Representation, Comprehension and Communication of Sets
the Role of Number (Parts 1-3)
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Parts 1-3 of Representation, Comprehension and Communication of Sets: the Role of Number (1978) Originally published in International Classification, 5, 1978, 3, pp. 126-133; 6, 1979, 1, pp. 15-25 ; 6, 1979, 2, pp. 92-103
There is a widespread tendency to formulate insights, proposals or principles in point form, namely as made up of a specific number of items usually presented as a list. Such items will be considered here as the elements of the set that they collectively constitute in any particular case.
This paper is therefore concerned with problems relating to the representation and comprehension of such sets--whether the elements in any given case are basic: human needs, human values, principles, concepts, problems, human rights, human responsibilities or components of a policy.
The paper explores the possibility that (irrespective of the nature of the elements in any such case) there may be different kinds of constraints on the distinctions and relationships between the elements, depending upon the total number of elements in the set. Clearly, the total number of elements in the set also affects the manner in which the set can be represented, communicated and comprehended.
Briefly, therefore, the paper argues that consensus on a 5-element set of human needs (or a 5-point programme) for example, implies certain kinds of distinctions and relationships between the 5 elements, depending solely on the number (e.g. in contrast with a 3-element or 10-element set). These may not have been met in a given case because the elements are either (a) inappropriately defined, or (b) appropriate to a 4-element or 6-element set (with the consequence that there are elements in excess or missing from the set). Inadequacies of this kind are of importance in themselves but also affect the representation and communicability of the set, and ultimately its role and viability in the psycho-social domain.
1. The following argument applies only to cases where the elements are conceived as making up a complete set. It does not apply when the elements have been selected (possibly as a sample) from a larger set. Where the elements are selected on a priority basis, as being the "most important", the argument only applies when this may be interpreted as implying most "fundamental" or "basic" .
Ideally the argument should also apply to any numbered list of points in an argument. But, since numbers are usually allocated for convenience to provide a simple structure to a sequence of paragraphs (and only indirectly related to the concepts developed), this is seldom the case. It should however apply wherever the author(s) declare that: "The following points apply", provided "including the following points" is not used or implied. The list of points should therefore have been elaborated through a "struggle" to get the best "fit"--a struggle which may have required much more than superficial reflection over a short period of time .
2. The sets under consideration contain elements which are essential to the ordering of an equilibrium state or an evolving process (especially in the psychosocial domain). As such each element is different and has a special part to play. Each complements the others and all are conceived as essential (e.g. in the case of human values or needs). There is a desire that such sets should be well-formed or well-ordered, even if some degree of "fuzziness" must be tolerated as the content is clarified through research and debate.
3. The elements in such sets should be equally distinct from one another or else the question arises whether two or more similar elements should not be redefined as one. This said, however, two cases must be distinguished:
3. Constraints on number of elements in a set
1. There is an implicit assumption that authors are free to include as many elements in a set (of the above kind) as they wish. In fact, 1-element and 2-element sets are seldom of interest to scholars, although there is a tendency reinforced by public policy considerations to identify 1-element sets (e.g. the fundamental value, need, problem, principle, etc.). At the other extreme, 1000-element sets are considered unacceptable, as are 100-element, or even 20-element, sets. The implication here would be that the authors have not made an ade-quate attempt to regroup the elements in the light of common characteristics. An apparent exception is the matrix, but even here the number of columns or rows becomes unacceptable (for other than special cases) in excess of 20, for example. In fact, the probability of encountering a set with a given number of elements seems to decrease rapidly when the number exceeds about 10. It would be interesting to see whether a survey  would show any relation to the isotope abundance curve (see Fig. 1) in which the peaks are approximately congruent with the atoms of highest structural stability .
2. Authors are therefore constrained, irrespective of the nature of the set, to reduce the number of elements to something in the region of 10. Each such element, however, may in turn be considered as a (sub)set within which a similar number of elements is admissible. In this way, any number of elements can ultimately be incorporated. This coding procedure is considered legitimate because it facilitates comprehension. The consequences of such a procedure have not been examined -- and yet it is this very procedure which produces the sets of values, principles, problems, needs, concepts, policy elements, etc. in terms of which attempts are made to order social processes and resolve their problems.
3. The objectivity by which elements are selected on the basis of scientific
criteria for inclusion in a set is therefore strongly affected by constraints
on the ability of the author/observer to comprehend the set as a whole
and to render it comprehensible to others. As Christopher Alexander notes
(ref.(2), p.5) it has been shown
4. This constraint is also reflected in the "embodiment" of such sets in social organization, namely in the limits on the size of an effective committee, on the one hand, or on any small encounter/therapy group, on the other (7). The limit to the number of subordinate bodies which a body can effectively control is of the same kind, particularly as evidenced by the number of divisions reporting to a coordinating or presidential office. Antony Jay has explored many organisational examples of such limits . Note that such organisational sub-division is carried out and limited irrespective of the complexity or diversity of the operations or problems with which the body as a whole has to deal.
5. The constraint is also "embodied" in the category sub-division of the thesauri which govern the manner by which information is obtained from libraries and information systems. Note again that this is so irrespective of the complexity or diversity of the subjects recorded in such systems.
