics: It is successfully applied to solve any kind of problem that has something to do with symmetry. 


    Rotations of a triangle, for example, form a group. These are shown above. The product of two rotations is a rotation, and there is an identity operation: leaving the triangle alone. And operations have inverses: e.g., rotating clockwise once and rotating clockwise twice are inverses since doing both is the same as leaving the triangle unchanged.

    About half a century after Galois' work and death, the Norwegian mathematician Sophus Lie (1842-1899; his name is pronounced "lee") sought to extend Galois' work on groups. He knew what Galois' theory of (discrete) groups did for equations, and so he invented (or discovered, as mathematicians would say) a theory of continuous groups, which would do for differential equations what Galois' discrete groups did for usual equations. It turns out that Lie groups are extremely useful in theoretical physics! 

    The simplest Lie group is called U(1) and it is the continuous group of all possible rotations of a circle: rotations by any angle whatsoever:


    It turns out that Maxwell's theory of electromagnetism enjoys a U(1) symmetry: when you rotate the model in an abstract mathematical space, the theory remains unchanged! This fact has been known but not understood well by physicists. Then in the early twentieth century, the German-Jewish mathematician Emmy Noether, who later escaped Nazism only to die in the United States from an abdominal tumor, proved two key theorems in mathematics, called Noether's theorems, which established a powerful link between symmetries captured through Lie groups and the all-important conservation laws in physics. For example: Symmetry through time gives physics the paramount law of the conservation of energy, meaning that energy can only change form (e.g., from mass to sheer energy, as per Einstein's most famous equation) but never be created or destroyed. 

    Einstein's general theory of relativity has a particular kind of symmetry called "general covariance," which gives the theory its power and validity.

    Quantum mechanics is especially amenable to the actions of continuous groups because of the superposition principle. Schrödinger's cat, as everyone knows, is both alive and dead at the same time: Until the box is opened and "the wave is collapsed," the poor cat lives in a superposition, hence a mixture of being dead and alive. The Lie group SU(2), technically called the group of special unitary 2 by 2 matrices, is a group that continually "deforms" one element into another, thus representing such a continuous mixture (continuous because the mixture can be 31% dead and 69% alive, or 23% dead and 77% alive, and so on). The cat can thus be modeled by an SU(2) group, and in fact this idea came from Werner Heisenberg when he tried to model the proton and neutron as a continuous mixture, noting that they were very similar in having almost the same mass. 

    In 1954, C. N. Yang and Robert Mills wrote a paper that developed the idea further and proposed what we call today Yang-Mills gauge theory (the term "gauge" comes from the mathematician Herman Weyl): a theory of continuous Lie-group symmetries defined independently at all points in space (a property called "local symmetry"). Yang-Mills theory underlies much of the theory of modern particle physics. To jump ahead in the story, the Standard Model of particle physics is modeled by a composite Lie group: SU(3)xSU(2)xU(1). As you may guess, SU(3) is the group that continuously mixes three entities; and here it is used to quantum-mechanically mix the three "colors" of the quarks. SU(2) in this model mixes an up quark with a down quark and the leptons: an electron with an electron-neutrino--the two particles that emanate together in beta decay (hence capturing the action of the weak force; the strong force being modeled by the SU(3))--and the same for the muon and the muon neutrino, and tau with the tau neutrino (and similarly SU(3) also mixes the higher-order quarks of the second and third "generations"). The U(1) part of the composite group can be described (simplistically, here) as representing electromagnetism.

    Part of the model, the unification of electromagnetism with the weak nuclear force (responsible for beta decay)--using the composite SU(2)xU(1)--was the work of Steven Weinberg in 1967 (and independently, in different ways, of Abdus Salam and Sheldon Glashow, who shared the 1979 Nobel Prize with Weinberg for the electroweak unification work). in his work, Weinberg used what he called "the Higgs mechanism" to break the original symmetry of the unified electroweak force when it split during the very early universe into electromegnetism and the weak force. When this happened, the Z and W+ and W- bosons that carry the action of the weak force inside a nucleus of matter gained mass.

