SYNDEX 1: Part II
General Systems Theory
Rhythm: The Music of the Spheres
Number & Jung's Transformational Process
The Diamond Body: Buckminster Fuller & the Qabala
N=50, V=6: A 56-Facet Stonehenge Mandalog
"Computation according to the wisdom of Anship, is the secret of the deity, the guarded knowledge of the expert. The informed may show it to the informed; the uninformed may not see it. It belongs to the forbidden things of Any, Eniii, and Ra, the great Gods."
Sumerian astronomical tablet from Uruk
Most of the population is so math phobic that they have little idea of what mathematicians do let alone how they do it. In fact, most mathematicians can't comprehend one another's work. Few realize that mathematics has become progressively more specialized (with over 3500 subcategories), until even most mathematicians cannot understand each others theories and proofs. There are extremely few qualified critics. They don't even know what one another are doing and generally only have a coherent pattern of development in their own narrow specialty.
There is no stated criteria for evaluating work in widely separated specialties, for example nonlinear wave propagation and category-theoretic logic. Probably no one knows what is going on in both fields, and 95% of all professional mathematicians couldn't understand either one. No one in all of mathematics attempts to decide what is important and what is ephemeral.
With all those subcategories, and over 200,000 theorems published a year, there is no assurance of their survival except "survival of the fittest," what is deemed useful rises like cream, and gets used and remembered. What is valuable in math is based on our notion of the nature and purpose of mathematics, itself. Most professionals have learned to bypass the ontological and epistemological question as irrelevant by cleaving to ontological materialism, and epistemological empiricism.
In practice, panels from many specialties are formed to determine grant funding. Since hardly anyone knows what is important outside of their narrow specialties, funding is political, rather than based on perceived merit. Generally, each specialist favors his specialty first, then each "area" or "field" gets its quota. "...no one has to justify his own field's existence, and everyone tolerates the continued existence of various other "superfluous" branches of mathematics."
There is work, then, which is wrong, is acknowledged to be wrong and which at some later date may be set to rights. There is work which is dismissed without examination. There is work which is so obscure that it is difficult to interpret and gets ignored. Some of it may emerge later.
There is work which may be of great importance--such as Canton's set theory--which is heterodox, and as a result, is ignored or boycotted. There is also work, perhaps the bulk of the mathematical output, which is admittedly correct, but which in the long run is ignored for lack of interest, or because the main streams of mathematics did not choose to pass that way.
In the final analysis, there can be no formalization of what is right and how we know it is right, what is accepted, and what the mechanism for acceptance is. As Hermann Weyl has written, "Mathematizing may well be a creative activity of man...whose historic decisions defy complete objective rationalization."
GENERAL SYSTEMS THEORY
Systems theory was proposed in the 1940s by the biologist Ludwig von BertalanffyGeneral Systems Theory, 1968). He and his colleaugues were reacting against reductionism and trying to revive a holistic perspective in science, lost since the expulsion of natural science as a unified, self-directed study of many subjects. His main thrust emphasized that real systems are open to, and interact with, their environments, and can acquire qualitatively new properties through emergence, resulting in continual evolution. (anthology:
Rather than reducing an entity to the properties of its parts or elements, systems theory focuses on the arrangement of and relations between the parts which connect them into a whole (holism). This particular organization determines a system, which is independent of the concrete substance of the elements.
Thus, the same concepts and principles of organization underlie the different disciplines (physics, biology, technology, sociology, etc), providing a basis for their unification. Buckminster Fuller was an avid generalist and proponent of systems theory and interdisciplinary interaction.
General Systems Theory was proposed by von Bertalanffy to provide a common meeting-ground for interdisciplinary studies. Fuller promoted generalism over the obvious alienating constraints of over-specialization, but he raised it to a high art in Synergetics, combining poetry, geometry, mathematics, architecture, etc.
Scientists and humanists had long lamented the isolation of their respective disciplines. Bertrand Russell pointed out:"It is noteworthy that modern Platonists, with few exceptions, are ignorant of mathematics, in spite of the immense importance that Plato attached to arithmetic and geometry, and the immense influence that they had on his philosophy. This is an example of the evils of specialization; a man must not write on Plato unless he has spent so much of his youth on Greek as to have had no time for the things that Plato thought important."We have dared to do so, for syndex theory is a platonic pursuit.
