PART ONE – NATURE OF SYSTEMATICS

PROGRESSION OF THE TWELVE

The twelve are the first twelve multi-term systems as articulated by John Bennett, according to the series of natural integers 1, 2, 3, etc. These numbers are not quantities as used in counting but taken in their qualitative or archetypal sense. Bennett’s approach had much in common with Carl Jung’s thesis that the smaller integers, which could be grasped somewhat ‘as a whole’, could symbolize or reflect fundamental types of movement towards order. The sequence of integers express an enrichment and deepening of wholeness, whether in human individuation or in an enterprise.

John Bennett	Carl Jung

    Though the integers look very simple, they are manifestations of the unconscious as creative spirit entering into consciousness and the knowable. This ‘unconscious entering into conscious’ can be taken in many senses - psychological, spiritual, social and historical – and one of the main features of Bennett’s method was that it encompassed many fields of enquiry. Number symbolism stretches back at least (in the west) to Pythagoras and crosses over many cultures and it has been suspected that an archetypal sense of number may go back at least 10,000 years. Bennett modestly called his own reflections on and articulations of number systematics and endeavoured to bridge the ages in uniting early number mysticism with contemporary holistic and systems thinking, and to form a bridge that could cross-communicate between religion and technology. 

The systems were seen as extended into a number of terms forming a complex of mutual relevance to each other and also intensively as a quality of wholeness. Every system could be contemplated and analyzed in its own right but the sequence of systems symbolized by the numerical series 1, 2, 3, 4, etc. expressed a progressive order that can be tied in with such contemporary notions as emergence and self-organization. 

Creative breakthroughs in art, music, science and mathematics have changed our very understanding of order and harmony and it is now commonplace to think of alternative realities and parallel universes. In the context of such innovation and its attendant uncertainties, the sense of meaningful principles at work becomes ever more significant.

The numbers are first of all pure form and prior to any image. This aspect, too, has a long history and shows itself most evidently in Gematria and the numbering systems of letters and words. Projected into images, the numbers become patterns in a kind of meaning space and we can see ways in which terms dance together or balance themselves one with another. Embodied in our circumstances they take on different colours and manners in different contexts. Every simple integer is an infinity of possible content and it is plausible to say that the qualitative numbers of systematics are actually the transfinite numbers infinity, beyond infinity, beyond the beyond of infinity and so on towards the absolute unknowable infinity of infinities the pious will call God.

The idea of twelve principles has many precedents. There are the twelve zodiacal signs and their spiritualization and transcendence in the twelve Disciples of Christ, later still reflected in the twelve Knights of the Round Table. Pantheons of gods from Egypt to Greece were often twelvefold.

Each system can be seen as a god – but a ‘god’ to be understood as a way of seeing and acting and not as some imaginary entity. The idea of a progression of systems is of some ultimate transforming energy that has to pass through all the forms to be completed. The progression is revolutionary but also cyclic. Every sequence of systems reflects this progressive energy or movement. The higher system in number is a meta- system to the lower system; there is always a deeper understanding to be reached.

Summarized here are Bennett’s definitions of the systems, with some modifications of our own. In addition, we adduce columns illustrating the systems in terms of natural phenomena and also energies, the material taken from Bennett’s work. This makes it clear that using systems to map knowledge from one area into another can be both illuminating and puzzling and both are essential if the approach is to be helpful and not a cul-de-sac. In our discussions of systems and systematics we will take illustrations and descriptions from sources ancient and modern, including myths and visual forms to help evoke an all-round sense of them as principle, image and function. The table below gives an example and includes the terminology of Bennett’s important writings on energies.  

Beyond binary thinking
Systematics is the study and application of ways in which a many can be seen to be or even act as one whole. When there are these kinds of wholeness, the many and the one are two perspectives on them.  This means that the one is seen in the context of the many and the many are seen in the context of the one (we will speak more of context later). There is not the many on the one hand and the one on the other. Constructions in words usually fail to capture the intimacy of the one and the many. At best we can say that the many emerge out of the one and the one comes to presence in the many.

At one extreme, we can posit as a limit collections of elements (a kind of many) that are entirely arbitrary. At another extreme we can posit as a limit a continuous wholeness in which any elements are ‘dissolved’ into oneness. It is in the region between these two extremes that we hope to find systems in the sense of Bennett’s systematics. In systems, the many is an articulation of the wholeness and the one is their union. In systematics, we speak of wholeness in the plural because there are as many kinds of wholeness as there are numbers and the number of the many is assumed to be the main determinant of the kind of wholeness that it can engender. This is the specific claim that systematics makes and which distinguishes it from most systems theory.

The main reason for this relative indifference in systems thinking to the number of elements is due to an underlying frame of thinking which considers, in effect, connections between only two elements ‘at a time’.  Systems diagrams, for example, are built up from chains of binary linkages. They can include ‘logic-gates’, which have some similarities to systems in our sense, but they also are restricted to binary thinking.

Nearly all established thinking considers only two elements together at any one time, whatever the form of togetherness in view (such as forces between particles). ‘At any one time’ includes thinking about any action or process or object that links or holds elements together. This is the principle of local action. Many physicists detest and reject any idea of a non-local or holistic action, in spite of its apparent necessity in quantum mechanics.

In dealing with interactions between more than two elements at a time, a problem arises known as the ‘three-body problem’. It is impossible to predict the motions of three bodies even though each pair, taken in isolation from the third, is perfectly predictable. If one attempts to visualise a three body interaction, it is possible to become aware of something missing from one’s picture. One sees that one does not see what is really going on. To come to grips with this, mathematical techniques have been developed and some have enabled the calculation of possible ‘families’ or orbits. These mathematical forms may reflect a world of form that is kindred to the domain of quantum potential and, in general, to what David Bohm called active information (see last part of Overview in Part Four).

Of course, the possible forms of motion of three bodies are taken to derive from the three binary interactions between them. The idea of a property emerging out of the interactions between bodies was applied by Ernst Mach in his view that inertia arose from the interactions of all the masses of the universe. The supposition that more global properties arise from the ensemble of local ones is usually favoured over the view that in some sense the global properties precede local ones. A debate continues that cannot be resolved because each side is assuming different versions of precedence or of time. The minority group of globalists – or ‘Platonists’ – look to some kind of precedence that is ‘beyond time’.
The three body problem illustrates how our picturing of situations is limited to binary relations. There are innumerable beliefs woven into our way of picturing things, especially about time and space. We picture things in what we suppose is an ‘objective’ three dimensional space and their interactions along a single line of time, one moment after or before another. The shortest or most immediate path between two things is a straight line and sequence in time is the underlying frame of causality.  It is easy to see that linear thinking is taken as the norm.

It is possible to see that linear, binary thinking

                        (+) ----------------- (-)

can be a collapse from, or a degenerate form of, a more complex kind. In physics, this is reminiscent of Feynman’s notion that all possible pathways connect two things but cancel each other out to leave only one. We can imagine that to some degree they may not cancel each other out and have an influence on what actually happens that cannot be predicted in terms of the single binary local connection. In a similar vein, William Pensinger has proposed that multi-valued logic is collapsed into 2-valued logic in our usual thinking. He also suggests that it is possible to train one’s mind to become aware of these higher order logics.

In contrast with the ‘norm’ of a collapsed and degenerate state are moments of ecstasy, sometimes regarded as psychic energy bursting the bounds of (habitual) thought, as in the experience of Dostoevsky’s ‘Idiot’.Such experience is marginalized in a 2-valued framework, which typically takes the form of objective/subjective, the subjective being taken as unreal or spurious. What is involved in ecstasy, or in its milder forms of aesthetic awareness, sense of living presence, or intuitive flow is rendered blank or treated as a negative. The world we picture – as in the representations used in physics – is denuded of ourselves and it is then a tautology that we appear as extraneous or meaningless.

