How the Totality of Mathematics Shapes Physics by Jonathan J. Dickau

Dear Jonathan,

I was waiting for an essay from you on this topic. I fully agree with Linas Vepstas that Mandelbrot's set has to be investigated from the point of view of the the modular group SL(2,Z) and modular fonctions

http://www.linas.org/math/sl2z.html

The Monster Group is the topic of my essay where the j-function is the hero.

And this entirely relates to a two-permutation group, a (modular) 'dessin d'enfant'.

Finally, I understand your feeling that the Mandelbrot set is as interesting as Moonshine for doing progress in the understanding of physics, and for shaping physics.

My best regards,

Michel

Dear Jonathan,

A nice read. I too share your sense of "oddness" about the 4 normed division algebras and that "one could equally well assert that sums of squares and properties of spheres are what defines the limit on the number of valid algebras". But I am further surprised by their central importance in physics. In my essay I note in passing that the conditions of causation for continuous fields with a size imposes on any maths theory the very conditions that define the normed division algebras. General Relativity and the Standard Model are in terms of NDA valued fields, but the NDAs seem to lie behind most theories trying to unify them as well. In physics, you can't seem to escape them ... because of their position in maths?

Many thanks for pointing me to this essay contest.

Michael Goodband

Jonathan -

A lovely essay and beautiful graphics. I appreciate your faith in mathematics as the form of nature and the eventual pathway for solving the current problems in physics, but wonder if you see any downsides. How do you explain, and/or avoid, the difficulties of self-referential systems and the Godel theorems?

I'd appreciate your thoughts on my essay when you have a chance.

Sincerely - George Gantz

Jonathan,

I'm glad you were able to get your essay in on time. You begin by stating "the totality of Math inspires Physics, but this does not exclude or deny the validity of views that math is primarily based on relations arising in physical systems, or change the fact that we could not be here to elucidate Math apart from the physical reality for us to inhabit."

So we're standing on the same ground. When one has both feet planted on solid ground, he can let his mind roam free. That is what I see you as doing. It is, after all, the essence of play.

One thing that sets your essay apart is the fact that it contains a figure of your own creation, the Mandelbrot Butterfly. It is quite interesting to see it side-by-side with the standard Mandelbrot diagram, as it makes quite clear that you have exhibited new structure in an otherwise opaque region of the figure. Very impressive.

I also like that, rather than take the position that all math has (somewhere, somehow) a physical instantiation, you ask the reasonable question, "What determines which mathematical fundaments find expression in physics?" But if you answered that, in any detail, I missed it. It's a very important question.

You make the interesting observation that the Real, Complex, Quaternion, and Octonian number types provide the only four well-behaved natural algebras [normed division algebras] and seem to assert that sums of squares and properties of spheres are what define the limit on the number of valid algebras, with each imaginary i projecting out into a new dimension, orthogonal to the others.

You describe the Mandelbrot Set as a "catalog of examples where symmetry preserved at one level of scale is broken for larger structures...", with accompanying diagram illustrating the point. I have noted in various comments that, at least for particle physics -- where symmetry has played a significant role -- none of the symmetries, from iso-spin to SUSY, are exact symmetries. This is especially interesting in view of Noether's association of conservation with each continuous symmetry. This implies to me that it is conservation of physical entity that is the ontological basis, while approximate symmetry is the epistemological descriptor. Any thoughts?

In short, I enjoyed your essay very much. Do you have any easy-to-understand explanation for the structure you have uncovered with your Butterfly?

My very best wishes, and appreciation for your insightful comment you made on my thread.

Have Fun!

Edwin Eugene Klingman

Dear Jonathan,

Whatever else might be said about Mandelbrot figures, their amazing intricacy is compelling. I watched your youtube video of the "Mandelbrot Butterfly Safari" late last night and it was a real treat - hypnotic and relaxing.

While I would not be surprised to find something like a Mandelbrot mechanism at a fundamental level, we seem to be in the same situation as the creators of the cosmic computer in "Hitchhiker's Guide to the Galaxy". Having used that computer to find the ultimate answer (the number 42) the creators found themselves having to create another computer to find the ultimate question, which had long been forgotten. If quantum mechanics corresponds to the incomprehensible answer, the question might be "where does quantum mechanics come from?" The answer to that question must surely involve the concepts from mathematics you discuss.