In sharing theories from other disciplines, light can be shed metaphorically on theories from another area of inquiry, for example using physics theories to create psychological models for therapy. This is the goal of the Society for Chaos in Psychology and the Life Sciences: to use chaos and complexity metaphors to model new therapeutic processes. Physics, has also been illuminated by spiritual metaphors, specifically works citing Buddhism, such as THE TAO OF PHYSICS (Capra, 19 ).
We call the union, through metaphor, of these seemingly disparate areas of research REFLECTAPHORS (Reflective Metaphors). Jungian Psychology has been one of the main proponents of this pursuit, and M.-L. von Franz, herself has invoked many a metaphorical model from physics and number theory to define her psychological concepts. Perhaps this type of interdisciplinary task force is what she had in mind when she suggested mathematics continue to be examined in fresh ways:"I would conjecture that the task of future mathematicians will be to collect their characteristics and analyze, when possible, every number in its logical relationship to all others. This research should be undertaken in collaborations with physicists, musicians, and psychologists who are conversant with the empirical facts about the structural characteristics of numbers in different mediums."Metaphors help identify how to apply discoveries to new situations, and may even guide efforts to develop new theories. At first, they may be a movement away from scientific rigor, but we need this flexibility in thinking. The interplay between rigor and metaphors can still be monitored to avoid confusion. We can redefine terms to give clear meaning to essentially untestable ones.
Metaphors are useful for discussing different ways of looking at process and behavior, but there is little effort to test them beyond limited cases. We can use them for epistemological, explanatory purposes, and in defining implications, for defining communication and mental clarification. In this regard, Syndex forms a comprehensible language for calculation and derivation.
Metaphors are tools for getting into and out of assumptions, which might not be obvious just from looking at assumptions. Metaphor is a language for communicating about phenomena, and can lead to nonobvious conclusions. We need innovative quantitative measures derivable from verbal content to apply our work on a deeper level than only numerical reports.
Immersing oneself in a concrete scientific thought, and then clearing the mind allows images to spring forth spontaneously which may be very relevant. The ego interacts with the deeper mind in wonder yet with logic and active images. Einstein had his two passing trains which evoked his theories about time and his famous thought experiment where he imagined riding on a beam of light. Watson and Crick had the two snakes entwining which stimulated the concept of the DNA double helix. Kekule dreamed of a serpent biting its tail, and discovered the benzene ring.
Rather than relying on the gifts of association, correspondences, and dreams, we can also create an atmosphere and situations which give us the opportunity to discover and create rich metaphors, which expand, deepen, and lead to valid concepts. In new studies, metaphor is a great aid to communication.
POST-QUANTUM PHYSICS AND THE 99 CYCLOFLEX
"Somewhere within that emptiness, we know, is a nucleus. We scan the space, and there it is, a tiny dot. At last, we have discovered something hard and solid, a reference point. But no--as we move closer to the nucleus, it too begins to dissolve. It too is nothing more than an oscillating field, waves of rhythm. Inside the nucleus are other organized fields, protons, neutrons, even smaller 'particles.' Each of these, upon our approach, also dissolves into pure rhythm."
Our first reflectaphor comes from the leading edge of physics. Most terms in physics are metaphors, including time and matter. When Max Planck originally worked out his theory of Quantum Mechanics, he drew inferences from his extensive knowledge about the nature of harmonics in music."The most important point for our context is the fact that the spin [of an electron] is always a whole-number multiple of the most important natural constant of the microcosm, Planck's constant. When I say 'whole number multiple,' it is clear that we are dealing with a harmonic phenomenon. As a matter of fact, Max Planck, who was very much interested in music, supposedly was inspired to work out his quantum theory by the well-known phenomenon that the notes in an overtone scale jump from one whole number to the next."Non-orthodox POST-QUANTUM MECHANICS is only about two years old as of this 1998 writing. In modern number theory, integers in their entirety are considered a field, an active informational field. Extrapolating from physics, we can suggest a "wave-particle" complementarity theory of the number field.
(Berendt, THE WORLD IS SOUND: NADA BRAHMA, p. 71).
This model reveals a key (here termed THE AURIC KEY) to the orderly distribution of prime numbers. It suggests a "nonlocal" connection between "distant" numbers. Post-quantum BACKACTIVITY is the key idea. PQM merely adds backactivity or a two-way process to all other QM processes. One of the main implications for this theory is the plausibility that not only does "mind affect matter" at the quantum level, but that "matter affects mind" in a backacting feedback loop. Proponents of PQM include David Bohm, Roger Penrose, Jack Sarfatti, Fred Alan Wolf, Nick Herbert, Basil Hiley, Stapp, etc.
