PART FOUR – SYSTEMATICS AS A GAME OF GAMES
They [the two sides of our brain] process information in radically different ways. This difference is most easily explained by a look at two words often thought to be a synonymous: order and structure.
Order, on the one hand, comes from the Latin ordo, ordini. It means "in a straight row," "in a regular series." Order implies linear, rule-governed activity. Order is imposed from without. Structure, on the other hand, comes from the Latin struere. It means "to heap together." Structure emerges from within. [Gabriele Rico http://volcano.und.edu/vwdocs/msh/llc/is/cm.html]

PREAMBLE

Bennett called systematics a ‘discipline of thinking’. This means that it has rules and principles. It does not tell people what to think but how to think. It has been often taken to consist of a set of ‘templates’ – the ideas of the systems – that have been collected and made available for people to apply to various situations. In this respect, it would be like TRIZ. As TRIZ correlates inventive principles with types of contradiction, systematics would correlate systems with types of situation.  However, there is a deeper aspect, in which systematics can be seen in terms of how systems are arrived at in the first place. We call this a ‘game’ because there are various components brought together according to rules and different people can interact with each other in making moves.

We can understand the game in terms of various levels of play, or sub-games. Every game consists of sub-games. The first four of these lead us into meaning games and the next four take us through into societies. The eight sub-games are dealt with here together, though they were presented in two parts at the Gathering.

It is important to bear in mind that although we take pains here to spell out what is meant by the various games, these conceptual descriptions and explanations are about what people can do instinctively without any apparent theory. In some ways, there is a strong dyad between what we can explain and what we can do. The explanations may be more complex than the experience of playing the games.

SET

Bennett’s definition of a system was: a set of independent but mutually relevant terms.

We first look at systematics in terms of sets and put to one side the idea of mutual relevance. A set is a ‘many thought of as a one’. It can be something completely arbitrary such as ‘the leaves blowing in the street this afternoon’ but it does entail that we can determine what belongs in the set and count how many it contains. This number is called the cardinal number. Different sets can be equated in terms of their cardinal number. It then does not matter whether they contain angels or motor cars, colours or digits, etc. because it is in this particular way that they are equivalent. For every member of one set, there will be a member of the other sets. It does not matter which member of one set is matched with that of another as long as there are the same number of them.

Systematics raises the possibility that different sets of the same cardinal number can have more in common than their cardinality. This can be taken to foolish extremes in certain kinds of obsession with numerology. For us, it is a valuable starting point. And people can play the game of putting two sets of the same cardinal number but divergent content side by side and asking what more might be held in common in them.

Though we put aside the idea of mutual relevance between the terms of a systemic set, it comes into play here in terms of mutual relevance between sets. What this relevance means is not to be spelled out beforehand. It is only a possibility. For example, we discover in China the system of 8 trigrams in the divination I Ching and also the teaching in Buddhism of the Noble Eightfold Path. Is this just a coincidence? Is there, for example, some meaning of eight that has special value in Chinese culture? If there is, does this have any significance for us? We know that ancient cultures often attributed meanings to numbers and in the case of the Chinese, 8 is associated with fortune. This can be dismissed a superstition, but is there some prior reason for it?

On the one hand, the association of one set with another is entirely arbitrary and signifies nothing. On the other, it has potential meaning. This meaning could lead us back to some earlier work done by creative groups as well as to current cultural accidents. There are some similarities here with Jung and Pauli’s concept of synchronicity. Two seemingly unrelated things are brought together in a way that implies that they are mutually relevant. Jung took this primarily in the sense of a coincidence between psychic and physical events, but it can be extended to our case of two sets that appear together because they are of the same number.
One of the forms of such a comparison is akin to triangulation, as when two points of reference are used to take bearings on a third. In taking two disparate sets, the base-line is due to their differences of content and the third point might be a ‘generic’ set, with the character then of a ‘system’. In other words, we could develop the idea of a system of N terms by considering together various disparate sets of N members. It is the bare idea of the system because it will not show us how it is composed, which concerns the next game.

When Bennett worked on the tetrad, he was much influenced by Aristotle’s Four Causes, as well as the Greek system of four elements. The two systems are reflected in much of his work, but they came together in his proposal of the systemic attribute of the tetrad as ‘activity’.

In various traditions, there are archetypal meanings associated with the integers. These are like ‘angels’ as ‘messages’ from the realm of higher intelligence. On a lower level there is magic, which is based on correspondences, folklore and superstition. For example, in China 3 is associated with ‘unanimity’ and 8 with good fortune.

SEQUENCE

Beside cardinal number, there is ordinal number. This is when we count as 1st, 2nd, 3rd and so on. In other words, the members of the set have a sequence. When we come to the game of sequence, we are able to match members of different sets with each other in the specific sense of an order. For example, in dealing with triadic systems, Bennett spoke of the 1st, 2nd and 3rd type of ‘impulse’ (simply as 1, 2, 3). Attention is on the character of the terms of the system and not just the attribute of the system as a whole. We can then look at triads such as the Chinese Heaven-Earth-Man in a comparative order, such that Heaven is matched with 1, Earth with 2 and Man with 3. Or the Christian Trinity with Father-1, Son-2 and Holy Ghost -3.