    The "Higgs mechanism," whose action is carried out by the then-hypothesized Higgs boson (now confirmed to exist by CERN) is what gave mass to these three bosons and, by implication, to all massive particles in the universe (save, perhaps, neutrinos, where another process may have been at play). What Peter Higgs, and his colleagues working independently of him and reaching the same results in 1964, Brout, Englert, Kibble, Hagen, and Guralnik, all did was to remove a technical hurdle from the process--that hurdle called the Goldstone-Weinberg-Salam theorem, which stated that massless bosons would appear when a primeval symmetry of the universe was broken. Higgs and the others proved in three independent papers that Yang-Mills gauge symmetry is not affected by the Goldstone-Weinberg-Salam theorem.

    In doing so they provided a mechanism by which the continuous Lie-group symmetry of the early universe--a tiny fraction of a second after the Big Bang--can be broken: the composite group SU(2)xU(1), representing a unified electromagnetic-weak force, breaks down into the separate electromagnetic and weak forces, resulting in a new U(1) symmetry (technically, it is not the original U(1) of the composite group) by interaction with the Higgs field. This is what gave us mass during the very early life of the universe--and it all goes back to the pure mathematics of continuous symmetries and the idea of a group developed by Galois and extended by Lie and originally proposed in order to understand why some equations can be solved with pencil and paper while others cannot!

    It eventually led to people building a huge, immensely technologically sophisticated, and very expensive particle accelerator--and the greatest discovery in physics history.

    Comments

    John Derbyshire writes in "Unknown Quantity" about Cauchy and Galois. It's a myth that Cauchy didn't even read Galois' letter (or manuscript). In fact, research has shown that Cauchy was impressed by Galois' results but asked him to polish his text a bit.

    Amir D. Aczel
    Hmm... I've never heard this before. I researched the story of Galois at the archives of the French Academy of Sciences in 2003 and was actually shown the original handwritten report by Poisson and Lacroix saying they didn't have enough information to evaluate the correctness of Galois's derivations. I saw many letters and manuscripts by Cauchy at that time (he was active till his last day) but nothing whatsoever about Galois. I think that if Galois really had had encouragement from a mathematician as prominent as Cauchy, he may not have descended to unproductive activities such as revolution, which ultimately led to his demise!  
    Amir D. Aczel
    Amir D. Aczel
    Okay, "Anonymous"--You made me research this further. I see online that someone named Rene Taton has "recently discovered a letter that proves that Cauchy has actually read the Galois paper." But I don't see anywhere proof of the letter itself. And Derbyshire seems on the attack with this kind of information, trashing a recent book by David Berlinski that deals with Galois: 
    http://www.johnderbyshire.com/Reviews/Math/infiniteascent.html

    This is all very interesting! But the gist of the matter, it seems to me, is that even if Cauchy read it--and this was, we now know, a work of absolute GENIUS--why would he just "encourage Galois to polish it" (and there are NO SOURCES at all for such "encouragement"--whatever this means) and not scream: Here is a great new theory, people!! Listen! We now understand why the mysterious quintic equation is unsolvable!!
    It is extremely likely that even if he indeed read Galois's paper, he probably either didn't care enough or didn't understand the theory. To "encourage to polish" (if it at all happened--and I doubt it) is worth little--as history has proven here!

    Amir D. Aczel
    The letter found by Rene Taton was not recent, it was published in the french journal "Review d'Histoire des Sciences" in 1971! This is 40 years ago! Here is the reference: "Rene Taton, "Sur les relations scientifiques d'Augustin Cauchy et d'Evariste Galois," Review d'Histoire des Sciences 24, 123 (1971)." I realize the article is in French, but it is published by a well reputed source, and I assume they did their due diligence.

    Among historians, the supposed "lapse" by Cauchy has been widely rejected, as in the award winning article by Tony Rothman in 1982 " Genius and Biographers: The Fictionalization of Evariste Galois" Tony Rothman, American Mathematical Monthly, 89, 84 (1982).

    What I don't understand is your linking to the Derbyshire review, who basically accuses Berlinski of the same thing anonymous accuses you of, namely relying too much on the romanticized story of Galois popularized by Bell, and not on facts. According to the review "Galois's most important ideas had in fact been submitted in a paper to the French Academy three years earlier. Cauchy, the greatest French mathematician of the age, had reviewed them, and thought highly of them. Nine months later Galois had submitted his work for the Academy's Grand Prix, very likely on Cauchy's encouragement. "

    No one wants to take away from Galois' achievements, and no one can deny the oftentimes petty actions of great scientists when it comes to others' achievements, but I think in this case the article makes some points that are too based on a history that has since been discredited. It could happen to anybody!