6. The constraint may also be noted in the sets of "key" or "fundamental" problems, values, needs, etc. which are identified as the basis for action programmes. Such a breakdown lends itself readily to institutional embodiment or reinforces institutional structures which already reflect (and are therefore unthreatened) by this structuring. The predilection for sets of 10 key problems is noted by the editors of the Yearbook of World Problems and Human Potential (ref. (19), see especially Appendix 3). An excellent example is Unesco's own exercise to identify the major world problems with which it is concerned. It found 12 and condensed them under 10 objectives in its Medium-Term Plan 1977 - - 1982 (Paris, Unesco, 1977, 19 C/4). Another excellent example is the Assessment of Future National and International Problem Areas (Washington, National Science Foundation, 1977, NSF/STP76-02573). This carries an illustration, reproduced here as Fig. 2, which shows admirably the nature of the process. The document concentrates on the 6 problems which emerge from this filtering procedure. (It is perhaps naive to ask what attention will be given to the 994 problems excluded by this procedure.) 
7. Such is the prevalence of this constraint that it is of interest to identify the conditions under which it is exceeded and the consequences of doing so for the communicability and viability of the set .
8. Another aspect of the constraint on the number of elements in a set emerges from recent explorations into the psychophysical significance of number as the common ordering factor of psyche and matter (9). Since this raises the question of the nature of the observer's relation to the observed, this is discussed separately below.
4. Representation of sets: introductory comment
Herbert Simon notes: "An early step toward understanding any set of
phenomena is to learn what kinds of things there are in the set--to develop
a taxonomy. The step has not yet been taken with respect to representations.
We have only a sketchy and incomplete knowledge of the different ways in
which problems can be represented and much less knowledge of the significance
of their differences." (5 p. 78)
It may be argued, however, despite the apparent ease of this approach, that widespread understanding of the many systems within which man functions (or with which he interacts) remains elusive. Indeed complaints about "increasing complexity" are now common. And studies of psycho-social systems have not produced insights to make them more manageable, in fact such systems appear to have become less manageable whilst such studies are produced.
There are three weaknesses in the conventional stress on the prevalence of hierarchical ordering. Herbert Simon follows the previously cited remark with: "Or perhaps the proposition should be put the other way round. If there are important systems in the world that are complex without being hierarchic, they may to a considerable extent escape our observation and our understanding." (5, p.l08). Such systems, possibly exerting a "field effect" or based on non-hierarchically ordered networks may indeed be at the root of our difficulties. It is interesting that the 1970s has witnessed a rapidly burgeoning interest in networks of all kinds and a suspicion of hierarchically coordinated social structures (13). The relationship between sub-sets of different hierarchies is recognized as being increasingly critical (e.g. in environmental systems). The problem of representing such complex patterns of relationship to facilitate comprehension has not been resolved .
A second weakness derives from lack of clarity on the nature of the set of which the hierarchical set under consideration is a sub-set--namely the super-ordinate set. Each discipline is responsible for its own hierarchical sets, none is responsible for the super-ordinate set (and the interactions between its sub-sets). This relates back to the first weakness. There is little understanding of what happens at the "top" of hierarchies and especially "above" them .
A third weakness derives from lack of clarity on the relation of the person creating or observing the set--to that set. Some aspects of this question are discussed separately below. It is particularly important where one or more such sets are expected to order the comprehension of the individual who therefore has the problem of "juggling" them into a suitable configuration in relation to his own psychic ordering . This raises the question of the iconicity of any representation which is discussed below.
In discussing the description of complexity, Herbert Simon makes a basic distinction between state descriptions and process descriptions . "These two modes of apprehending structures are the warp and weft of our experience. Pictures, blueprints, most diagrams and chemical structural formulas are state descriptions. Recipes, differential equations, and equations for chemical reactions are process descriptions. The former characterize the world as sensed; they provide the criteria for identifying objects, often by modeling the objects themselves. The latter characterise the world as acted upon; they provide the means for producing or generating objects having the desired characteristics.... Given a desired state of affairs and an existing state of affairs, the task of an adaptive organism is to find the difference between these two states and then to find the correlating process that will erase the difference. Thus, problem solving requires continual translation between the state and process descriptions of the same complex reality."(5, pp. 111-112).
5. Representation of sets: review of types
1. Lists: As implied above, the most favoured way of presenting a set is in the form of a list of items or points. Such lists may be unstructured or else items may be grouped into subsets. No other aid is provided for the comprehension of the set. It is assumed that any normal mind will be able to grasp the content in a satisfactory manner. Such lists do not identify the nature of the relations between the elements of the set (other than by what is implied by grouping into subsets).
2. Thesauri: As mentioned above, when there are many elements these are classified, with the aid of thesauri, into subsets at various depths within a thesaurus structure. Again little is provided to aid comprehension, the assumption being that a person knows which element is required and that the structure of the whole is of little importance. (There are a number of competing thesauri prepared by institutions -- themselves competing for resources.)
3. Tables /Matrices: The degree of order of a set becomes clearer when it is presented in the form of a table, of which there are various kinds (e.g. the periodic table of chemical elements). These blur into matrices as a more general form of tabular presentation, which may be multi-dimensional. But here again the mind has difficulty in comprehending the whole, although it may distinguish the parts. There is a limit to the tolerance for complex tables or matrices in policy-making circles, for example, and they are seldom suitable for media-oriented presentations.
4. Diagrams: As the variety of relationships between the elements of a set is recognized to be of importance a diagrammatic form of presentation may be used - even if it means sacrificing the precision of a matrix presentation. There are many kinds of diagrams (14), from the simplistic to the full detail of a system flow chart. But again the simplistic can only serve momentarily to introduce the set, they cannot carry the detail which a highly ordered set demands; whilst the overall significance of the detailed charts eludes the grasp of most minds [l6]. It is also interesting to note that there are constraints on the representation of such diagrams on paper due to the limited acceptability of lines crossing each other, multiple line coding, or the use of many colours.
