11.1 The purpose of this paper is to demonstrate the importance of number in the complete sets fundamental to social science and policy formulation. It is fairly obvious that formulation of a 2-term set of concepts (values, problems, etc.) establishes a dynamic for the advocates of each term, or those involved in any institutionalization of the dyad--namely a dynamic having any or all such aspects as: active/passive, right/wrong, we/ they, dominant/subordinate, conflict, complementarily. For example:
"By the very nature of scientific logic which is binary, intellectuals tend to form bi-polar structures with two opposed camps rallied under two paradigmatic banners. The polarisation often takes place even within each Of the two poles which then divide themselves into subpoles, and so on, and so forth" (49).
It is equally, obvious that promulgation of a l-term set (e.g. the problem, the value, the method, etc) gives rise to another kind of dynamic. It is however less obvious what kinds of dynamics tend to arise from sets with a larger number of terms. Yet sets with larger numbers are frequently produced and usually it is considered convenient to ignore how the elements of the set interact at the conceptual level or through organizations (departments, programmes, laws, information systems, etc) on implementation. This paper implies that, like it or not, certain interaction qualities are built in by the choice of the number of set elements. If ignored, they will erode or completely undermine the effectiveness of any action i based upon them. They define the problem to which the initiative is vulnerable and by which it will be counteracted, or nullified.
11.2 Implicit sets of a given number of terms usually ! engender particular styles of debate. For example: 1-term, promulgation and propaganda; 2-term, pro and con argument as in some legal, parliamentary and scholarly arenas; 3-term, mediatory and reconciliatory debate. Given that issues currently exceed the capabilities of such forms of debate or are exacerbated by them, other higher-term forms may be envisaged to contain and facilitate the interactions between a greater number of distinct viewpoints. This would also be relevant to the interactions within interdisciplinary teams and the design of the classification systems which serve them (98). A sense of issue configuration would stabilize understanding of the complete sets of "logically incompatible" problems which such teams are increasingly obliged to confront. This could lead to the emergence of methods based, on a non-dualistic complementarily. A need for an improved approach is becoming evident (132), even in unexpected places: "The mosaic theory of intelligence has focused attention on collection, the gathering together of as many pieces as possible for the analyst to work with. A more psychologically oriented view would direct our concern to problems of analysis, and especially to the importance of mental models that determine what we collect and how we perceive and interpret the collected data. . . there are important implications for the management of intelligence resources" (133).
11.3 Research on complete sets is required to clarify their nature and variety. Complementary approaches include: research on number, as advocated by von Franz; research on symbols in traditional cultures, of any well-ordered sets and their elements; and research on modern sets elaborated in scholarly and action-oriented texts. This should lead to better understanding of:
- (a) how sets can be formed and their elements classified,
- (b) how the relationships between their elements can be rendered comprehensible,
- (c) how the nature and value of higher term sets can be demonstrated, and
- (d) the nature of the totality, they are intended to encompass.
It is in the East that qualities and attributes have been so carefully distinguished and ordered, whereas sophisticated number-based frameworks have been elaborated in the West. This research should bring out the points of contact. An excellent point of departure would be the problems of "classifying" tones in music as explored in two complementary studies by a philosopher (134) and a musicologist (135) faced with the challenge of the alternative patterning possibilities within the Rg Veda:
"Rg Vedic man, like his Greek counterparts, knew himself to be the organizer of the (musical) scale, and he cherished the multitude of possibilities open to him too much to freeze himself into one dogmatic posture. His language keeps alive that "openness" to alternatives, yet it avoids entrapment in anarchy." (134, p. 31)
This appears to amount to a degree of order beyond that attained in classification today; the flexibility and the challenge to musical creativity are illustrated by Fig. 4. It is perhaps no accident that P A Heelan's work on the Logic of Changing Classificatory Frameworks (139) cites the Rg Vedic example and is considered of fundamental importance by these two authors .