Post-quantum mechanics uses the notion of a guide wave or quantum "pilot wave," which propogates backwards and forwards. We suggest, analogously, that there is an "EXEMPLARY WAVE" (99 CYCLOFLEX) which is a basic element of the structure of number -- a repeating cycle of 99. The wave permeates the field, and has no sources or other forms of dependence on the "particle positions" of integers, to the nth degree.
As in PQM, we suggest a two-way relationship between wave (cycloflex) and particle (integers). Its source, as in physics, could be "backactivity," the inherent tendency to intrinsic self-organization. Sarfatti (1998) informs us that, "Self-organization emerges when the direct backactivity of the particle suppliments the one-way relation of wave on particle on its guiding quantum pilot wave." There is a non-local connection between wave function and its corresponding physical particle.
He also states that, "the self organizational change of direct post-quantum backactivity is a self-referential kind of change." Since it is not a classical mechanical wave (such as a sound wave), "the quantum potential is independent of the strength (i.e. intensity) of the quantum field but depends only on its FORM."
A good description of the nature of mind (and also apparently rational numbers) requires non-random, self-organizing, post-quantum mechanics. In PQM, departure from randomness is the signature of sentience and intent. This statement is food for Syndex thought.
In SYNDEX THEORY, prime number distribution departs from randomness, indicating an "intentionality" in the rhythm. Thus, we can conceive of the number continuum as a self-organizing "living process."
We can pursue the analogy of a two-way relationship or feedback-control loop between wave (CYCLOFLEX) and particle (integers). Syndex theory reveals a symmetry structure, a reversible engine of spontaneous non-random, orchestrated self-organization. CYCLOFLEX is a feedback-control loop. Orchestration is from the two-way relationship between guide wave and integers feeding back like the loop between a conductor and his orchestra.
Classically, the quantum wave acts "one-way" as the "unmoved mover" of its manifesting particle without the latter acting directly on it in a self-organizing way. However, in PQM advanced influences propogate both forwards and backward in time from future to past. The past conditions the future causally, but there are also precursor waves travelling back in time from strange attractors in the future which "pull" us forward.
The wave affects the particle and the particle then affects and conditions the wave. This is a model for quantum mind over matter. If the "influence" acts as a signal, it has the additional property that it can be locally decoded at the receiver without needing additional bits from the "sender."
To summarize, PQM introduces the notion of backactivity of direct action of the "particle" on its guiding quantum wave. Sarfatti points out that this forms a closed feedback loop which replaces fragility in the system with a robust self-organization. Syndex is a catalog of operations designed to evoke operations of increasing order, resulting from just such feedback loops and self-organization..
RHYTHM: THE MUSIC OF THE SPHERES"Music is a hidden arithmatic exercise of the soul, which does not know that it is dealing with numbers, because it does many things by way of unnoticed conceptions which with clear conception it could not do. Those who believe that nothing can happen in the soul of which the soul is not conscious are wrong. For this reason the soul, although not realizing that it is involved in mathematical computation, still senses the effect of this unnoticeable forming of numbers either as a resultant feeling of well-being in the case of harmonies or as discomfort in the case of disharmonies."
Gottfried Wilhelm LeibnizRhythm is the basis of harmonic proportions and intervals, fundamental physical mechanisms. This harmonious equilibration was expounded upon by Pythagoras, father of Philosophy. Pythagorean thought formed the basis of the philosophy of Plato, and later Neo-Pythagoreans and Platonists, and greatly influenced the development of western science.
Pythagoras laid the foundation for a holistic science which integrated spiritual, ethical, and metaphysical, as well as practical techniques. Pythagoras is famous for his axiomatic viewpoint that "there is geometry in the humming of the strings. There is music in the spacing of the spheres." From Plotinus we hear, "All music based upon melody and rhythm, is the earthly representative of heavenly music." And from Sufi Hazrat Inayat Khan,
"When one looks at the cosmos, the movements of the stars and planets, the law of vibration and rhythm, all perfect and unchanging, it shows that the cosmic system is working by the law of music, the law of harmony; and whenever that harmony in the cosmic system is lacking in any way, then in proportion disaster comes to the world, and its influence is seen in the many destructive forces which are manifest there. The whole of astrological law and the science of magic and mysticism behind it, are based upon music."
Pythagoras systematized the laws which allow the creation of stringed instruments: musical scale intervals (octaves, fifths, fourths, thirds). He recognized that these fundamentally abstract relationships pervade all creation--even matter irself. In music, as in nature, a wave is a shape in motion. Each note has a wave-shape.