5. Yantras/Mandalas: One form of diagram of special interest, because of its deliberate orientation toward the observer, is the "yantra" (or "mandala", in its circular form). These have been used extensively in Eastern cultures to integrate many hierarchic levels of information detail concerning the universe in a form designed to be both comprehensible and to have a profound impact on the attentive observer. Indeed special practices have been developed for their preparation and use [l7]. Significant in the light of the weaknesses connected with hierarchical representations noted above, is the fact that here hierarchies are bound together within a common framework with detailed elements on the outer edge of the diagram and the super-ordinate sets linking into a common centre --the focal point for the observer [l8] through whose awareness (once refined) the disparate sets of experience are integrated. The challenge to the observer is to penetrate into and structure his awareness through the diagram. It is especially noteworthy that diagrams of this type contain a high degree of symmetry, as well as colour coding and symbols of various kinds. (These are in part designed to "trigger" the conditions required of the senses and awareness in order for the "programme" to work.) The symmetry features are of course constrained by the planar representation.
6. Other techniques: The paragraphs above would seem to mark out the current ability to represent sets, given the number of elements, the degree of their ordering, and the erosion of comprehensibility as the combination number/degree of order increases in complexity.
There are a number of other techniques of communicating the content of a set. Some are discussed in (16), but they tend to suffer from the defect of being unable to represent the set in a form which can be easily reproduced and which lends itself to detailed examination and review. It is also appropriate to note here that many authors do not summarize their insights as a set of points or insights and may well consider such a representation as damaging to the nature of the insights they seek to communicate. Indeed the pre-logical biases, identified by W. T. Jones (17)  against such a representation may in certain cases constitute an ultimate constraint on clearly distinguishing the elements in a set.
7.1 As noted above, diagrams in 2-dimensions are extensively used to represent sets. It is however very rare to see 3-dimensional representations of sets, partly for the obvious reason that it is difficult to see the internal structure of such representations. And, despite the considerably increased facility it offers, 3-dimensional representation creates a barrier to the linear verbal description so essential to the verbal and textual expression on which much research and decision-making is based . However there are techniques for handling the representation of sets in 3-dimensions, of which the most sophisticated are the graphic terminals used in computer-aided design (19, Appendix 6). But it is interesting that, despite much attention to hierarchical ordering in organic and inorganic systems composed of 3-dimensional entities, it is in terms of a 2-dimensional representation that such hierarchies are studied  .
This is so even though the champion of the hierarchical perspective, Lancelot L. Whyte, specifically notes that "the real need is for a systematic and exhaustive survey of the types of three-dimensional spatial ordering which characterize the more important levels in both realms" (ref. (10), p. 13). He also remarks that "Where a system is 'sufficiently ordered' end 'sufficiently stationary' (terms to be clarified) three-dimensional geometrical relations (i.e. lengths or angles) may play a fundamental role. . . It is conceivable, in principle, that under certain conditions everything is derivable from angles. It seems that theory may sometimes pass rather easily from central geometrical hierarchical models to the heterogeneous properties of static, stationary, or near-equilibrium systems, thus opening the way towards a physics of hierarchy" (ref. (10), p. 11). The equivalence in properties between physical and social systems has been repeatedly noted (20).
7.2 A further justification for moving to 3-dimensions is that it increases the iconicity of the representation, namely the degree of isomorphism between the structure of the reality represented and the structure of the representation. Where this is high, comprehension is considerably facilitated--which is why architects communicate new concepts to clients via models and not plans.
7.3 The question now arises as to what relation the cognitive elements of the set bear to their representation. This argument is based on the assumption that in the case of the fundamental elements under consideration, there is a strong configurational component to their comprehension as nested concepts. Many of the arguments in support of (and against) this assumption have been developed by Rudolf Arnheim (21), who states, moreover:
And also: "In the perception of shape lie the beginnings of concept formation." (21,p. 27). He defines "shape" to include 3-dimensional forms, though most of his examples are based on 2-dimensional shapes, especially sketches and diagrams. He does, however, imply that a third dimension (depth) enters into perception, when appropriate (as with pictures). It may therefore be concluded that under certain conditions man thinks in terms of 3-dimensional constructs, whether or not he also thinks in terms of words or 2-dimensional shapes.
7.4 In moving to 3-dimensions a highly significant constraint emerges. In 2-dimensions there is, conventionally , a certain freedom in that the planar surface may be extended and divided at will (within the limits of line and colour coding noted above). Whereas, in 3-dimensions, what are known as packing constraints become much more significant (23). The ways in which subsets can be nested within sets may then be severely limited.
The question is then whether such geometric constraints on representation bear any relationship to constraints on the interrelationship between subsets or their elements as concepts in the human mind. On a hypothetical 2-dimensional system flow chart, one can well imagine over 50 input/output lines drawn to a particular process box. There appears to be no restriction (although there must be electro-mechanical and computing limits to their control). But at the conceptual level, the number would be unacceptable (in terms of the constraints noted earlier) and the process box would have to be divided into smaller units. A process box with 50 input/output lines would not be a useful guide to thinking about the system. It is as though each such unit could only have one of a small range of "valencies", to borrow a chemical term (24).
Now in 3-dimensional representations the permissible valencies emerge from the manner in which the sub-components can be packed in contact together (e.g. packing small spheres into a larger one). In fact this is also true in 2-dimensions (e.g. packing small circles into a larger one), but at this level the number of relationships (i.e. points of contact) is more limited than with 3-dimensions. It can of course be argued that in many cases such a representation is adequate to the complexity represented. The search for improved tools is however stimulated by the failure of the existing ones to improve collective, operational understanding of the social condition; the assumption of adequacy may not in fact correspond to the complexity of the environment.