    It's not the coldest machine (see BEC's) and GWS does affect Yang-Mills, just in a different way.

    Amir D. Aczel
    jtsonberg: CERN tells you in all its official publications that the LHC is the coldest place in the universe. Liquid helium cooling the superconducting magnets is constantly at 1.9 degrees Kelvin (1.9 degrees celsius above absolute zero) while outer space is at 2.73 degrees Kelvin--the temperature of the microwave background radiation, which is the leftover cooled-down radiation from the Big Bang. Bose-Einstein Condensates are very much closer to absolute zero, but they are not a "machine" in any definition of what a machine is. And they are produced for extremely short periods of time in labs--in fact, one of them at MIT--and then they are gone. About the Goldstone-Salam-Weinberg theorem: I have no idea what you mean by "affects it in a different way." The question is: DO Goldstone-Nambu (massless) bosons materialize when a symmetry is broken, or not? In the case of a Yang-Mills gauge theory (which has local symmetries, an important aspect; and Lorentz invariance), these bosons do not occur, i.e., the Goldstone theorem FAILS to apply. READ the papers by Higgs (Phys. Lett. 12, 1964, pp. 132-133); by Englert and Brout (Phys. Rev. Lett. 13, 1964, pp. 321-323); and by Guralnik, Hagen, and Kibble (Phys. Rev. Lett. 13, 1964, pp. 585-587). For example, the last paragraph in that third paper, p. 587, reads: "Thus the absence of massless bosons is a consequence of the inapplicability of Goldstone's theorem rather than a contradiction of it." And Steven Weinberg, in a paper published in 2008 (in a collection called "Perspectives on LHC Physics"; World Scientific, Singapore, p. 140), wrote that: [Yang-Mills gauge theories]...just don't respect our theorem." 
    Amir D. Aczel
    Amir D. Aczel
    Dr. Kyle Jones: Thanks for your input. I still maintain that "very likely on Cauchy's recommendation" is neither proof nor good history. I tend to rely on my own first-hand research. I saw with my own eyes the original and shameful paper by Poisson and Lacroix--who were the decision makers in this case--and, when all is said and done, whether Cauchy has or hasn't seen Galois' work is a side issue and immaterial. The fact remains that Galois did not receive the recognition he deserved and largely left mathematics for activities that led to his demise. 
    Amir D. Aczel
    I'm not an historian, so I don't know what the words "proof" or "good history" exactly mean. But if you feel that's it's immaterial whether Cauchy saw Galois' work or not, they why do you mention Cauchy?

    A thorough analysis of the case would demand, I assume, an answer to many questions. Was the paper that Galois sent to Poisson and Lacroix clear and understandable by the norms of his day, or obscure? How many papers did the Academy receive from cranks who claimed to have a solution to intractable problems (the general quintic, squaring the circle, Fermat's Last Theorem etc)?

    You write about the "the original and shameful paper by Poisson and Lacroix". To quote Derbyshire again

    "(Poisson) found Galois' paper too difficult to follow, though he was not condemnatory, and suggested an improved presentation".

    Doesn't sound shameful to me. I can't detect malice or negligence, although, with hindsight, it's easy to accusse Poisson of stupidity. But if he was stupid, he wasn't the only one. Derbyshire also writes:

    "Galois' brother and friends copied out his papers and circulated them to big-name mathematicians of the day, including Gauss, but with no immediate result;"

    Why did none of the big-name mathematicians immediately "scream: Here is a great new theory, people!! Listen! We now understand why the mysterious quintic equation is unsolvable!!" Perhaps Gauss had his moments of stupidity too?

    dorigo
    Dear Amir,

    thank you for this very nice history of group theory and the connection to the Higgs. I found your article accurate to the right point -no popularization of science should be totally accurate, lest it loses its target from sight.

    And, it may surprise you if you read my bio (I have a blog here since three years ago), but I was ignorant of the ethimology of the word "algorithm". Thanks for that!