It is only now in this game that terms appear with distinctive characteristics in their own right. In the previous game, their sole property was as a member of the set and, in effect, one term was as good as another. Using analogy, we can think of the game of Set as in black and white while Sequence introduces colour. 

In ordering the members of a set as possible terms of a system we have to have some reason for the order we choose and one person’s preferred order may not be the same as another’s. So there has to be some consciousness of the reason for the order we choose in one set so that it is possible to look for the same kind of order in another set. Concern with sequence brings us into games of reasons.

One of the simplest but well known contrasts of order is that between clockwise and anti-clockwise directions for elements in a circle. In the system of the I Ching there is a clockwise order, which concerns how things unfold and an anti-clockwise order which relates to how this unfoldment might be predicted.

Bennett spent considerable time considering the meaning of different orders of the three terms of the generic triad, for which six alternatives are possible (in the case of the dyad, there are only two; in that of the tetrad there are 24). He developed the six different orders as six distinct versions or ‘laws’ of the triad.

There can be all kinds of reasons for various orders. One common kind is associated with sequence in a strict sense, as of operation in time. Another is associated with a hierarchy such as depicting different levels. The latter appears frequently in Bennett’s work. An example is where he took the transfinite numbers of Cantor and correlated them with the four worlds of Sufism (see Creation) with a hint of the Four Elements. The transfinite numbers (in fact, he started with everything that exists as a very large number N and then went on aleph-zero, aleph-one and aleph-two) have an intrinsic mathematical order. The worlds are in an order in terms of degrees of freedom. The ancient four elements are ordered in terms of degrees of subtlety. In other contexts, Bennett spoke of four mental energies which were strictly in the hierarchical order automatic-sensitive-conscious-creative. But, when it came to correlating this scheme with others there was often a sense of a transposition up or down. That is to say, we had the same relation of order in the two cases but they did not match in terms of levels and had to be adjusted to do so. This approach is evident in his many examples of twelve term systems, which were often treated as three sets of fours. Each set of four could be correlated with the others but were significantly different in the context of the total system in mind. Thus, there were four mechanical energies, four vital energies and four cosmic energies. We shall speak more of this later when discussing the game of SetN .
Discussion of the reasons for any order are important not only for being able to compare the terms of different versions of a system (or different sets) but also for entering into an understanding of the type or types of mutual relevance that obtain between the terms. Sequencing cannot though be a complete analysis of mutual relevance because it only deals with terms one after another. It is like a proto-version of mutual relevance.

In producing different orders we call upon our understanding, which embraces both some sense of physical laws and also of ‘inner’ meaning. There might also be very distinct orders for the same set of terms that reveal some underlying intention. A paradigm for this is the sequence of seven days of the week, which days have names that relate them to the seven terms of the solar system. Most commentators consider the orders of the days to be arbitrary, but if one takes the various ‘planets’ around a circle in terms of their angular velocity (as shown here) and then draws in the cyclic figure connecting them the inner sequence is the correct one. Here we see around the outside a visible order and inside the circle an invisible one (but we can ask, which really is which). We can infer the inner sequence but may find ourselves unable to understand why it was chosen. Perhaps some ‘message’ was intended to be conveyed by it, one that would only become apparent to those able to work it out.

This must of course remind us of the long-standing influence of encryption on the use of number. Cryptology translates one sequence of elements into another such that one has to have the code to decipher the message. This widely known principle was elaborated by Gurdjieff in his idea of legonomism. He proposed that ancient creative groups encoded their insights into works of art that were entertaining enough to be transmitted over generations (even millennia) to future wise people who could work out how to read them. According to his theory, the ancients altered the sequence so that what appeared would not be what was expected. By noticing this, intelligent people could detect that there was some hidden message they could decipher.

If we have two contrasting orders then one of them can appear as the ‘exoteric’ or outer meaning and one as the ‘esoteric’ or more hidden meaning. This is very evident in the case of the enneagram, which has one order of terms around the outer circle but two inner orders inside it.

Matching two sets in order might reveal interesting correlations. A possible example is that between the series of integers and the letters of the alphabet. There we discovered a curious feature that suggests a similarity of vowels to the prime numbers (see p.23). The method of sequencing – and matching up terms between different sets – is an heuristic device.

The game of sequence enables us to think in terms of ‘common ratios’, the latter word having original associations with reason. If we have a sequence 1st, 2nd, 3rd and 4th in two sets of four then we can look at how well the ratios between the terms match up. Is the step from 1st to 3rd in the one of the same kind as the equivalent step in the other?
                      

In this comparison, double arrows are cross-translations and single ones are steps. The comparison can help us understand both versions better. What has to be noted however is that our understanding of any of the terms might be very limited. In that respect, the comparative process can be imagined as encompassing a range of exemplars that includes very familiar as well as more obscure content. This was practised in Old Norse culture as what are known as Kenning games (see http://kennexions.ludism.org/old/kenning.html):

Kennings are an old Norse poetic device based on analogy. They're similar to Homeric epithets. Where the Greeks might say "the wine-dark sea" in their epic poetry, the Norse would say "whale road." This of course comes from the analogy "sea is to whale as road is to horse" or something like it. To use the standard shorthand, this becomes

sea : whale :: road : horse

You can also diagram it as

   sea      road

  ----- :: ------

  whale     horse

The key to the Kenning Game is realising that such an analogy provides four kennings possible (or at least permissible). In this case, we have

  sea   = whale road

  whale = sea horse

  road  = horse sea

  horse = road whale

Some of these seem a little strange, but we might make sense of them by positing that "road whale" for "horse" is the product of a culture of aquatic intelligent beings that ride whales the way we ride horses. Some kennings do come out strangely, but one thing we are after in art is the novel viewpoint.

