Dear Florin,

My impression is that our discussion becomes rather strange. I submitted an essay entitled "What Mathematics is Most Pertinent for Describing Nature?" and I argue that only a finite mathematics can be fundamental. I believe that my essay is fully in the spirit of this essay contest entitled "Is Reality Digital or Analog?" So I discuss mathematics. However, my impression is (maybe I am wrong?) that you are not interested in discussing mathematics but instead you try to convince me that my approach, which I call GFQT is unphysical or unnecessary or what? (I would prefer if you say explicitly what you want to prove). If this discussion has nothing to do with the essay contest then it's probably not correct to continue this discussion here; for example, we can communicate via email if you are interested.

As I noted in my previous response, our philosophies are almost orthogonal and, as a consequence, we have even a communication problems. For example, you think that my goal is to fix problems, which do not exist since they are well understood. For me your phrase "field theory is already on a solid foundation and by now we understand the infinities from both the mathematical and physical point of view" seems strange since if a theory has a solid foundation, it will not contain infinities at all. The arguments that QFT does not have a solid foundation are well known (do you know them?). They have been given not only by Dirac (whose remarks you treat as obsolete) but by Heisenberg, Wigner and other famous physicists. Even Weinberg, who contributed much to QFT, acknowledges in his textbook that infinities are still a big problem. Nobody has shown that those arguments are not correct but since QFT has achieved several impressive successes, the majority of physicists believe that there is no need to worry about foundations.

I have proposed a new approach and I believe that in view of the present situation in physics different approaches should be considered. I am not saying that you should like my approach. You believe that the existing problems can be better solved by the string theory or similar approaches and of course you have a right to think so. In that case could you, please say explicitly what you are going to prove in our discussion? If you think that my approach is incorrect, I am very eager to know why. If I understand you correctly, so far you try to convince me that the approaches you like will solve the existing problems more successfully. If this is your only goal then I do not see any further point for discussion since, as I noted, "De gustibus non disputandum est".

In my essay and papers I note that GFQT fully changes the status of such problems as particle-antiparticle, elementary neutral particles, conservation laws etc. You do not mention those problems at all. Meanwhile if you are going to prove that my approach is unacceptable then probably it is better to explicitly say why. Let me try to help you.

My original goal was indeed to get 8 correct digits with a solid mathematics (see [11] for a discussion). However, this naïve expectation fails for the following reason. In GFQT there are no independent irreducible representations (IRs) for a particle and its antiparticle but one IR describes an object such that a particles and its antiparticle are different states of this object. As a consequence, there are no neutral elementary particles, the electric, baryon and lepton charges can be only approximately conserved and even the notion of particle-antiparticle is only approximate. I believe that this is an extremely interesting result but you, probably have another opinion. For example, you can say that if even the photon cannot be elementary then GFQT is unphysical. Could you, please, tell me explicitly what you think about this situation? Let me also repeat that if I understand you correctly and you are interested only in discussing physics then the present forum is not an appropriate place for this discussion.

In probability theory, probability is defined as a measure of sets belonging to a sigma-algebra. Those sets can have only a positive nonzero measure. In physics, a definition of a physical quantity is a description how this quantity should be measured. So a question arises whether in physics it is possible to define probability in accordance with mathematics. Mathematics prompts us that we cannot define such a quantity as "the probability to find 5.0724" since the set containing only the point 5.0724 has measure zero and does not belong to the sigma-algebra. But we can try to define the probability to find a number in some interval. As I noted in the previous note, the only known way of defining probability in physics is that we should carry out an infinite number of experiments within and infinite time interval and this is problematic. Also, in quantum physics probability can be zero if there are superselection rules.

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    • Post #13

    Dr. Lev ... thank you very much for your thoughtful reply. One problem with being a nice person is that people tend to take advantage of you, as I shall do now. This is a quote from a paper by Brukner and Zeilinger (see cite below). I'm wondering if you agree or not, or agree/disagree in part ...

