Mandelbrot Mystery  


[Note that this discussion of the Mandelbrot began on the previous page, and it picks up again here.]

If you don't mind, I'd like to do an experiment, as a follow up to yesterday's exercise.  Here goes nothing and everything: 

Mandelbrot (217,000 hits).  That is a lot of web pages referring to a single mathematical object.  A couple of hours ago it occurred to me that I might be able to take possession of this object, that is if no one else chooses to claim it.  I will not even have to take it away from Benoit; just hear me out. 

Mandelbrot & mystery (3,600 hits).  So far so good.  What does this tell us.  It tells us that 98% of the websites dealing with this object do not recognize it as a mystery, nor, in fact does Benoit in a lengthy interview on his alleged baby.  If he cannot proclaim what it is, then his possession may be in jeopardy, after all. 

I will attempt to make the case that the Mandelbrot is the biggest mystery in the world.  If that were the case, then it would be very strange that at least 98% of the sites dealing with it neglect to mention its most important feature.  In that neglect and its reversal may lie the salvation of the world.  Do I exaggerate?  Perhaps. 

It may be more than a coincidence that my very first foray into a real-time mystery was with the crop circles, and that one of the best examples thereof was the Mandelbrot formation.  Was it perpetrated by Cambridge math students?  Perhaps. That coincidence, I notice, accounts for several of the first hits.  Let's look at the rest. 

I have glanced at the first 100 of the 3,600 sites mentioned.  I see nothing that is quotable.  So now it is my turn. 

There is no explanation for the Mandelbrot set.  I do not even see the beginning of an explanation.  It is surely the most surprising and one of the best known of all mathematical objects.  One might suppose that this object would be the very best way to pique the interest of students in mathematics.  Yet, my son, a senior in high school, has never seen it before.  The websites mentioned yesterday are excellent and are aimed at high schoolers, yet, Robert fails to remark upon the lack of explanation.  

Is there a lack of explanation, or should the Mandelbrot be taken to be just one more brute fact?  Let us compare Mandelbrot and Darwin.  The complexity of the world was claimed by Darwin to be explicable in very simple terms.  It was due almost entirely to the process of natural selection or survival of the fittest organisms.  This remains today the primary argument against God.  Almost one hundred years later we are confronted with a single object which exhibits complexity which almost seems to rival that of nature itself.  One might think, naively, that the proponents and opponents of Darwinism would be joining in argumentation as to whether the Mandelbrot set had any relevance to Darwinism, either pro or con.  

Mandelbrot & Darwin (2,800 hits).  Of the first 100 sites listed, the most relevant is Fractal Evolution, but I am unable to grasp the author's point.  Perhaps someone can help me out. 

Here is my point.  Should not the Mandelbrot set be used by the evolutionists as a demonstration that complexity can be sourceless.  You can generate an infinite amount of complexity by following the simplest of rules.  Indeed, the folks at the Santa Fe Institute are betting their careers on this alleged fact.  

There is a problem, however.  The Mandelbrot set did not evolve.  It sprang into the world fully formed, like Athena from the forehead of Zeus.  The Mandelbrot set, like the Monster group, is prima facie evidence for Platonism. Perhaps this explains why complexity theorists and evolutionists are reluctant to use Mandelbrot to support Darwin.  It is a sword that could cut both ways. 

Fractals abound in nature, reminiscent of the Monster group's ubiquity in physics.  



What does this tell us about nature?  That nature is random?  It tells us that there is self-symmetry and scale invariance.  This is something far removed from the simple randomness of, say, a random dot pattern.  

There is much more.  There is a plethora of patterns.  The patterns are not randomly located, but may be seen to be contiguously located so as to appear to 'evolve', one into the other.  Symmetry abounds, yet none of it is exact.  

Fractals do appear in the inorganic world, but the patterns in the Mandelbrot also exhibit a strongly organic and coherent quality.  People can play games based on this quality.  One player displays an odd pattern and then the others attempt to find it.  One gradually develops an intimate familiarity with the 'logic' of the Mandelbrot, and, yet, there also seems to be no end to the surprises.  

The Mandelbrot appears to be a self-organizing system, not unlike biological systems, and, yet, there is no self that need be postulated.  There is only the simplest of generating formulas.  What could be more impersonal?  Self-organizing should then be seen as nothing special, but rather as a ubiquitous aspect of a thoroughly impersonal world.  That, at least, is the Santa Fe creed.  Is that not also the creed of the pantheist: God without God?  It is the creed of the supply side, hidden hand, free marketing capitalist.  

