Essay Abstract

Mathematics is not physics. Rationally speaking, though, a physical result that corresponds one-for-one to a mathematical model forms a closed logical judgment on the truth of a falsifiable physical theory. Because it is not possible to make a closed logical judgment on physical probability for more than two possible outcomes of one event--no mathematical model can be one-for-one correspondent to the physics of probability, in a rationally complete scientific theory. We show that parity between all mathematical models and all physical results is possible if, and only if, probability exists independent of random events. Therefore, Max Tegmark's Mathematical Universe Hypothesis is probably true.

Author Bio

A student of complex systems, Tom Ray is a retired technical writer-editor whose newest contribution to the field of complex systems science, on net-centric logistics, was published by Springer in January 2015, in the book collection Conflict and Complexity. http://www.springer.com/physics/complexity/book/978-1-4939-1704-4

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Hi Tom,

Your essay contains fascinating information. Although many statements exist to the effect that experiments prove non-locality, the fact is that the experiments prove [or come very close to proving] that the experimentally found correlations agree with the correlations predicted by quantum mechanics. It is not physical experiment, but mathematics, that proves that Bell's models, as formulated, are incompatible with local realism. If Bell's model is correct, this seems to have physical implications for local causality, a.k.a. local realism.

Bell assumes as the basis of his model that the precession occurs in a constant field, which leads to a null result for Stern-Gerlach apparatus, and therefore an implicit contradiction, since 0 is not equal to +/-1; [zero being the result of the assumed constant field and +/-1 the result of the actual SG non-constant field employed in the experiment.]

In an attempt to avoid this inherent contradiction of Bell's constant-field model, I have analyzed the physics of the inhomogeneous field and constructed a local model that does produce the quantum correlation, -a.b, unless it is subjected to Bell's constraints.

This leads to the question, why does Bell impose these constraints, and I have attempted to answer this; Bell not being available to answer for himself. You are welcome to read my essay to see what I'm talking about.

It is for these reasons that I find your following statements fascinating [and find I must buy Solow's book, which I believe you have recommended before.]

You say:

"...Bell's Inequality - the formal mathematical statement of Bell's theorem - is only locally real. The issue of non-locality arises in the proof of the theorem and that proof is only by way of double negation."

"What double negation means is that the proposition A is proved by assuming it's negation (~A), and then proving A by contradiction without ever explicitly constructing A..."

As you remark "It's a perfectly legitimate proof technique; mathematicians use it all the time. However, in order to apply such a mathematical proof to physics, it would have to be explicitly shown that the physics is independent of the mathematics, which is impossible in the case of Bell's inequality ... the result being derived by inductive inference."

Although you state quantum theorists know this, if so, they keep it well hidden. I have never seen it so clearly stated. Thank you for stating it so clearly. As soon as I finish this comment I will go buy Solow's book, How to Read and Do Proofs. Thanks for that, too.

Finally, congratulations on your contribution to the Springer book.

Good luck in the contest, and best wishes,

Edwin Eugene Klingman

Hi Edwin,

Thanks for the kind comments. In fact, I had read your essay previously, and was impressed -- it's your clearest and most direct, in my opinion, of all your entries over these years. I am finally convinced that we're talking about the same thing.

Key to our agreement in principle is the value of a continuous function. Just as Einstein originally formulated mass-energy equivalence in terms of the Lagrangian, I think your energy exchange theorem in terms of the Hamiltonian underscores the scale independence of analytical continuity in a physical system.

Quantum theorists are not bothered by the double negation foundation of conventional quantum theory, because they consider it irrelevant. The Bell model is what topologists call multiply connected -- i.e., local pathwise connections are independent of global structure; the assumption of nonlocality is an unavoidable assumption of a multiply connected space.

That's how we get the inductive inference, "If nonlocality exists in the present, it does not not-exist in the future." One simply cannot discard nonlocality as an experimental assumption of Bell's theorem, because were the mathematical assumption correspondent to the experiment, the conclusion would have to be that nonlocality is an empty set. In Solow's chapter 11 ("Nots of Nots Lead to Knots") one finds that " ... to use the contrapositive method ... you must be able to write down the statement NOT B so that you can work forward from it. Similarly, you must know exactly what the statement NOT A is so that you can work backward from it." The statement A in Bell's theorem is never constructed; it is only found to be not not-A. Again, this assumption does not bother quantum theorists because counterfactual definiteness is exactly equivalent to, "If nonlocality exists in the present, it does not not-exist in the future." There's a problem with causality here:

Think of it in terms of the Schrodinger's cat gedanken experiment. So long as the cat is in a state of superposition, neither dead nor alive, there's no problem with the counterfactual. Dead and alive are equally likely. Doing a measurement on the box "collapses" the state into a classical bit.