The 2-dimensional model is not rich enough to reflect a 3-dimensional reality adequately (or with the compact elegance and symmetry that one may suspect comprehension of complexity demands). But it may also be argued that a 3-dimensional model is equally inadequate at reflecting higher dimensional realities. However there is little to suggest that man tends to think in 4 or more dimensions, even if some can think about them and represent their results in mathematical terms . To be comprehensible and widely so (in order to be of relevance to social change), "it seems safe to say that only what is accessible to the perceptual imagination at least in principle, can be expected to be open to human understanding" (21,p. 293). Hence the value of exploring the conceptual significance of 3-dimensional representation as opposed to other forms.
7.5 The point by Whyte cited above "that under certain conditions everything is derivable from angles" has recently been explored independently in a book by Arthur M. Young. He argues "a whole object or situation is divided into aspects (or, to use Aristotle's word, causes) and that these aspects have an angular relationship to one another" (25, p. xv). He asks: "Is my opening statement, 'All meaning is an angle', too abstract? Not if one accepts my allegation that meaning is in general a kind of relationship" (25, p. xv). Despite his unique understanding of 3-dimensions (as the inventor of the Bell helicopter), he only applies his approach to 2-dimensional cases. In a second book (26), published simultaneously, he explores related matters basing them on a 3-dimensional concept--but he does not link this explicitly to the angular concept of meaning.
7.6 For an extensive exploration of the meaning associated with the geometry of 3 dimensions, it is necessary to turn to R. Buckminster Fuller . His preoccupation, despite the subtitle of his book, is however with the architectural and concrete material implications of his work (of which one application is the geodesic dome which he invented). Nevertheless, in his work especially, and in that of others, stimulated by it  lie the basis for many generalisations in support of the argument here. In particular, as with Whyte and Young, he is also sensitive to the general significance of angle .
This is essential to his basic argument that the focal points for energy events in any system are linked into a closed pattern of relationships which can be effectively represented by an appropriate polyhedron (1, p.95 and 655). "All the interrelationships of system foci are conceptually represented by vectors. A system is a closed configuration of vectors. It is a pattern of forces constituting a geometrical integrity that returns upon itself in a plurality of directions." (1, p. 97). No reason is given why this should not apply to a system of conceptual elements constituting the kind of ordered set of interest here.
An attempt by a biologist has in fact been made to use the geometry of the 3-dimensional biological cell structure as a cubic framework in terms of which concepts may be ordered and interrelated (29). This has been extensively developed (using large-scale 3-dimensional models) as an experiential learning tool. Another very interesting approach (30), again using a cubic framework, has been considerably developed--from a model originating in the data-processing industry (31)-- in order to provide a way of structuring and representing ideas. Many points relevant to the argument here are discussed, as well as the transition from 2 to 3-dimensions. Whilst interesting and valuable as exercises, these raise further points discussed below.
8. Mathematical notations and N-dimensional representations: Much that is of interest with regard to sets and their elements is expressed and represented in mathematical notation which is meaningful to very few (including this writer!). This is the case with the highly relevant argument of Spencer Brown (18). It is also true of the very relevant insights of Rene Thom who leaves most social scientists, and policy makers behind at his point of departure:
He does however support the geometric representation argued above:
However rich the resultant insights, it is their significance and representation in 3 dimensions which is fundamental to their value for the comprehension and ordering of social processes.
6.1. Whenever it is convenient, there is a widespread tendency to avoid consideration of the impact of those involved on research or on the policy-making process in which they participate. Researchers correct for bias in experiments and aim for reproducible results. Efforts are made to balance the interests represented at policy meetings. Consequently, when sets of basic values, problems, concepts, or principles are generated by either, they are conceived to be objective. The relationship between any such objectively determined category sets and the thinking processes of those involved (or on whom those categories are subsequently "inflicted") is not open to rational discussion in the same arenas and may well be perceived both as impolite and threatening. And yet it is recognized that:
In justifying their own work, Bruner et al. argue:
This point was however made in 1956. Both in the research on which they report and in subsequent research, it would appear that the focus has been on categorization in the case of "laboratory problem" sets which are essentially trivial in comparison with the sets of fundamental concepts which are elaborated consciously in the course of research (or policy-formulation). The former are laboratory exercises requiring minutes or hours, the latter involve much reflection and a protracted "struggle" for the best "fit", possibly over a period of many months or years. In particular, to give the kind of "uncomfortable" example that is required, the research has not been applied to the sets and categories selected by those undertaking research in this very area, as an aid to explaining the differences of opinion which give rise to non-rational behavioural dynamics between the various schools of thought affected. Only "pointed", self-reflexive research of this kind, on the formulators of sets which are fundamental to social policy, can help to clarify the basis for the opposition between policies which tends to fragment society into hostile camps.
6.2 Laws of form: It is not sufficient simply to complain about the widespread tendency to avoid consideration of the impact of those involved in set formation on the sets which they formulate. The reason for such avoidance merits continuing study .
Part of the problem seems to lie in a missing link in the relation of mathematics to logic which has been provided, with the encouragement of Bertrand Russell, by G. Spencer Brown (18). Much of science (and that includes classification) makes explicit or implicit use of set theory based on Boolean algebra which was designed to fit logic--but in doing so detaches the observer from any involvement in the logical processes . Spencer Brown argues that: "nobody hitherto appears to have made any sustained attempt to elucidate and to study the primary, non-numerical arithmetic of the algebra in everyday use which now bears Boole's name" (18), p. xi). And again: "That mathematics, in common with other art forms, can lead us beyond ordinary existence, and can show us something of the structure in which all creation hangs together, is no new idea. But mathematical texts generally begin the story somewhere in the middle, leaving the reader to pick up the thread as best he can. Here the story is traced from the beginning." (18, p.v) And, according to Francisco Varela: "By succeeding in going deeper than truth, to indication and the laws of its form, he has provided an account of the common ground in which both logic and the structure of any universe are cradled . . ." (42, p. 6).
