Combinatorial analysis of relations between finite sets (Q-analysis)

Beyond Edge-bound Comprehension and Modal Impotence: combining q-holes through a pattern language (Part #2)

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1. Q-analysis is concerned with the theory and application of mathematical relations between finite sets ( 4, 5 ). The approach is based on the use of sets at different hierarchical levels: N-1, N, N+1, etc. A set at one level may act as a cover for sets at a lower level -although, if an element of the lower level set can only be found in one element of the higher partition.

From any one level, the relations between distinct sets are the subject of 0-analysis and give rise to two views of the activities between the elements of the two sets:

global, namely an overview of the multidimensional geometric structure which can be described as the "backcloth" or "space" which supports all relevant action. Since the global geometry is not the same in all directions, it is necessary to characterize it by two associated connectivity properties:

structure vector obstruction vector

These will be considered further in a later paragraph

local, namely relatively isolated properties on different parts of the surface. Of these, the most significant are gaps in the pattern of connectivity which have been termed "holes" and are discussed further below.

2. Interpretation of 0-analysis seems to suggest that the real experience of the community by individual persons corresponds very closely to the global and local structure features to be found there. It is argued that the multidimensionality of such spaces is a matter of common intuitive experience, the limiting case of three-dimensionality being necessary but insufficient for the full range of such experiences.

Physicists make use of various patterns of connectivity in the classical 3-dimensional space to represent physical phenomena of various kinds including the dynamics of systems. "Dynamics" can therefore he regarded as synonymous with patterns", provided these patterns are associated with a suitable unchanging geometrical geometrical backcloth. In the light of the Einsteinian formulations of relativity, however, the connectivity of this backcloth can be altered by special constraints so that the geometry only permits certain positions or paths. The modified geometry then has built into it the structure characteristic of the specialized patterns invented to describe the paths in a rigid backcloth. The paths (e.g. of particles) are then not those which supposed Newtonian forces induce but rather the "natural" paths which the geometry permits.

The changes of patterns are an expression of what, in the Newtonian sense, has been traditionally called a force. In this new framework, a change in the pattern can be considered as defining a force experienced in the static backcloth in which the phenomena are to be found. And since the pattern is graded according to the level of connectivity at which it is perceived, the force will also be characterized by the level of connectivity through which it is experienced. Thus a t-force is associated with q-connectivity. There are many instances where common English usage expresses an intuitive recognition of social forces or organizational pressure experienced by people living in a structure.

3. A relation between finite sets represents a simplicial complex and this can be intuitively appreciated as a collection of abstract convex polyhedra in a suitable multidimensional euclidean space. The polyhedra are then connected to each other by sharing "faces" (namely sub-polyhedra such as edges, triangular faces, etc.). Q-analysis is the process of identifying these pieces of the simpiicial complex which are q-connected, for all values of q.

4. Distinctions of level: Q-analysis provides a more refined approach the ideologically sensitive issue of how levels are to be conceived and interrelated in psycho-social systems. It also draws attention to, and clarifies, the distinction between:

• levels as dimensionality [e.g. higher dimensionality as higher level)
• level in hierarchy (e.g. in any organizational structure)
• level of sensitivity to connectivity (e.g. ability to comprehend a complex structure, of a given level of connectivity, as a gestalt rather than by sequentially scanning its parts)
• level of psycho-social force or "pressure" resulting from changes in the pattern of connectivity over time
• level of traffic, namely the connectivity of the traffic content which can be associated with a given level but not those "below" it.

5. Traffic and noise: Q-analysis helps to clarify the distinction between "valid" communication traffic and "noise" in complex organizations or in overlapping "invisible college" networks. The interesting point is that the boundary between the two shifts according to whether the emphasis is on "efficiency" or on psycho-social development. Creating the opportunity for "noisy" traffic may in some circumstances be more important than ensuring that the traffic has a minimum noise level. It is even possible that all traffic can be usefully perceived as noise, or vice versa, depending upon the level of connectivity to which one chooses to be sensitive.