That creates the illusion of observer determinacy -- the free will to look or not, and find the locally real state. Schrodinger's wave equation, though, is fully deterministic, backward and forward in time. The experimenter's assumption in preparing the experiment is that the cat is in a state called "alive" as opposed to the other binary state called "dead." The contrapositives not-alive and not-dead, however, are never constructed -- the experimenter assumes when placing the cat in the box that it is not not-alive. Try and make the assumption that the cat is not not-dead, and I think there is little doubt about the state one will find the cat in, when the box is opened. The initial condition makes the difference between the states -- yet the experiment assumes that not not-alive = not not-dead. It begs its own conclusions.

I'll get around eventually to commenting in your forum. Good luck and all best,

Tom

    Tom,

    Your essay is as usual, a Tour de Force of highly condensed mathematical acumen. I've given it two readings and find much I have to learn more about. In general, however, the theme is clear and is concisely stated where you say,

    "- if nothing exists in reality save random events, there is no natural correspondence of mathematics to physics."

    I was impressed by your carefully avoiding a philosophic character of argument, and keeping matters very much on message of the nuts and bolts of math needing an independent yet true correspondence to physics being about that which is physical, without any ad hoc contrivance of the math to match the physical empiricism. The point you make about QM conventionally constraining measurement space to the Bloch sphere provides a conceptual 'visual aid' when contemplating that a continuous function must exist in metamorphizing a spherical (curved) space and a cubic (flat) space. The 1500 year old edifice Hagia Sophia comes to mind, and with it the geometric sense of strength in physical spacetime.

    I think you have made a good argument that what distinguishes where math is exemplary of physical reality is where there is continuity, and your approach to the Contest Topic by way of probabilities is challenging. I still have my sticky note of your conjecture and think it has a good fit in your presentation. Good Luck, don't let the back-biters bug. jrc

      Hi John,

      It's been a constant source of pleasure and delight to me, that you truly get what I'm saying. And I don't just mean the words -- I mean the essential message of a committed rationalist.

      Thank you and all best,

      Tom

      Tom,

      The pleasure has been mine, I can only understand rationalists and have learned much from you. I was especially pleased to learn from your essay the logic of Euler's equation. It never bothered me that he once said that any first rate mathematician would immediately see it, because I've never imagined myself a mathematician. But it has pestered me that I couldn't see it. Forest for the trees sort of thing, I was hung up on the numerical values. The manner you introduced it in your argument displays the simple logic both geometrically of a 1pi rotation, and algebraically of 'e'; it is the real function of each, not the arithmetic values that are at work! My horizons have been greatly enlarged. Thank-you very much. jrc

      Hi John,

      I think it was Gauss who said that. You're right, though -- analysis is far more compelling as a physical language than arithmetic. We actually experience geometry, as 3-dimension beings with 4-dimension brain-minds. Einstein in fact made what seems a mystical statement on first blush -- that he experienced relativity kinesthetically. Rotation, though, is a physical sense in more basic terms than a simple discrete point or line.

      Best,

      Tom

        Dear Mr. Fisher,

        I will entertain your point if you communicate it to me telepathically -- i.e., without using words, pictures or sign language, all of which are abstractions.

        Can you do that?

        Best,

        Tom

        Tom,

        I think the last minute submission of your essay may have gotten lost in the flood. Not necessarily a bad thing given past experience with speculators whom seem to think the object is a sparring contest of pet peeves, and the quantum maniacal denial of realism.

        I've never had a fascination with probabilities, so I need taken by the hand and shown so to speak. Could you briefly explain the sqrt2 lower bound on information? Also I'm a little fuzzy on your construction of M=4P, though I see how it is necessary that four physical states must evolve from the possible outcomes of pairs of coin tosses.

        Perhaps in a while there will be comments by others with proficiency in both classical and quantum probabilities to promote a learning experience. Best jrc

        John,

        Haters gonna hate. :-) I know from past experience that a couple of knuckleheads will knock a new entry down without reading it, just to try and suppress competition. Along with some others, I have been critical of the 'peer' voting system; it does no service to science that the peer group is limited only by one's capacity to use a keyboard.

        You write, "Could you briefly explain the sqrt2 lower bound on information?"

        It's as old as the first proof of Pythagoras's theorem, which actually may be even older than the Pythagoreans. Independent of the respective values for the lengths of legs in a given right triangle, the hypotenuse gives up no rational information about its own length with the exception of square integrable values -- even though sqrt2 is an algebraic number, i.e. the solution to an algebraic equation, it is defined only in terms of the nonzero endpoints of its pair of legs. Take the smallest real integral result of a^2 b^2 = c^2: 3^2 4^2 = 5^2:

        It's pretty obvious that 9 16 = 25. The values are square integrable. If measurements consisted only of square integrable results, the theorem would be uninteresting. Not only are real 2-dimension measures generalized to every pair of straight lines forming a right triangle, however, all information of the hypotenuse is reduced to an irrational number. If one asks the question: what is the exact length of the hypotenuse? -- there will be no exact answer. It can only be given in terms of the squares of each side of the triangle; the area is bounded, even though the endpoints of sqrt2 are not.