The result of Spencer Brown's formal exercise to separate what are known as algebras of logic from the subject of logic, and to re-align them with mathematics is the explicit, and extremely elegant logical re-integration of the observer. His final chapter, entitled "reentry into the form" commences with: "The conception of the form lies in the desire to distinguish. Granted this desire, we cannot escape the form, although we can see it any way we please" (p. 69). It ends with:
Spencer Brown shares the concern of Buckminster Fuller and Keith Critchlow (22, 36) with the initial conceptualisation of a whole and its subsequent subdivision. He explores this using a powerful logical notation (18), whereas Fuller and Critchlow explore the structural implications in 3-dimensions. The latter would appear to be fundamental to representation and hence to comprehension. Jay Kelley, in considering the connection between man and his knowledge and the requirements for an adequate information system, arrives at similar conclusions .
6.3 Logical "curvature": Spencer Brown may have effectively established a means of encompassing the "curvature" of the logical universe of our science-dominated culture. In Part I it was noted that our culture was weak in its ability to handle anything "above" the top of the hierarchies of categories we care to distinguish. His work seems to offer a remedy. For it would appear that there is a "curvature" in the more fundamental hierarchies back to the (otherwise detached) person's involvement:
It is the observer/participant who links, through his own person, the top and the bottom of a hierarchy. Equally it is the observer/participant who links distinct hierarchies and is therefore challenged or fragmented by any conflict between competing coding systems to which his perception is subject.
Spencer Brown makes the point that "we cannot escape the form, although we can see it in any way we please" (p. 69). However all forms are not equally probable, as was argued above in the discussion of the numerical constraints on the subdivision of sets. His own work  explored the ordered emergence of certain forms. Rene Thom's (32) widely-acclaimed study is concerned with the stability of certain forms (in every domain of knowledge), of which the "islands of stability" encountered in the pattern of isotopes are a well-known example. His analysis extends to forms encountered in social systems and human thought [30, 31].
He argues that:
This "algebraic structure" (which he expresses in geometric terms) would seem to play a role in the human psyche which is functionally equivalent to the Jungian "archetype" . Although, even if this possible equivalence is invalid, this does not affect the argument below concerning such archetypes.
6.4 Self-reference and time: It is Francisco Varela (42) who has further developed the calculus of indications provided by Spencer Brown in order to deal with the many self-referential situations characteristic of our society.
In citing papers which address themselves directly to the self-referential nature of such systems, he notes that the topic is "normally avoided as undesirable difficulty (or circulus vitiosus)," and that such difficulties are rooted in language.
Consistent with the remarks of Rene Thom (above), and the preoccupations of von Franz (below), Varela argues that the duality of the producer and the produced (which embodies the producer, as in any category encompassing its user)
In his own extended calculus based on a 3-valued system, "self-reference, time, and re-entry (into form) are seen as aspects of the same third value arising autonomously in the form of distinction" (42, p. 21). Use of a third value enables the system to explore self-referential situations which are the basis for the limitations examined by Gödel (43). In his conclusion Varela describes his achievement as follows:
6.5 Number and time: Marie-Louise von Franz (of the C J Jung Institute, Zurich) has conducted an extensively documented study into the significance of number for mathematicians, in philosophy, and as symbols of psychological significance, in a deliberate effort to bridge the gap between psychology and physics. As she puts it, her remarks "balance to some extent on the razor's edge between philosophical-mathematical and numerical-symbolical statements" (ref. (9), p. 33 - 34). She deliberately bridges the gap between Western and other concepts of number, which is an aspect of a current debate into the wider interpretations of the concepts of science, space, and time, which have hitherto been supposed to conform conveniently to the Western versions (40) .
She notes that Niels Bohr has stressed that an important step had been taken toward realizing the ideal "of tracing the description of natural phenomena back to combinations of pure numbers, which far transcends the boldest dreams of the Pythagoreans" (9, p. I6). She argues that if we accept Wolfgang Pauli's contention that "certain mathematical structures rest on an archetypal basis, then their isomorphism with certain outer-world phenomena is not so surprising" (9, p. 19).
She sums up her argument as follows:
She concludes that:
In order to explore further, it is therefore necessary to return
Von Franz outlines the recommended programme as follows:
7. Number and logic
7.1 Beyond 2-term logic: multi-term systems: In the above argument the terms "set" and "system" have been used interchangeably since one of the characteristics of the sets of elements under consideration was identified as the complementarily of their elements. In discussing multi-term systems, a mathematician and director of industrial research J. G. Bennett clarifies further the kinds of sets to which these arguments apply (45, vol. 2, pp. 3-10). A set of elements taken without reference to any internal connections is called a class. A system is to be distinguished from a class, and he suggests rules for doing so [34, 35]. His summary of the characteristics of systems clarifies the definition of the sets considered here (see Annex 1).
