The Mandelbrot Revisited
[Please note that this discussion is a continuation from the previous page, and from an earlier page.]
The most obvious thing about the Mb is that it must be symmetric about the real axis. The next most obvious thing is that it is very asymmetric about the imaginary axis. It is intuitively obvious that there must be a filamentary process far enough out along the negative real axis, complemented by a channel on the positive side. One could then do worse than attempt to sketch the first few lemniscates.
One could note the connection between the Mandelbrot and Julia sets, and sketch a few of the Julia sets along the real axis. This would underscore the asymmetry along that axis. The general periodicity of the iterations could also be inferred with a minimum of calculation. (See here and here.)
At some point our intuition about fractals would kick in. We could easily speculate that the fractality of the main body would involve replications of the bulb processes, and that each bulb, by simplicity, would be as close to circular as possible. We could next infer that there would be points of discontinuity, such as where bulb touches bulb or filament touches bulb. The only way to handle such points would be to resort to infinite regress.
Next would come the problem of connectedness. It would not be difficult to discover that there are just two basic kinds of Julia sets: connected and disconnected. We might then wonder if the Mb could be composed of both kinds of sets. We could also speculate as to how the Julia set undergoes its transition between these two very distinct phases, i.e. from solid to gas. The only possible places where this could occur would be at branching/connecting points, and so there would have to be infinitely many such points, or, more technically, the branching points would have to be 'dense' in the set.
The next inference is that the Mb would be on the borderline between being connected and disconnected. It is only regressively connected, i.e. you cannot define a disconnecting set, but that it is not 'pathwise' connected. This latter point has not been proven, but I suggest that it may readily be inferred by inspection and intuition.
If I'm correct in this line of speculation, the Mb presents us with a mathematical reality, most of whose truths are open to intuition, but not to analysis. It this not a fundamental demonstration of Godel's theorem? I believe that this is a fundamental truth about our world in general. This is the whole point of emergent phenomena, downward causation, microcosms and Leibniz' Principle of Sufficient Reason. What analysis must leave incomplete, only God can complete. Each idea is very adequately represented here in what is at once both the most simply generated and the most complex acting of non-linear systems. How could the Mandelbrot be other than the Imago Dei, as posited by our beloved but fictional monk, Udo of Aachen? Here I gladly plant the flag of theism. Would anyone care to mount a challenge? If we can hold this beachhead on the motherland of analysis, to where can it retreat? Is this not the heart of materialism? Is it not the heart/cardioid of the matter?
At every juncture of reason we have only two choices: fall into an infinite regress of analysis, or look for the microcosmic mu-atom that is the Alpha and Omega of reason. We have dissolved the Mandelbrot Mystery. We either analyze it to our death, or we crank it up a few notches, recognizing that there is ultimately only one Mystery.
What can we do for an encore? This could well mark the end of our sojourn into matter. Materialism goes out with hardly a whimper. It just gives up the ghost in its machine.
It would be nice to get some idea of how the Mb mug shot was taken. Alright, here we go.
The Matrix abhors a vacuum. The Matrix is strongly biased in favor of the plenum over the void. Given downward causation, the highest possible cause has priority. The vacuum in question here is the failure of analysis, even in principle, to grapple with the overwhelming complexity. That leaves an opening for emergence. Yes, this is an emergence in the gap of analysis.
Given the fact that analysis fails almost everywhere in nature, should not the Imago Dei be plastered almost everywhere? You know what they say about faces in public places? My answer is that it is and isn't. For instance, you, me and the dung beetle definitely partake of the Imago. On the other hand, seldom do we see recognizable BVM's in the clouds. Clouds are functional fractals. Their job is mainly to rain on the plain. Taking on the shape of a the Pieta would be far beyond their job description, and would very likely interfere therewith. Here we use the PSR for a null result. When it comes to clouds and the formation of raindrops, atomism does very well, thank you. This same functional reasoning covers virtually all of the fractals to be found in nature.
Why then is not the Mandelbrot just some version of Cantorian Dust? That is what the Julia set becomes outside of the Mb. The beauty of the Mb derives in no small measure from its borderline status between the connected and disconnected forms of fractality. This makes possible an extreme richness of texture. Upon this richness we need to impose the universal condition of quasi-self-similarity. We have your basic cardioid with its attendant infinite set of circles, each with a filamentary process making up its beaded necklace of mu-atoms. This is the least constrained of all fractal systems. It becomes the mother of all fractals. Its infinite collection of Julia sets is just a drop in that bucket.
