Part of the telos of paving is just the prevention of cracking. The design of the materials and structures keeps that in mind. Indeed, this is where a notion of planned obsolescence arises. As we study the biology of ageing, we see evidence of similar 'planning' or programming in organisms.
OK, we know its going to crack and we'll have a pretty good idea of when, but not where. That will depend on any structural faults and on the history of insults. An analysis of the pavement would indicate stresses and strains that portend fractures. This is all part of materials science. We are still dealing with a generalized teleology. With proper design, as in safety glass, the fractures will be jagged, and in as random a manner as possible. The pavement acts also as a primitive recording device, in that it retains a partial record of the incidents that impact it. It is not designed specifically to do this, but it comes as no surprise. The fracture patterns, when they finally do appear, will have no further intrinsic meaning. They will not be memorable, unless there are additional circumstances as with the Liberty bell or the 'Abraham Lincoln' french fry. Once the random pattern appears, the PSR tells us that it will tend to retain its shape while the process of ageing continues. The only surprise would be if the cracks did not appear.
We are delving here in to the realm of noise, while the PSR is concerned mainly with signal. When the noise takes over from the signal, as it does in ageing processes, we may then usefully employ the mathematics of fractals. This is the science of 'fractures'. It will provide for us a range of examples of fractures that we are likely to encounter in varying circumstances.
So here we have a nearly complete science of materials, without resorting to materialism. The test will be to see if this line of reasoning will suffice to ameliorate my uneasiness when I next go jogging on the 'payment'. I'll be sure to report back here.
Noise happens, but we want to make sure that it can live comfortably alongside the PSR and the Telos. How does noise fit into the big picture? Noise is logically integral to metabolism. Metabolism relies on fluidics, and that circumstance brings Brownian motion to the fore. Those motions are quintessentially fractal. The diminutive size of the atoms speeds up that thermal motion and leads to an efficient metabolic transport of molecules.
The cell serves as a microcosm for the combinatoric processes occurring on much larger scales. Many social and economic phenomena have the appearance of stochasticity. Stochastic interactions are the primary function of our three-dimensional world.
Then there is synchronicity and serendipity. One function of the stochasticity is to conceal the synchronicity. Why conceal? This is partly how we are able to maintain our exaggerated estimations of our free will. This is how the vital hidden hand is able to remain hidden on all phenomenological scales. The vital and divine forces are able to operate effectively below the noise level. The stochastic dice are loaded. Subtle are the ways of God. The noise is what gives us and God room to maneuver. The noise acts almost as a lubricant for (the reconstruction of) the signal. The stochasticity delivers to us the pieces of the puzzle which we reconstruct to form the Telos, which then becomes our own. The virtual Telos is thus realized.
Without its fractal patterns, nature would not be natural. These fractal patterns are very easy to program into virtually any system. What we see phenomenally are seldom the pure fractals, but fractals that have been modified or constrained by their interactions with other processes.
How does matter maintain its identity without atomic objectivity? Water vapor mixes in with the rest of the air, but at the condensation point the water is able to reaggregate itself. How do we explain this without appealing to individual molecular identities?
If matter were, in general, not conserved, life would certainly not be possible. It would not be possible to observe such an unruly world. Atomicity bears directly on this issue. The problem is highlighted by the quantum fluctuations. All of physics would collapse if such were the case. This is somewhat like asking why the prime numbers are conserved. Otherwise there would be a collapse of logic and math. It is just the relationality and rationality of being which holds this 'physical' world together.
Numericity and atomicity are of the same ilk. That's my story, and I'm sticking to it. I don't think this is the first time I've made this observation (see above, at least); however, I do note that 'numericity' is new to my spell checker. This may be the crux of my story. We'll have to look into it.