6. Evanescence of memory: Q-analysis, though its discussion of levels of sensitivity to connectivity, makes it possible to discuss with greater precision the manner in which comprehension of gestalts may be eroded over time. Such gestalts may be problems, concepts, values, organizational structures, strategies, etc.

Thus an individual or group may at some moment be sensitive to a complex pattern of high connectivity. Subsequently, however, this level of comprehension of the whole may fade in parts, reducing comprehension to that of some connected parts, or possibly to the parts in isolation only. Comprehension of the whole may be recovered periodically, and cultural events and artefacts can play a significant role in bringing this about. Or they may never be recovered, as in the case of the irreversible decline of an individual with ageing, or a group or civilization after its "golden age".

7. Multi-level communication: Q-analysis gives precision to the recognition that traffic of different degrees of content connectivity finds (or creates) its appropriate level in any psycho-social structure. Communicable insights are level hound, especially where they are of high connectivity. In other words, at the level within which we can communicate, concepts cannot be anchored unambiguously into terms and definitions which "travel well". Precision introduces distortion which is only acceptable locally within any communicating society -although "locally" most be interpreted in the non-geographical sense in which all nuclear physicists are near neighbours, for example.

This draws attention to certain implications for the development of any such structure, whether an individual, a group or a society. The most important of these would appear to be the impossibility of eliminating "undevelopment" from any such structure if it is to evolve. Such structures must necessarily continue to have traffic of very low-level connectivity co-present with that of increasingly higher-level connectivity. The simplest illustration is that of infants who will, when resources permit, continue to be educated through to the level of connectivity to which they can respond. But there will always be communication at both low and high-connectivity levels, especially about socio-political issues. The question is then how communication at these different levels of connectivity can be woven together within a social structure.

8. Co-presence of connectivity spectrum of "developments": If, as suggested above, the communication problem necessarily segments comprehension of "development", there is consequently a multiplicity of concepts of development operative in society. Individuals and groups may "progress" from one to another, possibly with a general tendency towards those of higher connectivity. But other individuals and groups will emerge and find the concepts of lower connectivity more meaningful before moving on, if they do, to those of higher connectivity. (In this sense the "ontogenesis" of an individual tends to repeat the "phylogenesis" of his/her society.) Society in this sense is the arena within which individuals and groups refine their concept of development.

But how then is the development of that society to be conceived ? Seemingly it is less a question of some "super-dominant concept" of development, and more a question of how the different developments are interrelated.

9. Holes and objects: The major achievement of Q-analysis probably lies in its ability to give precision to discussion about a psycho-social phenomenon which is, by definition, sensed beyond the boundary of (collective) comprehension. These are represented by "holes" in the pattern of connectivity. It has been argued that holes in a physical structure are indistinguishable observationally from solid objects in the physical case (3). In the psycho-social case, such holes are necessarily less substantial without losing their reality.

"Generally speaking it seems to be confirmed that action [of whatever kind) in the community can be seen as traffic in the abstract geometry and that this traffic must naturally avoid the holes (because it is impossible for any such action to exist in a hole). The holes therefore appear strangely as objects in the structure, as far as the traffic is concerned. The difference is a logical one in that the word "q-hole" describes a static feature of the geometry S(N) whilst the ward "q-object" describes the experience of that hole by traffic which moves in S(N)." ( 3, p. 75)

As an "object" this phenomenon is an obstacle to communication and comprehension and obliges those confronted with it to go "around" it in order to sense the higher dimensionality by which it is characterized. As a "hole" this phenomenon engenders, or is engendered by, a pattern of communication. It appears to function both as "source" and "sink". It is suggested that in some way which is not yet fully understood, such object/holes act as sources of energy for the possible traffic around them.

From the initial research it would appear that such objects/holes are characteristic of communication patterns in most complex organizations. It seems highly probable that they can also be detected in any partially ordered pattern of communication. As such "societal problems", "human needs", "human values" merit examination in this light.

