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

              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