Atoms are physical; numbers are abstract, or so we are told. This dichotomy has been greatly exaggerated in the course of our sojourn into materialism. The early Greeks certainly did not adhere to this dichotomy. It is just this artificial dichotomy that leaves us so puzzled by the 'unreasonable effectiveness of mathematics'. Why have we become so resistant to seeing this linkage? It is the supposed dualism of mind and matter, the very foundation of modernity, that leaves us so susceptible to every possible incoherence. And why have we fallen, so unwittingly, into the incoherence trap? Has science been the cause or the effect? Or did it have something to do with monotheism?
Why is it that we have come to regard God as an artisan rather than an artist? Monotheism has come to be antagonistic to monism, which has become the province of pantheism. This is not the first time I have raised this issue. Why are we so intent upon separating ourselves from God? Is this not the basis of all duality?
Without atoms, there could be no numbers. Without numbers, there could be no atoms. Does this imply that neither exist in heaven? Perhaps. Is this the heart of the matter?
How did we come to suppose that there could be a Creator without a Creation? Is not creatorhood an essential attribute of God? In the MDZ scheme, I have set M&D above Z, and the Zodiac is a far cry from being monotheistic. X is the basis of my claim to monotheism. I hold onto a Quaternity. Otherwise there is a slide into pantheism.
The greatest duality in the BPW scheme appears to be that between Earth and Heaven. To get into heaven, we have to forget the multiplication table. There may be more to this than meets the eye. Down here it's atomicity, up there it's organicity. Atoms, get 'em while they last! What then is the analog of metabolism? It is something like D & Z and the ouroboric/psychic circuit. That is from whence we came and whither we shall go. We will have progressed from tail to head, from tribal to communal.
It is noteworthy the extent to which abstractions hold sway in this supposedly 'physical' realm. We may suppose that they will be less prevalent elsewhere. Our adherence to duality is a measure of our ignorance of metaphysics. It is part of the separation mandated by Creation itself. The duality is temporary and epistemic, rather than ontic.
I suggest the Deus faber came from Homo faber, which in turn came from the industrial revolution, or culminated therein. That economic reality, along with monotheism, was the main source of the dualistic impulse of modernism. It is only with postmodernism that we begin to overcome that dualistic worldview.
A question to consider: why is the complicated end of math, e.g. the MG, connected to the simple end of matter, i.e. the 'elementary' particles? Is there something ouroboric about this? Does this imply that simple math attaches to the complicated matter? In a sense, it does. Counting can be a very sophisticated feat. Something that animals decidedly do not engage in, nor would gods without access to atomic brains. Only the most complicated material structures can support simple numericity. Is there something passing odd in these juxtapositions of the simple and complex?
There seems to be something oddly ouroboric about these juxtapositions, but I'm having difficulty in providing any further specification of it. It is not making sense. The ouroboros does not make a whole lot of sense. It pushes rationality to the limit, something that is bound to happen when rationality attempts to chase its own tail.
It requires a great deal of abstraction to specify a simple logic. The simplest logic rests on the most complex of grammatical constructions. The very process of abstraction remains shrouded in mystery. OK, here is a possible analogy: mating act vs. mating ritual. Having the former without the latter is considered the most serious violation of the social order. This analogy might bring us back to the provenance of the Zodiac. There are involved here the profound issues of identity, along with the problem of the microcosm and of part vs. whole.
There is the simplicity of Fermat's last theorem vs. the complexity of its proof. It is in the self that the simple and complex are united to an unprecedented degree. This unification is the basis of all coherence, even of all being, with only the potency of being in the Matrix that is left unaccounted.
These juxtapositions are reminiscent of the self that is at the heart of the matter. Each of the other juxtapositions is a reflection of the self. If we understand one, we understand all. That would be something, indeed. The best understood conjunction is that of the FLT and its proof. First we might wonder if the proof is essential to the FLT. It might have been unprovable, and so could have been treated as an independent axiom, like the continuum hypothesis, which becomes a mere appendage to the mathematical corpus. This may well be the fate of the Riemann hypothesis, although the smart money says it won't be. This is certainly not how we envision the role of the cosmic self.