I wish to differ.  I see something more.  I see 'holism' at work.  And if I ever wished to use 'scare' quotes, then this is the time to be scared.  'Holism' may just be my biggest stick.  Pantheists do attempt to water down God, but do they succeed?  That is our question.  Just because God is everywhere, it does not logically follow that God is nowhere.  Yes, God is now here. 

My thesis is that there is intelligence in the Mandelbrot.  It is a chip off the block of cosmic intelligence.  And so are we. 

Here is another way to put it.  Here is the lesson of the Mandelbrot.  That which is not forbidden is inevitable.  And, finally, there is only one thing that is forbidden: the sub-optimal world.  

Consider again the ontogenesis of the Mandelbrot.  Consider z' = z^2 + c.  Does that generator cause the Mandelbrot?  I maintain not.  That generator permits the Mandelbrot.  The awesome beauty that emerges is our latter-day rainbow.  We can and should take it as a sign from God that, indeed, this is the one and only BPW.  Am I betting the farm on this?  We'll see.  If I'm correct, it is mine, is it not? 

Let me, then, reiterate my claim from above.  Besides you and me, the Mandelbrot is the most important demonstrable mystery in the world.  It cannot be explained without explaining you and me and God.  

There is a creativity, a vitality and even a playfulness in the Mandelbrot that could not be excelled by any human artist working abstractly.  

Our ability to aesthetically appreciate the Mandelbrot, its ability to speak to our sensibilities indicates a common ground and source between it and us.  It is a direct manifestation of the Matrix.  So are we. 

Is there any less profound explanation for this object?  I sure don't see it.  Instead, what I see is a lot of intelligent people, particularly at the SFI, studiously ignoring God's handiwork plastered all over their walls.  They have been left speechless.  They may even be afraid.  They should be.  I am here to loosen their tongues, to assuage their fears. 

On a more 'mundane' level, the Mandelbrot should help to explain the Monster.  Pray tell.  

The axioms of mathematics are like the above mentioned 'non-generator' of the Mandelbrot.  They create a special kind of logical vacuum into which Nature or the Matrix is invited to pour itself.  And that it does.  There is also some filtering that goes on.  There are boundary conditions to be met.  The quantum arena presents a similar challenge to nature.  In both cases, we end up with the Monster Group.  Should we be surprised?  


[10/21] a 

I am trying to rationalize both the Monster Group and its role in physics.  I am comparing the Monster and the Mandelbrot, claiming that neither one is causally explainable in terms of generators or axioms.  The source of their complexities is the Matrix.  Each is a boundary conditioned snapshot of the unlimited organicity of the Matrix.  Each of the non-linear fractal 'generators' acts as a special lens which allows us to peer into the Matrix.  Such fractals appear throughout nature.  Nature is the outward manifestation of the Matrix, which is the inward toti-potent Being.  

The Monster Group is a specialized projection of the Matrix.  The Monster is a more highly conditioned form of fractal.  Its symmetries are exact.  It has a finite and discrete structure.  

Ultimately the MG is conditioned by the cosmological anthropic principle.  The MG is an important part of that cosmological bootstrap.  On the other hand, it seems that the MG is to the cosmos as the Julia set is to the Mandelbrot.  The Julia set is not constant the way the MG is, but it is relatively constant over significant parts of the Mandelbrot.  It is not yet clear how to combine these two observations.  There are probably still several missing links.  

Maybe it is time to bring in the big guns: AORSAP.  I would further speculate that, probably in some accord with the Kabbalah, the pantheonic, AORSAP, consort of the Matrix creates an asymmetric vacuum, via the Zim-zum, which is filled in some sort of Mandelbrot fashion, with the MG serving as the Julia set in providing local, physical symmetry.  Pi serves as the numerical arbitrator, and helps to determine a local Euclidean geometry.  

Actually, allowing a virtual MG to vary as Julia does might be useful in a bootstrap situation.  There could then be a self-tuning of the MG into an optimal state for the various physical cycles.  We don't want to make things too easy for the Creator, but no point in busy work, either.  



Permit me, please, to excerpt a recent communication: 

There is a big difference between myself and the mathematical physicists. They are Pythagoreans. They put numbers first. You see that Saul-Paul wishes to build mind out of numbers:

> I would even say that Muses's speculation that consciousness  should be modeled via hyperdimensional mathematics is on the right track.

Thus would numbers be the 'roots of consciousness' and of all being.

On the other hand, I am putting the Matrix first. This is the vital potency, the womb of Creation, the fearful symmetry.

The closest I come to numerology, so far, is with the Big Six, AORSAP, which happens to include Pi as one of its members. This is the pantheonic consort, seminal seed, to the ovoid Matrix. It functions in Creation like the Kabalistic Zim-zum which starts the process of breaking the fearful symmetry of the Matrix. Recall that the Zim-zum is also the Logos or Sophia/Christ for the Christians.