The Mb is to mathematical fractals what the Matrix is to natural fractals. You, me and the dung beetle are the mu-atom pearls on Indra's necklace.
What stands out above all is the coherence of the Mb. It is a territorially based coherence, rather like an ecosystem. There are spheres of influence. Each mu-atom and each individual type of spiral has its sphere of influence. There is nothing that falls outside. Most every locus comes under the influence of multiple spheres. Every location may be uniquely identified. How are the potential conflicts and confusions mediated? Can we hope to reveal the hidden hand? Aren't there way too many cooks here? Is there a Darwinian survival of the fittest to fall back upon?
I submit that the Mandelbrot is forcing us to reexamine our conception of mathematical reality, just as the mind is forcing us to reexamine our conception of material reality. The results are bound to be related. In each case we must resurrect the notion of Pythagorean harmony, in partial contravention of the Platonic forms. To borrow political labels, the Pythagoreans are metaphysical populists, while the Platonists are the corresponding royalists. I come down on the side of hidden hand populism, with just a whiff of messianism.
In deference to the mind, I have railed against the idea of physical atomism. In deference to the Mandelbrot (Mb) and the Monster Group (MG), I rail against logical atomism. Logical atomism cannot hope to explain these objects, any more than physical atoms can explain the mind.
Is it not true, however, that there exists an analytic proof for the existence of the MG, laid out in 15,000 pages of equations? We shall see.
The Mb may be likened to a language. No word of the language may be understood outside the context of the whole language, and no language may be understood outside of how it functions in the whole world. Logic and numbers can only be understood inside a linguistic context. The Mb may also be likened to a grammar. Despite everything we are taught in school, there are no rules of grammar. There are, for the most part, only uncodifiable guidelines. There are also uncodifiable universal generalizations which allow every child potentially to become a fluent and intelligible speaker of every language, able to form intelligible constructions which she has never even heard before. I am sure to be guilty of more than a few such constructions, intelligible or not, on these pages. Intelligibility is not and cannot be idiosyncratic. Pace Wittgenstein, there is no such beast as a private language.
How then did language evolve? The same way that the Mb evolved: in short, it didn't. The longer answer is that it evolved in lemniscate fashion through a process of the gradual refinement of its primal/cosmic core, i.e. the Dialectic.
But wait, there is a perfectly analytic, finitely iterative procedure for separately calculating every point in the Mandelbrot. That is just the problem: how does this object of seemingly coherent beauty emerge from such a random process? Is the Mb a cosmic accident, or does it portend a deeper order? Fractals are supposed to be the exemplifiers of noise, disorder and chaos; and here we have the mother of all fractals, and yet it seems that its attributes are anything but. Is there not some explaining to do?
It is fair to say the the Mb is only semi-connected or disconnected. There is no procedure, not even an infinite one, that would connect or disconnect it.
If the Mb were subject to downward determination, then it would be overdetermined, since it is also subject to an analytic determination. This is the same problem we have with mind & brain. At some point, the physics has to be overridden. Or is there a middle ground? There is if the downward factor could be built into the analytic process, as if there were something like the quantum/observer effect.
One problem is our modern proclivity to think of the integer 8 as synonymous with 8.0000.... On the other hand, the computers that carry out the calculations can scarcely attend to questions of ontology. An uncertain computation is not to be tolerated in computer land. There ought to be, however, some Godelian self-reference effect that requires an observer in the loop. Perhaps the Mb/mind is just that. The Mb could be a telos that is operating for the whole number system. In that role it would be complementary to Pi and MG. We have noted that Pi is embedded in the Mb. Is it not possible that the Mb is embedded in Pi? In addition to the canonical value of Pi, there would be an infinite number of small variations on that number occurring throughout the Mb. Could it be that the Mb records the emergence of Pi from within the Matrix. This appears to be the significance of the zodiacal procession of the quasi-circular bulbs leading from and returning to 'elephant gulch' in the cardioid, and culminating in the perfect circle on the negative real axis. The Mb is the primal context of Pi. Pi is its pearl of great price. It seems to be telling us that Pi should be related (also here and here) to Fibonacci's Golden Ratio, phi. This page describes how the dwell times of points in the cusps are related to pi. A simple explanation is given here, relating the phenomenon to Euler's equation: e^i*pi = -1. Pi is the escape parameter for the Mb/Matrix.
