Dear Noson,

Your denial of "that beauty plays a role in either physics or mathematics" strongly contradicts to the history of fundamental science, from Kepler and Newton to Einstein and Dirac. Your quotation of Einstein is an example of an extreme misinterpretation by taking out of context. The only reliable source where I know Einstein mentions this 'tailor' comparison is his preface to "Relativity: The Special and General Theory" (1920). His words follow:

"In the interest of clearness, it appeared to me inevitable that I should repeat myself frequently, without paying the slightest attention to the elegance of the presentation. I adhered scrupulously to the precept of that brilliant theoretical physicist, L. Boltzmann, according to whom matters of elegance ought to be left to the tailor and to the cobbler."

I suppose that it is perfectly clear that the specific elegance left here by Einstein "to the tailor and to the cobbler" has nothing to do with neglect of the mathematical elegance or beauty. It is hard to say anything further from truth than to claim aesthetic negligence of Einstein in general and in the matters of theoretical science in particular. There are many clear statements of Einstein about the role beauty played in his own thought and in the history of science. Take for instance the following, where Einstein defines mathematical beauty, or "inner perfection" of theory:

"Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature." (Obituary for Emmy Noether (1935))

Thus, mathematical beauty/elegance is a unity of logical simplicity and richness of the content. In one or another way belief in the elegance of the laws of nature was expressed by many fathers of science, and this belief played a crucial role in the history of science. When Wigner wrote about 'unreasonable effectiveness of mathematics' he did not mean that laws of nature are somehow described by formulas; he meant that these formulas are both simple in form and rich in content.

The actual question, missed by many essays of this contest, including yours, is why 'the laws of nature are expressed by beautiful equations', as Wigner's brother-in-law put it. Symmetry is just a part of this logical simplicity. We may imagine a universe in which laws have nothing to do with any sort of symmetry and can be described by kilometer-long formulas only. Why the laws of our universe are so symmetric and so simple, which made them discoverable? This is the real question meant by Wigner, which your essay and so many essays here have completely lost.

Dear Prof. Yanofski,

Great essay! You offered a precise definitions of physics and mathematics and the relation between them. We seem to agree in many points, as my essay reflects, especially the importance of symmetry, and how regularities in nature allow mathematics to be so effective. I would be glad to take your opinion in my essay.

Best regards,

Mohammed

Dear Alexei,

Thank you. I never knew the origin of the Einstein quote and I agree it is indeed out of context.

But you cannot make your point by an "appeal to authority". Please give a definition of beauty/elegance/unity of ideas etc? Are such concepts quantifiable? Can a computer determine when a physical or mathematical idea is beautiful? (Can a computer tell when a painting is beautiful? ) If yes, please tell. If not, then we are talking about some vague wishey washy feeling that many great physicists had. That is fine. But it does not really say anything about the physical universe.

This is interesting stuff. Thanks!

All the best,

Noson

Dear Noson,

I enjoyed your crisply and clearly written essay very much. The fact that symmetries are lurking everywhere if we only look for them is a lesson that everyone who wants to understand the nature of reality should take deeply to hear. I had not previously thought of logical validity as a symmetry, or, for that matter, of the symmetry of applicability. While it is clear that the conserved "thing" in the former is truth, I am wondering what it is for the latter?

If I my give just one little piece of constructive criticism, I found ending the essay with an appeal to the anthropic principle a little bit of a downer. In its uncontroversial form it is a tautology, and if one wants to draw stronger conclusions from it, then it invites a host of questions the answers to which based on the strong versions stretch credulity.

However, you did mention at the very end that the anthropic principle does not answer all the deep questions.

On final note, I came across some reviews of your book "The outer limits of reason" which seems like a very interesting book and which I will add to my "to read" list.

Best wishes,

Armin

    The idea of describing the mathematical character of physical laws in terms of the abundance of symmetries, was also present in the essay by Milen Velchev Velev. See my reply there. See also the references of essays with effective descriptions of "what is remarkable about the success of mathematics in physics" I collected in my review of this contest, which your essay does not account for.

