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

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

    [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 :-).

    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

    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

    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

    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.

    Hi all,

    dear Felix,

    You are welcome.

    Indeed we have different points of vue(as many here on FQXi,the sharing of ideas seems essential), but the most important is this universality behind.

    I like finite groups, and I think that maths must be analyzed with the biggest rationality when we analyze physics in its details.

    I utilize algebras with an ocaam razzor,it exists several methods ,interestings and relevants.I add or superimpose them.

    But I don't rest in one method.

    In fact lie algebras, Clifford's alg.,.....are interestings when they respect the foundamental theorem of algebras.Now of course the physicality is the physicality.And the number is the number.

    I see the quantum entanglement a little as our universe.Now if the entanglement of spheres is specific....the volumes are important and the number is the same and finite as the serie of volumes.The begining is a fractal of the main central sphere.Now I ask me how is the serie between 1 and our number of cosmological spheres.My problem is about the spheres between the center and our planets.And between 1 and 2 and 3........the volumes decrease on a specific harmonous serie.

    Yes of course here is mine , a simple google mail.We can speak here you know I am transparent.

    Ps sorry for my poor litteral english.

    dufournybionature@gmail.com

    Regards

    Steve

    I like the idea of using the Galois fields in physics. Are you aware of the result that it is possible to construct a complete set of mutually unbiased basis for finite dimensional quantum systems if and only if the base is indexed by a Galois field?

      A direct generalization of mutually unbiased bases to the case of Galois fields is meaningless for several reasons. For example, ½ in Galois fields is (p+1)/2, i.e. a huge number if p is huge. In standard theory, probabilities are normalized to one but this is only a matter of convention since not the probability itself has a physical meaning but only ratios of probabilities of different experimental outcomes have (that's why Hilbert spaces in quantum theory are projective). In addition, as noted in my essay and papers, in theories over Galois fields the notion of probability can be only approximate. However, I believe that in situations when probability is meaningful, it is not difficult to modify the definition of mutually unbiased bases such that the main idea of the definition will be implemented. But the question that the base should be indexed by a Galois field is not clear to me. For example, when we have a finite dimensional linear space over a Galois field, we don't say that the basis elements are indexed by a Galois field, right?

      Hi ,

      Dear Felix,

      Like I said you in private, I have no publications,it's not my aim.

      I just work simply about my theory of spherization,and I improve it.

      I haven't finished my universities, in fact I have studied a little of all,I was in medecine, after in geology(there a little problem of neurology apparently,a little coma and a kind of epilepsy),after agronomy,.....and I continued my classments and works.I have even created an enterprize in horticulture and vegetal multiplication(at the age of 23) but apparently I am not skilling in business.thus brankrupcy in 2004,oh my god.well it's the past.

      I continue simply my spherization theory.Isolated I agree but that goes .

      About your essays, I see a very good knowledge about our foundamentals and its whole,about also our international language about sciences and maths and physics.

      I like also your rational pragmatism about our reality.Indedd only precise results are essential.That means a logical method.It's important, thanks for that.At this momment it's rare to see rationalists.

      In your conclusion, you say

      "We conclude that the very notion of particle-antiparticle is approximate

      and the electric, baryon and lepton charges are only approximately conserved quantities.

      The non-conservation of the baryon and lepton quantum numbers has been

      already considered in models of Grand Unification but the electric charge has been

      always believed to be a strictly conserved quantum number. The non-conservation

      of these quantum numbers also completely changes the status of the problem known

      as "baryon asymmetry of the Universe" since at early stages of the Universe energies

      were much greater than now and therefore transitions between particles and

      antiparticles had a much greater probability."

      Could you develop a little please why a much greater probability?

      If we take the CMD at low energy ....curves of Planck.

      At high energy ther origin is not thermic if I can say.

      Now let's take the annihilation of matter/anti matter and we see them on graphics with waves lenghts and ray in MeV.

      we can take also an other example with RX or gamma R.....See the interactions of relativistic electrons coming from galaxies and the photon in low energy in the cosmological deep sphere.ISOTROPISM

      How can we have a correct formalism interpreting a space time quadridimentional if the pseudo euclidian system smiles to Gallilei and Minkowski.I like the evolution and it's a main parameter,Fiedman and Robertson have understood this point of evolution, relativistic.

      3 DIMENSIONS AND A TIME CONSTANT BUT THIS SPACE TIME EVOLVES SIMPLY IT IS THE REAL RELATIVITY.

      MY ROTATING SPINNING SPHERES ANSXER TO .........GRAVITATION? QUANTIC ALSO....G c and h are linked .why because the sense of rotation has two main senses!!! TO MEDITATE.

      Steve

      Dear Florin,

      I have no problem in discussing any aspect of my approach; the only problem I have is whether the discussion is in the framework of this forum. If you think it is, let's continue and if we have a feeling that it goes out of the scope of this contest, we could continue the discussion via email if are interested.