However, it is striking that physics continues to play with ideas of higher dimensions and does not dismiss them as mere ‘imagination’. Higher – or simply more – dimensions imply that things that are seen as separate from a restricted view may be intimate when regarded in enriched dimensionality. It might also be the case that there are modes of interplay between things that are not restricted to binary relations but extend to three or more terms ‘at a time’. However, we would expect that predictability would mean something different from what it usually does. This is, in fact, suggested or implied in the ‘magical’ practice of divination that cannot yield determinate results but only offer a qualitative image of the ‘shape’ of possible events.  The connections possible in higher dimensions may appear as unexpected or even miraculous!

Evidently, there is a massive pull on us to think in binary ways.  Take the well known case of this drawing of a cube. It can be seen as a three-dimensional form in one of two ways. It is next to impossible to see both at the same time. It is even difficult to see the drawing simply as a set of two-dimensional lines. This perceptual constraint reflects into our thinking: it is this or that. If one can hold attention, it is possible to see how we lock onto one point and track to another along a line, or identify one plane and jump to another.

As we go from one thing to another, we cannot simultaneously take another path, e.g. via a third thing. We might have the feeling of more than one path at a time as what we call context. There are some well know physical examples of this action such as catalysis in chemistry and the role of the ‘presence’ of a third body in enabling a transition to be made between two things (an atomic nucleus can enable a photon to split into particle-antiparticle). There are numerous examples from the field of psychology such as the obvious ways in which the presence of a third person can influence the mutual conduct of two people.

For the most part, context is considered in a relatively undifferentiated state. But, it could be looked at in terms of a family of alternative pathways engaging with more than two elements at a time. There would be not only binary sets of connections but also triadic, quaternary, and so on.

Instead of drawing in series of linkages, we can bring out the shapes of the dyadic, triadic, tetradic and pentadic relationships.  The resultant figure then suggests that the dominant binary linkage is interpenetrated or influenced by a series of systems which make up the ‘context’. The identities of the two basic elements are not the same in each system. Pensinger referred to this property as identity-transparency.

It is also possible to see that features of the quantum potential may be represented here. An electron going through one slit as in the basic linkage is also going through another slit in the ‘relational’ system of the experiment. Human systems offer cases whereby we can see even more what Bohm would call implicate orders apply. A transaction between two family members may critically involve transactions that go back generations and may have a bearing on generations to come. In systematics, we consider both physical and human situations equally. Since we do this, we cannot build physical models for systems, since they will embody factors such as ‘intentionality’, ‘interpretation’, ‘values’ and the like – all such considerations removing the elements or terms of systems from the realm of objects.

It is generally appreciated but not understood that how we see influences what we see. This is usually taken to mean that we somehow impose filters and distortions on what we see; but this view stems from believing that that there are fixed objects as such in the first place. We can take another view, in which our various modes of interpretation serve to render us aware of the multi-valued nature of anything that enters our experience. In doing this, we are striving to articulate what context means. Systematics is one way of addressing this question.

Since we cannot make a physical model we cannot make corresponding calculations. In the place of calculations we put consideration of forms. Forms obviously relate to number and geometry but can enter the domain of aesthetic meaning, as in works of art. The ‘form’ of a piece of music exists on many levels. Form becomes subtle and elusive in the realm of feelings. These are not defects. The language of systems is that of symbols and Bennett proposed that symbols must always have multiple meanings. They embrace and transcend contradictions.  They cannot be resolved into any single ‘solution’. This must be the case, given that, as we have been discussing, we seek to transcend simple binary connections.

Words about words
We are somewhat aware that we use words in diverse, shifting, ambiguous and even contradictory ways. Different people associate different connections, images, feelings and memories to the ‘same’ word. We might consider words as elements with complex relations far beyond any simple A = B looked for in naive definitions. Some people are concerned to make definitions that enable them to pin down and fix the meaning of a word but the prime example of a text on the English language – the Oxford English dictionary – shows definition as only a small part of the process of clarification of meaning of words. The word ‘set’ has more than 250 definitions, which are analysed and illustrated by 95,000 words of text. To reduce all this to one sense is to greatly restrict the context in which the word is used. The more precise and narrow the definition, the less relevance it has to natural language; rather as Russell once said that mathematics develops to say more and more about less and less. To return to the example of the word ‘set’, it is found that most people can easily use the word in the right way in a variety of contexts without any recourse to consulting the dictionary!

In practice, definitions depend on complex frameworks. If one does not know the framework, the definition is useless and can even be misleading. Phenomenologically, no word ever exists or is used in isolation. It always stands in relation to other words and even whole families of words, including how its usage is exemplified in texts and conversations. The image we made of the possible ‘higher connections’ between things through various systems of meaning applies here. Although natural language is often devalued as being imprecise, it has the immense value of being used by everyone for a vast range of purposes in a way that largely works. We can consider there to be at least three main types of language: mathematical, linguistic and artistic. Each has its virtue. Systematics uses simple mathematics to clarify and enhance natural language and reaches towards the condition of art.

A starting point for considering what systematics brings to the use of words is to take a word and build around it other relevant words, each of which  can bring out or illuminate the meaning of the given word – by contrast, amplification, resonance, etc. In this example, the given word is ‘world’.

This is by no means a ‘good’, ‘best’, or ‘true’ compilation. It is simply a meaningful one amongst millions that could be compiled. An alternative one would associate from ‘world’ to similar words in other ways, such as via spelling, sound, symbol, etc. to produce a set including: word, whirled, weird, worried, etc. or even four (four quarters), evil (Gnostic view), oyster (‘world is my oyster’) also deriving from the ancient story of the Pearl, and so on and so on.

Now, not only can there be a vast number of alternative versions of a set but also of how the elements are arranged relative to each other. The use of such arrangements has been developed as meaning games.  Such games start from the following premises:

The consideration of arrangement is a prototype for the study of systems. Aspects of this can be found in certain management studies, such as the Cynefin approach developed by David Snowden and his associates. In this approach, it is assumed that there are four distinguishable realities: the known, the knowable, the complex and the chaotic. A working group generates MMs and then maps onto a two-dimensional space.  There is an implied 3 x 3 grid ‘behind’ the display space.

The A’s are for MMs that clearly belong to one of the four main realities. The B’s are for MMs that straddle two domains and have to be clarified further. The C is for where everything is confused. There are rules for clarification procedures and also for development of the meaning of the intermediary locations (e.g. in terms of pathways, boundaries, etc.)

The 3 x 3 arrangement used in these examples is an example of a meaning grid. Such grids can have any number of points or cells and take any kind of form. They provide a vehicle of container for the game. The overall shape of the grid can be changed as a game progresses, according to the agreements of the players. The 3 x 3 grid lends itself to thinking in terms of fours, or the tetrad; while the decadic format encourages triadic interpretations. The basic overall shapes are triangles, squares and circles. An ‘open grid’ provides an extensive space of points which can be populated from a variety of starting points leading to regions of confluence where meaning is to be negotiated.

To be noted is that a ‘region’ is any space defined by one or more MMs. If the MMs are single words, then we have the play of regions defined by pairs of words, triplets of words, quaternaries of words and so on. The meaning space of words as used in natural language may be multi-dimensional or indeterminate, so representation in two-dimensional space is only an approximation; but it is an approximation greatly enriched by playing a game, which gives every region multiple interpretations.

