Dear Jonathan J. Dickau.

In the time of Descartes, they did not think that speed had a limit. Now that we are confident that the speed of light is the highest speed and nothing can move faster than it, we should consider how space, which is matter, resists its movement relative to itself. Therefore, the physics arising from this should be called new Cartesian. I will read your essay again to comment on it from the point of view of a new Cartesian generalization of modern physics, which is based on the identity of Descartes's physical space and matter, in order to bring our views on physics closer.

However, I noticed this: "This is because M3,1 is a repelling point, which means that any point in its local neighborhood displaced from that spot is divergent. So it is not possible to enter a numerical value in that neighborhood that will repeat indefinitely. " A similar situation arises when I gave the opposite meaning to the Heisenberg uncertainty principle, which has now become the principle of definiteness of points in physical space. According to the new principle, an infinitely large momentum is needed to separate a point from other points.

Boris Dizhechko

Oh Jonathan,

I was cogitating on your observation that transcendental numbers are non-computable (in proofs) and last night there was a mention of Fermat's Last Theorem on a TV Sci-Fi program which made me wonder if Fermat's cryptic note in the margin wasn't referring to a proof of the non-calculable form of his theorem; but that the incalculable identity of the theorem was itself a proof that such mathematical entities as transcendental numbers were non-computable? Mathematics is, after all, an artifact of human intellect. best - jrc

Dear Jonathan,

your essay touches on many interesting themes. You start off with what I've heard described as the 'edge of chaos'---the site of interesting complexity where the strictly and sterilely lawful and the completely unstructured random meet, and where consequently interesting phenomena occur. Wolfram has extensively written about his 'Class 4'-cellular automata, which are capable of showing persistent, nonrepetitive, novel behaviors. It's conjectured that such behavior is necessary for supporting universal computation, for instance, and, as you note, may be what spurred the origin of life, or even supports its persistence.

Furthermore, you investigate the notion of fixed points: where maps, repeatedly applied, leave their objects invariant. There is an interesting connection here: fixed points---or rather, their absence---are at the heart of results like Gödel's theorem, or the undecidability of the halting problem (for which connection, if I am permitted some self promotion, see my own essay). It's not hard to see how they are related to self-reference: a map, placed on a desert island, if detailed enough, will have to contain a copy of itself, and a single (fixed) point where map and territory, so to speak, coincide.

You combine these notions via the fractal geometry of the Mandelbrot set. This, too, has regions of great complexity at the boundary of more 'boring' regions---an edge of chaos---, and whether a point lies within that region is decided by application of a self-referential formula.

It would be interesting, to me, to see this connection explored more in depth. When does self-reference facilitate the emergence of nontrivial complexity, of novel structures that nevertheless do not degenerate into mere chaos?

    I can see that you read for detail Jochen,

    It is a pleasure to hear your expert analysis and to come up not wanting more than I offer. I too like the edge of chaos notion. I tried to bring that out in my overlay showing the alignment of bifurcations with M at the same location I showed condensation happening earlier. Had I more space to write; I'd have devoted some discussion to Susskind's ideas about complexity being maximal at event horizons, but it was one more thing than I could add.

    I remember fondly a book by Briggs and Peat "Turbulent Mirror" where they do a very nice job with that idea and also that there is a far shore of the chaotic realm. I don't know if that was their intended meaning, but it seemed strongly implied. In any case the analogy of the band-merging point at M3,1 with gravity is strong. The trajectories gather to a point at the gravitational horizon/critical point of condensation, but behind it the degrees of freedom rapidly multiply as the trajectories expand.

    I would love to explore the connection you bring up in more depth.

    Warm regards,

    Jonathan

    I have begun to read your essay Boris...

    More about it on your page. It seems you have some interesting ideas. I'm not sure I see it all fitting together seamlessly. But such is the path of progress.

    We may well agree that Descartes was ahead of his time, and that some parts of his methodology still have value for today's Science.

    All the Best,

    Jonathan

    As I have commented below (Just a thought)...

    Some numbers have an exact algebraic value that can be written out. The long expression in the Endnotes is a number not a varying relation, and yet it would require an endless string of digits to render it in its pure form - so it is indeed a transcendental number in decimal form. That is; its algebraic form is finite though its numerical form is infinite.

    Cool stuff.

    Have Fun!

    Jonathan

    Hello again folks,

    For those who are interested to know more about or do some exploring themselves in the Mandelbrot Butterfly figure and its family of figures, I offer these links to content at Fractalforums.

    Jonathan

    The Mandelbrot Butterfly is a Spectral Manifold

    The Butterfly's inverse is also Spectral

    Formula files for Mandelbrot Butterfly family figures on Chaos Pro

    I hope this is helpful.

    More later,

    Jonathan

      Jonathan,

      Are you familiar with Keenan Crane's GPU code for ray-tracing the Julia set?

      https://www.cs.cmu.edu/~kmcrane/Projects/QuaternionJulia/

      About inverse ray-tracing, I'm not sure that would work well. You'd have to assume that the ray can penetrate the mesh many times, and that it's only the last (the outermost) penetration which counts. This is inverse of the regular method, where the ray only has to march to the first penetration point. I hope this makes some sense... I'm not a mathematician.

      Just for fun, I'd like to mention something about the Julia set: Normally one uses the quaternion magnitude to determine if a point remains within the set. The path followed by the quaternion not only has a magnitude, but also a total length and total displacement. These three properties of a path are all different in value (histogram). However, they all produce the same fractal shape. I would have thought for sure that the shapes would be different, but they're not. Weird.