The order of terms enables a correlation to be made between different expressions of a system. For example, Martin Lings in his book Archetype and Symbolism has a chapter on the triad of the primary colours red, yellow and blue that surveys correlations in Christian, Islamic and Hindu mysticism. Also, magical practices are founded on correspondences, such that a particular colour, flower, scent, image and so on are brought together to concentrate a particular influence. When Bennett spoke of the qualitative significance of number, he said that this was magic. Number corresponds to colour, scent, image, etc in schemes of magic. We may see this as purely reflective, a way of encoding information about a situation, but magicians claim that the practice can actually change the situation.

The relevance of systematics to magic and also to divination has hardly been explored, since systematics was largely taken up as a way of modelling situations such as in management and other ‘rational’ pursuits. But it is deeply rooted in instinctive intelligence and the reality that ‘mental’ forms and ‘physical’ processes are not separate phenomena, or that software and hardware are a coupled system.

We cannot leave the game of Sequence without taking a look at the basic symbolism of the series of integers used by Bennett. One of the questions we have concerns the intervals between them. We may assume that they proceed just by ‘adding 1’; but this arithmetical progression is only one out of several possibilities. There is the geometric progression in which intervals are defined not by addition but by ratios. This is what we use in the case of the musical scale. The note sol for example is ‘halfway’ in the octave from do to do’ in terms of frequencies (if do is given the number 1 and do’ the number 2 – doubling of frequency – then sol has the value 3/2 and is midway between do and do’). Intervals measured as ratios are different from those measured by addition. Another alternative progression is logarithmic. This is by no means just a human invention for aiding calculations; it manifests in the way we hear intensity of sounds and has recently been found in the way that pigeons can distinguish time-intervals. In experiments with pigeons, it was found that the boundary or transition point between 1 and 16 seconds was at 4 seconds and not at the expected 9 seconds. Bennett drew attention to logarithmic time in volume four of The Dramatic Universe.

The idea of intervals is present in Kennings. Just as we can equate intervals in music at very different regions of the audible frequencies, so in Kennings we can equate ratios of ideas. The span of an octave is crucial in defining a given whole that can be divided into several notes in various ways. In his book The Unanswered Question, Bernstein shows that various musical scales have evolved associated with different number divisions. The primary nature of the octave is deeply connected with the fact that men’s and women’s voices tend to be an octave apart. It is something primordial and can also be found in some shamanistic practice where the shaman ‘dialogues’ with himself in two voices, one high and the other low. In the next degree of scale, the ‘halfway’ note (sol in our usual scale) emerges and, according to Bernstein, this is the scale commonly used in chanting. The next common scale is pentatonic and used in folk music from all over the world. So we progress to ever-more complex divisions, which include our western diatonic scales (the major diatonic scale is the one used by Gurdjieff. Meanwhile, there have also been the six-fold scale of Debussy and the twelve-note scale of Schoenberg (based on the ‘chromatic’ scale of notes with intervals of a semi-tone).

In her writings on the Greeks in relation to Christianity, Simone Weil has dwelt on the significance of finding the ‘mediating’ or middle point between two contrary ideas. This was, for her, a powerful metaphor for the nature of Christ as mediating between man and God; but also, in general, as the idea of mediation.

Concern with intervals, common ratios and mediation is a concern for what is ‘between’ the numbers, or their mutuality. This is far richer in meaning than what can be contained within the numbers (or whatever the primary elements are) separately. In the game of Sequence, therefore, we begin to touch upon the inbetween. This is increasingly developed through the games.

SYMMETRY

This game, as we shall see, includes such operations as ‘splitting’ and ‘chunking’. It has to do with how we group sub-systems of systems into patterns and brings in a visual component. We use the word symmetry to emphasise how much depends on an aesthetic sense of order, which distinguishes it from the sense of order that we call sequence. Gurdjieff used the words ‘Form and Sequence’ to discuss how we can best learn.

A big factor is how we ‘chunk’ a number of elements into one thing, or one meaning. We are always looking for ways of doing this because there are limits on our mental capacity to hold several elements together at once. This has been identified with the phrase ‘seven plus or minus two’, which numbers (5-9) represent our maximum capacity. It is interesting that even these belong to the second level of our table of systems. For the most part, we can only grasp 1-4 items at a time. If we have lots of elements then we will only attend to some of them at any one time. This corresponds to the ‘figure-ground’ theory of Gestalt psychology. In LVT, we group the many MMs into a smaller number of ‘clusters’ and learn how to attend to cluster meanings to reduce the complexity.

Right at the beginning of the discussion, it is important to draw attention to the necessary vagueness of what it means to ‘see one meaning’. We have a sense of this but it is barely conscious.  We know that people such as master chess players chunk sets of pieces and moves and places into one meaning but even they may find it difficult to explain what it is they do or how it works. But this is also problematic when we take the simple case of a statement that consists of many words, but is read as one thing.