The supposition in meaning games is that we can translate from the multi-dimensionality of meaning into two-dimensional geometries and back again and that this translational ability is enhanced by having a group of people working together. There can be no rules in a strong sense for this process of translation, since the intrinsic realm of meaning we have called multi-dimensional is a priori beyond definition and codification.  We do not have on the one hand the realm of meaning and on the other the realm of representation on a page as if we could compare them. This is why the term archetype has been applied to the realm of meaning to signify forms we cannot directly apprehend but inform all our understandings.

Number Archetypes
The idea of the integers as archetypes prevalent throughout all kinds of search for meaning is now mostly associated with Carl Jung and his pupil Marie Louise von Franz.

After C. G. Jung had completed his work on synchronicity in ``Synchronicity: An Acausal Connecting Principle,'' he hazarded the conjecture, already briefly suggested in his paper, that it might be possible to take a further step into the realization of the unity of psyche and matter through research into the archetypes of the natural numbers. He even began to note down some of the mathematical characteristics of the first five integers on a slip of paper. But, about two years before his death, he handed the slip over to me with the words: ``I am too old to be able to write this now, so I hand it over to you.'' --- Marie-Louise von Franz, from the preface of Number and Time.

Paul Schmitt has given the following etymology of the word `archetype':

The first element ‘arche' signifies `beginning, origin, cause, primal source, and principle,' but it also signifies `position of a leader, supreme rule and government' (in other words a kind of `dominant'); the second element `type' means `blow and what is produced by a blow, the imprint of a coin...form, image, copy, prototype, model, order, and norm,'...in the figurative, modern sense, `pattern, underlying form, primordial form' ( the form, for example, `underlying' a number of similar human, animal, or vegetable specimens).
Citing Von Blumenthal, Van der Hammen has argued that the meaning given to `type' as `the impression made by a blow' is incorrect, and he derives `type' from the Greek noun `typos', which originally referred to a mould (a hollow form or matrix). (6) There are numerous in stances of the use of the term `archetype', or its Greek form, archetypos, or the Latin form, archetypus. The term was used in the metaphysical sense of Idea, namely as the original in the Mind of God of which all things are copies, by Philo Judaeus (first century) and in a more or less similar way by Plotinus. Apparently, Jung took the term `archetype' from two sources, namely the Corpus Hermeticum and Dionysius the Areopagite's De Divinis nominibus... Use of the term also appears in Irenaeus's Adversus haereses, and its Latin equivalent, `ideae principalis', can be found in St. Augustine's De diversis quaestionibus, and later in Agrippa von Nettesheim's De occulta philosophia, Libra tres. Instances of its use also appear in De Dignitate Hominis of Pico della Mirandola (1463- 1494), from which its later use by the Cambridge Platonists, in particular Henry More (1614- 1687) can be derived. In the 16th Century, Johannes Kepler used the term `archetypus' to refer to ideas or forms pre- existent in the Mind of God which are geometrical in nature. Because the human soul is, according to Kepler, the Image of God, the human is capable of discerning the archetypal geometrical forms according to which the world is structured. Other usages of the term ‘archetype' can be found in Rene Descartes' 1641 printing of his Meditationes Prima Philosophia and later by John Locke, in Books II and IV of his Essay Concerning Human Understanding.
[‘The Emergence of Archetypes in Present-Day Science and Its Significance for a Contemporary Philosophy of Nature’ by Charles Card (http://www.goertzel.org/dynapsyc/1996/natphil.html)

Jung himself wrote:
Again and again I encounter the mistaken notion that an archetype is determined in regard to its content, in other words that it is a kind of unconscious idea (if such an expression be permissible). It is necessary to point out once more that archetypes are not determined as to their content, but only as regards their form, and then only to a very limited degree. A primordial image is determined as to its content only when it has become conscious and is therefore filled out with the material of conscious experience. 
The archetypal representations (images and ideas) mediated to us by the unconscious should not be confused with the archetype as such. They are very varied structures which all point back to one essentially ‘irrepresentable' basic form. The latter is characterized by certain formal elements and by certain fundamental meanings, although these can be grasped only approximately. The archetype as such is a psychoid factor that belongs, as it were, to the invisible, ultraviolet end of the psychic spectrum. It does not appear, in itself, to be capable of reaching consciousness. 

In his discussion of the relevance of archetypes to modern science, Card says:

A further indication of archetypal order in quantum phenomena may be inferred from the prominent role played by symmetry properties and principles in the formulation of quantum mechanics and in the description of elementary particles. The correspondence of the concept of abstract group with its particular realizations to the concept of archetype-as-such with its archetypal representations has received attention from several authors: Jung himself initiated this comparison when he asserted that the archetype, "might perhaps be compared to the axial system of a crystal, which, as is were, preforms the crystalline structure in the mother liquid, although it has no material existence of its own." Werner Nowacki has pursued the relation ship between archetypes and groups further, asserting that symmetry groups may be thought of as primal images:
Symmetries are formal factors which regulate material data according to set laws. A symmetry element or a symmetry operation is in itself something irrepresentational. Only when...it has an effect upon something material does it become both representational and comprehensible. As primal images the symmetry groups underlie, as it were, crystallized matter; they are the essential patterns according to which matter is arranged in a crystal....The analogy between symmetry elements and the archetypes is clearly unusually close. This is the pivot of the structure of reality.

We will have reason to return to the theme of symmetry later on.

What is important for our discussion is to realise that Marie Louise von Franz largely limited herself to just the first four numbers. 

The archetypes primarily represent dynamic units of psychic energy. In preconscious processes they assimilate representational material originating in the phenomenal world to specific images and models, so that they become introspectively perceptible as "psychic" happenings. (48) In Number and Time, von Franz has discussed in particular detail the qualitative aspects of the four archetypes called the quaternio. While the quaternio are naturally associated with the first four integers, their archetypal nature gives them a much more comprehensive role. von Franz has given a summarizing statement of their archetypal behavior:

Numbers then become typical psychological patterns of motion about which we can make the following statements: One comprises wholeness, two divides, repeats and engenders symmetries, three centers the symmetries and initiates linear succession, four acts as a stabilizer by turning back to the one as well as bringing forth observables by creating boundaries, and so on. (loc. cit.)

Card also cites von Weizsacker, who spoke of ur-phenomena based on two; we can also think of Charles Sanders Peirce whose metaphysics was based on three (Oneness, Twoness and Threeness). What is apparent here is that key figures in thinking about the meaning of the integers as archetypes have restricted themselves to just the first 2, 3 or 4. In contrast, Bennett set himself to investigate the first twelve integers, even though, as we shall see, he really did not go beyond eight.

Arnold Mindell, the ‘process’ psychologist, has argued that different people operate with different number-bases. We are used to the decimal number base, which uses ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The symbol 10 for ten signifies that we ‘start again’ in counting up to ten, the ‘1’ signifying ten because of its position. If we had a number base of 7, then 10 would not signify ten but seven, 21 would not signify twenty one but fifteen and so on. In the modern world we are also used to the binary system where, for example 10 signifies two, 21 signifies five and so on.

‘Starting again’ in our count is psychologically equivalent to reaching the limits of our discrimination and adopting another cycle to deal with higher numbers. This enables us to count however large a number we want but it does not mean that we can see larger wholes. The number-base signifies the limits of our mental embrace. The difference between counting and seeing is of primary importance. In counting we can deal with one thing at a time while in seeing, while in seeing we need to grasp the whole ‘all at once’. Obviously, counting relates to the many and seeing to the one.  We must emphasise that this does not mean that we have to take counting as one thing and seeing as another!