The seven-fold spirit is probably a deeper aspect of the Matrix than is the Six, which would form later in the Creation. Thus would the Six be a later, more explicit, version of the Seven. There is obviously nothing very subtle about the Six.

The whole motivation for my excursion into numbers has been to defeat the Pythagorean program of mathematizing reality. My main challenge is to tame the Monster group. Let it be clear that the Monster is the created tool of God and not the other way around. My exhibit A is the Mandelbrot. That provides a relatively unfiltered peek into the Matrix, and the Monster is just a more highly structured and filtered projection of the Matrix.

All of mathematics is just the Anthropically filtered, numerical projection of the Matrix, which, in turn, is the ground and source of all being.

Mathematicians like to think of their subject in Platonic terms. It is a very orderly abstract construction, evolving from clear and simple axioms. The Mandelbrot is then a great embarrassment. They see it only as an aberration. I see it, rather than the axioms, as the source. The axioms are just a latter-day, feeble attempt to rationalize the Matrix.

What we see being practiced by Jack and Saul-Paul is the Patriarchal, sanitized version of mathematics that has come down to us from Pythagoras and Plato. They see mathematics as a way to root out the evil Apeiron, which was the Greek term for the fearful symmetry of the Matrix, or prima materia. They are simply denying the Mother.

Christianity may be seen as a step toward a balance between the male oriented prophetic tradition, and the feminine mystique of the East. I am attempting to give that balance a more rational, coherent basis. No mean feat.

This by way of providing background information for the present excursus.  



I suggest that the Mandelbrot serves as a microcosm with respect to our scientific investigation of the world, with the computer serving in place of our scientific instrumentation and infrastructure.  There is complexity, coherence and continuity wherever we investigate.  One thing missing is a time dimension, but that need not detract from the analogy.  Admittedly this bare analogy needs to be fleshed out.  In both cases the Matrix has its work cut out for it. 

This analogy pushes us further in the direction of comparing the world to a virtual reality in which fractals would play a major role.  The problem would then be to create a virtual reality with distributed intelligence, as with parallel processing and object oriented programming.  The time dimension would be largely taken up with cycles operating on all scales.  

Our world may then be likened to a Mandelbrot set projected from the Matrix and shaped by the pantheonic AORSAP.  The A&O cycle expands to include the reproductive cycles (R) of the 10^10 distributed human intelligences that participate in the creation and maintenance of the world.  The foundational Matrix ensures coherence throughout.  The sun, atom and pi (SAP) are the more explicit, pantheonic, sources of coherence.  



I have been referring to the Six as a pantheon.  A more conventional term would be 'archetypal'.  I suggest then that the archetype to be associated with Pi is the ouroboros.  In that case, Pi represents the Matrix among the Six.  Previously I have associated the ouroboros mainly with the A/O cycle, and it could also be associated with the reproductive cycle (R) in general.  Now, however, with Pi in the loop, it may be advantageous to consider the feminine trinity of Pi, Matrix and ouroboros, along with our mathematical trinity of e^i*pi.  Pi may be seen as the nexus or distillation of the cyclical character of Creation.  Pi, especially in conjunction with 'e' and the iota, may also be viewed as the seminal seed of mathematics, thereby representing the Six in that realm.  The Six taken as one, Pi in this case, are also Matrix-like.  From a strictly numerical view, {0,1} are also seminal to mathematics, but they do not fit this broader context. 

Taken by itself, Pi does not seem to manifest the duality or dialectical property that is usually associated with the ouroboros, as in its Janus or yin-yang guise.   I suggest that this duality is to be found in its qualitative vs. quantitative aspect.  Thus will Pi serve as the primary bridge between these realms, and become the primary source of psychism in the mathematical domain.  The modern equivalent is Godel's transformation of arithmetic syntax into semantics.  



I am working on a postmodern Just So story for Creation.  In particular I examine the primal personification of the archetypal Six, AORSAP.  The Six emerge from the relatively undifferentiated Matrix through a process of psychological individuation and projection.  This process is only partially spontaneous.   The potentiality of the BPW exists in a background state that is gradually materialized or substantiated in the manner I am attempting to describe in metaphorical and quasi-mythic language.  Even the notion of temporal sequence must be taken as metaphorical in this pre- or supra-temporal regime.  Time comes about in a bootstrap fashion with the metanarrational asymmetry of the Alpha and Omega.  The present attempt to download the metanarrative into real time is one part of the teleological Creation process. 