    Cheers,
    Tommaso
    Amir D. Aczel
    Dear Tommaso,
    Thank you so much!! That was very nice to hear! I did think the barrage about Cauchy was a bit too much and of course missed the point of the entire article. I am glad you liked the "algorithm" origin!
    I look forward to reading your posts.
    With best wishes,
    Amir   
    Amir D. Aczel
    Yes I enjoyed this post as well. Nice job tying together such a range of sub-topics (and even disciplines) so coherently! (To somewhat echo Tommaso) It's refreshing to see something straightforward (i.e. not over-hyped or 'B.S.-ey') and yet readable.

    Amir D. Aczel
    Thank you, Eloheim! (after the "God particle," I assume...)  :)
    Amir D. Aczel
    Mentioning Ancient History, Plato in his Timaeus refers to traces of the 4 [ i.e. implying elemental massive quantities by my consideration in the form of a 4 line spectrum ] elements existed before Creation [ in disharmony] and after being harmonized resulted into building the the UNIT "Stereoid Bond" , the "Somatoides" i.e that which has matter, via the use of two orthogonal triangles, the Isosceles and the Scalene one, which he referred to as the "MOST BEAUTIFUL TRIANGLE [ and finally building the 5 SOLIDS].
    Ref: Pl.Ti.53, 54 , 55
    http://www.stefanides.gr/pdf/BOOK%20_GRSOGF.pdf
    http://www.stefanides.gr/Html/Photo.htm
    http://www.stefanides.gr/html/Root_Geometries.htm
    http://www.stefanides.gr/html/All_triangles_derive.htm
    http://www.stefanides.gr/pdf/PROPOSED_GEOMETRY_OF_THE_PLATONIC_TIMAEUS_G...

    Up to now parts of this Plato’s work may have been considered as “Dark”. But are they?
    Should this work be reviewed and reinterpreted?
    Regards from Athens,
    Panagiotis Stefanides
    http://www.stefanides.gr

    fundamentally
    The H-boson seems to explain why particles feel inertia. It does not explain why these same particles are surrounded by a gravitational field that uses the same property for the strength of the source of this field. I also have not seen why a single H-boson can deliver such a huge variety of masses to the particles that are supported with inertia by that H-boson.
    Inertia can also be understood by investigation of the interaction between particles in the paper of Dennis Sciama "On the origin of inertia" that was written in 1953, ten years before the papers that published the Higgs mechanism.

    I also wonder why nobody followed the opportunities offered by Constantin Piron a.o. who proved that Hilbert spaces can be taken as quaternionic Hilbert spaces, such that as a consequence wave functions can be quaternionic probability amplitude distributions (QPAD's). It means that the wave functions are carriers of a scalar field in their real part and a vector field in their imaginary part. It also means that the quaternionic nabla  of a QPAD can be interpreted as a continuity equation. This interpretation throws completely new light onto quantum physics, because it turns QP into quantum fluid dynamics. This approach makes it possible to study the geometry of the undercrofts of physics in a multi-dimensional way.

    This approach also brings to the forefront the importance of the sign flavors of continuous quaternionic distributions. They are normally hidden in the gamma matrices that also hide that the Dirac equation in fact stands for two coupled quaternionic continuity equations, one for the electron and one for the positron. These sign flavors are the reason for the existence of a variety of elementary particles. The electrical charges, the color charges and the spin of the elementary particles can be attributed to the sign flavors of the fields that constitute them.

    See http://www.e-physics.eu for details
    If you think, think twice
    blue-green

    Is there a typo here: “the Goldstone-Weinberg-Salam theorem, which stated that massless bosons would appear when a primeval symmetry of the universe was broken.”

    Don't you mean to say massive bosons? Or vector bosons with mass? ….

    Although the phrases “broken symmetry” and “symmetry breaking” are popular, I think it is more instructive to use “hidden” or “cloaked”.

    The notion of “asymptotic freedom” at high energies is somewhat intuitive and could help explain what is meant by symmetries being fully revealed.

    ((By the way, I really wanted to read your book on Fermat's theorem when it came out .... I never made time .... the topic always seems relevant .... and yet I can't put my finger on why. Thanks for showing up here. I hope you didn't make a typo, because it has been copied all over the Internet.))