Bennett made significant contributions to our understanding of the meaning of the integers from 5 to 8. In particular, these two numbers were given special treatment. For the number 7, he developed ideas derived largely from Gurdjieff. Gurdjieff in his turn concentrated almost exclusively on the numbers 3 and 7. As we said, Bennett aimed to deal with the integers from 1 to 12 but did not in fact do much for the last four numbers. For 9, he took the Enneagram symbol from Gurdjieff. For 10 and 11 he made vague statements. The number 12 is used extensively, but scrutiny will reveal that he dealt with it largely as the compound 3 x 4.

We can make a grid of the first twelve numbers, taking into account the strong influence of fourness. Bennett himself tended to group the numbers or systems into fours, a trend which began in Volume I of The Dramatic Universe. This simple arrangement gives an interesting picture, in which some patterns may be seen.

Just from a cursory glance, we notice that the third column is all primes. It is also the case that the transitions from 3 to 4 and from 7 to 8  are the ‘critical intervals’ Gurdjieff describes in his explanations of the octave, which might lead us to suppose that there is some equivalent for the transition from 11 to 12.

The first four numbers are always taken as universal, applying to everything. They are the most abstract. In his treatment of the natural sciences, Bennett correlated these numbers with ‘hyponomic existence’ or that which was subject to law; in common parlance, the material world. The first four systems can be thought of as rudimentary. Everything that exists must include them.

The next four numbers mark a significant change and indeed Bennett ascribed significance as such to the pentad. In his early scheme, these numbers correlate with the world of life, or that which is autonomic and has its own meaning. The number 5 relates to a virus, while the number 8 relates to the human self. Bennett went on to relate the number 6 to the present moment and events, while he related the number 7 to transformation. In life-terms, 6 relates to the cell and 7 to the organism.

The last four numbers are akin to framework conditions or ‘cosmic’ totalities. Bennett related 9 to cosmoses, or ‘worlds’, while 12 he related to ‘fulfilment’. Associating from the systemic attribute of 12, we might consider 8 and 4 to represent degrees of fulfilment. Bennett called 8 ‘completedness’. The number 4 appears in all cultures as the number for the totality of matter, as in the four elements; or for the ‘world’.

Looking to the columns, we notice that the second one contains all those systems that are strongly symmetrical or concerned with balance and complementarity.  This is in contrast with the third column, which contains the systems that are most dynamic and concerned with change. What then of the first column? One striking thing is that Bennett often spoke of the pentad as enabling us to identify the monad: whereas the monad itself is like a collection, the pentad shows a self-sufficient whole. With the pentad, the monad discovers its ‘name’. The systems in the first column are all starting points. In the number-base of 4, they signify a new cycle or new beginnings. The set of columns then signifies commencement, complimentary and completion. There is a meta-pattern.

The three rows can be see as universal, emergent and containing; but such terms are at best the crudest approximations to their meaning.

It is tempting to consider a 4 x 4 grid. After all, in Bennett’s scheme there is no inherent reason why the systems should stop at 12, the duodecad. 13 is a significant number, e.g. in the Jewish tradition. 14 and 15 often appear in lists of ‘principles’ and 16 is the development of 4 as 16 = 4 x 4. It is also the number of the next N-gram after the Enneagram in the series defined as n = 1, 4, 9, 16, 25, etc. the series of the squares.

The placement of the systems within a meaning grid draws attention to them taken as a higher totality. The meta-pattern portrays a ‘meta-system’ or system of systems. This is a natural extension of taking seriously the fact that known treatments of the numbers as archetypes only encompass a limited range of numbers and gives this number significance.

The Enigma of the Alphabet
The origin of alphabets as we know them today goes back to the 17th century BC, when people in the Sinai area developed a series of signs based on Egyptian hieroglyphics, but operating in a very different way.  The word aleph meant ox and the sign for an ox, like an inverted A, was used to represent the first sound of the word, namely ‘a’. Similarly, the word beth meant house, and its sign was used to represent the first sound or ‘b’. In other words, signs were adopted and used to represent sounds and not meanings.

The alphabet we are familiar with originated with the Phoenicians in the 12th century BC. One branch went on to become the Hebraic alphabet and another to become the Greek alphabet by the 9th century BC. The Greeks added signs for the vowels. The Greek alphabet gave rise to the modern European ones. The Greek alphabet has 28 signs whereas our English one, based on the Roman version, has 26. The alphabet for Hebrew has 22 signs and does not include any vowels.

For centuries, people have ascribed archetypal meaning to the series of letters in the alphabets. This is strongly exemplified by Hebrew as expounded in the mystical system called Kabbalah. The sequence of letters we have in the English alphabet has come down to us over thousand of years. A question is whether we can discover any sense and meaning to this sequence, which may have been subject to many historical contingencies.

First we should note the almost universal appearance of the first three letters; if we allow our ‘c’ to stand in for the Greek g or gamma (the original letter was from the word for camel and this is taken over, it seems, into our English letter ‘c’). In some sense, the first suggests the insemination of the spirit and the second container or womb (which relates to beth as house); with the third as of the nature of becoming.  There is even some correspondence with Peirce’s Firstness, Secondness and Thirdness and we have the saying, ‘Simple as A, B, C’. Now, if we adopt the device of ascribing the first three letters to the number 1 and then go on counting, we find the following result:

The vowels A, E, I, O, U fall on the prime numbers 1, 3, 7, 13, 19. The primes 5, 11 and 17 do not match vowels; but Y, which can act as a vowel, does (23). Of course, sometimes 2 is taken as a prime also. It is striking that E - which here corresponds to 3 - is the letter most used in the English language, since 3 is the dominant prototype of a system, as in the influence of the Holy Trinity, dialectical materialism and dialogue.

The prime numbers can only be divided by themselves and in this sense are unique. It is impossible to exactly predict when a prime number will occur in the series of natural numbers. Some commentators have therefore proposed that the numerical archetypes should be restricted to the series of prime numbers.

            1   2   3   5   7   11   13   17   19   23 29   31    etc.
            1   2   3    4   5    6    7     8      9   10 11   12   etc.

We have placed the sequence of integers under the primes to suggest a correspondence in ordered sequence. From one perspective, 10 may be the equivalent of 23 while from another it is obviously different. How different things can be seen as ‘equivalent’ is an important aspect of systematics. The correlation with primes can be extended to consider even more extraordinary numbers such as the transcendentals. A few examples are given here.

φ appears in all matters of design and proportion, e is used in logarithms,  pi is universally known and delta is a number of great significance in complexity theory. Though we have found that the appearance of the prime numbers in the sequence of integers has some correspondence with the appearance of the vowels in the sequence of letters in the alphabet, this ‘proves’ nothing. It can only be suggestive, in the manner Jung spoke of synchronicity; which is the property of correspondence of forms of two things that has no apparent reason but can be found to be meaningful. We can explore the correspondence in question further by ringing into the picture such things as Gurdjieff’s enneagram, in which we see a sequence ‘punctuated’ by special elements.

            0 – 1 – 2 – 3 – 4 – 5 – 6 – 7 – 8 – 9

We note of course that the enneagram has a much simpler and symmetrical form than that of the alphabet with its consonants and vowels. But the vowels are like ‘energies’ or qualities that are given form by the constraints of the consonants just as the stages of the enneagram provide a ‘container’ for the energies introduced at the critical points. The vowels have always been given special significance, as described by Joseph Rael in his explanations of chanting as something that transcends particularities of different languages. If we are to consider numbers as signifying archetypes, then we can also consider letters as signifying archetypes. This can at least open the way to appreciating yet other manifestations of systems in even more subtle forms as in painting, where we might take the colours as ‘vowels’ and the lines as ‘consonants’.

The realm of natural language is rich in content and form but it is usually left to the poets to realise this. For the most part, we skate over its surface and are rarely conscious of its depth. Thinking in words need not be linear and we restrict ourselves in this way by a narrowness of awareness. Language involves an active attention to every distinct element (so that they can be ‘counted’) but also a receptivity to the wholeness from which meaning comes. Between these two we participate in the making of meaning (perhaps a third force in the dynamism).

