Consider that there are vastly more genetic combinations than there will exist humans. Does this unrealized human potential create an ontological problem? We don't normally associate any object with that unrealized genetic pool. Perhaps the Mb has less claim to objectivity than the MG. There are also many poems that will never be written. We don't usually suppose that there is a Platonic heaven for unwritten poems. That are mathematical hypotheses that will never be decided. Riemann's could be one of them. Are there hypotheses that will not be hypothesized? That need not be an idle question. Even in the BPW, the greatest symphony might never be composed or performed.
Does the Mb fail the microcosmic test? Or is its incompletion essential to it and us? Will there be no undiscovered dinosaur fossils? Is that any more or less certain than the undiscovered minibot? The Mb forces us to wrestle with the immaterialist ontology. Is there life on other planets, other suns? Are there other planets? What actually constitutes discovery and existence? The existence of the Mb can only be relative, like all else. The reality of the Mb and its parts will finally be optimal. What coheres, lasts. The minibots are like everything else, you find them when you need them. If your presidential reprieve is lost in the mail, so be it. The celestial host are there when we need them, and they need not be from another planet.
Is the Mb infinitely complex? Potentially, yes, but actually it is optimally complex, the same as the cosmos. That optimality will be determined largely by us.
[3/11]
I was attempting to enlist the Mb into the BPW cause, but then it threatened to take over the cosmology by dint of its potentially infinite complexity. We had to beat it back with the stick of relativity. Where does that leave us? I have made a case for the microcosmic status of the Mandelbrot set, but the case remains open. The larger problem is the role of anthropics in mathematics. Such a role runs against our deeply ingrained Platonic view of numbers. It may be the additional role of the Mb to help disabuse us of that illusion. This may be why mathematicians are reluctant to fully engage this beast. It is just too obviously organic/holistic. On the other hand, mathematical Platonism runs up against the unreasonable effectiveness of mathematics in physics. This is especially true in regard to the MG and its kin. I have previously remarked that the more we discover about math, the more we have to appreciate its interconnectedness. This fact ought not to keep surprising us, but no one is keeping score. Platonists tread softly around the problem of relativity. We physicists had to learn these things the hard way. Mathematicians want to be the last of the Mohicans. They fail to appreciate that the BPW will be the eschatological hurrah of Ionia. Plato was just an Ionian backslider. He tried to put his cart of absolutism before the Ionian donkey. Ionian truth did not need a Platonic pedestal. Socrates knew that truth would have to be hashed out in the Agora. Cosmology would ultimately be consensual. The Mandelbrot smacks of the Agora. It is the Agora of the minibots -- too much democracy for the Philosopher King. The organicism is sua sponte. The Pythagorean harmony of numbers was subverted into a Platonic mausoleum. Plato had certainty. It will take a hundred Godels and a hundred Mandelbrots to wean us away from the comfortable lap of certainty. That, or perhaps one more Srinivasa Ramanujan.
Can we extend the minibots' democracy to their numerical cousins? Let the hidden hand have its way. How do we go about this? Will the minibots divulge their secret? What have they in common with numbers? The mini-b's wear their auras on their sleeves. The numerals are much more discrete: 'there ain't no moss on us'. The moss is hidden in the elliptics. There lie the filaments. The elliptic is the cardioid. The numerals are the nearly circular bulbs. We just can't quite see the deviations that portend the greater harmony. Harmony is seen in the syzygys. It is seen in the RH. Recall (and here) the nearly integral values of the elliptically generated Ramanujan constants of the form e^pi*sqrt(n). We suppose mathematical complexity is derived from simplicity, but that may not be the case. This is the analytic illusion. By the same token, we used to think that the perceived world was generated from sense data. Quine began the deconstruction of this illusion. Much deconstruction still lies ahead. Numbers, like atoms, must be emergent entities. The same goes for logic. We haven't quite understood the measurement problem for numbers. Physicists speculate about pre-geometry. We need to speculate about the pre-numerical. Pre-geometry is usually thought to be a logical network of discrete entities. This may be too analytical. According to Leopold Kronecker (1823-91), 'God created the integers, all else is the work of man.' We have to turn that around. Let us recall Watkins, in this regard.