The emergence of Pi will be one of the more difficult parts of this story to recreate.  Pi might even be the crux of the story: it sits at several significant crossroads.  It has to be our main bridge from quality to quantity.  In that capacity it is also the foundation for the 'unreasonable effectiveness' of mathematics, which is what keeps our trains running on time, so to speak.  It is the warp and woof of our physical infrastructure.  Pi may just be the buckle of our bootstrap.  We shall see. 

As a for instance, the Mandelbrot cardioid is the second order distortion of the complex unit circle defined by e^i*x (0 < x < 2*n*pi).  As we have seen, the Mandelbrot set is a numerical/geometrical projection of the Matrix, noting that Pi is also a bridge from the numerical to the geometrical and spatial systems.   Its role in cyclical process indicates that it is instrumental in relating or even generating space and time.  

In myths we see the personification of the archetypes.  The logical and psychological regime of personification exists in proximity to the Alpha and Omega, which, in turn, have their own archetypal personification.  

Pi also sits at the crossroads of the macro and micro cosms: between the Sun and the planetary Atom. It helps to define the wave-like character of atomic physics.  In so doing it lays the foundation for the metabolic cycles that make up the all-important, archetypal Reproductive cycle.  Thus does Pi quantify and reify the ouroboric nature of the world from the A&O to the Atom.  Pi is seen to embody the primary projection of the Matrix.  We need to explore the (social) psychology of this projective Creation.  This is of more than academic interest.  This is our ticket to ride the eschaton.  This is where the rubber meets the road, to once again invoke an aspect of Pi. 

Pi is the prime Logos, the Zim-zum, the Dialectic, the Om, the seed, the Matrical/metrical mirror of Creation.  No mean feat.  We are simply asking for the source of the seed.  A seminal, Zen-like question, this.  In other words, what was Pi before it became Pi? 



A recent communiqué: (a

The Matrix is the relatively undifferentiated ground of being. It is potentiality.

Only by self-limitation can it create. This is spoken of in the Kabbalah, I believe. Pi is and represents the self-limiting, 'mark of distinction' (N.B. the ouroboric logo). Pi yields the unit circle in the complex plane. A projective mapping can either carry the infinite plane into the circle, or the other way around. The Mandelbrot set is an actual image of the Matrix under the aegis of its self-limitation by Pi. 

But Pi does more than this. It is also the self-projection of the Matrix into the realm of quantity and logic. Therein Pi acts as the glue and the seed for the entire system of mathematics. It is the 'fermentation' of this Pi 'yeast' that results in the 'Monstrous Moonshine' for which Richard Borcherds recently won the Fields Medal. The resulting Monster Group contains all the symmetries to be found in particle physics, it is, in fact, the 'M' in M-theory. 

I think it is not difficult then to relate the dynamical unfolding of Pi to the dynamics of the X-event, i.e. the Incarnation. We see there the same dialectic of self-limitation and self-transcendence, do we not? This dialectic is perhaps best represented in the symbol of the Ouroboros. That same dialectic is present in the self-replicating figuration of the Mandelbrot. Christ stands at the crossroads of the cosmos and microcosm. He is the Alpha and Omega, etc. He is the Six (AORSAP) in One.

I don't think there is any lack of material here that might beneficially be explored.

Explore it we will. 

An important item that comes to light here is the historical misconstrual of the trinity.  The Catholic church attempted to balance the gender of the trinity by appending its Mariology, yielding a quaternity.  This is awkward, and is not even good numerology.  The trinity must be mother, son and spirit.  Feminists were on to this early on, well B.C.  The matriarchy never actually disappeared.  Father and son are redundant from a cosmogonic perspective, viz. Oedipus.  I have mentioned the e^i*pi = -1 trinity.  Therein I switched the roles of 'e' & pi.  All too glibly, I assigned to pi the female role.  Euler's natural logarithmic constant 'e' best represents the relatively unformed fecundity and potentiality of the Matrix.  Pi is the quintessential form.  Pi projects and transforms, in Mandelbrot fashion, the Matrix into mathematical structure.  'i' retains its ghostly, hermetic essence. 

Christ/Pi then is the first fruit of the Matrix.  Thus we have the Christ/Adam connection.  The Anglic homophone, Adam/Atom, may not be an historical accident.  I have already alluded to the Pi/Atom connection.  Most of chemical physics can be read right off of the spherical symmetries and harmonics latent in e^i*pi.  In that capacity Pi stands to become the buckle of our Anthropic bootstrap, in a manner which I am attempting to grasp.  I need to relate the observer principle specifically to Pi.  The historical foundation for all our mensuration is to be found in the lunar/solar syzygys.  Only now has the atom taken precedence for measuring time as well as space.  In every cycle there is a connection drawn between space and time.  