When we have more than four elements, we probably start to look for ways of splitting the whole set into chunks that makes it easier for us to hold all the elements in mind. The easiest way is to think of a square made of four of the elements and the fifth as in its centre. A more aesthetic way is to draw a pentagram which arranges the terms symmetrically. When we come to six elements, we are tempted to put them into two triangles: on one level we have just two symmetrical elements; on another we have two sets of three. In the case of the heptad, we can use two squares though in this case one term is common to both. Again, on one level we have just two elements but on another we have two sets of four and, finally, a special case of one element.

Splitting the system mathematically corresponds to producing partitions. For example: 4 = 3 +1 = 2 + 2 and so on. However, a visual component assumes importance also, because we can use shapes to help chunk items together. These shapes – such as triangles, squares, circles and so on – support a sense of wholeness. Symmetries help us to ‘compress data’ into simple forms. In making visual patterns with symmetries we can evoke new aspects of the meaning of the system.  These are aspects that are difficult to spell out in words! The shapes are symbolic of wholeness and relate to the continuum in which systems appear.

When we represent an Octad as two squares, this is certainly an efficient way of holding the 8 elements together in our minds. But it implies that we have a reason for this partition and symmetry and presume it can tell us something. Of course, we might be led to make the picture simply because of aesthetic feeling but the choices we make in grouping the terms can be significant. It can lead us to ask questions about the two sets of terms: why are they partitioned in this way and are there correspondences between terms belonging to the two sets? (see my essay on the Octad at http://www.systematics.org/journal/misc/OCTAD.pdf)

The game of symmetry brings out sub-systems. We have referred to different levels. In the Octad as two squares, we have one set which has two members and two sets which have four members. In the enneagram symbol we have two sub-systems, one of which has six and the other three members (actually the former has seven members and we have to take into account the special term – at the top – which is common to both). For the Decad we have used the triadic form as shown here for maximum symmetry. On one level there are four sets of 1, 2, 3 and 4 members respectively ; on another there are two sets of six and three members, but there are also many sub-sets of three members (see the smaller triangles). All these various views of sub-systems can provide us with a way of looking in-to the meaning of the system in terms of mutual relevance of terms. Every visual sub-set carries a ‘chunk’ of meaning.

Appeal to symmetry raises the question of asymmetry. When a more asymmetrical form is chosen for our representation of a system, this too is significant. Why would we break symmetry? Here we can reflect that symmetry evokes in us a sense of perfect balance, or an ideal state. The breaking of symmetry can then lead us to think of the ‘imperfect’ way in which systems may be realised in real life. The enneagram symbol embodies asymmetry as well as symmetry. There is a reason for this and it has to do with Gurdjieff’s idea of ‘shocks’ having to come in to keep the system going in a right way and not fall apart, which Bennett interpreted as a set of corrections to overcome hazard.

The breaking of symmetry is in general an indication of the way in which the Ideality of the system fails to be realized in actual circumstances but it also suggests ways in which we might correct for this.

The breaking of symmetry can also carry a story or narrative. This may be the case in the Tree of Life symbol of ten terms used in Kabbalah, as it was developed in (probably) the 12th century. It is both symmetrical and asymmetrical. It is full of sub-systems. Some connections are filled in but not others. These are, incidentally, correlated with the Hebrew letters and the design may have been influenced by the intention to find such correlations. In a word, it is at least as complex as the enneagram and the two have often been compared. In front of this diagram we have to ask: why are the terms arranged like this? Here is at least a story to do with its history that can be looked into (even though it is difficult to find documented explanations from the period).
Symmetry and its adjunct asymmetry give rise to symbols and we could equally well have chosen the word ‘Symbol’ for this game. Our general argument is that symbols show a pattern for a system that can sometimes suggest ways in which we have to put something into them to make them work. This ‘putting in’ symbolises putting work into actual situations to improve their Ideality.

One of the greatest symbols of our culture is that of the Crucifixion. This can be taken as an act of suffering needed to heal the divorce of eternity (vertical) and time (horizontal), or the Ideal and the Actual (reminiscent of Bennett’s version of the tetrad as: Actual, Practical, Theoretical and Ideal).
There is an obvious tendency for our perceptions to organise around symmetries but this may reflect natural phenomena:

It was Wolfgang Kohler who, impressed by the gestalt law of simple structure in psychology, surveyed corresponding phenomena in the physical sciences in his book on the "physical gestalten," a naturphilosophische investigation published in 1920. In a later paper he noted:
In physics we have a simple rule about the nature of equilibria, a rule which was independently established by three physicists: E. Mach, P. Curie, and W. Voigt. They observed that in a state of equilibrium, processes-or materials-tend to assume the most even and regular distributions of which they are capable under the given conditions. (R. Arnheim at http://acnet.pratt.edu/~arch543p/readings/Arnheim.html#1.1)

SET N

In speaking about the game of symmetry we had to introduce the idea of sub-systems as appearing within the system. In the next game, we include the idea of any system as operating in the context of ‘all’ the systems. The ‘all’ will not be infinite. In the case of the Jungians, we concluded that their effective set of sets was four-fold. In the case of Bennett, it at least eight-fold. In the case of Peirce, only three-fold. The effective set of sets defines the repertoire of systems involved in the interpretation of any one of them.
 

