Dear Torsten,

Thanks to you I discovered the exotic world of manifolds. I fully agree that mathematics is the driving force for science as you perfectly showed. We have much to share in the near future and I intend to work hard in this direction. My rate this year is eight. New questions to you in preparation.

Best,

Michel

    Thank you so much Torsten,

    I started to read your book

    http://www.maths.ed.ac.uk/~aar/papers/exoticsmooth.pdf

    I am also doing mathematical experiments on 3-manifolds

    http://magma.maths.usyd.edu.au/magma/handbook/text/742

    Another mathematical result of interest

    "that every finitely presented group can be realized as the fundamental group of a 4-manifold"

    http://mathoverflow.net/questions/30238/constructing-4-manifolds-with-fundamental-group-with-a-given-presentation

    Of course, I just enter your field that I consider a pandora's box.

    Best,

    Michel

      Torsten,

      Very readable essay that cogently presents your 5 basic ideas. The integral function of math I see as connected with computers and modeling, augmenting those mental weaknesses we have and requiring peer review (BICEP2, for example) to get it right.That math is a unifying force for all sciences I see and relate to the new field of quantum biology, DNA studies, and the LHC.

      Many of your ideas I mention but cite more of the pragmatics and less of the integral connections you represent.

      Your essay traces well the historical to the modern. Because our length is limited, you didn't seem to have time for the quantum needs and connections in physics.

      Thanks for giving us the opportunity to share your views.

      Jim

        Jim,

        Thanks for your words (and rating?). Your essay is also on my reading list.

        Certainly more later

        Torsten

        Jim,

        thanks for writing this essay. It contains a lot of ideas and conclusions to agree with. As you know from ym essay, I'm really interesting into the relation between the disciplines like biology, sociology, physics, math etc. Your essay covered all these question.

        It reminds me on a discussion with a biophysicist about consciousness and quantum mechanics. New experiments seem to imply that quantum mechanics is needed to get consciousness and higher brain functions. You explained it also at the example of birds finding their route.

        Therefore you will also get a high rate from me.

        Best

        Torsten

        Torsten,

        Thanks for taking the time to read my essay and for your kind remarks.

        Jim

        You wrote:

        "... numbers as an abstract count of objects was the beginning. ... But math is in particular a relational theory. Let us consider Euclid's geometry. One needs some obvious basic objects like point, line or surface which is not defined. Then the axioms are given by the relation between the three objects (like: the intersection between two lines is a point). In principle all axiom systems are of this kind."

        Euclid's math was built directly on modeling structure in the world of phenomena, and therefore has phenomena as it's "referent". The same cannot be said for much of math that comes since, although it certainly has been adapted (with great effort and creativity) to the task of modeling phenomena.

        Before you can claim otherwise, can you answer the question: what is a number?

        Also, Euclid's axioms and postulates have the quality of encoding the law-like behavior of phenomena. Does that get carried forward into any subsequent math?

        You might want to check out the entry "The Mathematics of Science" by Robert MacDuff.

          Dear Torsten,

          You didn't confirm agreeing agree on that alephs in excess of aleph_1 didn't find any application in science.

          What about non-Dedekind but Euclidean (Maudlin's) numbers?

          Regards,

          Eckard

          What is a number? Honestly, I don't know. Counting of objacts in reference to a numer ia an abstract process. Human done it but it don't answers this question.

          Here I can answer woth Kronecker: the natural numbers are made by God. But all the rest belongs to Humans.

          You are also right, also Euklids geometry contains terms like line point etc. which cannot be defined or explained. The same is true in set theory.

          I will have a look into the essay of Robert MacDuff.

          Best

          Torsten

          Michel,

          thanks for your interest in exotic smoothness.

          In the last years we found some interesting relations to quantum mechanics for understanding decoherence or what is a quantum state geometrically.

          You follow me on ResearchGate and find all relevant papers there. One interesting result for you could be: a quantum state is a wild embedding (see Alexanders horned sphere or Fox wild knot) and we showed that a quantization of tame embedding (a usual embedding) is a wild embdding.

          This result is of curse connected to exotic smoothness: conider an exotic S^3xR then a S^3 insider of this space must be a wild embedded S^3.

