What is Beyond Reckoning? by Jonathan J. Dickau

Dearest Jonathan,

As usual, you wrote a remarkable and interesting Essay. In particular, based on our previous conversations, I am not surprised by your using the fascinating Mandelbrot Set as a fundamental ingredient of this nice Essay. I also appreciated your stressing the importance of the remarkable formula r = 1, which represents, among other things, the foundations of trigonometry.

Summarizing, this is an excellent Essay deserving the highest score. Thanks for sharing it and good luck in the Contest.

Cheers, Ch.

Glad to hear from you again, Jonathan,

"r=1 also defines a unit sphere"

given that there exists no universal reference of scale for either time or space, that deceptively simple statement is key to rationalizing a proportional base volume in theoretical criteria that can be then scaled to the standards of empirical measure chosen (I like cgs). best as always jrc

Dear Jonathan J. Dickau, I was impressed with your statement, "Questions arise both in the physical and abstract reality of what is decidable or undecidable, what is computable or uncomputable, and which events are predictable or unpredictable." So I argue that it is necessary to distinguish geometric space from physical space. Geometric space is an abstraction of physical space. Physical space moves relative to itself, because according to Descartes it is matter. Arguing in this way, I showed that the probability density of states in an atom depends on Lorentz abbreviations: length, time, mass, etc. I believe that this is the unifying principle of modern physics, which will reduce the level of unsolvability, uncomputability and unpredictability in it.

I invite you to discuss my essay, in which I show the successes of the new Cartesian generalization of modern physics, based on the identity of space and matter of Descartes: "The transformation of uncertainty into certainty. The relationship of the Lorentz factor with the probability density of states. And more from a new Cartesian generalization of modern physics. by Dizhechko Boris Semyonovich »

It interesting your essay. A good essay.

You use a simple formula to study a simple condensation (simple to obtain using a computer), then you can use the results on this set to obtain general results for condensation.

I think that each condensation phenomenon, with a general use of critical exponent, that come out from a statistical mechanics study of simple system (for example Ising model), and experimentally tested, to obtain same results (condensation of cluster from short range interaction); then, I have a problem: in a Mandelbrot set there is not a interaction from points, in a lattice, so that a statistical analysis is not possible; it could be possible using different initial points, and consider a swarm of moving points, but there is not interaction (so that the statistical phase transition is improbable); but it is possible that I am wrong.

I understand that the Mandelbrot set is interesting because of the dynamics is unpredictable a priori, but a Conway's Game of Life (a hypothesis) or the Ising model (a classic analysis), with many different interacting patterns have the semplicity and the statistical complexity of a physical system.

I tkink that it is possible to use the chaos theory to reduce the dimension of a space dynamics (for example Hausdorff dimension for chaotic system), but almost every differential (or discrete) dynamics system reduce the dimension of the space.

Domenico

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?

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