      I appreciate that you are interested in my approach. Indeed I have strong claims and agree with your statement that "strong claims deserve strong scrutiny". So I would be grateful for any criticism. My expectation is that first of all any criticism should say whether or not MY RESULTS are correct and physical. However, so far you did not discuss my results at all. You expressed your opinion that the string theory, or something like that have greater chances to solve the existing problems and that so far you see the application of my approach only in the Landau pole problem. These questions fully satisfy a principle that "De gustibus non disputandum est" and here I don't see any point for discussion. So I propose to discuss the results of my approach.

      You advised me to calculate the 8 digits in my approach. Many years ago it was indeed one of my main motivations. The idea was to replace irreducible representations (IRs) for the electron, positron and photon by their modular analogs and obtain a finite theory. However, this naïve attempt fails for the following reason: in a theory over a Galois field, one IR describes a particle and its antiparticle simultaneously and there are no IRs for neutral particles. In particular, even the photon cannot be elementary. Could you, please tell me your opinion on this situation? For example, a possible point of view is that if even a photon cannot be elementary then GFQT is unphysical.

      Happy New Year! Felix.

      17 days later

      I read your paper with interest. I do have a couple of questions about the idea of Galois QFT. The cyclotomic numbers of F_4 z = e^{i2πn/3} describe the root space of D_4. F_4 is the Dynkin diagram for D_4 ~ SO(8). The D_4 root lattice is the dual of the F_4 and a subring of Hurwitz quaternions. In this way Galois groups can characterize the symmetries of a QFT, or a YM gauge theory.

      Cheers LC

        Thank you for your interest to my essay. Here I argue that any fundamental physics can be based only on a finite math and consider an approach based on a Galois field. So I cannot have e^{i2πn/3} or SO(8). Also, I cannot use Dynkin diagrams for describing Lie algebras over Galois fields since the latter are not algebraically closed. Your first F_4 is not a Galois field; probably you mean a Galois group. Galois groups are used for describing field extensions and in general the fields are not assumed to be necessarily Galois ones. Since you are talking about cyclotomic numbers, you probably mean extensions of Q. So, in my understanding, your questions refer to standard theory but not to my approach.

        Best regards, Felix Lev.

        Hi to both of you,

        Happy dear lev to see this rationality, indeed the finite systems must respected their own limits.

        We can superimpose but with rationality of course.

        Finite maths....galois field.very relevant indeed , very relevant.

        Steve

        Dear Lev,

        in your essay you make the clear point that finite mathematics (such as GFQT) is the most pertinent choice for describing physical reality.

        However, you do not seem to take an equally clear position about the ultimate nature of reality: is the universe (discrete and) finite or infinite? In fact, are both possibilites still open, under GFQT?

        Do you perhaps envisage a third possibility, namely that GFQT works very nicely for just making accurate experimental predictions in a QM setting, without still resolving this finite vs. infinite universe puzzle, which is perhaps only of philosophical relevance?

        If the question sounds indeed too philosophical (but that's essentially the title of the contest...), I could reformulate it as follows: would your theory be compatible, incompatible, or neutral, with a statement such as 'there are 10^234 atoms of spacetime in the universe'?

        A second question. You talk about parameters p and n, defining the size of the GF, as universal constants. Would it make any sense to rather imagine them as changing, I mean on a cosmological scale?

        Thanks!

          Cyclotomic numbers are roots of a cyclotomic field, as I recall the term, but you are right it is a Galois extension of the rational numbers.

          I was wondering if there is a connection to standard quantum field theory. It seemed to me there should be an underlying Lie group theory with this.

          Cheers LC

          Dear Steve Dufourny,

          Thank you for your comments. I wish you success in developing your approach.

          Best regards, Felix Lev.

          ==========================================================

          Dear Lawrence B. Crowell,

          In standard theory (over C) we typically have a one-to-one correspondence between representations of Lie algebras and representations of corresponding Lie groups (for finite dimensional representations this is well known; for infinite dimensional ones this is usually a case for reasonable assumptions). Representations of Lie algebras by Hermitian operators have a clear physical meaning since the representation operators describe physical quantities. At the same time, one might ask whether all representation operators for representations of Lie groups are meaningful. For example, are temporal translations by 10^{-1000}sec. or spatial translations by 10^{-1000}m meaningful? However, in the theory over complex numbers we must agree that they are meaningful since we have infinitesimal transformations and exp and so we can construct any representation operator of a Lie group from a representation operator of its Lie algebra.

          However, in a theory over Galois fields, we don't have infinitesimal transformations and exp. So in this case there is no notion of a Lie group over a Galois field corresponding to a Lie algebra over a Galois field. I believe that this by no means indicates that a theory over Galois fields is unphysical. In many cases we can have finite transformations, which approximate transformations over C with a high accuracy. However, the absence of Lie groups considerably changes the theory. In my essay and papers these problems are discussed in details.

          Best regards, Felix Lev.