11.4 It is not recognized, when advocating or imposing the use of particular sets (e.g. of values, needs, etc), that these effectively compete as functional substitutes in traditional societies for other sets of qualities represented by hierarchies of gods or spiritual beings governing those qualities (or some of them). The fundamental sets society now attempts to generate are indeed designed to perform many of the regulatory functions previously ascribed to supernatural beings or potencies. Given the relative rapidity with which such sets are now formulated compared to the long cultural refinement of a pantheon it is not surprising if they are viewed as superficial, "bloodless" and unrelated to the cultural refinement of the traditional sets. These are so meaningfully represented (with nested levels of interpretation) through richly decorated beings and memorable tales exemplifying their relationships - to the point that the quality and its representation are difficult to distinguish in a particular culture. The lack of success of public information programmes of national and international agencies, in substituting modern intellectualised versions (of somewhat ersatz quality) using product marketing techniques, is understandable. The new versions lack credibility and durability even if the traditional versions are destroyed by the process [69, 70]
11.5 Comprehension of the qualitative characteristics encompassed by higher-term sets has been shown to be no easy matter despite their vital importance for a more adequate grasp of our current social crisis . Problems of classification, comprehension, memory aids and representation need to be considered together. There is every indication that conventional methods do not have an adequate degree of complexity to embody, and reflect for comprehension, the complexity of multi-term systems [72, 73]. Research is required:
- (a) on the generation of iconic symbol sets of high mnemonic value,
- (b) on the consequence of disposing them in configurations so that the pattern of relationships may be comprehended as a whole, and
- (c) on any paradigm shift or change of awareness which this may facilitate.
There is no reason why this should not include an investigation of the traditional memory technique and its intimate relationship to classification systems . To what extent were traditional symbol systems, or associated numbers, successfully used for their powerful mnemonic value?
11.6 Intriguing lines of investigation emerge from recognition of the intimate relationship between brain operation and classification. Varela notes: "the contents of our reality are truly a reflection of the recursive biological and cognitive computations, in contradistinction to the more commonsense view that our knowledge is a map of the out-there. From this point of view, there is more a construction than a map. These are tantalizing possibilities for a cross-connection between epistemology and science, for the design of knowledge representation systems, and for management and societal problems." (106) . This is related to current investigations of the transformation of the categories of conscious experience associated with shifts in characteristic EEG frequencies.
For example, it is suggested that: "the felt shift and the reorganisation of conscious experience is a multi-level phenomenon, involving a reorganisation of concepts, a choice of principles consistent with these concepts. . . as well as the appropriate reorganisation of all lower levels of the hierarchy consistent with these changes .... The transformation, then, is not merely a reorganisation, but at a deeper level is a re-creation" (107) . EEG data may even provide a link between characteristic frequencies (1- 3, 4 - 7, 8 - 12 Hz), the preferences mentioned in Part 1 for sets of a given number of elements, the ability to comprehend them, as well as the quality of that comprehension.
A better understanding of the conventional separation of subject and object can be obtained by exploring, as does R. Fischer, ecstatic and meditative states in which "the separateness of object and subject gradually disappears and their interaction becomes the principal content of the experience. . . meaning is "meaningful" only at that level of arousal at which it is experienced, and every experience has its state-bound meaning" (136). Relevant to the "concept triangle" question (see Part 2 and Fig. 5), Fischer in a section on "sign-symbol-meaning transformations", discusses evidence of the transformation of sign to symbol in the visual realm "where the constancies of space and time are replaced by geometric-ornamental-rhythmic structures", namely hallucinatory form constants. These are visible metaphors, otherwise uncommunicable, within a structure of symbolic logic and language whose non-visual equivalents also govern the order of poetic and musical rhythm in such experiences. Once again the importance of number becomes apparent. This question is set in a wider framework in studies initiated by Erich Jantsch'(137, 138), to which the argument of this paper links at points too numerous to mention here.
|Fig. 5:Convergence of concept triangle elements: Diagram indicating a particular condition of "adulteration"of the absolute distinction between the pure elements of the concept triangle. (Such diagrams are used to indicate the variety of equilibrium conditions of 3 elements in physical chemistry.) Such a presentation may be used to clarify the nature of other kinds of "blurring" of distinctions between the three elements.|
11.7 This paper attempts to show the basic role of number and configuration in overcoming limitations to man's ability to perceive (and denote through classification schemes) the patterns which affect him and in which he is embedded. Biologist Gregory Bateson's central thesis is:
"The pattern which connects is a metapattern. It is a pattern of patterns. It is that metapattern which defines the vast generalisation that indeed it is patterns which connect" (112, p. 11).