THE QUALITATIVE NATURE OF SYSTEMATICS

Since systematics draws attention to the number of terms in a system, we have to think about how we discriminate and count them. Terms are not things just hanging around for us to enumerate and collect. Here, we come across a fundamental paradox or ambiguity in the fundamental nature of systematics: it is predicated on understanding the qualitative significance of number. This amounts to saying that it concerns the qualities of quantity! Let us think about this in terms of a discourse, such as a text or story or conversation.

A discourse does not come in discrete chunks, though there are words, sentences and so on. Its meaning is as much continuous and unbroken as it is in parts. We could extract every distinct word and every distinct sentence and take these as indicators or carriers of meaning. The former is used when we make an index, which is one standard way in which we ‘bring out’ what the discourse contains. The latter is rarely used, though it would be a reasonable approach, since every sentence is supposed to convey one idea.  In practice, there is some ambiguity as between sentences and meanings and the one cannot be identified with the other.

If one person summarised the discourse into a set of meaning chunks, another might do it differently. We call chunks of meaning ‘molecules of meaning’ (MMs) to signify that they will always be complex in their own right. Various attempts, such as that of Descartes, to abstract ‘atomic’ or irreducible units of meaning from e.g. philosophic discourse have failed. Just as particle physics has split the atom and revealed a great complexity of yet ‘more fundamental’ particles, so we find that ‘a’ meaning can always be resolved into relations between other meanings. In natural language, ‘bootstrapping’ prevails.

It is important to point out that the complexity is not hierarchical. There are not ‘smaller’ meanings which are made into ‘larger’ ones. There is always some kind of judgement of what is essential. This judgement relates to the nature of the discourse and the person who is articulating the MMs it contains. If a reader went through the previous paragraphs, he might come up with a different set of MMs than would the author. There could, however, be considerable overlap, which would mean that the reader and the author could carry on a coherent conversation.

If we remember that meaning is both continuous and discrete (wave and particle if one likes that analogy) then we should realise that any identification of a set of MMs not only contains a set of discrete elements but also, at least by implication, some aspect of the continuum. This is represented in the method of logovisual technology (LVT) by having the discrete items written onto separate hexagons but then placed on a display board so that we also have the ‘gaps’ between the MMs  to signify the plenum of continuous associations between the MMs and even so to say ‘containing’ them. The initial blank display signifies the pure continuum before any act of discrimination.

Putting to one side the detail of how similar or dissimilar the sets of MMs that different people produce might be, in systematics we are concerned primarily with their number. Let us also say that the task in hand is to bring out what are the ‘essential’ MMs. By preference, some people will go for a smaller number and others for a larger. There is at work something like an inherent ‘plane of reference’ or way of ‘cutting through’ the whole complex such that different planes of reference are giving different results. A plane of reference relates to what one is tending to look for.

Going back to the paradox of quality and quantity, we can add a third term related to plane of reference as form. This idea gets its meaning from considering that we could have N objects but they could be arranged into a shape and their shape though visible cannot be counted as the objects can. Described mathematically, a shape can be precisely defined but this description is relational rather than arithmetical. It leads us into algebra. The idea of ‘shape’ is then extended into a more general idea of form. The word ‘form’ has been used since the time of Aristotle in contrast with ‘matter’, when it meant that which was intelligible and could be thought about. In some Scholastic schools of the Middle Ages, matter even needed the form of quantity to be measurable! Our use of the word ‘form’ is intended to mean any kind of shape, however we see what shape might be and we will be extending its meaning to embrace ‘images’. 
Let us imagine a discourse as a sphere. This does not mean that it ‘really is’ a sphere; it is simply a means of thinking about some of the questions we have raised. We are using a shape to think with. Now, let us imagine that we take various cross sections through the sphere. We could start at the equator and work our way up to a pole. Picturing this, we can see that a cross-section at the equator will contain ‘more’ than one near a pole.

Now, the image of a sphere is intended to convey the idea that discourse is not only a sequence of elements but also a holistic ensemble. This means that any ‘region’ of the discourse reflects the whole. Thus, the various cross-sections are not of different parts of the discourse but relate to different ways of seeing it. The different cross sections will yield different numbers of MMs but yet each will represent the whole.

In this picture, every region (close to a point) on the surface will represent the whole in a monadic way. The set of all such points would be a monad. Thinking of them as ‘on the surface’ signifies that there are no essential discriminations in the monad – it is simply a collection. The regions within the sphere are more essential. We can then think in the following way: the cross sections as we move from pole to equator signify higher and higher multi-term systems. The equatorial section then represents the highest order of system we can reach. The polar region is simply the monad as represented by one term.

The last statement needs explanation: any element in a monad can stand for the whole of its content. This is, in fact, how wholeness is understood in phenomenology.

We can reduce the image to a simpler form by picturing just a circle. In this image, we add the different cross sections as relating to different planes – or systems - of representation.

The monadic plane is now not the outer surface of a sphere but a line through the circle, which divides it into two halves. We can take this plane as a boundary between what is consciously articulate and what is not. The lower half of the circle is then a symbol of the unconscious and inarticulate.

The portrayal of successive systemic cross sections is shown as at different angles to the monad. This is to signify that they progressively enter into the realm of the unconscious. As they are shown in the diagram, they are asymptotic to a line in the vertical direction. This is to suggest that they have no finality, only a limit. It would have been equally possible to shown them as continuing round to eventually coincide with the monadic boundary. In which case, the presumption would be that the set of cross sections or systems would be finite and forms a cycle.

The ‘conscious’ part of a line would signify articulation of an increasing number of discrete terms.  The ‘unconscious’ part would signify an increasing inclusion of continuous meaning. This representation agrees with the picture Jung gives of integrating unconscious elements into conscious experience. It does not mean that the unconscious elements become ‘known’ in any obvious sense. It could well imply that understanding a higher term system would require a greater participation in the unconscious, rather as Gurdjieff suggests in his expositions of cosmic laws in Beelzebub’s Tales.

The image can also be amplified by reference to ancient thinking about the cosmos. The lower half would then be the cosmic ‘sea’ or that which is below the plane of the ecliptic. This how Santillana and von Dechend speak of it in their masterwork Hamlet’s Mill. They interpret world wide references to a Great Flood as signifying not an onrush of water due to terrestrial climate but a shift of the ecliptic relative to the celestial equator. The constellations visible in any epoch do not remain the same and some descend or ‘drown’ over time while others emerge. As a new one comes into the ascendant, another goes out of sight. (Some helpful explanations can be found at http://fusionanomaly.net/hamletsmill.html which also quotes from William Sullivan). The realm of visible stars represents the intelligible realm of what is knowable. It gives us a ‘celestial architecture’ and, for the ancients, was their instructor in number. 

Santillana, Sullivan, Stewart and others have spoken about the ‘shock’ that may have been experienced by the ancients when they discovered that ‘heaven’ was changing. According to Pete Stewart ( see his ‘Architecture of the Spirit’ at http://www.duversity.org/Pete.htm), this must have been the origin of the idea and sense that time and eternity were broken apart and there needed to be some way of ‘mending’ them again. One way was to find the pathway to the gods, and another was to gain immortality.

A third way came down to us in Christianity and the idea of the Redeemer. We rightly associate Christ with the cross which is an obvious symbol of the divorce between the eternal (vertical) and the temporal (horizontal). When Christ is nailed onto the cross to suffer and die, he is taking on the ‘sins of the world’. We should remember that the word ‘sin’ originally meant not immorality but missing the mark.