If this is true about the escape parameter, this does shed a new light on the role of the Pi/X archetype. X leads us out of the Matrix into God's kingdom, which is what we approach now. Then the eschaton involves our return to the Matrix. This is hardly the orthodox view of the salvation economy. Does this then motivate us to reinstate the Millennium? We shall see. It is orthodox to suppose that Jesus was saving us from the world, but this Mb model suggests that he is saving us into the world. He then is our philosopher king. Pi indicates the perfection of the economy. It is a perfection, however, that is necessarily finite in time, being delimited by our eschatological return to the Matrix, cosmic womb or nirvana. That would tend to put our Srini2, Y2C, X2 event/factor more into the logical role of the anti-X within the cosmic dialectic. Or perhaps we should say that X is the anti-Matrix, while Y2C is the synthesis. That might make more sense. This scenario is closer to Owen Barfield's three stages of participation. It adumbrates the positive side of history, something that distinguishes theism from pantheism, but which the theists are usually quick to downplay.
The bootstrap and dialectic are ideas that we need to be reconsidering at every opportunity. And certainly more must be said about how the Mb comes under a teleological influence. It may go back to numerology, and the peculiar notion of the evolution of numbers. Just about now, the skeptics out there ought to be thinking that the Mb will be my tar baby, but I'm thinking 'briar patch'!
Pi could be the pre-spatial telos of the Mb, which, in turn, is the telos of the number system. However, none of the structure of the Mb is visible on just the real axis. We need to consider the holism of the Mb, and how that holism is reflected in numbers generally. We could look at the Mb as the primal numerical object: a zim-zum, etc.
It would be helpful to explain the branching process for the Mb. To what does it relate?
I may have missed something previously when I reported a dearth of professional interest in the Mb. The question of branching led me to the topic of Hubbard trees which play a significant role in the phenomenology and taxonomy of 'complex dynamics', and from there to 'experimental mathematics' (here and here). The Mb is a recurring structure in many types of 'complex' systems. 'Complex' here refers mainly to the presence of iota, but also to the sheer complexity. The Mb is the touchstone for an important and expanding branch of mathematics. No wonder and much wonder.
After Douady and Hubbard, Robert Devaney appears to be the primo professional focused on the Mandelbrot and its kin. There is some analysis mixed with much phenomenology, and most of the analysis comes directly from D & H. Let us not overlook John Milnor. Then here it is that Gregory Chaitin makes his argument against Leibniz' PSR:
And here is where the concept of algorithmic information can make its surprising contribution to epistemology, to the philosophical discussion of the origins and limits of knowledge. What if we can find mathematical facts that are true for no reason, where would that leave our philosophy, what would it do to us?
Yes, what indeed?
I'm not sure what to make of these unreasonable or brute facts. How does this relate to the Mandelbrot? How might they be reconciled with holism? Are these atomic facts just swerving in the dark as some suppose of their physical counterparts?
Is this not just additional support for a constructivist view of math and physics? Are we not unnecessarily falling victim to what Whitehead sees as our proclivity for misplaced concreteness? Are we not being frightened by our own shadow, i.e. by our own abstractions? Does not the Mb demonstrate the taming of randomness? There is ever only a pseudo-randomness, and it is ever only of our own making. The same may be said of the quantum when we factor in the observer principle.
Is not experimental math founded on construction and intuition? Gregory's use of absolutism is an ill-founded justification for construction. Kurt Godel's defeat of Hilbert's absolutist agenda was purely constructive. Now Gregory seems to be returning to absolutism via the back door. Our Platonist/absolutist proclivities die hard. It is our final idolatry.
The Mb exhibits aspects of the halting problem, relating to its bounded and unbounded orbits, given a finite cut-off.
We are still being thrown back on the question of how the vital/bootstrap/dialectic manages to infiltrate the computer logic, if that is what the Mb points to. There is here a stark contrast of mechanism and organicism. Computation cannot be an absolute. But neither are we looking for a god of the gaps. We can only settle for a holistic God. How do computation and mechanism take holism onboard? Where is the leverage?