The special value of doing so is that it can clarify why action/discussion in connection with them tends to be "circular" in the long-term, however energetic it may appear in the short-term. As such it shows how social change is blocked by the way in which conceptual traffic patterns itself around the sensed core issue which is never confronted as such because the connectivity pattern is inadequate to the dimensionality of the issue. This would explain why so many issues go unresolved and why the process of solving problems becomes institutionally of greater importance than the actual elimination of the problem.

This approach also draws attention to the probable presence of holes/objects of even higher dimensionality than those whose presence can be sensed relatively easily. Such phenomena, it may be supposed, are of great significance to long-term development.

10. Configurations of holes: Before being able to examine whether a particular concept of development is more"or less adequate, further understanding is required of how such a concept functions as a more or less stable hole/object in relation to other hole/objects corresponding to other concepts. How can such holes exist in relation to one another ? What is necessary to permit transitions from one to the other ?

The question is how configurations of holes can be identified and/or designed. It is the configuration of the holes which provides the minimum structure to stabilize and give form to the co-presence of the differing concepts of development. Such configurations, in order to fulfil their function, must presumably exist within two boundary conditions:

• the connectivity between elements bounding holes must no be so great as to erode or destroy the identity of the holes so connected
• the connectivity between elements bounding holes must be great enough so that the integrity of the configuration as a whole is maintained

A further question is then the manner whereby "better" holes are to be identified or reached within such configurations. Now from one point of view it is necessary to avoid introducing an element of evaluation, because from each hole the perception of other holes will he distorted so that no communicable assessment can be usefully formulated. On the other hand, it may prove to be the case that, at the level of the configuration as a whole, more than one such configuration can be identified/ designed in order to interrelate the perspectives associated with the set of holes. And at this level, without privileging any particular hole, more adequate inter relationships between the elements making up the holes can be identified.

Expressed differently:

• introducing evaluative judgements into the relationships between the holes within a particular configuration can only contribute to the dynamics between such holes in terms of perceived advantage/disadvantage. Excessive emphasis on this runs the risk of tearing the configuration apart. In this sense the configuration as a whole functions as a kind of "macro-hole" around which such traffic/noise circulates
• the identities associated with the holes can be respected in each of the configurations in a series constituting progressively more adequate or richer formulations of the relationships between "developments". The transitions between these successive configurations can be described with some precision such that continuity is in effect maintained. Although this "series" can usefully guide evolution of the "development set", there is a sense in which the more "primitive" configurations in the series are as valuable as those which-are: more-complex.

The characteristics of basic configuration of holes are examined in separate papers on tensegrity structures (From networking to tensegrity organization).

11. Human development: Q-analysis provides another way of discussing and interlinking certain aspects of human development. It opens the possibility of defining the individual in terms of overlapping sets of characteristics whose interrelationships can be explored with greater clarity at a new level of significance. This is of particular importance as a new language with which people can understand themselves experientially and communicate that understanding. A great advantage is that it provides pointers to those aspects of human development associated with greater connectivity ("integration") and the sensitivity stages whereby any intellectual understanding of it can be "experientialized".

The concept of holes/objects provides a very useful way of clarifying the manner in which an individual's "internal dialogue" can get locked into certain "circular patterns of reflection", or onto seemingly unresolvable issues. As discussed above, however, rather than attempting to favour the development of some of these at the expense of others, the dimension of human development which merits emphasis is that associated with "richer" configurations of these localized features.

It is the progress through the alternative perspectives provided by such configurat ions which constitutes the "changing self-image of man". The range of such configurations suggests the interesting questions: In how many distinct ways can man usefully perceive himself ? How are less useful transitions between configurat ions to be distinguished ?

