Mathematics as unifying force for science by Torsten Asselmeyer-Maluga

I like your section: "MATHEMATICAL STRUCTURES FOR A QUALITATIVE UNDERSTANDING OF SCIENCES"

Do you think that one can "forecast" the future accurately without mathematics?

Good essay.

Dear Torsten Asselmeyer-Maluga,

Love it when you talk about models "Physics describes the dynamics of simple objects. Even at the beginning of physics, there were simple models which served as universal models to understand the underlying processes. The harmonic oscillator is one model which is used in mechanics (pendulum), thermodynamics (heat bath), electrodynamics (oscillator of Hertz) and quantum physics (harmonic oscillator, free field quantization in quantum field theory)."

And I feel point 3 of your conclusion is very important "Math changed around 200 years ago and in particular after the Second World War. Now it is the science of structures like groups, modules, categories etc. instead of elegant calculations...". The importance of more complex structures leads to whole new understanding of physics and better predictive power.

Good work, enjoyed the read.

Torsten,

Thanks for the encouraging word on my essay. I am going to try to read yours sometime today. Your topic appears related to mine in some ways.

Cheers LC

There is the prospect that the universe in effect computes things. The universe as a quantum compter, an idea I am partially supportive of, would tell us the universe involves states that are the logical outcomes of certain elementary computations. I am interested in 4-qubit entanglements of 8-qubit systems that are E8. The structure of four manifolds involves a construction with Plucker coordinates and the E8 Cartan matrix. This seems to imply, though I have not seen it in the literature, that for 8 qubits there is not the same SLOCC system based on the Kostant=Sekiguchi theorem. However, I suspect that the structure of 4-spaces might hold the key for something analogous to KS theorem and the structure of 2-3 (GHZ) entanglements that are constructed from G_{abcd}. If the universe has this sort of discrete structure via computation, then it makes some sense to say the universe is in some ways a "machine" that functions by mathematics.

Your essay makes a nice overview of the subject. You avoid some of the metaphysical aspects of this question, such as Platonism. I mention this somewhat briefly, but put this in a certain metaphysical category.

Cheers LC

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

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

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

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