        This mathematics only becomes interesting, when raised to a four-dimension Pythagorean theorem (Riemannian geometry). That's beyond the scope of my essay.

        "Also I'm a little fuzzy on your construction of M=4P, though I see how it is necessary that four physical states must evolve from the possible outcomes of pairs of coin tosses."

        Remember, we are talking about relativity. The coefficient 4 is a constant that includes the observer as a physical state, just as the constant c^2 in Einstein's equation defines rest energy directly proportional to mass. The whole set of coin tosses for all time is encoded in the relation -- as Tegmark says, his MUH will be refuted if there is fundamental randomness in the universe. M = 4P tells us that there is no fundamental randomness. As I show in the essay, only pairwise initial conditions have a definite Truth Value -- so there is only classical, i.e., binary, probability and no fundamental randomness.

        Best,

        Tom

        Dang it. My posts keep appearing outside the thread that I thought I placed them in. Sorry.

        Tom

        Dear Tom

        I liked your quote of John Barrow's interpretation of Gödel: "if mathematics is a religion, it is the only religion that can prove it is a religion". As I discussed in my 2012 essay I have found such a proof. The problem is that we have a religion that doesn't want to admit the truth that it is a religion. The expedient way of dealing with such a proof is to deny it exists. As you discuss in your essay, Bell-like analysis on the border between maths and physics is not straightforward. It is possible to make implicit assumptions that undermine the generality of the result - as you are saying about Bell. In my essay I adopt a physics realism approach to the same questions on the basis that the true physical dynamics is hidden by simply being too fast for any experiment to measure. This option isn't covered by Bell, and I find that quantum theory results can be reproduced. Where my essay was cut short by the word count, is on the issues of probability, locality and non-locality you discuss in your essay. I would be interested in your view of my new approach to Bell-like analysis, and what it implies for probability and non-locality.

        Regards

        Michael Goodband

          Dear Tom,

          I am so sorry that you apparently cannot understand written English.

          Remedial courses are available at very low cost.

          Congenially,

          Joe Fisher

            Thanks, Joe. It's just all those abstractions that confound me. Words, you know.

            Tom,

            I must confess that I have not yet mastered the erudite arguments you pose, but knowing a small bit about Tegmark's MUH, I wondered about the "independence of humans" aspect of the ERH. Current physics theories are the math equations and structures describing the theory and their concepts, thus explaining the connections to our observations. The equations are built by humans and are their baggage. I don't see the separation.

            My "Connection of Math, Physics and Mind," I know, seem mundane, but I don't see the physical world as completely independent of humans.

            What am I missing? Many scientists say that humans adhere to the ERH concept.

            Jim

              Dear Tom,

              I am having an awful time with my credentialed fellow essayist. Several of them have reported my post as being inappropriate and had it removed.Thank you for your gracious humorous response to my comment.

              Thankfully,

              Joe Fisher

              Joe, you're not wrong. There's just no possible framework in which you could be proved right, that isn't self-referential.

              As Popper said, "All life is problem solving." The problem here is that life can't apparently communicate with sentient life other than by using abstract symbols and signals. How do you know, in fact, that you aren't communicating with an abstract being (me) through the abstract symbols on your keyboard that you are using? How do you know that I am 'real'?

              I don't know your level of knowledge or interest in philosophy or philosophical problems; however, if you agree with Wittgenstein's view, there are no philosophical problems at all -- just "language games and forms of life." At the end of the day, that may be a great truth, and it's still a philosophy I reject outright -- for the same reason that I reject your claim that the world includes no abstractions:

              Your conclusions, and Wittgenstein's, are based on inductive inference -- "Seeing is believing."

              I am a rationalist, though. In order to solve a problem, one must identify it -- even a guess is good -- and find the logical correspondence between the problem and its solution in order to consider it solved. I quote J. Bronowski often: "All science is the search for unity in hidden likenesses."

              Rationalism unites the world. Inductive inference divides it.

              All best,

              Tom

              Well, Jim, all I can say is that if you agree with Max's ERH table (slide 14 in this PowerPoint: www.fqxi.org/iceland/images/Iceland%20Talks/tegmark.ppt) you'll find my view at the extreme of "less baggage."

              I am a rationalist. An external reality and metaphysical realism are fundamental assumptions.

              Thanks for the note. I'll get to your essay when I can.

              Best,

              Tom

              Hi Michael,

              I'm ashamed of myself that I've had your book for a couple of years now, and haven't penetrated it -- though I know we have so many ideas in common.

              Please let me beg off commenting until I read your essay -- and thanks for dropping by!

              Till later, all best,

              Tom

              Dear Tom,

              "All science is the search for unity in hidden likenesses." If the abstract likenesses are hidden, how are you going to prove what they are likenesses of? There are no hidden abstract likenesses in reality." Therefore, all of science as you know it is erroneous. My contentions that real light is the only stationary substance in the real Universe and there is no physical space show that it is reality that is unified.

              Warm Regards,

              Joe Fisher