    "Clearly, a number of important questions remain open. Of these, we mention here two. The first refers to continuous variables. The problem there is that with continuous variables, one has in principle an infinite number of complementary observables. One might tackle this question by generalizing the definition of (3.4) to infinite sets. This, while mathematically possible, leads to conceptually difficult situations. The conceptual problem is in our view related to the fact that we wish to define all notions on operationally verifiable bases or foundations, that is, on foundations which can be verified directly in experiment. In our opinion, it is therefore suggestive that the concept of an infinite number of complementary observables and therefore, indirectly, the assumption of continuous variables, are just mathematical constructions which might not have a place in a final formulation of quantum mechanics.

    "This leads to the second question, namely, how to derive the Schrödinger equation. ...."

    from: "Quantum Physics as a Science of Information" (2005)

    http://tinyurl.com/26dwfel

      Dear Rick P,

      Thank you for this reference. If you read my essay you could see that it is in the spirit of these remarks.

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      • Post #15

      Dear Dr. Lev,

      Thanks again. I certainly thought so, which is why I made the connection. There's plenty of material out there to choose from but very little of it anywhere near as apposite. But of course the devil (or God, if you're Mies van der Rohe) is in the nevertheless-not-entirely-spiritual details.

      I know that both of these guys are Community members and possible contest voters so not to press you further.

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      • Post #16

      Dear Felix,

      You did a nice exploration and exposition of a possible application of finite fields in quantum theory. Your article on arxiv shows that you developed extensively this idea. Since I do not know any field in fundamental physics which is closed, or at least which accounts for all observations, I think that we should not demand new-born theories to be perfect and answer all questions. Let's let them grow up so that we can really compare them with others which were developed during one century by so many scientists. I think it is good to question them and to compare them with experiment even from the beginning, but I don't think that their value should be judged before their maturity.

      I will ask some questions about your essay, if you don't mind. Please, if you feel that we disagree at some points, consider my questions as a proof of interest and curiosity.

      Best regards,

      Cristi Stoica

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      • Post #17

      Dear Felix,

      I confess that I do not perceive the standard mathematics as being wrong, and the discrete or finite one as the only justified. I don't think I have enough information to decide whether our world is discrete or continuous. This is why I salute both directions of research, and I am interested in the arguments or evidence of each of them.

      You said: "Standard mathematics is based on axioms about infinite sets (e.g., Zorn's lemma or Zermelo's axiom of choice), which are accepted without proof. Our belief that these axioms are correct is based on the fact that sciences using standard mathematics (physics, chemistry etc.) describe nature with a very high accuracy."

      This triggered in my head the following questions (I would be pleased to receive answers from other readers too):

      [?] As far as I know, Zorn's lemma and the axiom of choice are independent of the other axioms in set theory (although they are equivalent for example in Zermello-Fraenkel's system). Would it be possible to interpret one of the known experiments, or to devise a new one, so that we can check if they are valid from our world?

      [?] All mathematical physics uses mathematics based on some axioms. Were some of these axioms tested directly, or only through their consequences (predictions)? Would it be possible, at least in principle, a physics based on axioms which are tested directly?

      Best regards,

      Cristi Stoica

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        • Post #18

        Dear Felix M. Lev,

        Mathematics is a blind tool designed mainly to follow physics and describe the products made by physisists, but not a tool to discover something in physics; There are a few examples only when mathematical methods discovered something in physics, but a many thousands of erroneous mathematical papers and false mathematical "proofs". It is dangerous for physics because a lot of people mask their false and erroneous papers under mathematical formulas and mathematical theories.

        I have examined some your papers in order to find what this mathematical instrument can really discover in PHYSICS; However, the most of your papers deals mainly with Galois fields. Even in your paper "A POSSIBLE MECHANISM OF GRAVITY" I don't found any physical mechanism of gravitation - mathematics only; Gravity is a manifestation of Galois fields? Can you explain how this manifestation of Calois fields can curve spacetime and slow down time? Also some your papers repeats the same information, for example the figure "Relation between Fp and the ring of integers" I saw in 3 your different papers.