While the initial reaction to the realisation of cosmic change may have been in terms of catastrophe, there eventually came a complementary sense of progress. Loss of the initial vision of agreement between heaven and earth was reflected in the Biblical story of the Fall of Man and the expulsion from Paradise; but this was also linked to eating from the tree of knowledge.

We have used various images including Dali’s famous painting of the Crucifixion and made mention of various ancient narratives. This illustrates the contention that between quantity and what can articulated and enumerated, and quality which must remain more a feeling than a thought, there is a realm of form that is essential if we are to be able to think about number in a qualitative way.

Underlying our discussion has been the prospect that what is to hand in discourse, such as words and sentences, does not directly translate – at last by any simple formula – into terms of systems. A term is as enigmatic as the system to which it belongs. We cannot discern the terms without awareness of the system and we cannot identify the system without awareness of the terms.

In their turn, terms and MMs are related. An MM is any discernible and expressible meaning, whereas a term must belong to a system. In LVT, we can produce terms by using shape, as in an arrangement of MMs. The words ‘syntax’, ‘shape’, ‘form’ and ‘image’ shown in the diagram all refer to ways in which many elements are related together. In the early descriptions of systematics we had only terms and systems. Then we began to appreciate the relevance of form. With the development of LVT, we next realised how terms could come out of meaningful elements (MMs) through their arrangements or collective ‘shape’. This led us down into ‘syntax’ or the coherent use of words.

Because we are making a picture, the various terms are shown separate from each other, and the diagram even suggests some kind of hierarchy. It is this sort of thing that requires us to distinguish the term ‘image’ from such things as diagrams. Image is what is seen in the mind and not just what is on paper. We call a picture a ‘work of art’ if it transcends this distinction. The picture in the diagram is as false as it is true. Anyone looking at it needs to create the image it means.

Wholeness is the realm of continuous meaning. It stands in contrast with the realm of words in an important way: words are the way we are able to split things apart. This is their virtue as well as their vice! Ultimately, words are just marks on paper or sounds in the air, while wholeness is likethe air itself.
Any picture has elements of syntax, shape, form and image and it is up to us to discriminate them. The nearer we can get to the image, the better for our capacity to see meaning. The shape and form of the diagram easily evokes a resonance with the system of four elements. But, would it be of value to explore the four realms of the diagram in relation to Greek ideas about earth, water, air and fire? Alternatively, we see that the diagram combines threes and fours, or consist of seven (or potentially nine) components: could we then explore it as a duodecad, heptad or even ennead? Again, there is a suggestion that our four realms might correspond to the ‘four worlds’ as discussed by John Bennett in his collection of talks published as Creation.

Systematics can speak of the qualitative significance of number (quantity) because it brings into play the meaning of form. Form is not yet another ‘thing’ to add to ‘quality’ and ‘quantity’ but is a way of seeing how to relate them. But it brings into play a whole new realm of thinking that extends into William Blake’s Imagination. We may invent the term structural images to signify pictorial forms that are intended to represent systematic patterns.

Finally, we return to the original starting point of taking into account both the discrete and the continuous in discerning terms of systems. According to the French mathematician Francoise Chatelin our scope of understanding is bounded by random and discrete elements on the one hand and continuous wholeness on the other. We cannot go below the former or above the latter. By using Bennett’s technique of partition and blending we can derive four main regions. From the top down:

In the diagram, the extremes of ‘randomly discrete’ and ‘holistically continuous’ are marked by bold lines. This picture lends itself to portrayal as an ‘octave’. This is to play with terms and forms.

Do’        the holistic continuity of all
-----
Si           images as in art
La          systems as in systematics
Sol         form as related to ideas of togetherness
Fa          terms as of systems
----         shape of visual representations
Mi          MMs as in LVT
Re         sentences as representing syntax
Do         associations of words without any syntax, shape, etc.
 

DOSTOEVSKY
The Idiot pp. 241-3
He remembered among other things that he always had one minute just before the epileptic fit (if it came on while he was awake), when suddenly in the midst of sadness, spiritual darkness and oppression, there seemed at moments a flash of light in his brain, and with extraordi­nary impetus all his vital forces suddenly began working at their highest tension. The sense of life, the consciousness of self, were multiplied ten times at these moments which passed like a flash of lightning. His mind and his heart were flooded with extraordinary light; all his uneasiness, all his doubts, all his anxieties were relieved at once; they were all merged into a lofty calm, full of serene, harmonious joy and hope. But these moments, these flashes, were only the prelude of that final second (it was never more than a second) with which the fit began. That second was, of course, unendurable. Thinking of that moment later, when he was all right again, he often said to himself that all these gleams and flashes of the highest sensation of life and self-consciousness, and therefore also of the highest form of existence, were nothing but disease, the interruption of the normal condition; and if so, it was not at all the highest form of being, but on the contrary must be reckoned the lowest. And yet he came at last to an extremely paradoxical conclusion. "What if it is disease?" he decided at last. "What does it matter that it is an abnormal intensity, if the result, if the minute of sensation, remembered and analysed after­wards in health, turns out to be the acme of harmony and beauty, and gives a feeling, unknown and undivined till then, of completeness, of proportion, of reconciliation, and of ecstatic devotional merging in the highest synthesis of life?" These vague expressions seemed to him very comprehensible, though too weak. That it really was "beauty and wor­ship," that it really was the "highest synthesis of life" he could not doubt, and could not admit the possibility of doubt. It was not as though he saw abnormal and unreal visions of some sort at that moment, as from hashish, opium, or wine, destroying the reason and distorting the soul. He was quite capable of judging of that when the attack was over. These moments were only an extraordinary quickening of self-consciousness—if the con­dition was to be expressed in one word—and at the same time of the direct sensation of existence in the most intense degree. Since at that second, that is at the very last conscious moment before the fit, he had time to say to himself clearly and consciously, "Yes, for this moment one might give one's whole life!" then without doubt that moment was really worth the whole of life. He did not insist on the dialectical part of his argument, however. Stupefaction, spiritual darkness, idiocy stood before him conspicuously as the consequence of these "higher moments"; seri­ously, of course, he could not have disputed it. There was undoubtedly a mistake in his conclusion—that is, in his estimate of that minute, but the reality of the sensation somewhat perplexed him. What was he to make of that reality? For the very thing had happened; he actually had said to himself at that second, that, for the infinite happiness he had felt in it, that second really might well be worth the whole of life. "At that moment," as he told Rogozhin one day in Moscow at the time when they used to meet there, "at that moment I seem somehow to understand the extraordinary saying that there shall be no more time. Probably," he added, smiling, "this is the very second which was not long enough for the water to be spilt out of Mahomet's pitcher, though the epileptic prophet had time to gaze at all the habitations of Allah."

PLANETARY ARCHETYPES
Richard Tarnas, Cosmos and Psyche: Intimations of a New World View, New York: Viking, 2005)http://www.matthewstelzner.com/Essay_RT_Planets.aspx

The concept of planetary archetypes, in many respects the pivotal con­cept of the emerging astrological paradigm, is complex and must be approached from several directions. Before describing the nature of the association between planets and archetypes, however, we must first address the general concept of archetypes and the remarkable evolution of the archetypal perspective in the history of Western thought.
The earliest form of the archetypal perspective, and in certain respects its deepest ground, is the primordial experience of the great gods and goddesses of the ancient mythic imagination. In this once universal mode of consciousness, memorably embodied at the dawn of Western culture in the Homeric epics and later in classical Greek drama, reality is understood to be pervaded and struc­tured by powerful numinous forces and presences that are rendered to the hu­man imagination as the divinized figures and narratives of ancient myth, often closely associated with the celestial bodies.
     Yet our modern word god, or deity or divinity, does not accurately convey the lived meaning of these primordial powers for the archaic sensibility, a meaning that was sustained and developed in the Platonic understanding of the divine.

