Dear Torsten,

Great essay! I learned from you. Very enjoyable reading, brief and yet packed with information. You covered Turing, Plato and many others. I also covered Turing and Plato. I also covered number theory briefly. I also concur with Pythagoras that "all things are numbers" and in KQID, it is Einstein complex coordinates Ψ(iτLx,y,z, Lm). However, I started with a premise that everything, yes everything is infinite qbit(00, 1, -1) or Qbit(00, +, -). The infinite contains both finites and infinites. Similarly, finite contains infinites. Because both are governed by infinite law(KQID). Finite law cannot govern infinite entity like the Qbit. Furthermore, as you pointed out everything including the Qbit or our Creator is evolving. Please review and comment on my essay.

Best wishes for the contest and I vote your essay highly,

Leo KoGuan

    Dear Leo KoGuan,

    thanks for reading my essay and for the vote.

    Your essay is a little bit unusual but interesting. I like your first law. Information is really conserved and your claim is logical.

    I also vote your essay highly.

    Best

    Torsten

    Dear Torsten,

    You took the risk to go outside your main domain of expertise and I admire you for that. I intend to give you more comments in a next post.

    Meanwhile, you mention the use of dessins d'enfants in your work. I am eager to know if it is related to your remarkable papers on exotic smoothness.

    Cheers,

    Michel

    Dear Michel,

    in the last two years I went more deeply in hyperbolic geometric (hyperbolic 3-manifolds). Then I found many interesting relations to finite groups (of course much of it is also covered by a book of Kapovich "Hyperbolic 3-manifolds and discrete groups"). Together with my coauthor Jerzy, we calculated the partition function of a certain quantum field theory and found quasimodular behavior. Then we started to go into it more deeply and again found interesting relations to finite groups (Fuchsian groups). Then we managed to find a folaition of an exotic R^4 and this foliation is given by tessalation of a hyperbolic disk. Here, I found also your picture.

    Your essay opened my eyes and it was like a missing link to fulfill another goal of us: to get a geometric description of quantum mechanics (right along your way).

    For my there are many really deep thoughts in your essay and I certainly need moer time to grasp them.

    Very good work,

    Excited greetings

    Torsten

    Dear Torsten,

    I read your essay and I like it very much. As you commented on my wall, we are in "boring agreement" :) Yes, I feel the same as you that "Mathematics (in short: math) is not only driven by logic and formal systems of axioms but rather by intuition and creativity." And I agree with the idea that mathematics = understanding structures to forecast the future. Your essay is filled with interesting historical information which exemplifies your point of view and is instructive in the same time.

    Best wishes,

    Cristi Stoica

    Dear Torsten Aßelmeyer-Maluga,

    After I realized that the letter ö in Schrödinger is correctly written, I tried the letter ß in Aßelmeyer.

    You wrote to LC: "I mostly agree with Gödel: the numbers is (God-)given but the rest belongs to us. I cannot imagine that we only discover mathematics."

    Kronecker referred to the natural numbers. I am not familiar with Gödel. What did he mean with "the numbers"? Did he include G. Cantor's transfinite numbers too?

    I should avoid hurting the feelings of almost all mathematicians who firmly believe in set theory. However, when Cantor claimed having got CH directly from God, I don't believe this.

    Hopefully we can agree on that alephs in excess of aleph_1 didn't find any application in science.

    Regards,

    Eckard

      Dear Eckard,

      at first thanks for the correct spelling of my last name. It is absolutely unusual to spell Asselmeyer like Aßelmeyer. Secondly my first name has also a misspelling. Thorsten is correct (Thor from the nordish god of thunder, sten measn stone -> Thunderstone, the stone that makes the thunder).

      But now toyour question (or statement): You are right I used implicitely a quote from Kronecker. But Gödel also thought in that direction: the natural numbers were discovered but all the rest is made from us and is not given in some 'world of ideas' (Platon).

      From the experimental point of view, you are absolutely right: there are only countable numbers to express the measured values. But the continuum is at least good as a model.

      Regards,

      Torsten

      Dear Torsten (or Thor stone),

      Your essay is a very good survey of the comparative recent history of maths and physics and how one arrived at a " cultural change in our thinking" needed by our specie to adapt the environnement. You are clearly closer to Darwin than Plato and Tegmark.

      Can you explain your strange conclusion that "the relation to physics is mainly caused by the simple calculable problems in physics" that seems to contradict your main thesis?

      From the Clay Institute's official problem description of Yang-Mills theory by Arthur Jaffe and Edward Witten:

      " [...] one does not yet have a mathematically complete example of a quantum gauge theory in four-dimensional space-time, nor even a precise definition of quantum gauge theory in four dimensions. Will this change in the 21st century? We hope so! ".

      Reading your papers like

      http://arxiv.org/pdf/1006.2230v6.pdf

      I understand that the theory of four-manifolds including the exotic geometries has something to say. This pefectly fits our topic.

      Best.

      Michel

        Dear Michel,

        thanks for your words.

        You are absolutely right, this conclusion is strange. Actually I used the wrong tense and interschange math and physics. The corect statement is:

        "the relation to math was mainly caused by the simple calculable problems in physics"

        I think then it made more sense.

        Thanks for the quote. Yes it is my intention. Our new paper about foliations of exotic R^4 gives also a relation to quantum field theory (we found a factor III_1 algebra which is typical for a QFT)

        My remarks about dessins d'enfants were a little bit cryptic. A central point in the construction of the foliation is the embedding of a tree in a hyperbolic disk (here one has a Belyi pair i.e. a polynomial). A central point in the 4-manifold theory is the infinite tree giving a Casson handle. Of course one has finite subtrees. Here comes the dessins d'enfants into play: the embedding of these finite trees are described by this structure.

        Currently we try to relate this Casson handle to Connes-Kreimer renormalization theory. If our feeling is true then the action of the absolute Galois group (central for the dessins d'enfants) must be related to the cosmic Galois group.

        Of course the whole approach must be related to the interpretation of quantum mechanics too. Even in your essay you presented this relation. Certainly I have to go more deeply into your ideas.

        Best

        Torsten

        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