What is the 'proof' of the self? Well, the proof is in the pudding. The proof of anything is in the coherence that it brings to the world. The FLT, CH and RH combine to amplify the coherence of mathematics. The self is a concept of proven social utility. So too was the concept of the tribe, in its time, but has that construct not outlived its usefulness?
Creator and creation is another juxtaposition of this type. It is generally supposed that downward causation also flows from the simple to the complex, as teleology is translated into functionality. And then came love. What sort of juxtaposition have we there? It is a vital and creative force. All this seems far removed from the FLT: a^n + b^n /= c^n for n>2. Evidently the FLT serves as an organizing principle. It is also telling us that sporadicity is a rare feature. The MG and its finite family may be the only examples. The powered triple has either infinitely many solutions or none. Having a finite number would violate some version of the PSR. That the proof is so convoluted is pointing to the profundity of sporadicity. The BPW is sporadic. It is the organicity of the number system that ensures this. And there is a trinitarian component to this organicity as witnessed in the power of the elliptic functions. Between the quadratic and the cubic lies all of creation.
Then there is the self-referential quality of the construction used by Godel in his incompleteness theorem. I doubt that the meaning of the construction could be recognized by any mechanistic system: it does not compute. This construction has become a mainstay of logic. Embedded in math is a rudimentary concept of self.
There cannot be identity without identicality, and there cannot be identicality without a finite quantum basis with built-in space-time symmetries. Observation and self-reference are intimately involved, as we have seen. The numericity of identicality will not be supported without sufficient metabolic complexity. The ouroboric circuit is closed with complexity of the MG matching the metabolic complexity. What qualifies this as the optimal degree of complexity remains undetermined. God's self-referential identity depends on this outworking or outsourcing, if you will.
The juxtaposition of the simple and complex may be a manifestation of the microcosmic construction of the world. Even the most simple items, e.g. numbers and particles, must reflect the complexity of the world. This is the basis of relational being.
The FLT may be interpreted as a statement of the fractal dimensionality of the world. This is a bit like Bekenstein's holographic bound for entropy, which also gives space a fractal dimension. Fractality does epitomize the juxtaposition of the simple and complex. Recall that the Mandelbrot set is generated by z' = z^2 + C. The notion of the microcosm is holographic in spirit. This is also the statement that the world is a projective construct. The particles generate the MG, and the MG projects the particles. The holograph provides a bootstrap. I'm not sure how the observer fits in.
The Creatures have to hold up their end of the Dialectic. Then how does the dialectic enter into the other juxtapositions? Can anyone see a dialectic component of the Mandelbrot? There is at least something very compelling about the Mandelbrot. There is an aesthetic component to it which resists mathematical analysis. Fractals have been very effective in lending a natural quality to virtual realities.
There is something organic about the Mandelbrot. There is a peculiar aesthetic unity to the beast. It seems to possess a kind of ecological or functional coherence, while every piece of it retains a unique identity. At the same time, each piece contains a microcosmic recapitulation of the whole. The natural and unnatural, the simple and complex reside together in a perfect harmony. Then there is the mystery. From whence did this beast come? What is its provenance? But once we finally see it, we begin to recognize it everywhere. I claim that the mystery reigns. There is more than pantheism in the Mandelbrot. There is also coherence. Coherent pantheism = theism. The Mandelbrot is one very high resolution snapshot of God, caught in the act of creation. It is a vector to the higher intelligence. It is a cairn on the path of transcendence. Materialists claim our world is flat in some mechanistic sense, but with the Mandelbrot, we come to the end of mechanism and keep on sailing. Imagine finding this gem in the midst of a computer. Never discount the divine ubiquity.
But how is the Mandelbrot something other than merely self-generated (sua sponte! Fenn)? How does it connect? How do we defend monotheism in light of the Mb? The Mb is born of God, as are we. Ergo, we do not need to be baptized. Suggesting otherwise is the sin of the Christians. Has Jesus not saved us? I say that Jesus is the lynchpin of our salvational/eschatological economy. He does his job, we do ours. You may take that for what it is worth.