A cycle necessarily involves self-similarity, constituting a primitive sort of recurrent self-measurement.  If one were to postulate panpsychism, cyclic processes would be of particular interest.  Or, from the perspective of  immaterialism, cyclic processes might even constitute a suitable ground of being.  As much as any quantity can, Pi incorporates this notion self-similarity and total symmetry.  This is a significant aspect of the Matrix.  It should not be a surprise that Pi is intimately involved in defining and limiting the complete spectrum of symmetries that comprise group theory, and even the finite groups up to and including the Monster. 

Pi represents an ideal form, as no other number can.  It is the numerical incarnation of both ideality and of the Matrix.  



I have been pressing the case that Pi incorporates ideality, and so perhaps intelligence, in a unique fashion amongst numbers.  If this be true, then so also do the important structures of math and physics, quite apart from arguments pertaining to the quantum observer principle.  Since Pi is the seed for much mathematical structure, that structure is not mind independent.  Does this explain our anthropic bootstrap? 

Yes, it does explain the bootstrap, provided that we understand the link between math and physics.  The quantum is the obvious link.  The quantum does double duty by linking both math and the observer to physics.  The quantum and Pi have to compete to be the buckle of our anthropic bootstrap.  We have noted the connection between the two in the context of quantum 'wave mechanics' where e^i*pi plays a primary role. 

Admittedly there has been an immoderate amount of hand-waving leading up to this point.  However, the conclusion, if it can subsequently be strengthened, justifies all the hand-waving in the world.  We have undertaken nothing less than an explanation of the anthropic principle without resorting either to many worlds or to an omnipotent deity.   Simply by ascribing a suitable source to the evident organicity of mathematics, we can gain considerable insight into the logic of Creation.  We take Pi to be a principle one of the six archetypes associated with our world.  It dominates the quantifiable, measurable aspect of nature.  The replicable, replicating patterns of nature must maintain a teleological coherence.  Pi is the de facto, and allegedly the de jure, nexus of that intelligent coherence.  Its relational, rational power ensures that the psyche is the alpha and omega of nature.  It is the organizing center of mathematics, and thus for much of nature.  It supplies the patterns that are the physical and logical basis of life.  The Mandelbrot set is an example of its implicit powers.  Pi encapsulates the holographic principle within mathematics.  We cannot understand math without understanding Pi, and vice versa.  

Pi is the proxy for the psyche, for the Matrix in the physical domain.  It portends the symmetries of the Monster Group, the symmetries that provide the adequate, logical basis for life.  



A significant monist thesis is that numbers and nature are not disparate.  On a dualist or pluralist view, the evident effectiveness of mathematical physics can only be seen as anomalous or 'unreasonable'.  On the one hand, we have Platonic dualism; on the other hand, we have the more monistic views of Pythagoras and Aristotle in which the Platonic forms are actively present in our world.  The upshot is that nature is more formal and the forms are more natural than we had once imagined.  The increasingly evident organicity of mathematics is just the flip side of the increasing mathematicization of physics.  You can't have one without the other, and it is all part of the underlying holistic nature of reality. 

The organic, holistic nature of the world can only be the result of its self-organizing ability.  This is neither a top-down nor a bottom-up process.  It is rather an ouroboric, bootstrap dynamic.  We seek to obtain a handle on the bootstrap.  That is the purpose of the six archetypes, AORSAP.  The archetypes are the primary organizing principles of our world.  We have been focusing on the last one of these, Pi.  Pi is taken to be the organizing center for the system of numbers, i.e. mathematics.  Pi is the ultimate source of order within mathematics.  

I have said very little about the source of the Six, except to point to a psycho-social cosmogonic dynamic.  Admittedly this was only intended as a suggestive labeling of what remains a formidable mystery.  The strategy herein, in keeping with the cosmic structure itself, is circular.  As we repeatedly traverse the cosmic circuit, it is to be hoped that the spiral will gradually tighten up, so as to direct our attention to the key organizing principles, and then to see how they hang together within the ultimate source, or Matrix.  

The focus on Pi has been motivated, in part, by the fact that mathematics is the best understood among the fields of knowledge.  Understanding the organizing principle of mathematics should be instructive elsewhere.  If our Six have been properly selected, then they should be mutually supportive, epistemically as well as ontologically.  

Perhaps our best handle on the psycho-social dynamic is to be found in e^i*pi.  Referring to this as a 'trinity' (and here) was intended to be more than just a figure of speech.  Comparing Pi to Christ might easily be seen as being grossly sectarian, grossly sacrilegious, or both.  In any case, it is not a matter that I take up lightly.  Rational theism is indeed a rationalization of God.  Theists guard their deity jealously.  I am proposing here a postmodern gnosis.  While I don't expect to be immolated as were my pre-modern colleagues, neither do I expect to be embraced by my relatively 'agnostic' fellow theists.  I can only point out to them that clearly we have not seen the end of God's self-revelation.  The only issue then is one of truth.  Pity be upon those to whom truth becomes a threat.  History has never smiled on such folk. 