        Sincerely,

        Constantin

          Dear Cristi,

          Thank you for encouraging words about my works.

          I am not saying that standard mathematics is wrong. The question is whether we

          i) accept a principle that only those statements have a physical significance, which can be experimentally verified (at least in principle) or ii) we agree that some statements (axioms) can be accepted without proof (for some reasons). Since you pose two questions [?] in your note, you probably think that we should accept i), right? But then we should acknowledge that standard axioms cannot be verified. For example, how can we verify that a+b=b+a for any natural numbers a and b?

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          • Post #20

          Dear Felix,

          could my question "Would it be possible, at least in principle, a physics based on axioms which are tested directly?" be answered positively by a physics based on finite fields?

          Could the habitants of a finite universe know everything about their world, just because there is a finite number of things to be known? It seems to me that they are "more finite" than the knowledge about their world. So, I would incline towards the second possibility you mentioned.

          Dear Constantin,

          You write: "Mathematics is a blind tool designed mainly to follow physics and describe the products made by physisists, but not a tool to discover something in physics" and so on. This is really nonsense.

          I noticed that sometimes the authority argument works better than a more rational one.

          Isaac Newton was Lucasian Professor of Mathematics at the University of Cambridge. His contemporary and archenemy Gottfried Wilhelm Leibniz was a mathematician as well. His Monadology, considered to be utterly unphysical, started the research program which ended with the invention of the computer and of the www (namely the http).

          Coming back to Lev paper, it is certainly nothing wrong, IN PRINCIPLE, to suggest that a physical phenomenon "is a manifestation of Galois fields", because, look, Newton himself had to invent differential calculus in order to establish his theory.

          (Leibniz invented the same mathematical "blind instrument" simultaneously and better than Newton, for reasons which were independent of physics.)

          Dear Constantin Leshan,

          In my understanding, you think that my works contain only mathematics but not physics, right? For example, in my essay I describe the following results of my works. In contrast with standard theory, in my approach based on finite fields there are no independent irreducible representations (IRs) for a particle and its antiparticle but one IR describes an object such that a particle and its antiparticle are different states of this object. As a consequence, there are no neutral elementary particles, the electric, baryon and lepton charges can be only approximately conserved and even the notion of particle-antiparticle is only approximate. In your opinion, these results are only mathematical or they have something to do with physics?

          The problem of describing gravity is out of the scope of this contest; we could discuss this problem via email if you are interested.

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          • Post #23

          In addition to what Marius said, I would mention:

          - Riemannian geometry preceded its applications to general relativity

          - Hilbert spaces preceded their applications to quantum theory

          - Clifford algebras and spinors preceded their applications to relativistic quantum theory

          - connections on fiber bundles preceded their applications to Yang-Mills theory

          - holonomy groups preceded their applications to gauge theory, to Wilson loops and to loop quantum gravity

          - representations of Lie group preceded their applications to particle physics

          - topos theory preceded its applications to quantum theory obtained by Chris Isham

          - the particular Kaehler manifolds named Calabi-Yau manifolds preceded their applications to string theory

          I agree that mathematics originated from practical necessities, which come from the physical world. But mathematicians are playful species, and they like to explore platonic worlds as well. For some reason, their explorations anticipated many of the necessities of physics. Or maybe physicists find easier to borrow from mathematics, rather than making their own tools ;-). Or when they do, the tools are often full of divergences and singularities, and are inconsistent. As John Baez said once, it is the job of mathematicians to eliminate these inconsistencies. So I would say that both physicists and mathematicians have their equally important role.

          ___

          Lebniz's monadology finds also applications in Haskell (programming language). It also influenced Whitehead, and through him some applications to quantum theory.