How then does God manage to intervene in the Mb generating formula: z' = z^2 + C? How does God manage to intervene in any conception? It is coherent vitalism. It is the dialectic in action. As I pointed out on the Mb page, using the term 'generator' is a misnomer. It is more like an aperture that is allowing the vital coherence to shine through. One could argue about whether there is a finite or infinite amount of information contained in the Mb, but no doubt it is vastly greater than the information in the 'generator'. What then are we to surmise concerning the provenance of that information? Perhaps I have misspoken. Could not the origin have been the Matrix? With minimal coherence in the Mb, only a minimal participation of God is required for the Mb. The MG is another matter. The anthropic principle requires more coherence on the part of the MG. Perhaps the generator is to the Mb as God is to the world. God places the optimal constraints on the natural effusions of the Matricial potency. One of God's tools in this process is the MG. We creatures are the other 'tool' of Creation. We are simply the co-Creators.
Benoit Mandelbrot, one of the world's mathematical giants on fractals, said that "fractal shapes of great complexity can be obtained merely by repeating a simple geometric transformation, and small changes in parameters of that transformation provokes global changes." In essence--through a predictable, orderly process the "simple iteration appears to liberate the complexity hidden within it, thus giving access to creative potential."
Thus, in that misnomer called chaos theory, mathematicians and physicists have discovered an *underlying order, a kind of memory operating in non-linear, evolving systems. Fractal geometry illustrates that shapes have self-similarity at descending scales. In other words, the form, the *information,* is enfolded--already present in the depths of the cosmos. So this is reminiscent of the Implicate Order. Iteration liberates the complexity hidden within it. It is not dissimilar to Bohm's law of holonomy: a "movement in which new wholes are emerging."
Ostensibly, our brains also provide an aperture into that 'implicate order', referred to here as the Matrix. This may be particularly the case if one is under the influence of an hallucinogen.
Speaking of apertures, there was a time when I subscribed to a 'quantum aperture'. This was back in the early days of my metaphysical sojourn, 1975-'84, when I initially adopted a Cartesian/quantum dualism of mind and matter. Lately, the idea of a quantum aperture has gone mainstream under the rubric of the quantum mind (19,800 hits). On this view, instead of Descartes' suggested pineal gland, it is the non-local strangeness of the quantum realm that provides a connecting link between our minds and bodies. Now that I have adopted the much greater coherence of monistic immaterialism, I see the quantum as just another symptom of immaterialism, rather than having any direct functional role. It is now ours and God's creative love and rationality which is the aperture between the Matrix and the rest of Creation. Under the influence of drugs, or in some other altered state, we individually confront an unfiltered Matrix.
Where does this place us with regard to the Mandelbrot? There is still the question of how there could be an external influence upon the seemingly closed system described by the Mb generator.
The Mandelbrot lies on the tenuous boundary between structure and chaos. This is the same borderline inhabited by living systems, and so the Mb exhibits its organic quality. Novelty reigns in its interacting patterns as one form mutates into the next. There is much replication, but it is never exact. It is themes and variations. A good symphony or poem has these qualities. One wonders if there is any conceivable pattern that could not be discovered in the MB or one of its cousins. I don't believe that anyone has discovered anything resembling a face. Prove me wrong! Invertebrates are represented almost to the exclusion of us vertebrates.
Here is a worthy attempt at rationalizing the Mandelbrot. Significantly the site is maintained by an extropian, shades of transhumanism. (This is one of the best, aesthetic, Mb Explorer. (This one computes externally, most of the others are applets.) Beware, though, that it rounds off the edges as compared with this explorer, rendering it of dubious analytic value!)
It is noteworthy that there are many claims concerning the infinite complexity of the Mandelbrot (420 hits), and I see no attempts at refutation, beyond those implied by Robert Munafo noted above. There is no doubt that there is an (uncountable) infinity of patterns contained in the Mb, none of which is derivable from the others. And what are we to make of this? Here is Robert Curry's suggestion:
I do not know exactly how to convey the degree to which this entire harangue utterly fails to impress anyone who has ever taken time to explore a few portions of the Mandelbrot set, which is a mathematical model of infinite complexity derived from a simple (one might say "mindless") recursive process.