Christianity is properly considered to be a heresy against monotheism.  On that score it is an idolatry.  The fact that 18 GI's were killed today in Iraq is not totally unrelated to our being viewed as idolaters in most of the world. My thesis is that Christianity was a very big step from monotheism to monism.  The Trinity is essential to monism, not to monotheism.  Many lives have already been lost in the confusion between the Father and the Matrix.  Many more will be, before this is resolved.  I strive to hasten that resolution.  If I am correct about the Pi/Christ connection, then we have just made a significant step toward that resolution.  On the lighter side, Robert Mapplethorpe came very close to a world shaking revelation, it's just that he mistook pi for pee.  I'm glad to be able to set the record straight. 

The social psychology, or small-group dynamic, of the trinity is the best handle we are likely to get on our cosmogenesis.  The issue before us then is the transformation of the Three into the Six.  Our numerical trinity may, hopefully, provide a clue.  On the simplest level it is just a case of boxing the compass, if the compass is the complex unit circle implied by e^i*pi.  The zodiacal pantheon does a similar job, as a form of celestial psychology and small-group dynamic.  

Time and space are effectively defined by the foursome of AOSA: first/last & big/small, respectively.  That leaves with the archetypes of R & P, reproductive cycle and pi.  



At this point we bring on the ouroboros and e^i*pi, with the hope of connecting R & P to each other and to the other four. 

The reproductive cycle is the primary bootstrap for nature.  The r-cycle is similar to the A/O circuit, and it employs atoms for its metabolic base, with pi as the cyclic number.  This leaves out S, the Sun.  In mythology the celestial sun is often juxtaposed with the spiritual sun, which may be likened to the trinitarian Son.  That is a very roundabout link between S & P, substituting e^i*pi for the trinity.  Another trinity to consider in this context is earth, sun & moon, in which case the moon would replace the iota as the hermetic link between earth and sun.  Using peripheral vision one may barely distinguish yet another trinity by judiciously pairing the Six: A/O, S/A & R/P, taken as matrix, son and Hermes/spirit, respectively; or as time, space and cycle.  Any cycle entails both time and space components.  Thereby, however, pi is then linked closer to spirit than to son.  Such blatant shape shifting should not be unexpected as we delve back into the Matrix.  That is a condition of monism. 

Right now, pi is most abstract of the Six.  There is an abstraction gap.  The atom is the next most abstract.  If we made the substitution quantum -> atom, this would provide more balance, yielding AOSRQP.  Writing this as SAORQP would take us from least to most abstract.  We have, however, lost the pairing noted in the previous paragraph.  If we try to put this into a circular form, there is a discontinuity between S and P.  



The circular form of SAORQP may be of some interest.  Cycles and circuits are then predominate in the archetypal structure, and so is the son.  Previously we have seen the second member of the trinity identified with Sun and Pi, as well as the Alpha and Omega.  The Six then manifest a structure similar to the A/O circuit.  On this circumstance, Pi would be the new god consuming Sol, the old god, making use of the ouroboric form.  What are we to make of this strange figuration?  

If we distinguish two threesomes, SAO/RQP, there is a juxtaposition of the macro and microcosms, with the latter somehow representing the new order.  There is also the semblance of a contrast between low and high tech.  This does not fit with my previous understanding of a Millennial order.  Are there shades of transhumanism here?  Perhaps we had better check the archetypes again.  Or will we have to take Ray Kurzweil more seriously?  Before jumping to conclusions, we might compare this Six with Gurdjieff's Nine, the enneagram.  Let us also not forget the I Ching, Sephiroth and Zodiac.  




Where do the archetypes come from? In his earlier work, Jung tried to link the archetypes to heredity and regarded them as instinctual. We are born with these patterns which structure our imagination and make it distinctly human. Archetypes are thus very closely linked to our bodies. In his later work, Jung was convinced that the archetypes are psychoid, that is, "they shape matter (nature) as well as mind (psyche)". In other words, archetypes are elemental forces which play a vital role in the creation of the world and of the human mind itself. The ancients called them elemental spirits

For now, I'm not obtaining any new insights from the traditional archetypes.  Let us then continue with the trinitarian themes.  With the macrocosm, AOS, we might have mother/father, son and lux/spirit.  With the microcosm, RPQ, we could have mother, son and spirit; while with e^i*pi, it was mother, spirit and son.  The archetype of the trinity is our most persistent one.  RQP and e^i*pi are closely fitted.  ASO does not seem to fit as well.  If we were to replace sol with zodiac, however, it might fit better: AZO.  This leaves us wondering about the then putative Q <-> Z connection.  How much of a stretch is this?  Is there not a temptation to try another sequence, QRP, then R and Z are more obviously connected.  The ambiguities being encountered here are part of the shape-shifting endemic to monism.  Q would take on the role of matrix in its threesome.  The fact that quantum physics used to be called matrix mechanics is, unfortunately, beside the point.  