      - Shawn

      P.S. Haha! You have The Beauty of Fractals too! Page 61 is my favourite version of the Mandelbrot set!

      8 days later

      Dear Johnathan

      Thank you for your kind comments on my essay. You are correct re the way I have structured it. My overall point is that I think there does need to be cojoining of physical theories with ontological landscapes. A new mathematical-physical-philosophical language formal in its nature that allows us to better categorise but also expose the landscape of what we think we know and what we do not. I look forward to commenting on your paper before the deadline.

      Also, I have gotten into Seeger after reading your Bio. In particular the songs 'little boxes' and 'what did you learn at school today' are remarkably profound. So many people live narrow, controlled, social lives, but as they are the vast majority those outsiders who sit on mountains although wise, are lonely and frustrated. Being right or being a fool... how should one live.

        Thanks Jack,

        Little Boxes was actually a collaboration with Malvina Reynolds inspired by the comment of local resident Fred Mercer, an architect, who described the row houses in Levittown as 'ticky tacky.' And yes; we need to build those bridges connecting Physics back to Ontology. Some try to do that, but plenty are content to 'shut up and calculate.' I'd rather find some meaning; thank you.

        All the Best,

        Jonathan

        I don't think the universe is dead yet Shawn...

        But we might be, and just not realize it yet. I tend to be a little less gloomy, although I am told my 'Professor Snape' imitation is spot on. And my 'Boris Karloff' is right up there with the best. But seriously; how about 'all symmetries are broken eventually.' That's a bit closer to the truth.

        Thanks for reading.

        All the Best,

        Jonathan

        I do have Crane's Quaternion Julia program...

        And in addition to "The Beauty of Fractals" I have "The Science of Fractals" "Fractals Everywhere" "The Fractal Geometry of Nature" and a few other Fractal or Chaos Theory related or included books. I like and would recommend "Turbulent Mirror" by Briggs and Peat because it explains a lot of stuff relating to the edge of chaos and what lies beyond its boundaries. Fun stuff.

        More later,

        Jonathan

        A point of interest...

        I mention in my paper that there are 3 types of Misiurewicz point in the Mandelbrot Set. At all Misiurewicz points; the scale factor goes to zero where patterns of repeating forms get smaller and smaller coming to a point, and then exhibit one of 3 behaviors.

        Type 1 are branching points where one path in then splays out into multiple equally spaced (near the center) branches. Type 2 are what I call inflection points, having one path of entry and one path of exit (2 external arguments). Type 3 are terminal points, where repeating forms get smaller and smaller along one thread - and just stop.

        The Mandelbrot Butterfly I discovered about 32 years ago is useful in the study of these points because Type 2 Misiurewicz points are rather innocuous and hard to spot, but they have great importance for understanding formation. The Butterfly figure lets us identify, label (by pre-period), and classify these points which would otherwise remain hidden.

        I am writing something up about this.

        All the Best,

        Jonathan

        Dear Jonathan,

        Please forgive me if I see you first of all an artist who intends "trying to help the human race harmonize with Mother Earth and heal our planet". If only we all were children-like like you. I feel guilty for even blaming Fourier wrong.

        Eckard Blumschein

        Thanks so much Eckard,

        I think that play gets to the heart of learning. No need to feel guilt over your critique of Fourier either, because it's what brains do - one half always putting things together and the other taking them apart.

        Glad for your visit,

        Jonathan

        Jonathan,

        You quickly noted my posting. Hope you get a chance to read it. I am printing out yours, as I plan to do for several. It looks weighty in message so I will do my best on it.

        Later.

        Jim Hoover

        Dear Jonathan Dickau,

        I was about to comment on your essay when I realized that I was not reading your essay, I was reading another of your papers, "Gravitation by condensation." I've now read the essay and have a few remarks.

        You explore a large conceptual realm, and most of what you discuss cannot be proved or disproved, a point you make up front. In this regard you observe that

        "Things requiring proof or explanation by one person may be intuitively obvious to another..."

        I meet somewhat regularly with a physicist who is adept at proof and seems to have no intuition. It seems those who have weak intuition do not value it much. I find your Mandelbrot map an excellent metaphor for math. One can infinitely branch in any direction, but without intuition, it's just turning the crank - only intuition can drive the boat - mere calculation is endless (turning computers loose to calculate endlessly as in Mandelbrot is quite interesting, and hopefully enlightening.) You've certainly become good at making analogies and you seem to have honed your intuition in this regard. Once the data has been generated, our pattern recognition can reduce the dimensionality and discern features and regularities. For example your remark on the interplay of local symmetry and global asymmetry in M is possible only after one has produced the map. One would probably find it difficult to predict from the generating equations, but after seeing it in the data, one can probably identify relevant aspects of the equations. Also quite interesting is structure with pre-period 3 (the delay before repeating) then it repeats every time.

        As we have discussed, I believe that focus on convergence and condensation is potentially extremely valuable.

        You note that "the main difficulty theoretical physicists face in crafting a unifying theory of physics is that in relativity time is flexible, while in QM it is absolute." My essay will deal with this, and I think you might find it interesting.

        I'm encouraged that Foundations of Physics published three papers in November 2019 analyzing the mismatch between relativity and intuition, so the topic is not dead.

        And like you and Rick Lockyer, I suspect 0HCR is telling us something important, but I'm not sure what.

        Best regards and good luck,

        Edwin Eugene Klingman