Mathematical proof is less disputable and far more reliable than the ravings of any creationist lunatic: complexity can and does derive from simplicity, just as simplicity can and does arise from complexity.
The issue is one of self-organization or autopoiesis vs. externality. There seems to be no way for there to be non-physical or non-mathematical influence upon objects of biological or mathematical provenance. The mere existence of the mind, however, ought to cause us to think twice about this issue.
Here is an excellent Mandelbrot show and tell. Be sure to see this analytic (lemniscate) approach to the Mb. The point is that ultimately the Mb, for all its depth of structure, is neither analyzable nor computable. Are we left to contemplate our navels? There is no mechanism to point to an explanation. There was no forewarning when we fell into this monster. There was this much warning: Gaston Julia. (More fractal art.) Here is an applet showing the iteration process for the Mb. A more erudite collection of pages may be found by searching on Fatou & Julia & theorem (4,300 hits). THE MANDELBROT MONK is a must see:
While Udo himself is little-known, one of his works is far more familiar. This 13th century German monk was the author of a poem called Fortuna Imperatrix Mundi (Luck, Empress of the World) in the collection of mediaeval underground verses now known as the Carmina Burana.
[...] In Salus, Udo describes how he used these numbers: "Each person's soul undergoes trials through each of the threescore years and ten of allotted life, [encompassing?] its own nature and diminished or elevated in stature by others [it] encounters, wavering between good and evil until [it is] either cast into outer darkness or drawn forever to God."
When Schipke saw the translation, at once he saw it for what it was: an allegorical description of the iterative process for calculating the Mandelbrot. In mathematical terms, Udo's system was to start with a complex number z, then iterate it up to 70 times by the rule z -> z*z + c, until z either diverged or was caught in an orbit. [*4]
Below the description was drawn the first crude plot of the Mandelbrot, which Udo called the "Divinitas" ("Godhead"). He set it out in a 120x120 frame he termed a "columbarium" (i.e. a dovecote, which has a similar grid of niches) and records that it took him nine years to calculate, even with the newly imported technique of ‘algorism', calculation with Arabic numerals rather than abacus.
"It tends to be taken for granted," Schipke says, "That the Mandelbrot is too calculation-intensive to be done without computers. What we have to remember is the sheer devotion of the monastic life. This was a labour of faith, and Udo was prepared to work for years. Some slowly-converging pixels must have taken weeks."
This anecdote ought to give us pause. [2-25: Yes, indeed! Although the story is still being carried by ABC News, it is apparently too good to be true.]
Here is a further iteration of the lemniscate approach to the Mb, taking N to 25. (Here is a another iteration scheme. On this one you can set the iterations from 5 to >500K.) Is this the closest we can get to an analytic version of the Mb? How much does it give away? To what degree is the Mb an artifact? How do nature's versions of it differ from the computer's in terms of the generators? Robert Munafo provides a synopsis of the computing algorithms. (Yet another viewer, about the best general viewer. )
(Pardon me while I note an area w/o structure: -0.7528656865, -(+?)0.0418538316i from this last viewer. The size is 10^-8.) Usually it is possible to recognize even a minimal pattern. This seems to be a dead zone between patterns. There should have been a four-fold pattern with the usual minibot, but it was split in three parts. It took me about an hour to relocate this section. It's just on the other side of the lake, not quite one width east of the upside-down minibot. The bulb on the right points to it. There are at least two similar areas just to the north and east. Each non-pattern has a minibot near its center of gravity, and there is some repetition at smaller scales. Across each of the eight lakes surrounding the central minibot are similar areas of confusion. This area is repeated, along with the central minibot, on repeating unit of this large spiral deep in seahorse valley. On a sister spiral, several clicks above, there are similar areas surrounding the central minibots.