The quantum is the Pandora's box out of which the microcosm emerges into a definite form, Pi/Atom/Adam/Christ, when mediated by the reproductive cycle, R, within the A -> O evolutionary circuit; which in its turn is mediated by the celestial cycles and precessions, as we proceed to the telos, O.  There seems to be ample flexibility built into this archetypal system to allow for future evolution.  We may recall the evolution of these archetypes: SAAO -> SAAOM -> AORSAM -> AORSAP -> SAO/RQP -> AZO/QRP.  I had been assuming that the Sun would have been the first created celestial orb.  These modifications suggest otherwise.  This does exacerbate the problem of the initial and final celestial appearances.  You'll notice that I am not rushing out to tackle this one.  The outlook remains hazy. 

I should note that biological enzymatic cycles, ~R, function as efficient quantum measuring devices.  The juxtaposition of QRP suggests that the telic anthropos, i.e. us, could be the outcome of a prolonged 'delayed-choice' quantum 'measurement', with pi serving as our psychic proxy in this case.  This suggests that Adam and Eve might better be associated with the Omega than with the Alpha, as given in Genesis. That our mythos might contain such anachronisms is not a surprise.  Zodiac-style zoomorphisms might better be associated with the Alpha.  The iota, sqrt(-1), in e^i*pi, shows a similarly ambiguous identity.  Morphogenesis could be substituted for Darwinian evolution.  All of this remains most speculative, of course.  



The bottom line with the archetypes is phenomenology.  Eventually we idealists must produce a coherent phenomenology for the Alpha and Omega regimes.  We have inhabited the middle regime for millennia.  The Millennium brings us into Owen Barfield's 'final participation' (and here, here).  Using Owen's terminology, 'original participation' or 'alpha thinking' is followed by 'beta thinking', or ordinary ego consciousness, in the middle regime where we are still stuck.  In original and final participation the archetypal forms flow and come alive.  Panpsychism becomes explicit.  In order to participate, we must anticipate the resurrection of the forms.  In the Millennium we will, to a degree, be back-engineering the Creation.  We must understand the system of archetypes in order to properly modulate the flow of events so as to effect our optimal participation in them.  

Of immediate significance is the nature of the initial departures from ordinary physicalism and from the ego consciousness associated with it.  It is possible and even likely that aspects of the 'new age' movement portend such developments.  Need we mention lucid dreaming?  We should make note of any new developments in holistic health.  Even the rise of autism could be a negative premonition, and there may be more than a few such negativities.  Drug fads may partially mask deeper, spontaneous psychic shifts.  The varied phenomenology of 'close encounters' deserves attention in this regard.  The reenchantment of the world will be a bumpy road, and it will behoove us to anticipate or quickly recognize the more significant of the bumps.  

Science, more or less wittingly, will necessarily participate in the phenomenology.  Consider the opening in the heavens rendered by Hubble.  It would be quite premature to say that astronomy is not currently involved in finding its way back to some of its archeo-astronomical roots.  The Internet and virtual reality are among the modern technologies that seem to be playing into and guiding our shifts in consciousness away from 'beta thinking'.   

By the time we awaken to the above developments, we may realize that they are already far advanced.  It will be very difficult, even impossible, to effectively distinguish between subjective and objective elements of a shifting phenomenology.  That difficulty will only make us more keenly aware of the monistic nature of the world.  The shifting norms of epistemology quickly subvert purely ontological schemes. 



A notable feature of AZO/QRP is the pervasive duality with respect to linear and circular.  A&O mark the end points of our linear time frame, while Z represents cyclical time and a spherical sky/earth.  Pi, of course, is the ratio of line to circle.  The quantum, Q, evolves in linear fashion until a measurement, e.g. R, is performed.  What I'm lacking at present is a psychological analog for this duality.  I would note, however, that causality is usually taken to be linear unless a specific feedback mechanism is introduced, as in the case of self-organizing systems.  Non-linearity is the natural state for most processes.  On the other hand, cyclical process are the mark of biological systems. 