The arrangement of systems shown above is a form of symmetry that groups the systems vertically and horizontally. In this guise, SET N is similar to the previous game. However, it has further implications. It says that work done in any one system will be influenced by what has been done in other systems. This leads, for example, to look for consistent ways of interpreting the systems.

The arrangement leads us to look at the series of systems in different ‘periods’ and we note that 5 – 8 is a new cycle of the thinking that goes into 1-4. Bennett made original contributions to our understanding of pentad to octad. His pentad is a unique interpretation that combined insights from the natural sciences with insights into Gurdjieff’s Diagram of Everything Living. The natural sciences gave him a sense of transflux equilibrium, while Gurdjieff’s ideas gave him the idea of essence classes. Putting these two together was a master stroke. (Incidentally, it was closely paralleled by his relating the energies of natural science to psychological energies and also to the idea of the ‘divine operations’ of the Eastern Church.) The systems 5-8 evoke the sense of living systems. They appear to be more individualised and unique. And they seem to require a more specialised creation.

There is a story about how Bennett came to the Octad. When he met Idries Shah, the latter was claiming to represent the source of Gurdjieff’s teaching and, naturally enough, Bennett asked him about the enneagram. Shah dismissed this to one side, claiming that the Octad was of greater importance in Sufi tradition.  In typical style, Bennett went away and thought about this and consequently produced his complex interpretation of the eight-term system. His explanations are nowhere to be found in any Sufi document!

As far as the hexad is concerned, Bennett added to it greatly by associating it with events and with the present moment. He also added value to the heptad by seeking to integrate its two aspects – akin to a spectrum on the one hand and a sequence of steps on the other – discernible in Gurdjieff’s writings. Both these systems became embodiments of diverse ideas brought together in new ways. It is also important to note that particularly in his treatment of the heptad, he was addressing different ways in which we can interpret any system; since every system can be seen in both ways as we touched upon in talking about the game of sequence.

The vertical resonances suggested in the table above are suggestive. In Bennett’s writings they appear in manifold ways. In one, for example, the principles of the triad are developed to give a heptad of seven worlds of will. In another, he spoke of the pentad as yielding the ‘name’ or essential character of the monad.  The duality of fact and value is replaced by the coalescence of the hexad as the present moment.  The octad is seen as two tetrads. And so on.

It is interesting to reflect that the period of systems 5-8 can be seen as disturbances of the previous ones, 1-4. A metaphor for the pentad is that of the grit in an oyster from which a pearl can grow. They raise new questions and show the previous systems to have been incomplete. The third period contains the systems in which intervention on our part is needed.

We must remember the thesis that different people have different temperaments or capacity and hence that the N in SET N is different for different people. This has been discussed by Arnold Mindell, a Process psychologist. Even though the higher term systems for any N may be directly addressed, nevertheless they have an influence. The N set is like a framework. At the lowest level, if it is possible for someone to make a move into a higher term system this is different from not being able to do so. In the latter case, it leads to efforts to ‘recycle’ lower term systems to accommodate complexities rather than think in different parameters. 

The grid form used in the table is significant. It is not an obvious symbol. It is a reversion to the simplicity of sets and counting. We can give two examples which show how this form can influence thinking about systems.

The first concerns the nine-term system.  Instead of using the enneagram symbol we can show the set of numbers (which an be taken not just as terms in a sequence but as also representative of the systems the numbers represent) as follows:

Looking horizontally, we see a division between three sets. These divisions are ‘located’ in correspondence with the idea of three octaves or processes, which Gurdjieff spoke about in relation to the enneagram. 

Looking vertically, there are three sets of three also. In Richard Knowles’ book The Leadership Dance he describes three types of leadership in terms of these three sets. Strategic leadership’ corresponds to equates to 1-4-7, ‘control and command’ to 2-5-8 and ‘leadership in self-organization’ to 3-6-9.
Thus, the simple grid contains much of the significant information of the enneagram symbol. The next example is taken from Bennett’s The Dramatic Universe Vol. I where he presents twelve levels of existence. He groups these into three sets of four: the mechanical, the living and the cosmic. The simple table of the levels shown here suggests divisions or boundaries between the sets and, in fact, he proposed that there were critical transition regions between them. The first between 1-4 and 5-8 he called ‘active surface’ and for the second between 4-8 and 9-12 he used the term ‘biosphere’. In an analogous way, the region between 9-12 and 1-4 might be called the ‘creation’ (boundary condition of ‘our’ universe).

These examples illustrate the principles of periods and boundaries. The periodic principle says that a form can recur at various levels or depths. The boundary principle is that transitions from one period to another have interfaces.

Bennett himself had a similar notion to SET N  in his concept of construction: “A construction can be understood as a situation where the mutual relevance of systems is significant.” (DU Vol III p. 230). According to Bennett, systems are the most abstract representations of structure we can have. Structures can be seen in terms of combinations of systems, which then include such things as the enneagram (and N-grams in general).

SET N jumps from previous games into three levels of meaning:

and all three are mutually involved with each other.

The SET N format of a matrix of all the systems of a given range of systems is only the general case of structure and can be considered more properly as framework. In any specific case, some systems will be ‘stronger’ or ‘more relevant to the purpose’ than others. As we saw, structure enters in when there is a breaking of symmetry. As specificity becomes important – the ‘more than’ general – structure becomes more complex but, at the same time, its elements become more significant. 