          Currently I try to understand quantum mechanics from this point of view.

          Best

          Torsten

          Dear Torsten,

          Please find Euclid's famous, plausible, and compelling definition of a point as "something that has no parts" via Ref. 1 of my essay. Euclid summarized the still useful definitions and axioms of ancient mathematics.

          Naive point set theory was logically untenable and therefore substituted by competing among each other i.e. rather arbitrarily chosen systems of axioms (NF, ZF, ZFC, NGB, ...) that were fabricated with the only intention to avoid paradoxes, cf. Fraenkel 1923 and 1984.

          Regards,

          Eckard

          Dear Torsten Asselmeyer-Maluga

          You wrote very exhaustive presentation of mathematics in physics. At this search it is also importantly to find the most precise words, which describe our intuition. One good example of your precise words is: ''Without abstraction, our species with a limited brain is unable to reflect the world.'' Thus math is a process of abstraction. Thus, my conclusion is that the essence of math in pyhsics is to be abstract and simple as much as possible. Because foundations of physics should be simple, the task of math is to describe quantum gravity on a t-shirt. Or, answer, why universe exists, should be short one. This would also confirm trend in physics until now. Smolin is also naturalist, as I am, but he think that elementary physics is not simple. What do you think about this?

          My essay

          Best regards,

          Janko Kokosar

            Dear Torsten,

            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

              Dear Torsten,

              I just red your post to me about a wild embedding and a corresponding quantum state. Although my familiarity to your field is weak at the moment I am fully confident in your approach.

              Cheers,

              Michel

                Dear Janko Kokosar,

                thanks for the comment. I also had the chance to have a look into your essay.

                Interesting mixture of topics. I remember on a discussion with bio-physicists. Now there is more and more evidence that Consciousness (as caused by thehuman brain) is strongly related to quantum mechanics. The quantum nature of some processes in the brain is maybe the root of Consciousness.

                I think that at the end elementary particle physics can also explained simple. Currently we work on a topological model (based on the braid model of Bilson-Thompson). Maybe it is a way in this direction.

                I rate your essay with seven.

                Best

                Torsten

                Dear Michel,

                I think that we both have the same goal: to understand quantum mechanics from a geometrical point of view. At the end, our approaches will be converge.

                BTW, there is a new Springer journal Quatum Studies

                (they send me an email). Maybe interesting for you?

                Best

                Torsten

                Joe,

                the boundary of a 3D object is a surface. In this point I agree with you. Of course this is the reason why we see only surfaces at the first. But at the other there is a lot of experimental evidence for three (space) dimensions. I would expect that it is part of reality too.

                I'm quite sure that at the fundamental level (around Planck length) the world is 2D. But I remember on former discussions...

                Torsten

                Dear Torsten,

                Yes: Quantum studies: mathematics and foundations.

                The editor in chief Yakir Aharonow writes in the preface:

                "Finally, there is the approach championed by Dirac and repeated successfully by Feynman and later by Freeman Dyson, namely "playing with equations" as Dirac puts it. This approach sometimes causes equations to reveal their secrets as in the Dirac equation. Dirac took this approach and created results that mathematicians and physicists are still digesting. Feynman, first with the Lagrangian approach to quantum mechanics, the so-called path integral approach, and later with QED and most of the subsequent papers he wrote, operated in this manner. The same could be said of what Dyson did when he "cleaned up "QED into a methodology usable for calculations. Playing with the problems of quantum mechanics often leads to the creation of new mathematics."

                and "Think, reconsider, explore, create deep questions, use paradoxes as a tool for understanding, and finally: publish in this journal!"

                A priori this is a good journal for us. My own essay has quotes from Dirac and Dyson, and implicitely to Feynman that anticipated quantum information theory: "There's Plenty of Room at the Bottom" (in 1959). May be I can submit my Monstrous Quantum Theory and you?

                Best,

                Michel

                  Dear Torsten

                  Thank you for 7 points, but I gave you 10 points yesterday. I did not send message, thus this is in rules of FQXi. :) Thus, this that you give points to me today, is a coincidence.

                  The main reason is because you used right words, that I continue to describe relation between math and physics.