Bennett notes that: "The properties of systems are usually studied in terms of their inner connectedness, but there is no general doctrine of systems based upon the properties that are associated with the number of terms by which they are constituted. This is strange, for philosophers have always been deeply concerned with the question whether or not there is a fundamental number system in the basic structure of reality." (45, vol. 2, p. 4) Such systems have however always had to be studied by using the conventional two-term logic. "Our usual language, though full of inconsistencies and ambiguities, can be adapted to the description of two-term systems. When the meanings of words and sentences are defined with special care, a logic is constructed that turns out to be the law of two-term systems. . .
The ambiguities and inconsistencies of our ordinary speech are not a defect, and recognition of them is a reminder that experience has more dimensions than logic. Analytical and sceptical philosophers have, during a hundred generations, exposed the barrenness of two-term thinking, and it becomes necessary to examine the possibilities latent in higher modes of thought." (45, vol. 1, p. 25)
In support of this investigation Bennett quotes Bertrand Russell on classical two-term logic: "The extension of the subject-predicate logic is right as far as it goes, but obviously a further extension can be proved necessary by exactly similar arguments. How far it is necessary to go up the series of three-term, four-term, five-term relations, I do not know. But it is certainly necessary to go beyond two-term relations." (46) Bennett draws attention to the widespread dualism in thought, feelings and instinctive reaction  and comments, as an example, on the difficulty of triadic (three-term) thinking. "Contemplation of the triad is not merely recognizing a third idea as the reconciliation of two contradictories, but rather seeing in the union of the three an exemplification of the fundamental relationship by which all experience is governed. So long as nothing more is at work than the primitive associating mechanism, to speak of the 'unity of the triad' conveys little meaning. In order to perceive this unity directly, a power of attention is required that comes only with a change of consciousness." (45, vol. 1, p. 26) He notes Russell's view that it appears to be beyond the ordinary power of man (47), although clearly both Bennett, and others believe that there is a way around the limitation (42, 48, 49).
As an indication of the route to be followed, Bennett remarks that:
The point to be emphasized as a result of the above argument is that the sets fundamental to the social sciences and policy-formulation constitute systems whose characteristics merit investigation irrespective of the nature of the terms in any particular case. Namely a 5-term set of values (concepts, principles, problems, etc.) has characteristics distinct from a 7-term set, irrespective of the values selected in either case. And, furthermore, such characteristics are solely dependent upon the total number of terms in the set.
7.2 Logic of inter-paradigmatic dialogue: In proposing a deliberately non-western complement to the Aristotelian logic of western science, Kinhide Mushakoji (49) introduces a third pole in the dialogical process to destabilize the intellectual equilibrium which exists between two paradigms dividing a given intellectual community into two opposing poles. He then argues, in the light of complementarily in physics, that:
This "logico-real" problem of the relationship between the logical and the reality level calls for a study of the morphogenesis of paradigms. Catastrophe theory helps us here since it sheds light on the different logical positions in the morphogenetical space. A major difference between the two levels of "significant" and "signifié" lies in the fact that the former is composed by discrete concepts while the latter is a continuous space. Therefore, it becomes necessary to apply a catastrophe theoretical model relating the continuous reality (i.e. the "signifié") with the discrete set of concepts (i.e. the "signifiant"). "
His reference to catastrophe theory, formulated by Rene Thom (32), relates this argument to that on logical "curvature" (above). Mushakoji then argues for a nonformal logical model developed in oriental logic on the basis of four lemmas (affirmation; negation; non-affirmation and non-negation; affirmation and negation). Such lemmas are concerned with the modalities according to which the human mind grasps reality rather than how human intellect reasons about it (51). He considers the lemmic approach to be a breakthrough in view of the possibilities it provides for overcoming the static ontology of the West inherited from Parmenides and highlighting the limitation of means-end rationality.
Mushakoji's concerns are shared in part by Sallantin (48) and Varela (see above), although they both elaborate on 3-term systems in much greater detail. The relationships between these three is elusive and a broader framework touch as Bennett's) raises questions: (a) of the possibility of 4, 5 or higher term systems, (b) of why the three authors are seemingly insensitive to the qualitative attributes of systems higher in the series and (c) of the implication for classification.
7.3 Number and N-term systems: In order to make further use of the programmes that Bennett and von Franz respectively set themselves, it is necessary to link the concept of N-term system (Bennett) and that of number as studied by von Franz. What these and other authors have each attempted, in one way or another, is to identify the qualitative characteristics to be associated with each term in the series:
Bennett argues the case as follows:
Sallantin also addressed this point (48).
Bennett argues that there are several other ways in which we can think about number, such as lead to cardinal or ordinal numbers. In addition the 'arithmetic quality', based on the inner relationships of a group, may be used to distinguish prime and composite numbers, for example. But even so
Bennett joins von Franz in recognizing that: "The search for the concrete significance of number is very ancient... At some unknown period ... man had already become convinced of this concrete significance, and must, therefore, have seen how a number can enter directly into events as experienced by himself." (p. 28) And: "If we are ever to free ourselves from the limitations of logical thinking, we shall have to discover a new significance in number; for number and logic, as we know them today, are inseparable." (p. 28)
8.1 Problems of comprehension: It is appropriate to note that work in the well-defined field of "multi-valued logic" does not seem to have had any impact on these concerns37. Nor does that on the "theory of numbers"33. It is only more recently in studies which face up to non-quantitative considerations with propositions for 3 or 4-valued logics that the nature of the link begins to emerge (48, 49). The reason for the lack of progress would appear to be that in both fields named above the problems of comprehension, and the status of the observer, are ignored despite the early efforts of Korzybski on general semantics (54). It is here that the questions of self-reference (see above) and the wider implications of complementarily are now significant in legitimating further investigation  (55).