SIGNIFICANCE

In the previous games, the emphasis was on the terms as determined by the systems, in that the number of the system was primary. In the game of Significance, the relation is inverted and it is the content of the terms that becomes primary. This can also be thought of as a ‘bottom-up’ approach because it starts from raw material and builds into structures.

The game begins with thinking about what other complex wholes are relevant to a given one. It is like considering the family or kin of the given whole. Later on, we will be taking systematics as the complex whole in question and then looking for kindred disciplines or ways. If we were to take a critical experience in our lives as a start, then we would look for other experiences. Whatever the nature of the given complex, we look for other things that are similar to it in kind. We also look for significant items that have their own power and depth.

This leads us to the technical term ‘molecule of meaning’ (MM). This term was chosen to describe significant elements that have strong meaning in their own right, without any reference to any system.  MMs are not ‘terms’ because we do not begin with any system. MMs lead us to structures, while systems lead us to terms. When we assemble MMs, we are paralleling the making of a monad. The two are complimentary in many ways. In making a monad we ‘flesh out’ what we are thinking about, finding what it contains; while in assembling MMs, we explore what it relates to (the ‘family’ to which it belongs). The idea of a ‘molecule’ of meaning is that it is a whole world or monad in its own right and could, in principle, be transformed into systems of its own.

When there is a group of people, each will have their own repertoire of MMs. Ones chosen by one person may be unfamiliar to the others. The experience and knowledge of an MM may widely differ amongst the members of the group. It follows that, in the assembly of MMs there can be much discussion, explanation and illustration. For convenience, MMs are usually stated briefly and often consist of just a name. Again, one person may see a strong mutual relevance between a given MM and the initial one (which in our game was systematics) while others may not.

In both basic systematics based on terms and structural thinking based on MMs we look for ways of understanding something in terms of what it is related to. In basic systematics (the first three or four games) we look for internal relations while in structural thinking (the next three games) we look for external relations. However, the distinction between internal and external should not be rigid; the one informs the other. They both share in the property of finding understanding through mutuality.
Sense of mutual relevance is the underlying source of method. We do not translate one thing into a composition of other things but look for their mutual relevance. The principal game is to bring apparently disparate elements together to enable a new meaning that can lead us into understanding. It is an art.

We can consider the assembly of MMs to make a ‘mosaic’ or ‘fabric’ and use other such metaphors, but only if we are also able to change the relative positioning of the MMs to each other. The process is like weaving rather than cutting up material. If we imagine the MMs in the state of mutual relevance, then certain possibilities emerge:

1. The MMs correct for the defects and biases of each other. This is one of the major potentials of the dialogue process in general.
2. Their juxtaposition draws attention to mutual relevance, rather as the interactions between particles can be seen as arising out of a ‘field’. The meaning of mutual relevance is beyond the meaning of the separate MMs. It is more subtle; perhaps more like a feeling than a concept.
3. Each MM leads into and out of the others. There is a movement and/or ‘energy’ of this movement. It is this ‘energy’ that makes possible new ways of thinking.

A good example of this game from the realm of group analysis is the social dreaming matrix as developed by Gordon Lawrence. In this process, the MMs are dreams reported by members of the matrix. These dreams are not taken as material peculiar to the persons who had them but as source material for the thinking of the group as a whole. By association and amplification, the mutual relevance of the dreams is brought out and often leads to new thinking about a situation relevant to them all. It is clear that the MMs in this example are often rich and complex. And it is also clear that it is by looking into their mutual relevance that something new can emerge rather than by taking them one by one.
The first of the three possibilities outlined above has a special case of some significance. This is when nearly all of the MMs cancel each other out leaving only a few or even just one. The overwhelming tendency is to add and accumulate and to operate by cancellation is rare. Yet its importance is evident in any investigation which is looking for specific answers.

SERENDIPITY

Serendipity is the ’happy accident’ of two or more things coming together that gives a new insight. In this game, we have to provide some means for MMs to be brought into conjunction. This is the ‘game board’ for which the MMs are ‘pieces’ to be positioned and moved. The structure of the game board can be arbitrary or involve just a few elementary considerations. The two main features of the game board are:

1. The number of positions. If the set of MMs is 100 and the number of places only 10, this is too big a limitation. In broad terms, the number of places should be at least half the number of MMs. There will be a dynamic relation between the subset of MMs that are brought into play and the remaining subset of those that are not.
2. The shape of the game board is important. This relates to the number of players. A triangular board suits three players, while a square one suits four players. Rectangular grids are useful for many players.

Placing an MM on a grid makes a representation of its mutual relevance with other MMs already in position. Replacing one MM with another signifies that it is regarded as less mutually relevant than its replacement. There are two main considerations. First of all, whether an MM should be in the game or not. Secondly, where it should be in relations to the others. 

Representing – and evoking – mutual relevance by relative positioning is called toponomics (topos – place, nomos – rule). We may want to have some kind of grammar to tell us what to do; but it is neither possible nor desirable to have such. We do not have to know what the rules are in advance of actual play. Instead, we discover them as we go; and may never be able to spell them out. This is not mysterious because the playing of the game entails a dialogue that reflects in consciousness the underlying unconscious process of making and applying rules. It is a self-organising process. Indeed, playing such games is a good way of experiencing and reflecting on self-organisation in human systems.
There are some explicit rules to define what an allowable move in the game is. These will be discussed when we describe the particular game we played at the Gathering. But it is important to bear in mind that the few simple and explicit rules leave completely open the higher level rules that come into operation when we ascribe meaning to the relative positions of MMs on the game board.  Part of this is easy to grasp in general terms. The diagram shows MM A and four other MM positions. We can explore by experiment or implication various types of meaning according to whether an MM is above, below, left or right relative to A.

