I wrote the following in my essay blog:

In the end there is a bit of a duality here, or a dialectic of sorts. I think that what is measured in physics is discrete. We measure certain observables that have finite values, and quantum physics in particular bears this out pretty seriously. The continuum aspects to physics is pretty much a mathematical issue. Experimental data does not have any reference to infinitesimals or infinities. The calculus is based on the limit where the difference between two points becomes infinitesimally small. Physical experiments have not direct bearing on this.

It is the case that homotopy does involve curves that are smoothly deformed into each other, but this is used to get the value of the homotopy group that is usually Z_2 or Z, where Z could be interpreted as just unbounded and infinity is avoided. The homotopies are then more directly related to the actual measured aspects of physics.

Spacetime is a bit odd with regards to this. The Planck scale does indicate that one can't isolate a qubit in a region smaller than sqrt{G徴/c^3}. The Heisenberg microscope argument indicates that if one tries to isolate the Planck unit of area a quantum state is contained that it will scatter violently. This illustrates that using a large value of momentum to isolate particle demonstrates that spacetime has a discrete structure. This has an interpretation in the generalized uncertainty in string theory. On the other hand the FERMI and Integral spacecraft measurements of distant burstars found no dispersion of photons predicted by loop quantum gravity. This is a discrete form of quantum gravity, and it appears to be in trouble. In this experiment a very large ruler (measurements out to a billion light years) found that spacetime appears very continuous. This suggests a more general form of the uncertainty principle, where at one limit spacetime is continuous, and on the other limit discrete.

The problem is that physics is not completely discrete or continuous. One of these FQXI essay contests went into this. The main thrust of my essay though is that the physical observables we measure, and physics is an experimental science, are discrete. Mathematics has what I might call a "body" and a "soul." The body is what is computed, and can be computed on a computer. The soul is all of the continuum stuff, calculus, infinitesimals etc, which have a weaker connection to experiments. I am not committed to any metaphysics about whether the soul exists or not. That is to say I have no belief or lack thereof with respect to what some might call Platonism.

Cheers LC

    Dear Joe Fisher,

    Oh I see your point. As I point out in my essay, I see abstraction as a tool to describe the complex world around us with our limited brain.

    But of course we depend on our sensors. You have a more exaggerated point of view. But why not. I would agree with Lawrence answer to you.

    Best wishes

    Torsten

    Lawrence,

    Ok I see the point. Of course the outcome of experiments is not a real number but as you also point out, one has problems to confirm the discrete structure of spacetime.

    I see one reason in the underlying topological nature of physics. You also discussed it in your essay. I will illustrate it in a an example:

    If two curves intersect then we measure the number of intersections (a discrete number, gauge or diffeomorphism invariant) but in most cases we are not interested in the coordinates of the intersection. Even sometimes we have problem to determine the coordinate system.

    I see the measurement values in physics in this fashion. But then one has a dichotomy between discrete (number of intersections) and continuous. The measured values are in principle discrete but you need the continuum to express the probabilites of quantum mechanics.

    I don't see any contradiction in this picture. Of course you will never measure that spacetime has a continuum structure but you can measure a discrete structure. And as you correctly point out: every experiment failed up to now.

    In principle I agree with you very much. In particular I like your body-and-soul picture

    Best

    Torsten

    Lawrence,

    Interesting idea to use 4-manifold topology to say something about qubit entanglement. As far as I understand your approach, you have the E8 manifold in mind (which is not smooth). I have to think about it....

    In my essay I also made some remarks about the Platonism (but more implicitely). I mostly agree with Gödel: the numbers is (God-)given but the rest belongs to us. I cannot imagine that we only discover mathematics. Even model theory showed us that mathematics is in principle free to use any logical system (but it should agree with the structure of our mind).

    Torsten

    Hi Torsten,

    It was a great pleasure to see you here again. I have nothing to add to your essay's conclusions.

    I know and appreciate your other publications referring exotic smoothness structures of the spacetime. In your last one: "How to include fermions into General relativity by exotic smoothness" you wonder: "But then one has the problem to represent QFT by geometric methods (submanifolds for particles or fields etc.) as well to quantize GR. Here, the exotic smoothness structure of 4-manifolds can help to find a way."

    I take the opportunity and propose the answer in my essay engaging the set of Thurston geometries (that you have mentioned in your essay) with metrics. I treat them as a space-like, totally geodesic submanifolds of a 3+1 dimensional spacetime. In three dimensions, it is not always possible to assign a single geometry to a whole space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed into pieces that each have one of eight types of geometric structure, resulting in an emergence of some attributes that we can observe. As you know, Thurston geometries include: S3 the geometry of constant positive scalar curvature, E3 the flat geometry, H3 the geometry of constant negative scalar curvature, that all three are homogeneous and isotropic, and five more exotic Riemannian manifolds, which are homogeneous but not isotropic (S2 ﾃ-- R, H2 ﾃ-- R, the universal cover of SL(2, R), Nil geometry and Solv geometry). The constant curvature geometries arise as steady states of the Ricci flow, the other five arise naturally where the dynamics of the Ricci flow is more complicated and where topological changes (neck pinching or surgery) happen. I have tried to attribute the geometries to interactions and fermions, except of five exotic ones.

