Thank you for this brilliant essay. In nine pages you have summarized several thousand years in the evolution of math and physics and done it with (relative) completeness and consistency. (Far better than I did in my essay - The Hole at the Center of Creation.)

We are left with some very serious problems, as you point out, but you seem quite confident that the intuitions of physicists will find solutions: "Any new mathematical forms used in future physics will have to be interpreted through meaningful concepts. The applicability of mathematics to reality hinges on human ingenuity, not on the objective existence of mathematical forms."

With the disarray you note in the foundations of mathematics and the apparent paradoxes in physics, and even the inadequacies you cite in AI, one might conclude that we have reached the limits in the human capacity for intuiting the nature of reality. What does your optimism rest on?

Sincerely - George Gantz

    Dear Professor MacKinnon,

    Your intro of "a schematic account of the way scientists and others came to use mathematics in physical accounts." leads to a very well written historical account of many of the major concepts linking math and physics today. In your conclusion, I like the statement "The application of mathematics to reality presupposes a conceptual structuring of reality." and could not agree more. Enjoyed the chance to read a well throught out conclusion based on the history of physics.

    As an aside, I have recently run into a youtube video of a Dirac lecture in 1975 here, although somewhat difficult to hear (you need headphones), it provides remarkable insight into the history of physics in the early 1900's, thought you might enjoy it.

    Regards and best of luck in the contest.

    Ed Unverricht

      Hi Ed,

      Thanks for the information. I tried to follow the link but got nowhere. However I heard Dirac lecture on the state of physics at Yeshiva grad school some time earlier

      Edward,

      You say, "The mathematics of quantum physics clearly goes beyond physical reality as structured by classical concepts. Yet, its physical interpretationrequires using classical concepts while recognizing the limits of their valid applicability."

      I agree with most of your concepts but the above hit me in what I see as the normal practice of attributing quantum behavior to the classical like Schrödinger's cat. I marvel at new connections of the quantum with quantum biology, atomic orbital math functions feeding into the molecular, for example, applying quantum aspects to the molecule.

      Jim

      Dear Edward McKinnon,

      A little bit in contrast to yours, my essay follows historians whose explanation of the sudden progress around 1500 focused on details that were in particular of interest to me for quite a simple reason: I was an engineer who got aware that the real-valued cosine transformation doesn't convey less information than the Fourier transformation except for the arbitrary choose of point zero. That's why I am after the origin of John Wallis' block time/scale from minus infinity to plus infinity. Fourier had dealt with heat conduction within a ring of metal. I consider an accordingly closed loop of time at best an unwarranted speculation.

      Incidentally is Wallis' the correct Saxon genitive of Wallis? I just live in Saxony-Anhalt. English is not my mother tongue. I wondered; why did you write (on p.3) Leibnitz's and not Leibniz'?

      Being not a mathematician, I did not yet found "a calculus based on [point] set-theory. I accept aleph_0 and aleph_1 as a reasonable distinction between rational numbers and the reals of continuum, but not as a justification for cardinality. So you may me identify as heretical. My arguments always arose from practically relevant logical inconsistencies that so far nobody could explain away.

      Sincerely,

      Eckard Blumschein

        Dear Edward McKinnon,

        First of all I have liked your lucid account of the co-evolution and path-breaking (Laplace and Cauchy) history of MP. En Passant wrote here a short essay about "qualities could admit of quantitative representations, an essential step in developing the mathematical relations and laws needed in physics". I also fully agree that language is the new ingredient needed by QM and fully recognized by Bohr. Finally, my experience as an engineer and a mathematical physicist is just like the Dirac quote "A great deal of my work is just playing with equations and seeing what they give" most of the time I use the magic found by clever mathematicians, ask the computer to pursue the play and try to use my "human ingenuity", "not the objective existence of mathematical forms" to improve on the existing knowledge.

        To develop further what you are saying about QM and language, a better understanding of human cognition seems to me essential. In this direction, I recently found (following Grothendieck) that the language of permutations is a needed ingredient for further progress. This is explained in my essay: a free group with only two letters, a permutation group (two permutations), a corresponding Riemann topology, a point-line incidence geometry, an algebraic curve over the rationals "objects so simple that a child learns them while playing" (say Grothendieck).

        Thank you for a very pleasant and relevant essay.

        Michel

        Addendum:

        "Relativity theory and quantum physics contravened this simple correspondence [between classical reality and mathematical expressions]". Did you read the essay by Phipps and also my reply to Matt Visser?

        Eckard Blumschein

        20 days later

        Dear Edward,

        I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

        All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

        Joe Fisher

        13 days later

        Hi Edward,

        Thank you very much for your very well written essay. To follow the development of mathematics and physics near to the proponents is a nice experience. And I fully agree with your final Bohr like interpretation of quantum physics. Which seems not so popular any more according to the rating.

        I tried to find Heisenbergs comment on your reconstruction of his quantum formulation in the Sources for the History of Quantum Physics, but I couldn't. Could you be more specific on where to find it?

        Thanks and beset regards

        Luca

          Hi Luca,

          Thanks for the comments. Heisenberg sent me a letter, dated July 12, 1974, commenting on the first draft of my HSPS paper. I made a copy available to the UC Berkeley center for the Sources. I presumed that they made copies available to all the centers. However, I could send you a copy of the letter if I had your email address.

          Ed

          Hi George,

          Thaanks for the comments. I am slow replying because I have been traveling. My optimism does not rest on any theory of intuition. There are so many people trying to go beyond the standard model that someone is bound to make a breakthrough sooner or later.

          Hi Edward,

          I would be delighted. Here's my adress:

          luca.valeri.zi@gmail.com

          Thanks a lot

          Luca

          Edward,

          Shark time as they pull you down, so I am revisiting essays I've read to assure I've rated them. I find that I did not rate yours, and am rectifying. I hope you get a chance to look at mine: http://fqxi.org/community/forum/topic/2345

          Jim

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