      Dear Noson,

      I enjoyed very much your essay. One can't deny the success of mathematics, as you illustrated by predictions like those by Le Verrier, Maxwell, Dirac, Kepler, etc. You criticize well the positions of theologians and Plantonists, and especially the standard response, which presents mathematics as relative to the human experience. Indeed, the role played by symmetry must be fundamental, both in physics and mathematics, and may explain the connection between the two of them. I plan to read your references on the symmetry in math. I am particularly interested since some years ago I identified a symmetry in mathematics that connects many different structures under one umbrella, but I never find time to finish that paper (I think I thought that not many would consider it to be very useful to worth the effort). Of course, the idea is different than yours. I like your definition "a statement is mathematics if we can swap what it refers to and remain true" and "anything that satisfies symmetry of semantics, is mathematics". I agree with "Mathematics is invariant with respect to symmetry of semantics. We are claiming that this type of symmetry is as fundamental to mathematics as symmetry is to physics." Congrats for a so well written essay, which truly addresses the question of effectiveness of mathematics in physics.

      Best wishes,

      Cristi Stoica

        Dear Armin,

        Thank you for the nice words.

        Applicability is the property that is preserved by symmetry of applicability. Let me explain. If a certain rule works with an object of a certain type and you swap that object for another object of the same type then the rule will still be applicable for the new object. Agree?

        I agree with you that the anthropic principle is true but there is a feeling of cheating with it. As you say, I pointed out its shortcomings.

        Please let me know what you think of my book when you read it.

        Sincerely,

        Noson

        Hi,

        MVV did not really address symmetry in mathematics.

        I have a hard time seeing where my essay sits in your idiological oppositions chart.

        Noson

        Dear Noson,

        I read your essay a while ago but I realized I forgot to comment. My sincerest apologies for this, because I enjoyed your writing very much! I found that there is much similarity of thought between us.

        This is an extraordinarily precise and accurate analysis that you are making. You are saying that the problem only exists if "one considers both physics and mathematics to each be perfectly formed, objective and independent of human observers". This is absolutely true! Unless one starts with the assumption that they are unrelated, the problem disappears. You are going to the heart of the problem when you bring into discussion the symmetry and conservation laws of physics, because it is this reducibility to invariant quantities that is characteristic to both math and physics that makes them work together. Can the symmetry of syntax that makes math be a good support for physics be called a conservation law as well? I am thinking here on the lines of the internal logical consistency of math.

        Wish you the best of luck in the contest!

        Alma

        Dear Professor Yanofsky,

        Your example of James Bond leads to the obvious question whether you are saying that mathematical regularities, like well-known fictional characters, are in the end items of inter-subjective human agreement. Many people would say that mathematical entities, such as pi or the integer eight, appear to have a definiteness which is independent of human perception and agreement. In this way, the mathematical entities seem unlike fictional characters. I think a similar dissimilarity holds in the case of mathematical truths as contrasted with truths about fictional entities. But, as you say, the issues here deserve more thought. In any event, thanks for your stimulating essay here and for your book, "The Outer Limits of Reason."

        Best wishes,

        Laurence Hitterdale

        Dear Noson,

        first let me say that I enjoyed very much reading you book "The Outer Limits of Reason". It is one of the few books that I read more than 60% scratching my personal notes on it and highlighting the most interesting parts. The reason is that you penetrate the epistemology of physics and the structure of a scientific theory with focus and magnetizing style. In your nice essay I can recognize your style of the book.

        I completely share the idea that the key to the solution of why mathematics works so well in physics is group theory and the notion of symmetry, as I also emphasized in my essay. I came to the conclusion that a lot of physics can be derived in terms of group representations. For example, we recently gave a definition of reference frame and boost that is purely group theoretical (without space, time, kinematics, and mechanics!) corresponding to a general invariance that only in a limiting situation (the so-called relativistic limit) leads to Lorentz invariance of QFT, but more generally is a nonlinear version of Lorentz group. This allowed us to extend the notion of boost to a discrete Planck scale.

        My best wishes

        Mauro

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