          Best regards,

          Cristi

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          • Post #24

          Dear Marius Buliga and Cristi Stoica,

          I agree that there are a few examples of successful predictions in physics made by mathematics, against the thousands of wrong predictions on the other hand; I can show that about 70 percents of all theoretical papers made by mathematicians in physics are wrong. For example, about hundreds of different theories of gravitation have been published in the academic journals, but it is self-evident that one or two similar gravitational theories only can be true at the same time, but not hundreds of theories. It is self-evident that 99 percents of all published gravitational theories are erroneous. Meanwhile all these erroneous theories have the "PERFECT MATHEMATICS" and "mathematical proofs" and are accepted by peer reviewed journals and physics community. Also the same situation arise in other areas of physics: about 70 percents of all theoretical papers in physics made by mathematicians are wrong. For example, let us analyze this paper this paper published in Physical Review Letters: I found tens of errors here whereas this paper has been supported by Physical Review Letters and NASA. You see, the authors try to prove their erroneous papers by help of mathematics; Thus, the mathematical proofs in physical theories must be in doubt. All the Standard Model is a mathematical model only that can compute only but explain nothing. The invasion of mathematicians will stop the development of Physics. That situation arise because peer reviewed Journals accepts papers with mathematical content only. Journals should accept that a physical logic (reasoning) must have equal rights with mathematical proofs.

          Dear Felix M Lev

          Yes, your mathematical approach tries to describe the EXISTING already physical phenomena only. Your results confirm my point of view that mathematics must be a tool to describe quantitatively products (theories and phenomena) made/discovered by physicists only, but it is not a indicator for physicists what they must do. Because physics is directed by mathematics, it is a cause of modern crisis in physics.

          Sincerely,

          Constantin

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            • Post #25

            [I apologize to Felix, I do not want to monopolize this thread. I would kindly ask a FQXi admin who validates the comments to move the discussion to a different thread, if it is off topic. Or perhaps to move this comment and the father comment as children to the discussion opened by Constantin Leshan, so that the discussion gets collapsed and does not occupy too much of this page.]

            Dear Constantin Leshan,

            you say "I agree that there are a few examples of successful predictions in physics made by mathematics"

            The examples I gave cover a very wide part of fundamental physics, and I think that we can go on with such examples to cover most of it. But I did not claim that those mathematical theories which found applications in physics were predictions. Well, in some cases they are, for example Riemann, Hamilton and Clifford intended to obtain a mathematical description of space and time, although the result was not exactly as they expected. But most of them - for example, the Hilbert space - were not made with the physical applications in mind. It was only discovered later that they can be applied, probably, as I said, because physicists realized that these tools can be borrowed and used with success.

            You say "I can show that about 70 percents of all theoretical papers made by mathematicians in physics are wrong. For example, about hundreds of different theories of gravitation have been published in the academic journals, but it is self-evident that one or two similar gravitational theories only can be true at the same time, but not hundreds of theories."

            Should I understand that these hundreds of different theories of gravitation are published by mathematicians and not physicists? I was thinking that physicists are those publishing them. If you are right, then it is simple to find the correct theory of gravity: just look at the resume of various authors, exclude the theories invented by mathematicians, and keep those discovered by physicists. If they are one or two, it should be easy to identify them.

            My guess is that the percentage of wrong theories, let's say 70% as you say, although I think it is larger, is the same for physicists and for mathematicians. Or it would be so if mathematicians would be interesting in making theories of gravity.

            My viewpoint* is that physicists are those doing physics. Mathematical physicists develop the theories discovered by physicists, or try to express them in different mathematical formalisms, in order to find the best fit. Mathematicians which are not particularly interested in physics, develop and generalize and solve various particular cases and classify the solutions etc., without caring about the applications. From time to time, a purely abstract mathematical theory is found to provide a good formulation of a concrete physical problem.

            I apologize if I let the impression that I claim that all the work in physics is done by mathematicians. This is far from truth, and I would not do such a discrimination. In fact, most of my heroes in science are physicists rather than mathematicians.

            Best regards,

            Cristi

            __________________________________

            * Oversimplified and stereotypical, of course, but if I would like to be correct in detail, I should never speak :-).