As discussed above, and elsewhere (4 ), an important goal of human development would seem to be associated with the ability to shift flexibly between configurations rather than with irreversible shifts towards configurations of higher connectivity. (In traditional tales, the sage or the hero retain a special skill in communicating with children)

The structural language of Q-analysis also provides a useful means of discussing such diverse issues as: identity-and individuation [in terms of structural eccentricity), mystical or ecstatic union (as a limiting case of high sensitivity to high connectivity), rites of passage (transitions between configurations), oriental emphasis on breathing exercises (an experiential metaphor for a cycle of transitions between configurations), psychedelic drug experiences (uncontrolled shifts to alternative configurations of higher or lower sensed connectivity).

12. Social development: As in the case of human development, it is the concept of holes/objects which provides valuable insights for social development. It is probable that most continuing societal problems should be seen as holes/objects, especially given the well-established record of unfruitful action in response to them -however vigorous and dedicated. Typical examples are: peace/disarmament, development, human rights, environment, etc.

In such cases Q-analysis could provide understanding of why any action tends to be drawn into a vortex of futility, however much it satisfies short-term political needs for visible "positive" action. The participants in the action find themselves circulating" around a central concern of which they are unable to obtain an overview, due to the geometries of the overlapping conceptual and organizational structures through which they work (or which they somehow engender).

The term "futility" used above is however only appropriate if the sole consideration were the elimination of such problems. In fact the existence of such problems is extremely important to the organization of society, to social development, and to the direct or indirect employment of many people. Just as the "defence" business is vital to the economy of many countries, so is the "social problem" business vital to many sectors of society. Eliminating social problems would be a disaster for many people, especially problem-oriented intellectuals, or the employees of problem solving agencies.

As in the case of human development, it is possible that a more fruitful approach is to identify the configuration of such problem-solving bodies around the holes which engender their activity. The configuration may itself be seen as engendered by the macro-crisis hole which absorbs the development initiatives of society at this time and gives rise to the immense volume of action/communication traffic around the surface of the configuration.

Whilst it is probably neither feasible no desirable to eliminate such configurations in the name of efficiency, it is possible that the emphasis could be shifted towards alternative configurations which respect the original geometry but are more attractive because they develop it by elaboration and enrichment. A great advantage is that Q-analysis provides a measure of the obstruction to changes in the pattern of connectivity at any given level.

A major hindrance to social development is that initiatives are elaborated on the basis of the perception that some people are "right" (well-informed, objective, tolerant, etc) whereas others are "wrong" (misguided, intolerant, self-interested, etc). Effort are usually made to contain, repress or eliminate the latter. In fact as the record of every such initiative shows, those who considered themselves "right" had significant blindspots, and the arguments of those who were "wrong" were not without validity. It is even possible that lasting social development only results from the interaction of such initiating and constraining tendencies, but only when some"creative compromise is achieved to correct for the self-righteousness and bigotry of each tendency.

Through its use of "anti-vertices" Q-analysis offers a powerful tool for handling such essential differences with greater clarity. This appears important, not only in the light of the preceding paragraph, but because of the way in which social development is fuelled by difference. The drive to social development is fuelled, paradoxically, by the differences which that development (as presently conceived) strives to eliminate. Once eliminated the society would be in danger of stagnation or of instabilities in reaction to that stagnation. For this reason a more creative attitude is required to those ever-present conditions which engender difference, namely: ignorance, apathy, intolerance, etc. They can usefully be considered as "fuel elements". But energy can only be usefully generated from them if they are bound into a suitable configuration with those elements which react to them. The question is how to detect or design configurations which can channel and focus social energy in this way.

The above argument implies the existence of one kind of social "energy" which is vital to the life and development of society. Analysis of configurations may well bring to light other kinds of social energies which have to be kept in balance. Oriental traditions highlight the existence of many distinct sets of energies "nested" in relation to one another ( ). But the difficulty is to sense the complementarity between them. Without this understanding, the manner in which they function as a "team" is unrecognized and those identifying with particular energies are trapped in the dynamics of the configuration in question.

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