Comprehension may be considered purely as a problem of "pattern recognition" in non-verbal data. This is now receiving considerable attention in some branches of information science concerned with the man-machine interface. It is a quantitative response to complexity and is of limited relevance here (although the illusion that the conventional quantitative mode is neutral and "value-free" is now being widely attacked (56, 57). A much more subtle problem is associated with the comprehension of qualities, and as such necessarily involves the observer actively to a greater or lesser degree.
The question is how qualitative distinctions can be comprehended and communicated. Clearly the problem does not even become apparent until differences in interpretation create difficulties. Even then it may be disguised by explaining differences as characteristic of different schools of thought, social backgrounds, educational levels, or cultures. The effect of such perceived differences on the medium used to portray the quality in question may even be the focus of appreciation, where the preoccupation is primarily aesthetic (painting, music, poetry, etc.), thus again disguising the problem. Where deliberate efforts are made to use words to define the meanings to be conveyed by other words, obvious discrepancies can be resolved whilst subtler ones remain. The systematic approaches currently explored by COCTA and INTERCONCEPT (58) may further reduce the problem.
Nevertheless, even when the ideal has been achieved of an agreed definition for a qualitative attribute (available in a universally understood language), a core problem still remains. The word-ensemble constituting the definition will be comprehended in different ways according to the capacity and inclination of the reader--even if the definition triggers a single gestalt perception of the quality rather than a serial perception of its multiple facets. It would be naive to expect that the ultimate definition of "beauty", "justice" or any other quality could be formulated in 1979--thus depriving the future of any possibility of comprehending, describing or expressing them more appropriately than is now possible. Similarly both a child and an adult may share a verbal definition of 'peace"--but their comprehension of it is likely to differ, especially if one has experienced the realities of war. Further elaboration of a verbal definition does not eliminate the difficulty and is quickly counterproductive.
The nature of the challenge to comprehension can be illustrated by the simple sequence of numbers: 1, 2, 3, 4, etc.:
(a) where 1 denotes any single entity isolated from its context, no demand on comprehension is made; higher numbers merely provide an arithmetic total. The totality is never more than an aggregate and the relationships between the elements are irrelevant.
(b) where 1 is used to denote a totality within which no element has been isolated, then use of higher numbers indicates successive degrees of subdivision of the original totality . With each higher number the total pattern becomes increasingly difficult to comprehend.
(c) where 1 is used to denote the totality encompassing the universe as experienced, then the comprehension demanded is associated in traditional cultures with a supreme being. Higher numbers then reflect hierarchies of such gods, each governing qualities of an appropriately lower level of abstraction .
(c) where 1 is used to denote the totality encompassing the universe as experienced and including the experiencer, the state or level of consciousness of the observer is necessarily affected. Higher numbers then denote successively more multi-faceted levels of consciousness at increasingly lower levels of abstraction. The focus of this paper is on complete sets which in some way aim to encompass a structured totality. These may raise problems of comprehension of type (b), (c) or (d) depending on the level of abstraction of the set elements and the degree of "insulation" of the observer.
There is currently great faith that when verbal descriptors are used as identifiers for such set elements, they will carry some universally understood meaning (e.g. peace, justice, human rights, development, democracy, etc.). As argued above, and as ongoing investigations are demonstrating (59, 60), this is far from the case. Such fundamental characteristics elude complete or even adequate definition by any particular set of words . Clearly the definition or label merely points towards a comprehensible experience. It is not the comprehension of that experience. ("The map is not the territory.")
8.2 Comprehension, remembering and mnemonic aids: The special problem in comprehending complete sets lies in the relationships between the interdependent elements. These are seldom explored in any verbal definition, thus detracting from its adequacy. But even if the member elements can be comprehended singly or in groups in serial fashion, remembering them is increasingly difficult and their relationships are lost as is any grasp of the totality.
It is at this point that various mnemonic aids are used in describing such sets in order to provide some reinforcement to memory. The crudest and most prevalent is a simple numbering of elements. At the other extreme are sophisticated diagrams showing their relationships. Attention has even been drawn to the advantages of interactive computer graphics as an aid to maintaining creative "thinking momentum" and obtaining a grasp of a total pattern . But as noted in Part 1 (Section 5), it is the mandala-type representations which constantly stress this mnemonic function. It is the manner in which they are designed to be used which recognizes the challenge to comprehension and causes attention to be focused (as with an optical lens) through the member elements disposed in an appropriate configuration.
The last sentence and other comments in the paper link back to the discussion of the previous section. But the technique is said to have been used only "intuitively and almost unconsciously" by S. R. Ranganathan in developing the Colon Classification.
8.3 The quagmire of number symbolism: the past: As a contrast, those preoccupied with number symbolism over the centuries have been quite deliberate in their attempts at seeking to associate numbers and concepts . Such investigations have from time to time been very fashionable, whether from Pythagoras onwards in the West , or in the East, as Ranganathan writes in his Prolegomena:
But although there have been many investigations and the literature is vast, the result is a veritable quagmire into which many have ventured and from which few have returned unsullied. This is not to deny that many of the eminent intellects who have been attracted to this question have not come up with valuable insights, but rather that it is now difficult to filter the significant insights from a rich smorgasbord of culture-bound speculations and outright nonsense which have accumulated over many centuries.
The investigations of von Franz and Bennett of the qualitative attributes of the simple systems are therefore to be welcomed because they successfully disassociate themselves from number mysticism in its more unfortunate traditional forms. Indeed, working independently, they provide the necessary complementary perspectives of psychologist and physicist which is von Franz's objective (see above). She herself explores material concerning the first four integers. Bennett ambitiously explores up to 12-term systems  - thus inviting misunderstanding, however due to the ever present problems of comprehension to which he himself draws attention.