The octave comes from exactly doubling, or halving the string length, that is in 1:2 proportion, while the harmonious fifth has a 2:3 ratio and the fourth 3:4. There is also the less obvious 4:5 interval of the third, and even less obvious consonances.
Any tone in the overtone scale is higher than the preceding tone by precisely one whole number. These are the so-called harmonics. The lower the proportions of the numbers, the stronger the consonance, the more harmonious the sound of the two tones together. The primal polarity ratio of 1:2 is the most harmonious to our ears which are biologically geared to seven basic laws of harmony based on the primal law of whole-number quanta (which prevails in physics as well as music):
THE SEVEN LAWS OF HARMONY1. the overtone scale
2. the interval proportions
3. the division of the octave into twelve semitones
4. the difference between consonance and dissonance, the consonance growing as the proportion of the numbers gets smaller
5. the difference between major and minor, the major proportion being the most frequent by far
6. the predominance of the 1:2 polarity, the octave
7. the law of the Lambdoma (a column of numbers written in the form of the Greek letter lambda, whose right leg consists of whole numbers going from one to infinity while the left leg contains the fractions of these same whole numbers, so that the coordinates of the open isosceles triangle follow the scale of overtones or undertones). There are correspondences in physics, acoustics, arithmatic, geometry, crystallography, cybernetics, theology and philosophy, the genetic code and I Ching.
Harmonics are fundamental to our human biology as well, according to Haase:
"It has been found that the rhythmics of the human organism function utterly harmonically--that is, the frequencies of pulse, breathing, blood circulation, etc., as well as their combined activities. We can observe that these rhythms are strictly coordinated, primarily in terms of the numbers one through four, which are able to form the intervals octave (1:2), fifth (2:3), fourth (3:4), twelfth (1:3), and double octave (1:4)."
These realizations led to the contemplation of the abstract ratio of numbers and geometric figures, and were instrumental in the discovery of the harmonic mean (pitch ratio between two neighboring musical notes). In general, means and proportions have special significance as symbols of the integral order of the universe.
Ancient Greek music employed the Diatonic Scale, which progressed downward in pitch. It has weird quarter-tone modes and modalities and was used in plaintive songs of temple and church, sung as haunting chants. The Hindus, on the other hand used an unbridled scale of 22 alternative notes of variable sequence. The Arabs used various modes in quarter tones, while other cultures explored third-, sixth-, eighth-, twelfth-, and even sixteenth-tone music. Our modern, western ears prefer the tones of the "Equable Keyboard."
However, even though offering few perfect intervals and triads in keys other than its own, the diatonic scale is interesting because all its tonal differences in frequency are multiples of the prime number 11. Still, it failed utterly as a practical all-around tool of music. So much for old models. A useable scale is, after all, just a tool of music.
We will return to some of the basics of number theory in the thought of these philosophers later. First, we will show an example of music related to our theory of backactivity in the number continuum. For this, we need to visit a composer with an ear for physical theory.
The RETROGRADE FUGE, or CRAB CANON, was arguably J.S. Bach's most significant contribution to the science of musicology, at least in terms of Syndex. If properly understood, it makes valid the term "musiconomy" as a legitimate term, (the laws of rhythmic structure).
The parallels of nature's scenarios to the Retrograde Fuge are endless. That it took 6000 years to produce a musician they would once again use the retrograde function in designing music is incredible. Ancient cuneiform musical notation has revealed extensive use of this fundamental rhythmical format, as discovered by a Berkeley musicologist in 1979.
We can conjecture that the music of the spheres would naturally be an example of a point at which the orbiting planets would unfold in a perfect reversal of cyclic patterning. Otherwise, what rhythm would be significant at all?
The structures of number behavior in the so-called baseten system reveals a retrograde function in itself. This has been demonstrated in the synchrographic representation of Holotome E which is a geonumerical picture of the number 2520, the first and lowest number divisible by all eight base digits (one is not a number).
Base ten, then, is actually an octave with a 9th null event which occurs between nine and eleven or at number ten where the zero first comes into effect, by enspiraling the number continuum.
Since nine times eleven equals 99 and 49.5 is one half of 99, a perfect retrograde pattern occurs through the interaction of nine and eleven or 99 and its multiples.
What is musical notation without an instrument with a full octave range? Twelve-tone is the dominant alternative system.
(Crab Canon pages)
major sixth 3:5
major third 4:5
minor third 5:6
minor sixth 5:8
minor seventh 5:9
major second 8:9
major seventh 8:15
minor second 15:16