Then we might look further to the possible meaning of the placement of MMs such as B and C in relation to A. These meanings are more ‘triadic’ in that they involve yet other MMs and not just A.

In discussing the game of Sequence, we introduced the significance of between, ratio and interval. The game of Serendipity brings these to the fore. Our placement of MMs in relation to each other reflects (implicitly) our evaluation of relative meaning. We look for what can be placed between A and B (see diagram above) so that it is not biased towards either but is ‘in the middle’ of them. At the same time, this placement alters our perception of the meaning of A and B.

Thus it is that, from relatively primitive feelings for the representation of the mutual relevance of two MMs, we can build into an understanding of the mutual relevance of three or more MMs.
Also, by utilising a two-dimensional representational space, we introduce a multiplicity of directions of mediation, which radically distinguishes what we do here from the game of Sequence, which is played in only a one-dimensional space. In principle, we could at least make use of three-dimensional representational space, perhaps as shown below, where there are 27 ‘places’. In practice, however, this would be difficult to handle: how would we place MMs; how could we read them, and (as we shall see in the next game) how could we change the structure as the game developed?

By being forced to place MMs in a restricted way (on a game board increasingly occupied by MMs) we can ‘accidentally’ produce conjunctions that suddenly yield new insights.  There is a quasi-sequence:

            Primitive mutualities – Complex mutualities – Serendipities

A serendipity will tend to change the sense of the total configuration, because it ‘concentrates the energy of the game’ in a new way. In other words, insights come into play which act as organising influences on the structure and dynamics of the game play. In our description of the game we played on understanding systematics we will point out some of the serendipities that emerged.

SYNERGY

Though we begin with a certain game board such as a grid, this is not binding or fixed. The games we are looking at now are, as we said, of a ‘bottom-up’ kind. This means that the content drives the form. The shape of the game board can change as the game develops.

A structure can emerge out of the interplay of various organising complexes arising from serendipity. These complexes are centred in regions of the game space which have begun to self-organise in their own right. One version of this state of affairs is that the different self-organising regions represent different systems. The Synergic game is then to combine the various systems into a structure in which each its place. This can be seen as a recurrence of the game SetN . But, instead of having a simple grid to ’contain’ the systems, we have the systems working together (= synergy) to ‘agree’ on an integrative structure.

What is now foremost in the players’ mind is the emergent shape of the game, or structure. We may have started with an arbitrary game board, but this now evolves into a new design that expresses the meaning of the whole.

The new design will have an archetypal character of its own, perhaps an organic form such as that of a tree or the human body. It will embody the group’s realisation of itself. In a strong sense, it will mirror what the people have brought to the game, though now in a relatively conscious form.

SOCIETY

If we ordered the various games according to Gurdjieff’s idea of the octave we would have:

Do’       Society                               
---
Si         Synergy                              
La        Serendipity                       
Sol       Significance                      
Fa        SET N                                                  
---
Mi       SYMMETRY                           
Re       SEQUENCE                          
Do       SET                                        

where the marks between Si and Do’ and between Mi and Fa signify critical transitions or changes in character.  The transition to the eighth game could then be a major one. We said that the previous games were a new beginning in being (a) based on MMs and not terms, and (b) concerning structures rather than systems.  The new kind of step restores us to the beginning (Set) but in a radically new way, which nevertheless relates to the previous game in its possibility of producing a ‘self-realisation’ of the group.
Bennett himself proposed that societies would come after structures which, in their turn, came after systems. The movement is towards greater concreteness. This word cannot be equated with materiality. To put it in a terse and enigmatic way, concreteness has more to do with will than with matter. For this reason, we propose that the game of Society concerns individuals. In the first games, we had terms, in the next series MMs, but now we have individuals. A fourth level of events is also included which will be explained in the next section.

 The togetherness of individuals is a communion (see Blake-Blake theory below). It is formed by the agreement of the individuals to be together and precedes any process, interaction or negotiation. An important aspect of this theory (from theoria – to see, related to ‘theatre’) is that any group of people coming together for dialogue implies such a communion even when it neither becomes conscious nor manifest in the course of the activity of dialogue. In other words, even when people actualise in argument and stupidity they nevertheless ‘imply’ a communion.  In a way, the communion is more real than their actual behaviour. Bennett himself often placed great emphasis on the difference between actualization (in observable behaviour) and realisation (to be known only through participation).

Every society (in these terms) is unique because not only is it a case of consisting of a certain number but also of its unique members. This relates to the idea of the systems as best symbolised in the transfinite numbers. The members of a society are transfinite in quality. They are not terms. They go beyond relations. It may well be that the realisation of true societies is extremely rare and that when they come about they create archetypes. Instead of thinking about systematics along the line of general laws, it is possible to understand it more in terms of a reflection of unique forms. Societies are ‘more than creative’ and can be associated with Bennett’s concept of the unitive energy, or the theological idea that the medium of the will is love. Here we might also remember Bennett’s comments on sex in the book of that name, where he refers to the beits or ‘dwellings’ which are degrees of union. It is possible that music provides the best medium for understanding this game, as in terms of harmony. The roots of the word ‘harmony’ relate to fitting together and it is closely related to arithmos or ‘number’, since numbers both set things in order and fit them together.