    Amir D. Aczel
    Hi Blue-Green, No, it's "massless" as stated: You want the massive ones...it's the massless ones that don't exist and that's why you don't want them there (they are a theoretical artifact that doesn't apply to Yang-Mills theories).
    I'm glad you are interested in Fermat! and thanks for the asymptotic freedom comment.
    :)
    All best, Amir
    Amir D. Aczel
    blue-green

    We appear to be in an unpleasant Dr. Jekyll and Mr. Hyde situation. No matter how pretty, unified and Jeklly a grand theory may appear, it has hyding within it dual natures that manifest themselves with the slight introduction of new chemicals, circumstances or even blind chance. The symmetric vision is dashed as soon as it crashes upon the shoals and meets real matter.

    Just finished your entanglement book and loved it.
    You got some serious writing chops Sir.

    Off topic..but i wonder could you tell me what you think of this:
    http://quantummechanics.mchmultimedia.com/2011/quantum-mechanics/009-dis...

    I am a barely conscious moron, with almost no detailed understanding of such things. They simply give me awe.
    One or two words back would be such an honor.

    Please keep up the great work Sir...Life is magnificent for us who benefit from gentleman like you.
    Blessings,
    David Daniel Savage

    Amir D. Aczel
    Dear David,
    Thanks so much for your comment, and I'm really glad you enjoyed my "Entanglement"! I watched both videos (the one in your link and the next one that follows) carefully. This is certainly intriguing, and done by a professor at McGill--a prestigious university and one I am familiar with very much. Two things stand out right away, which may give one pause. First: Prof. Brian Sanctuary is a chemist--not a mathematician or theoretical physicist, so maybe his technical understanding of Bell's theorem is not as perfect as it could be. And the second is that usually results that are as "earth shattering" as the one he is proposing should probably be presented in papers in refereed journals--so peers could read them and review them and determine validity. He does quote others--but I haven't looked at papers. If indeed what he says were to be accepted by the physics profession, this would be such major news that it would be announced on the front pages of major newspapers after physicists would have validated it in conjunction with an appearance of an article in a refereed publication. So I am very skeptical right away. But let me explain why and how it may have happened that a serious scientist at a major university could be fooled in this way. 

    Bell's theorem is a mathematical result, and once a mathematical theorem is proved then--unless the person who proved it had somehow made not only a key error, but one that later eluded all professional readers of the theorem--the theorem is correct. Unlike science, in which theories can come and go, a mathematical result will very, very rarely "go away." This is why mathematics grows by accretion while science can grow by substitution. The question is: Does this mathematical theorem apply correctly to the real world?--and this is the key idea about applying purely mathematical results to the real world of physics. In the case of the Higgs, for example, the Goldstone-Weinberg-Salam theorem was of course mathematically perfect;y correct--but did not apply to the physics in question (leading to a Higgs mechanism): so this can happen. 

    Now, getting to the nitty-gritty: The concepts involved in the EPR thought experiment, Bell's theorem, and the subsequent experiments of Clauser-Horne-Shimony et al., of Alain Aspect and his collaborators, of Nicolas Gisin in Geneva, and finally of Anton Zeilinger in Vienna, are extremely SUBTLE. Without writing a very long blog on it, let me suggest a paper by David Mermin (you'll have to look it up--it's in a major science publication such as Scientific American) that gets into the subtleties and shows what quantum entanglement really is--and what it is not. The concepts are so complicated and require deep thinking that even great physicists have failed to grasp them: George Loschak was a student of the great French physicist Louis de Broglie, and even he made an error in understanding Bell's theorem! (See the book "Speakable and Unspeakable in Quantum Mechanics," by John Bell, Cambridge Univ. Press, 1993 for an answer to Loschak and to others). And let us not forget: All the experiments by the people mentioned above DID lead to positive results!

    But let me do one more thing. I will be meeting Alain Aspect in Paris in late September--of course he is the one experimentalist who successfully performed the most exacting experiments on EPR and Bell's theorem--and I will ask him what he thinks of this. Then, if the editor of Science 2.0 agrees, I will write a blog post on that.

    Thanks again for a very though-provoking question and reference.

    With best wishes,
    Amir  

    Amir D. Aczel
    Sir,

    Wow. I am delighted. On several levels.
    Your generosity, passion, and rigor are infectious!

    ( Certainly I will read Mermin / Bell / Speakable.. etc )
    I Hope someday to thank you in person.

    Most Sincere Thanks.
    Blessings,

    David Daniel Savage