I can only think that habituation trumps logic and idealization. The vital principle enters the system through this process. The only other consideration is numerology, and this probably amounts to the same thing. It is in the difference between 8 and 8.000.... Are we just talking ordinal vs. cardinal vs. nominal? Pi and the Mb become the de facto touchstones of computation. Without their tangibility and specificity, computation is mere abstraction.
The good news is that we evolve. The bad news is that so do numbers. The good news is that Pi and the Mb provide road maps for that mutual evolutionary path. This is how the world hangs together. It can do none other. This is the foundation of the BPW: ergo, the eschatological telos.
Yes, Gregory, Pi and the Mb are brute facts. They are the brute facts of existence and life. Any other facts are figments of our imaginations. We might just as well wonder that they can be computed at all. Small miracles in a bigger miracle, like e^i*pi = -1. If we can believe that tri-unity, we ought to be able to believe all the rest.
Teleology rules, in the depths of matter, and in the bowels of our machines, simply because it is the logical alpha and omega of all being. That is the most brute of all facts. We had best get used to it. We'll never get over or beyond it. Asking how teleology works is a bit like asking why 1 + 1 = 2. It is the only way that being can be.
Does this make the job of Y2C any easier? Pi and the Mb can lead us to the water, but they cannot make us drink.
Facts: united they stand, divided they fall.
This is my day: a fool for God!
Looking back at the original mention of Mandelbrot on these pages, I notice there was a context of functionalism. I take my functionalism in the spirit of Leibniz' PSR.
Functionalism is just relationalism from a teleological perspective. Things are functional because they are related, and they are related because they are functional. This could be the first law of holism. Numbers, logic and computation are no exception. Functionality we take to be the epitome of mundane existence, in all its slimy, greasy glory. This seems just the opposite of Plato's supra-mundane habitat of numbers and forms. But in that exclusivity, Plato was abrogating the spirit of the Pythagorean harmony of being. Descartes followed Plato's felix culpa, but the mathematical physicists are trying to bring us back to Pythagoras.
Numbers exist to function, thus the UEM, the unreasonable effectiveness of mathematics. Numbers could not exist without the Telos. They cannot and do not exist in the splendid isolation of Plato's heaven.
Being the mother of fractals, the Mb is the logical venue for numbers to most freely display their functional organicity. And do they ever! Bless their little cardioids.
One glance at the Mandelbrot ought to cure the most cold hearted of us of our reductionist, deconstructionist, absolutist proclivities. That it has not yet done so, only indicates that the Mb lacks its proper champion. Can the Y2C/X2-event fail to fill that role? I've been looking high and low. Is there a better candidate for Exhibit A? Should it not suffice? Mandelbrot & God (36,900 hits). I'm not the only fool out here today! Imagine if we ever got our act together. Let the good times roll!
The Mb, how can it be? Nay! How could it fail to be? We must allow that explanatory burden to be shifted. The organicity of the Mb is the alpha and omega of numericity. We should be teaching our kids to appreciate the Mb before teaching them to add and subtract. What surer way to set our materialism on its head? It is the irreducible building block that has been sorely neglected by modernity. Where are those tie-dyed t-shirts of the '70's?
And how does the seemingly singular, solitary perfection of Pi/X emerge out of that Pandora's box? Somehow through the operation of the cardioid. What the X-bulb points to is the holistic perfection of the of the whole organic creation, without which, Pi/X would not exist. All is in harmony. Every hair is numbered. Why are we so slow to grasp this? When we use our analytic skill to calculate Pi to billions of digits, we are short-circuiting Creation. With the Mb, Creation bites back. Everything in its proper place and time. Our most strenuous efforts to thwart creation only play into its ever loving hands. Won't it be a marvel when we stop beating our heads against all fate? We realize that God's love can know no bounds.
The Mb is a hard act to follow....
Perfection and imperfection are all relative and all in the eye of the beholder. The perfect eye can only see perfection. All our eyes can and will be perfected in the Telos. They are already perfect in their context, it is the context that will shift.
God said to the Numbers, 'Be fruitful and multiply.' And Lo, the Mandelbrot! In point of fact, it was likely the Mb that taught the numbers their multiplication tables. It is the primordial/universal multiplication table.
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