    If you are interested you can find details in my essay.

    http://fqxi.org/community/forum/topic/2452

    I would appreciate your comments as you are an outstanding scientist. Thank you.

    Jacek

      I wrote my essay with a certain perspective in mind. I am not locked into any particular perspective on this matter. That spacetime is continuous and smooth seems very much in line with the NASA Fermi and ESA Integral measurements, which involves a large ruler, or equivalently nearly zero energy, measurement of spacetime. Conversely a high transverse momentum measurement would result in chaotic fluctuations. A high energy of a particle in a tiny region of spacetime would by virtue of the Heisenberg microscope argument demonstrate a very chaotic or maybe "foamy" fluctuating structure. The quantum fluctuation is then a manifestation of the stochastic nature of quantum measurement.

      The measurement of the smoothness of spacetime by this method is rather indirect, which is by virtue of no observed dispersion predicted by a foamy non-smooth structure. The measurement of wild fluctuations in the metric by high energy or transverse momenta would be more direct.

      This reflects something odd about quantum mechanics. The founders touted the idea of a wave-particle duality, but in time it became clear that the particle is what is measured and the wave sort of "flaps in the breeze." Feynman even proclaimed quantum physics to be entirely about particles and not waves. It has only been recently that by weak measurement techniques with electrons entangled with an EM wave function have aspects of the quantum wave started to appear empirically. There is this odd asymmetry to nature. The field, wavy or continuum aspects of nature are difficult or more indirect to measure that the particle nature.

      The "body-soul" duality I tend to advocate is something one can "wear" as needed. I might by virtue of some argument want to invoke a mathematical objectivity of sets, continuous spaces and even to the point of Platonism. At other times I may put this entirely aside. In my essay I largely put this aside.

      Garrison Keillor has a feature on his show "Prairie Home Companion" on Guy Noir with the opening line, "On a dark night in a city that knows how to keep its secrets, one man seeks answers to life's persistent questions; Guy Noir, Private Eye." That is about my sense of the question about the relationship between physics and mathematics. We may never know for sure. Further, the universe may have a kernel of structure, symmetry and order to it that appears in a fractal-like form at different scales, but where nature also has this inherently chaotic or disordered nature to it as well, which I think is distilled down to the stochastic nature of quantum measurement.

      Cheers LC

      Lawrence,

      thanks for the answer. It is very interesting and I agree with it. Yes I know the foam agrument was always used to introduce the discrete spacetime structure.

      I also agree with you about the chaotic nature of quantum fluctuations. Currently I work on a geometric understanding of this point using McMUllen's Field medal work about complex dynamics, fractals and hyperbolic 3-manifolds.

      So at first, thanks for the discussion

      It help me to understand your point better

      Best

      Torsten

      Hi Jacek,

      at first I skimed over your essay and found it interesting. Actually I used the exceptional geometries (NIL, SOLV, SL2) in my work. In a previous paper I described the interaction with them (see the paper). The idea is simple: the connection between the knot complements (=fermions) are torus bundles and there are only 3 types of torus bundles which can be identified with interactions (weak, strong and EM). In this paper you will also find the identification of bosons to the geometries.

      Currently I think about the SL2 geometry and the Ricci flow.In the past I thought that the Ricci flow is something to do with the measurment process in quantum mechanics but now I changed my mind. I have to go over your essay more carefully. Thanks for your words.

      Best Torsten

      The smooth 4-manifold is dual to the quantized discrete manifold. This duality is STU, or in the case of T duality with R --- > a/R, the large R involved with measuring spacetime across billions of lightyears is dual to a measurement near the Planck scale. Alaine [link:arxiv.org/abs/1411.0977] Connes [\link] has found an interesting structure that has an 8-fold structure to it, similar to Bott periodicity, that also looks a lot like quantized geometry. This appears to be similar to the idea of quantum foam, or where a superstring is wildly fluctuating as a graviton.

      The E8 manifold is a four manifolds with intersection form given by the Plucker coordinates and E8 Cartan matrix. This forms the "exotic" 4-manifolds that have E8 lattice or group structure. This all again seems to point to some strange relationship between continuous and discontinuous structures. As I think that geometry is a manifestation of quantum entanglement I think it means an observer with access to quantum bits in a large number of entanglements would observe geometry in effect 'breaking down." The STU dual to that would be the observer that does not have access to this level of quantum information, but observes physics commensurate with a smooth realization of geometry.

      LC

      Hi Torsten,

      Could you please give me the link described above as "(see the paper)" as it does not work and I am really interested in.

      Thanks,

      Jacek

        Hi Jacek,

        try http://arxiv.org/abs/1006.2230 and download the PDF (it is the published version). Secion 8 discussed the gauge group and explains the geometry.

        Unfortunately, I see now that I don't mention the geometries (I think the referee don't want to see them and ask to remove them).

        Here it is:

        finite order = E3 (Eucledian)

        Dehn twist = NIL

        Anosov = SOLV

        Best Torsten

        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