[3/12]
What does the Mandelbrot teach us of holism? It tells us about the microcosmic and relational nature of existence. This is the logical foundation of holism. In contrast to holism, there is atomism. Holism comes naturally; it is grounded in the shining present. Atomism we have to strive for. It is the end product of our analytic skills. It is based on the making of distinctions.
The ultimate distinction is between nothing and everything, between the void and the plenum. Buddhism is the religion of non-distinction. Buddhists identify the void with the plenum. It is nirvana. All problems are solved, all distinctions are dissolved. One could do a lot worse. Could we possibly do any better? The answer is a definite 'maybe'.
I am a bigger fan of the plenum than of the void. I say, 'Give the plenum a chance.' If there is going to be a plenum, it ought to be the best possible one, and so we have the Best Possible World (BPW). Why settle for less?
I make the case for the BPW. How do we get from atoms to the plenum? We consider distinction. The primal distinction is between nothing and everything. In numerical terms, this is the distinction between the null and the infinite. They are reciprocals: 0 = 1/infinity. Infinity = 1/0. That is the mathematical proof of Buddhism. To arrive at the BPW we have to look at this distinction more carefully. We can go step by step in a temporal sequence. The logical starting point is unity. From the previous construction we see that unity is the half way point or the mediator between the null and the infinite. Unity is the numerical emblem of monotheism and the prophetic tradition. It describes the ultimate state of grace or apokatastasis/restitution wherein creator and creation are reunified: the theistic nirvana.
There is a dualism implicit in the unity. Under the operation of multiplication, unity is the dividing line between the null and the infinite. If we start with, say 1.000...0001, repeated multiplication will eventually take that number to infinity, whereas its infinitely close neighbor, 0.999...999 will be sucked into the void. The numerical unit becomes the mark of distinction between the plenum and the void. In the next breath we will be speaking of heaven and hell, but let us not go there.
We continue with the mathematical foray. It is the operation of multiplication, x' = x*x, that leaves us with three distinct elements: 0, 1 and infinity. Yes, this could be one version of the Trinity. The Trinity is the simplest way to defeat the dualism inherent in monotheism. Recall from high school algebra the parabolic function y = x*x or x^2. Its general form is y = a*x^2 + b*x + c. What can we do with that? About the only thing we can do is to simplify it by finding its two roots: y = (x - r1)*(x - r2). Recall then that r(1\2) = (-b +\- sqrt(b^2 - 4*a*c))/2*a. This notorious 'quadratic formula' confronts us at once with three earthshaking 'realities': negative, irrational and imaginary numbers. The capitalist system is based on our acquiescence to the notion of (+\-)$. Greek rationalism was wrecked upon the irrationality of sqrt(-2). Quantum uncertainty is founded on the fact that the 'commutator' of [position, momentum] = sqrt(-1)*(Planck's Constant), which simply tells us that position and momentum cannot be determined simultaneously.
But this is not all. A fourth shaking of the earth comes from the quadratic. We generalize it to z' = z^2 + C, where z = x + i*y, with i = sqrt(-1), is a 'complex' number made up of real and imaginary parts. The single unit no longer serves as the boundary, under multiplication, between the finite and the infinite. The complex values of C lying on the new boundary in the complex plane define the Mandelbrot set, the most complicated object in mathematics.
[3/13]
I have not yet run across an explanation of the most basic fact about the Mb: how it manages to replicate itself. Mandelbrot & self-similarity (22,600 hits). Self-similarity is a defining property of fractals, but this hardly constitutes an explanation for the case of the Mb. In this case we are dealing with quasi-self-similarity. I note that the minibots or 'mu-atoms' occur at places where the filaments appear to intersect orthogonally, as with the real and imaginary axes intersecting the main body. They never occur at the branch points of the filaments where there exists only an infinite regression. One can get a feel for this property by seeing how the Julia set becomes disconnected near the apparent junctions.
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Topical Index
1/5/05
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.
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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?
[3/15]
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.
[3/16]
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'!
[3/28]
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.
[3/29]
It would be helpful to explain the branching process for the Mb. To what does it relate?
[3/30]
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?
[3/31]
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? |