N-GRAMS

Structures as Combinations of Systems

In Bennett’s account of systematics, he introduced the concept of structures. Structures were combinations of systems, and Bennett said that these were more realistic than single systems alone.
Bennett’s background included use of such a structure, called the enneagram. This structure exemplifies the properties of N-grams as they have been investigated by Sigurd Anderson and Anthony Blake. Understanding of the properties of N-grams has been slow to spread because of the widely held belief that the enneagram is a unique case.
Even when first introduced (by Gurdjieff circa 1911) it was emphasised that the geometrical figures depicted in the symbol form of the enneagram were based on the number base of 10 and derived by the application of the numbers 3 and 7. Obviously, 10 = 3 + 7 is significant. The two distinct figures within the symbol are derived by dividing 1 by 3, and also by 7. (What may not be easily apparent is that the top point, labelled 9, belongs with the hexadic cyclic figure and is a seventh point; and it also belongs with points 3 and 6 in being derived from 1/3).

The simple generalisation of this case is as follows:

If B is the number base, then B = P + Q, where P and Q are the numbers of simple systems. A simple system is a system that depends on only one number.

B represents the wholeness number of the compound system or structure.

P and Q represent the numbers for different systems. P and Q represent the division of the whole into aspects. (This terminology is adopted from the archetypal case of the sevenfold spectrum as the splitting of white light into its component colours).

N = B – 1 and is the effective or visible number of the structure.

If P and Q are different from each other, we obtain more complex diagrams than if they are equal (as they can be in the cases that B is even). For any given B, there will be a range of possible P and Q and, the larger B, the greater the range.

The Generations that include the Enneagram

Within the totality of possible forms obeying the rule B = P + Q, we can discriminate particular ‘generations’ by applying other rules. Another rule the enneagram obeys is that, if P < Q, then, P2 = B –1 = N. In the case of the enneagram, B = 10, P = 3 and P2 = 9 = B – 1 = N. This means that we can have a sequence of structures defined by the two rules:
                        P2 = B – 1                   and                              B = P + Q
The sequence is generated by taking P = 1, 2, 3, 4, etc. The figures that derive in this way are shown below.

B = 2; P = 1, Q = 1	B = 5; P = 2, Q = 3

B = 10; P = 3, Q = 7	B = 17; P = 4, Q = 13

It is obvious that complexity greatly increases as P increases. The pattern that holds is that the value of P is represented by a ‘static’ figure in each case. In the monagram, P = 1 and B = 2 and so Q = 1 as well. There is only one point. In the tetragram, P = 2 and B = 5, so that Q = 3. The value of P is shown in the vertical line between points 2 and 4. In the enneagram, P = 3, B = 10 and so Q = 7. The value of P is shown in the triangle. In the 16-gram, P = 4, B = 17 and Q = 13. The value of P is shown in the quaternary. And so on. In each case, the form derived from P represents that number and is static.

In the cases of P = 2 and P = 3,  we show the figures derived from Q with arrows. These arrows depict the sequence of the points the figure includes. Calling this figure dynamic is pure convention. In the case of P = 4, the figure for Q (= 13) is composed of two hexads and the top point.

Brief Descriptions of the Simpler N-grams of the Family N = P2

Monagram: this is the archetypal absolute unity. In Gurdjieff's cosmology, it represents His Endlessness (the point at the top of the circle) residing on the Sun Absolute (the whole circle). According to Gurdjieff, this state of affairs was 'threatened' by the progressive diminution of the Sun Absolute, forcing His Endlessness to fill the circle with an inner life.

Tetragram: this comes out as the traditional form of the quaternary. It represents the universe as an ordered process. The original monadic point at the top now plays the role of an ideal pattern or 'form'. Its complement, point 2, at the base of the circle represents the universe as it has been actualised, or 'created'. The two horizontal points, 1 and 3, are between the state of 'creator' and 'creation' and are shown in reciprocal interplay. This interplay is the harbinger of the full hexadic circulation we find in the enneagram. It is the bare form of what Gurdjieff called 'reciprocal maintenance'. The prime 'law' in the TetraGram is the 'law of two', which is the dividing of 'above' and 'below'. The secondary 'law' is the 'law of three'. This last law translates in the figure into an exchange between two points. This signifies the 'blending' of different states. Creation then appears as 'partition and blending'. The third factor of the law of three remains invisible. As Gurdjieff implied, we are 'third force blind' at the level of P = 2.

Enneagram: here we have the diagram that is most familiar. The law of three is shown fully. With Q = 7, we have linear and temporal process, as in the days of the week, which reflect the 'days' of creation. The central idea is that of the creation entering into further creation that modifies the starting point, or the 'creator'. The 'end' can be more than the 'beginning'. The reciprocity of the TetraGram develops into a full-blown circulation. This signifies the great universal process that Gurdjieff called 'trogoautoegocrat' or the way of eating and being eaten.

16-gram: in this more complex figure, we see the twelve-term system of the zodiac. In tradition, the four elements (as signified in the four terms linked in the primary 'law of four') map into the twelve houses. The realisation of further creation is emphasised by the appearance of two hexadic figures. The hexadic figure coming from below suggests a 'counter-creation'. This counter-creation is associated with individualised intelligence and learning and represented by the figure of Beelzebub himself: banished to the solar system, he is able to learn about the modification of the universal laws in practice as well as theory. The top-down hexadic figure then represents higher intelligence, which intelligence is limited to dealing with the general case and not specifics.

25-gram: this complex is based on the primary law of five. In Bennett's cosmology, the five represents the interweaving of the essence-classes and the way in which existence is spiritualised and essence realised. There are three hexadic figures, indicating a yet further capacity for intelligence. These may represent a further stage in our understanding of the three foods necessary for the maintenance of a cosmos. At this stage, every cosmos is a reality in itself and has equal value with any other.

36-gram: based on the law of six. The four-fold structure of the Tetragram appears at a greater depth with four hexadic cycles. We are into a super-ecology. We have the universe as capable of giving rise to alternative versions of itself.

49-gram: based on the law of seven. The emergent picture becomes ever more detailed and transformative. All that has come before now appears as an abstraction. Ideas of the beginning and end of the universe now seem to be too primitive. Ideas of a creator and a creation are too limiting. What emerges is now the source of what it emerged from.

64-gram: based on the law of eight, this may reflect the system of the I Ching. It is the Language of the present moment. It includes the Sufi octad as a symbol of the primary law. The six cycles of six represent all possible actions.

HyperSystematics
These speculations are tentative at best. They serve mainly to suggest that there are different orders of N-term systems. The progression in the value of a, the primary informing law: 1, 2, 3, etc. is reflected into a representational space defined by the secondary reflecting law Q (to use the terminology of Sigurd Anderson). In the traditional systematics of John Bennett, the representational space is by default taken to be of the same order as that of the primary informing law. This means that the basic systems are self-reflective and have to be looked at as operating on themselves.

A further point is this. The constituents of the informing law of three in the enneagram are taken to be of three kinds.

1. As points, in the sense of 'shock points' or 'portals, etc.
2. As lines, in the sense of the logos of 'commands'.
3. As octaves, in the sense of the three interweaving processes.