Our concept of the game of Society is intended to be approached as a limit or ultimate extrapolation from the maximum extension of both individuality and wholeness. In this way we se that we have come to a stop or limit in our scale of understanding. Any games beyond Society will therefore be implicated in it and we cannot distinguish them. What follows is then merely an abstract exercise, which may or may not lead to substantive insights in the future.

SYMBIOSES

In moving to Significance we made a something of a fresh start, introducing MMs and structures in place of terms and systems. We implied a Kenning game (see under game of sequence above) in that

systems : terms =  structures : MMs

We might think of yet another new beginning now, especially since Bennett tended to speak, though in vague terms, of a transition beyond societies to symbiosis and history. The Kenning game is extended to read:

systems : terms =  structures : MMs = histories : events

The word symbioses is used here to give a name to the next four hypothetical games, but we expect them to realise ‘structural history’, culminating in a fully intentional history.

            SYNCHRONICITY
            SACRIFICE
            SACREDNESS
            SUPERNAL

The theme of structural history lends itself to describing this realm in terms of events instead of MMs or terms, and history instead of structures or systems. Symbiosis is then a word for the mode of operation that renders events into history, our usage differing somewhat from Bennett’s and emphasising the time-like character of this domain of realisation.

We leave the series of games in this vaguely suggestive way because we do not know how to play the higher games, if they exist at all. There are many powerful associations to explore, such as to the Abode of the Gods or the Hidden Directorate.  These are mentioned because Bennett alluded to histories as coming after societies. The higher games would constitute what we might call ‘higher intelligence’.An important suggestion lurking in systematics is that the higher systems we are not able to operate with are still real but in the domain of higher intelligence. We experience and understand in a bandwidth of meaning. It is somewhat removed from the realm of life and also from what Bennett called Demiurgic Intelligence.

In the table below, we label the second level of games as ‘dialogues’ because these are games requiring several players which bring into play the potentials of mutual relevance. The progression to the hypothetical third level dissolves the distinction between elements and mutual relevance.  The first level designation ‘models’ relates to Bennett’s attempts to distinguish various kinds of collectivity.
An implication of putting ‘events’ beyond ‘individuals’ is that symbiosis (in the very special sense we are using that term) concerns the making of a total human soul: history is the way we contribute to and participate in That.

To illustrate the third level of systematics, we adduce the main propositions from our “Blake-Blake Theory of Communion”

PROPOSITIONS

1. Reality is made of Communions.

2. A Communion of Individuals is such that every Individual is in a State of combination of Individuals of that Communion.
2a. There can be an Individual that is in a State of combination of every Individual of the Communion (including the ‘fallen’ – see below). This is the Plenary Individual.
2b. There can be a ‘symbolic form’ (such as ancestral totem pole) in place of the Plenary Individual.
2c. The symbolic form is ‘God’. The Plenary Individual is ‘prophet’.

3. Individuals who are in a State of combination only of themselves are ‘fallen into sin’.

4. Sex consists of all States of combination of two Individuals in the Communion.

5. Individuals of a Communion can be in a State that includes the Plenary Individual. Such States are called ‘participation’; but they are only partial.
5a. A symbolic form of a participation is called a ‘church’.

6a. The States of combination of single Individuals (‘in sin’) are ‘conscious’.
6b. The States of combination of two individuals (‘in sex’) are ‘creative’.
6c. The States of combination of three or more Individuals, including the Plenary – i.e. in participation – are ‘unitive’ (“When two or three are gathered together in My Name, then am I with them”)
6d. The States of combination of Individuals which belong to different Communions are ‘transcendent’.  

7. A Communion is defined by its inclusion of a Plenary Individual or symbolic form. Hence such are religions, faiths, tribes, ways of living, etc.
7a. Individuals who are included in two or more Communions are called ‘peace-makers’.

8. Reality is without boundaries.
8a. The Individuals of a Reality cannot be counted.
8b. The States of a Reality go beyond experience.
8c. The Communions of a Reality are unknown.

9. States resolve into subjective and objective aspects in that single-valued Individuals are most like objects and Plenary Individuals are most like subjects.
9a. It is likely that this gives much the same results as e.g. Kashmiri Shaivism.
9b. The theory of Communion contains Whitehead’s concept of organic prehension (as States) and Leibniz’s concept of monads (as Individuals).

10. The theory does not involve communication or any transfer ‘between’ Individuals. We regard communication as a poor theory of communion. In Communion, there is no need for any exchange because different Individuals are not separated in the States they assume.

11. A divine messenger is transcendent
      A prophet is unitive
      A saint is creative (lovers = one saint)
      A sinner is conscious (“Hell is oneself” T. S. Eliot, taken from Blake)

12. In a Communion, ‘many’ is always ‘one’, and ‘one’ is always ‘many’. When one = many, there is a State. All States are ‘images’ of the Communion.

(see http://www.duversity.org/articles/theory_of_communion.doc for the article as published in the DuVersity Newsletter).