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            • Post #26

            Dear Cristi Stoica,

            The border between mathematicians and physicists is very thin; therefore by papers of mathematicians I mean papers were the percentage of mathematics is more than 40 of volume, or all proofs are mathematical; From this point of view, almost all gravitational theories are mathematical theories, made by mathematicians. Also, I do not accuse all mathematical world; I say that all false physical theories have mathematical proofs, consequently mathematical proofs in physical theories must be in doubt. Therefore, it is mathematics' fault that the most of published physical papers are wrong.

            It is because journals accept papers with mathematical content only. I'm sure that the percentage of false theories may fall, if the journals allow physical reasoning instead of mathematical proofs. Thus, Journals must allow to physicists to publish their papers with physical proofs instead of mathematical. Since the Standard Model is more mathematical model than physical, therefore I accuse mathematics. We'll never find any Higgs boson because SM is a mathematical model only that may fall in nearest future. The future Physics will be based on physical reasoning rather than on mathematical proofs.

            About moving this comment to the discussion opened by Constantin Leshan: soon I'll send my essay to FQXI. I invite you to find logical errors in my theory.

            Sincerely,

            Constantin Leshan

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            • Post #27

            Dear Constantin,

            I am sympathetic to your viewpoint that these day physics is too much math without physical content. I stop now because I took too much space from Felix with my off topic comments. We can continue by email. Good luck with your forthcoming essay.

            Cristi

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            • Post #28

            Dear Felix,

            Sorry for the delayed answer, I was caught up in a lot of work recently. Let me start by clarifying my intention. First and foremost I am interested in understanding your approach because I work in a different number system for QM myself. Second, I am interested in your essay entry. Let me repeat that I find your essay interesting, otherwise I would not spend my time trying to understand your ideas. Also you have strong claims, and strong claims deserve strong scrutiny, IMHO.

            I am puzzled by your statements: "Let me also repeat that if I understand you correctly and you are interested only in discussing physics then the present forum is not an appropriate place for this discussion. " and "I believe that my essay is fully in the spirit of this essay contest entitled "Is Reality Digital or Analog?" So I discuss mathematics." First, this is a physics contest and FQXi is mostly a physics organization. Second, mathematical statements without physics support are irrelevant to deciding if nature is digital or analog. Mathematical (or any other kinds of) statements without agreement with reality are just marks on paper.

            To me, discussing from the physics point of view it is the only thing which makes sense and interests me. But if you find this inappropriate, I will respect your wishes and not continue to ask questions. But if you want to continue the physics discussion, I am available.

            5 days later

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            • Post #29

            Hi all,

            Congratulations for your beautiful essay dear Felix,The finite groups of Galois are relevants in my humble opinion when we want calculate rationally the quantic number and all its proportionalities.This system has a finite serie at my opinion.

            To all, very relevant discussions.Don't stop dear Friends, hhihihi Laplace, Poisson and Gauss shall be happy to see these discussions and they shall say,; don't forget the theory of errors and the dispersions.....a kind of precison and sorting appears in the same rational logic.Like an Occam Raozr applied to maths for rational physics.

            That permits to see better the serie towards the Planck scale and its finite number.

            The infinity , the 0 and the - must be rationalized in the pure physicality and its pure laws in 3 Dimensions and a time constant of evolution.I d say even ,they doesn't really exist, if we add them yes, but not in our pure uniqueness, and their finite system and their pure number.

            We can for example add or multiplicate our cosmological spheres, that doesn't mean that their number changes...their pure number inside an evolutive Unievrse rests like it is.It's the same for our quantum number, we can add or multiplicate them ,their pure number rests.It's a little like a proportional approximation in fact with rational limits.

            Regards

            Steve

            Dear Steve,

            Thank you for encouraging words about my essay. Some your remarks are not clear to me and, probably, we have different opinions on some issues (e.g. on the role of geometry, whether the theory should be based on finite groups or Lie algebras over finite fields etc.). We could discuss them via email if you are interested. Happy New Year!

            Felix.