We might presume such complexity to apply to the other N-grams as well. If we consider just the third category, we have to accept that what we mean by a 'process' may have to be stretched in meaning. There are hints of this already in the enneagram. Though Gurdjieff uses the device of three interlocking octaves, in fact the enneagram only contains one complete octave and the others are truncated. The second 'octave' can be said to have six points (3, 4, 5, 7, 8, 9) and the third, four (6, 7, 8, 9). (If we treat the first as seven, then the second is five and the third, three.)  The second 'octave' then maps into the inner lines and the third into the triangle. The enneagram then appears as a summation of itself, starting from three independent ingredients (circle, inner lines and triangle) and then integrating them in progressive order. The three 'octaves' are then three different things.

We might extend this thinking to other N-grams. In the tetragram, the informing law is 2. Are there, then, two 'processes'? In fact, we see this in the diagram of the tetragram as the vertical and horizontal forms. Reading the diagram in this way, we can think of a reciprocity between 'higher and lower' and another reciprocity between 'left and right'. The latter might be thought of along traditional lines as the interplay between 'male and female'. Another way is to think of Gurdjieff's repeated concept of 'world creation and world maintenance', the former represented in the vertical order and the latter in the horizontal one.
Let us now take into account the reflecting law of 3. We might, for the sake of argument, ascribe the six 'laws' of the triad in the following way:

Vertical:                Involution (top-down)               Creation
                             Evolution (bottom-up)
                             Order (their mutuality)

Horizontal:            Identity (left-right)                    Maintenance
                            Interaction (right-left)
                            Freedom (their mutuality)

Of course, it might be counter-argued that this is to import foreign concepts on an ad hoc basis. We would have to say that each set of three triads ought to be considered as a whole and not distinguished.
Yet another way of considering a two-fold process is provided by John Bennett's scheme of Creation as 'partition and blending' (see chapter 33 in The Dramatic Universe Vol. III). Partition would relate to the vertical and blending to the horizontal.

We can see that, whatever the interpretation, we do not have processes as actions structured in time. In the case of the monagram, where we have only one informing element, the reflection is also into a one-space. We have already said that this is a picture of His Endlessness residing on the Sun Absolute.

In general, we would like to propose, the informing law is time-like and the reflecting law is space-like. In the 16-gram, where we have four informing elements, each of these should be taken as 'time-streams'. Charlotte Bach calls them 'ritual streams' and uses them extensively in his writings on the quaternary. The reflecting law is 13-fold which, strangely, approximates recent speculations about hyperspace.

What we suggest, therefore, is that the scheme of N-grams I first proposed in The Intelligent Enneagram and which has been further developed by Sigurd Anderson, represents another order of systematics. In doing this, it makes a bridge between the classical form of systematics and the more recent explorations of many-term systems (such as the 30-term system of 'team syntegrity' offered by Stafford Beer). The potential of N-grams resides in the fact that they offer a structural taxonomy for complex systems such that any insight from one such system can feed into and assist insight into other such systems. It is just this mutual informing that is of value in the whole conception of systematics. Systematics is not a theory of the world per se but of the ways in which we can think about the world (and alternative worlds!).

William Pensinger defines consciousness as arising from an operation of time on space. Taking this into account, we can conceive of an hypersystematics of consciousness. This is also to open a new chapter in the writing of the Dramatic Universe!

We propose that hypersystematics views number-systems in various band-widths.

1 - 12: the elementary systems
12 - 144: the N-grams with both informing (time-like) and reflecting (space-like) laws
144 - 1728: organic forms
1728 - 20736: genetic forms
and so on. Our purpose here is to emphasise John Bennett's primary contention that
            God is the infinite-term system

KABBALAH

Carlo Suares Cipher of Genesis, http://duversity.org/suares

FIRSTNESS, SECONDNESS, THRIDNESS

"Careful analysis shows that to the three grades of valency of indecomposable concepts correspond three classes of characters or predicates. Firstly come "firstnesses," or positive internal characters of the subject in itself; secondly come "secondnesses," or brute actions of one subject or substance on another, regardless of law or of any third subject; thirdly comes "thirdnesses," or the mental or quasi-mental influence of one subject on another relatively to a third." ('Pragmatism', CP 5.469, 1907)
"... I was long ago (1867) led, after only three or four years' study, to throw all ideas into the three classes of Firstness, of Secondness, and of Thirdness. This sort of notion is as distasteful to me as to anybody; and for years, I endeavoured to pooh-pooh and refute it; but it long ago conquered me completely. Disagreeable as it is to attribute such meaning to numbers, and to a triad above all, it is as true as it is disagreeable. The ideas of Firstness, Secondness, and Thirdness are simple enough. Giving to being the broadest possible sense, to include ideas as well as things, and ideas that we fancy we have just as much as ideas we do have, I should define Firstness, Secondness, and Thirdness thus:
      Firstness is the mode of being of that which is such as it is, positively and without reference to anything else.
      Secondness is the mode of being of that which is such as it is, with respect to a second but regardless of any third.
     Thirdness is the mode of being of that which is such as it is, in bringing a second and third into relation to each other." (A Letter to Lady Welby, CP 8.328, 1904)

http://www.helsinki.fi/science/commens/terms/firstness.html

VOWELS

Taken as a set, the vowels (in English) are a pentad and Joseph Rael in Being and Vibration relates them to the four directions and the centre:
A (aah) Purification. Direction of the East: Mental body
E (eh) Relationship. Direction of the West: Emotional body
I (eee) Awareness. Direction of the West: Physical body
O (oh) Innocence. Direction of the North: Spiritual body
U (uu) Carrying. Center of the medicine wheel

LANGUAGE

Beelzebub Tales to His Grandson  pp. 15-16, G. I. Gurdjieff
Man has in general two kinds of mentation: one kind, mentation by thought, in which words, always possessing a relative sense, are employed; and the other kind, which is proper to all animals as well as to man, which I would call "mentation by form."
The second kind of mentation, that is, "mentation by form," by which, strictly speaking, the exact sense of all writing must be also perceived, and after conscious confrontation with material already possessed, be assimilated, is formed in people in dependence upon the conditions of geographical locality, climate, time, and, in general, upon the whole environment in which the arising of the given man has proceeded and in which his existence has flowed up to manhood.

Creation is a Language, from VALIS, Philip K. Dick
In Summary: thoughts of the brain are experienced by us as arrangements and rearrangements - change - in a physical universe; but in fact it is really information and information processing which we substantialize. We do not merely see its thoughts as objects: how they become linked to one another.
But we cannot read the patterns of arrangement; we cannot extract the information from it - i.e., it as information, which is what it is. The linking and relinking of objects by the Brain is actually a language, but not a language like ours (since it is addressing itself and not someone or something outside itself).
We should be able to hear this information, or rather narrative, as a neutral voice inside us. But something has gone wrong. All creation is a language and nothing but a language, which for some inexplicable reason we can't read outside and can't hear inside. So I say, we have become idiots. Something has happened to our intelligence. My reasoning is this: arrangements of parts of the Brain is a language. We are parts of the Brain; therefore we are language. Why, then do we not know this? We do not even know what we are, let alone what our outer reality is of which we are parts. The origin of the word "idiot" is the word "private." Each of us has become private, and no longer shares the common thought of the Brain, except at a subliminal level. Thus our real life and purpose are conducted below our threshold of consciousness.
From loss and grief the Mind has become deranged. Therefore we, as parts of the universe, the Brain, are partly deranged.
Out of itself the Brain has constructed a physician to heal it. The subform of the Macro-Brain is not deranged; it moves through the Brain, as a phagocyte moves through the cardiovascular system of an animal, healing the derangement of the Brain in section after section. We know of its arrival here; we know it as Asklepios for the Greeks and as the Essenes for the Jews; as the Terapeutae for the Egyptians; as Jesus for the Christians.

STRUCTRUAL IMAGES

Example: The Doctrine of the Golden Mercuric Sulphur - The Fire of the Unitive Path

http://duversity.org/elixir/