"How is logic, the classical procedure for making chains of ideas, related to an outside world of things an creatures, parts and wholes? Do ideas really occur in chains, or is this lineal structure imposed upon them by scholars and philosophers? How is this world of logic, which eschews "circular argument", related to a world in which circular trains of causation are the rule rather than the exception? . . . we shall see as every schoolboy ought to know that logic is precisely unable to deal with recursive circuits without generating paradox, and that quantities are precisely not the stuff of complex communicating systems" (p. 20).
"as of 1979, there is no conventional way of explaining or even describing the phenomena of biological organisation and human interaction". It is through study of number-governed qualitative configurations that responses to Kelley's related questions should be sought: "And the ultimate question is, what nature of pattern or system of patterns will enable the human mind to retain familiarity with the maximum number of patterns? And what is the maximum number of patterns the human mind can hold if the patterns are of this type? What other attributes of patterns are conducive to greater retention by the mind?" (41).
Finally, how is this related to the level of awareness or maturity of the observer? (107) [77, 78]
11.8 The ability of the mind to retain elements of information long enough for it to form memorable patterns with other elements (e.g. of the set) can be enhanced by the use of mnemonic aids. Whilst these may be viewed with disdain by those familiar with the subject matter, it must be recognized that classification schemes are not memorable to the uninitiated (e.g. the public, its representatives and those from other disciplines) who ultimately determine through the democratic process whether resources will be allocated to the maters ordered by such schemes. The same applies with regard to any argument presented in a linear sequence in an article or book. There is a strong case for interrelating the points made in a non-linear presentation. This goes beyond the seminal mnemonic serial structure described by Neelemeghan (63) . Furthermore, in view of the increasing resistance to written arguments of any length there is a case for investigating the possibility for their partial replacement by mnemonically structured diagrams which may provide the detailed pattern for dramatized portrayal necessary for communication to a wider audience. Three dimensional centered mnemonic structures may offer possibilities for memory reinforcement and comprehension beyond those of the two dimensional variety.
11.9 In considering contemporary efforts in the West to allocate qualities and attributes to multi-term systems  one is particularly struck by the "bloodless" nature of the resulting categories (however innovative the exercise, such as in the case of Bennett). Such frameworks are generally conceived as mutually exclusive, the advocates of each ignoring the others in favour of their own particular slant on reality. There is much misplaced confidence in the ability of words to label qualitative concepts without ambiguity . It is not recognized and that, as such, each constitutes a representational aspect of a more subtle and more comprehensive framework (cf. René Thom's approach). In fact, however apparently distorted or inadequate the attempt, its degree of "distortion" identifies the location of its advocates in relation to other perspectives, challenges, and problems of comprehension. Such relationships are governed by numbers indicative of qualitative distinctions.
11.10 It is to be hoped that this paper has demonstrated the importance of a new approach to representation and the possibilities for it. It may indeed be argued that Johan Galtung's emphasis (56) on the need to switch from the conventional "facts-theory" to a "facts-theory-value" (i.e. from 2-term to 3-term) approach, should be extended to "facts-theory-value-representation" (4-term), or beyond . The dynamics resulting form facts-theory are too well-known, but the difficulties are not eliminated by his 3-term suggestion. Basically, if insights cannot be meaningfully represented, they are incomprehensible and therefore irrelevant to the period in which they are formulated.
11.11 Finally in the words of Bennett:
"For a long time, men have looked for ways of getting beyond the dyed: but mankind as a whole remains bound by sentiments of exclusion and contradiction. Meanwhile the progress of science and technology is leading us towards structured notions of greater and greater complexity. The same is true of nearly all branches of life: psychology and sociology, art, history and religion; all are moving away from naive expectations of simple unstructured solutions to human problems and towards the recognition that we and the world in which we live are an organized complexity that can be understood even to the limited extent that we do understand--only by discerning the structures that bind us together". (45, vol. 3, pp. 74 5).
Or in the words of Bateson: "Break the pattern which connects the items of learning and you necessarily destroy all quality". (112, p. 8). Unrelated set elements break patterns and therefore destroy quality.