I'm thinking there is a missing link.  We have a micro and macrocosm, but no mesocosm.  In esoteric studies the mesocosm is usually identified with the anthropos, or the primordial or universal human.  In the prophetic tradition this is Adam/Eve, or Adam Kadmon in the Kabbalah.  However, from the pervious discussion (1, 2, 3), it might make more sense to take this archetype as the alleged incarnation.  We then have the following scheme: AZO/X/QRP.  The missing element now is the Matrix.  It remains as the source of being.  The third member of the trinity also remains implicit in this particular representation.  We'll have to see if this move from 6 to 7 buys us anything.  



I wish to explore the pantheistic, or, more accurately, the pantheonic aspect of AZO/X/QRP.  Z is just a pantheonic version of the zoomorphic zodiac.  The second person of the trinity is often referred to as the Alpha and Omega.  However, the Alpha can also be seen as the Creatrix or Matrix, or as the Father.  R may be seen as the microscopic clone of Z.  It represents the various biochemical cycles.  Both Q and P partake of the dialectic or ratio between linear and non-linear or circular.  This renders them more ouroboric.  As the most formal or formed element of the Six, Pi may be positively compared with X.  Relative to the Q, however, and in the terminology of the Sephiroth, it would be on the pillar of 'severity', along with the father.  There is a bit of the paper/rock/scissors type of contextuality here, lending itself to the cosmic ouroboric psycho-dialectic.  

The contention is that the Seven may be seen to have spontaneously evolved in a bootstrapped, psycho-dynamical manner within the Matrix.  The teleological constraint is the overall optimization leading to the BPW.  Thus do I attempt to rationalize Creation.  


[...] the infinite circle of its body is a boundary between what is and what isn't. To the Egyptians it represented the end of time when Ouroboros, the creator god, turned back on itself.

Stated explicitly here is the duality of the ouroboros: it is the symbol of both continuity and discontinuity.  [also see 12/3



It is the nature of the psyche to be both inclusive and exclusive.  The ability of the ouroboros to express this duality impressed Carl Jung.  Also expressed in the same symbol is the self-reflexivity, self-organizing power of the psyche.  

The first and last threesomes of what are now our seven archetypes of cosmology, AZO/X/QRP, strongly incorporate the power of the ouroboros.  This quality provides an adequate basis for panpsychism and, then, immaterialism.  The anthropos, X, and the ouroboros cover the macro, meso and microcosms.  The anthropos is necessarily the buckle of the ouroboric bootstrap.  

Even the cellular nature of biology should be implicit in QRP.  The cell is to the organism as the anthropos is to the cosmos.  R expresses the metabolic imperative of life.  Q & P express the symmetric and atomic basis of any metabolic system.  The organizational hierarchy of the cosmos then follows, which is to say its microcosmic and holographic structure.  Is this to say that the psyche is necessarily embodied?  Only in general.  The psyche is nothing if not creative.  There cannot be a Creator without a Creation.  Creatio and Imago are not the same.  The Creation is nothing if not creaturely and participatory.  It is intensive while the Imago is just extensive.  

I cannot claim to comprehend the intensivity of creation.  It seems to be intensive approximately to the degree that it pushes the envelope of monism.  The BPW is as pluralistic as the law of monism possibly allows.  The pushing of that envelope may have something to do with the objectifying of pi. Our seemingly endless fascination with that quality/quantity may not be epiphenomenal in this scheme of things.  Pi comes as close to mind-independence as one can get.  In the context of e^i*pi it is a proxy for the mind-independence of atoms.  The logical fact that it cannot fully exercise that proxy is a crucial statement of monism. 

It appears, however, that Pi is more than a proxy for the mind.  Well, wait.  It is the philosopher's stone in the universe of quantity.  It provides the organizing principle for most of mathematics.  It is, then, equally the proxy for mind in math and for math in mind.  In linking math and mind it replicates the function of the anthropos in linking creator and creation, and in linking macro and microcosms.  The observational and normative/functional quality of Q & R, respectively, completes the close assimilation of QRP to X. In much the same manner the metanarrational A & O are also assimilated to X, leaving us with Z.  

Z manifests the dynamic equilibrium of the psyche.  Z is our constant ouroboric link back to the Matrix.  It is the necessary proxy (and metronome!) for the still remote A&O archetypes.  The dual function of Z in both linking us to, and separating us from, the Matrix is stunning in its extremity, I readily concede.  It pushes the same envelope and to the same extent as is pushed by Pi.  Our understandings of the organic abstractions of Z & P should be mutually reinforcing.  The sky was to be the first frontier of our minds and the final frontier of our materialism.  It was long anticipated that our fascination, and then familiarity, with matter in all its forms would finally bring us to glimpse our own reflection in the reflective potency of the material world.  We would finally grasp the significance of the Six and then see ourselves as the Seventh.  We see through the glass darkly.  




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