Best regards,

Colin

Dear Jonathan,

I am glad you were able to finish your fine piece. I totally agree with you especially on Hooft's explaination. I totally agree with him except I do think Qbit is the smallest possible and the largest possible Multiverse. Qbit is the atomic unit that describes its bigger self. To quote your quotation from the great physicist Hooft: "What does the calculating? Do we need Planck-sized atoms of space?" And he said "We don't need atoms of space or whatever, because the laws of nature do the calculating for us." I do believe infirmation is everything and information is describing information. Alternatively, math describes math universe. It is a self referential system of information that is bootstrapping itself into what we feel and see as our Existence. You have been the last 30 years as an independent researcher to view this new paradigm of math.

I invite you to review mine and as usual you wrote a great essay and I rate it accordingly.

Best wishes,

Leo KoGuan

Hi Jonathan ,

no worries, it certainly does meet with approval. Your enthusiasm shines through even if you don't feel your presentation is as polished as you would like.I enjoyed your descriptions of mathematics in the universe itself. I have been thinking of it as mathematics 'in the wild' controlling what can and does happen, being the relationships at the heart of everything. I also like the idea that you mentioned of lots of kinds of mathematics co-operating. Bringing an ecosystem to my mind- Organisms doing different things but somehow all working together as an entirety.

I'm not sure it is necessary for the mathematics, the relationships, to be distilled from reality and placed in an abstract theoretical space or to assume that the mathematics precedes the concrete universe. It can be distilled into pure maths but that seems to me a bit like distilling the psychology of a man and speculating that it somehow exists separately from the body of the man and the environment in which the man finds himself. (Sensory experiences, thoughts and knowledge can be considered a different facet of reality but they are still within the minds within the concrete reality influenced and shaped by it.)

Thank you for sharing your work with us, I do appreciate it. Kind regards, Georgina

Dear Jonathan,

Thank you for the reference to Tiozzo's thesis.

It is interesting that what he defines as a quadratic interval looks the same as what we measure in the "superheterodyne calendar" of my paper [see Fig. 2 and eq. (17)] from continued fractions

http://empslocal.ex.ac.uk/people/staff/mrwatkin/zet

a/planat6.pdf

At least the starting point and the connection to Thurston's "quadratic minor lamination" is encouraging. I would also like to recognize a possible link to the f Farey fractions of hyperbolic polygons (tesselations of the upper-half plane, or of the conformally equivalent unit disk) that I mention in Sec. 3 of my essay.

We know what we have to work on. Thanks.

Michel

Jonathan,

Perhaps I don't understand what you mean by "pure Mathematics." I think math does reveal many elements of structure and form utilized by nature because scientists model them that way. I agree that using math to model physical systems is a very effective way to do physics.

Perhaps my view is simplistic in seeing a integral connection between math, the mind, and physics. What are your thoughts here?

Jim

Jonathan, the mandelbrot set arises in chaotic systems as a natural and universal structure so it is related nicely to my work too. It seems to be a common feature of universal structures that they have a fractal self similar structure as the mandelbrot set does when you zoom in. This is close to what I see happening when iterated quantisation is used to form a universal structure emerging in mathematics. Quantisation itself is then the transformation under which the laws are self similar like a fractal structure. It would not surprise me if the Mandelbrot set turns up in physics at the deepest level in this way just as you suggest.

Your butterfly idea adds an extra level of originality to the use of fractals. Thanks for the mention too.

Jonathan,

I am a little late reading your essay, though I had been anticipating it. I was not surprised that you addressed the Contest Topic in terms of the Topic! While many entrants in any FQXi Essay Contest quite naturally piggy-back their own special interest on any given topic, the harness generally needs an "evener", which is a common accoutrement in Amish Country where I come from. A team of horses is seldom matched in all proportions so a clever piece is built to couple the harnesses and even the load, and spare the less powerful animal being subjected to a greater part of the pull. No such devise is necessary for your superb essay, it IS your specialty!

"The question of whether ideal forms predate their physical representations, and to what extent all physical forms are a representation of their mathematical ideal, remains open." And well it should.

Perhaps there is some thing we might call geometry where the finite value obtainable for rectilinear space is also obtainable for curvilinear space, and we only lack a better way of trying to compute 'pi' that returns a finite ratio. But what we observe physically argues well that we can trust our foundational maths as being inherent to reality. The only difference between space and time might be in that dichotomy of finite vs. infinite geometries. and the origin of energy in a continuum of creation. If 'pi' reaches a finite limit, would energy cease to exist? Is it physically possible to devise a mathematic form that finds easement to directly equate flat geometry with curved geometry? If it were, wouldn't all be symmetric and the universe a singularity?

Your fascination with fractals and the asymmetry at scales is communicated well in your essay, and it should be appreciated by all that you invite the totality of mathematics to the party. Well done, Jonathan! Best wishes, jrc