The nature of the danger is illustrated by his results which are summarized in Annex 2. Although it is the most systematic and disciplined attempt (in the West, at least), its main weakness lies in the verbal descriptors. These are really only useful as tentative signposts lacking any indication as to how the referent is to be experienced. The problem, as with all number symbolism in the past, is that it is only too easy for a reader to assume that his own comprehension of the descriptor as defined is as complete as that which is intended (leaving aside questions of Bennett's own limitations and difficulties of comprehension). Although more limited in scope, the same reservations must apply to the verbal descriptors of seminal mnemonics .
This is the basic difficulty with verbal descriptors and their definitions however much effort (à la Academie Française) is made to govern their usage and significance. It is worth considering the possibility that those such as Pythagoras were aware of this problem, as well as of the others indicated above: level of comprehension, superseding dualistic logic, the need for mnemonics, and self-reference (including the distinguisher's relationship to the denotative mark). What better way could they use to embody the subtlety of their insights than through numbers and their interplay, specially since the above questions are all number-related? The use of numbers does at least continually confront each user with his responsibility for any verbal descriptors he chooses to associate (temporarily and according to circumstance) with the concept. More important, it continually challenges him to greater levels of understanding of that concept and the manner in which it relates to others. It also leaves the future free to reinterpret the concepts in other ways or to greater depths--a process which is full of pitfalls and discontinuities when culture-bound verbal descriptors are used as at present.
However numbers are sufficient only to the very few. They do not provide a concrete image for those incapable of sustained thought at that level of abstraction. They can however be readily associated with archetypal figures (divinities, etc.) which each constitute a highly complex composite of qualities comprehensible as a whole at many levels of understanding and according to ability (9). Such figures are characteristic of many cultures for which their nature is powerfully clarified by many vivid tales and myths concerning their relationships. Their value lies in their ability to clarify or intensify ideas or emotions through appeal to sense experience. By such symbols, "the abstract may be brought into the realm of the concrete, where it is immediately recognizable and meaningful" (66, p. vii). And indeed studies of the symbolic nature of medieval thought and expression "reveal in the medieval mind a web-like structure of abstract ideas and concrete realities so closely interwoven and interdependent that no serious gap was felt to exist between them." (66, p. vii)
8.4 The quagmire of number symbolism: the past resurgent: This is of course essentially a sympathetic assessment and it cannot be denied that much that was produced within this context, if not most, now appears at best as fascinating nonsense--but this is a predictable consequence of using the culture-bound verbal descriptors of interpreters feeding endlessly on one another's products. But it would be a great mistake to believe that modern society has completely resolved the issues to which number symbolism responded.
On the one hand very many scholarly or administrative papers now enumerate lists of fundamental issues, principles, values, problems, etc. (to be compared with the medieval predilection for N virtues, sins, principles, etc.) as discussed in Part 1. It is then the task of the classifier to prescribe some meaningful order. But for various reasons, society now faces a crisis of meaning which the plethora of studies is instrumental in aggravating rather than alleviating. And, despite the efforts of classifiers (who themselves have various preferences for number-governed ordering systems), such studies tend to achieve quicker oblivion than their medieval counterparts, and are just as meaningless to the uninitiated. Meaningful synthesis is rare and of limited relevance to societal problems. Comprehension and integration tend to result in spite of current enumerations and classifications and not because of them.
On the other hand, in an effort to render meaningful the nature of the complex issues which face society and the importance of the values by which changes should be guided, government agencies, social-change movements and educational authorities are now obliged to resort to symbols which can be satisfactorily communicated through the available media .
This requires that such symbols be easily comprehensible, coherent, and that they bear only a simple meaning (irrespective of the complexity of the issue). Because of the low status of "public relations", such symbols are only indirectly linked to the weakly ordered substantive items enumerated in agency programmes or in the scholarly studies on which they may be based. Where complex abstract notions must be communicated (e.g. concerning ecological systems), cartoon "personalities" are often used, appropriately adapted to each culture. Were it necessary to embody the characteristics of justice, beauty, love, etc. which were a preoccupation of the past, it is not unlikely that public relations would need to resort to symbols of updated versions of the superhuman beings on which that period so successfully projected its beliefs. (As it is, cartoon characters and film stars define the limits of our subtlety.) Were it considered necessary to show the relationships between the concerns of the different United Nations agencies (e.g. education, justice, agriculture, health, etc.), it is not unlikely that public relations would have to resort to an interplay of such personalized symbols in a modern series of "myths". The quagmire of the past has not been avoided. It is in process of being re-evoked, because the problems from which it arises have not been recognized.
8.5 Encyclopedic memory systems: Strangely enough it is only through the recent remarkably insightful work of historian Frances Yates (68) that the role of memory in relation to knowledge classification has been subject to a preliminary investigation. She demonstrates that: "The history of the organization of memory touches at vital points on the history of religion and ethics, of philosophy and psychology, of art and literature, of scientific method" (p. 374). She makes it clear that not only did the Greeks (and possibly the Egyptians) possess an art of memory, but that this art was widely practised, extensively developed up to the Renaissance, and instrumental in the growth of the scientific method. Briefly, the art involved the memorization of an ordered set of "places" (loci or topoi ) which effectively constituted a permanent system of filing locations--whether based on a building, a town, or a set of divinities . Onto these the user "impressed" images (imagines agentes, "corporeal similitudes") which would trigger access to the things or ideas to be remembered--a technique reminiscent of that described by memory and calculating prodigies in recent years. It is somewhat disconcerting that this lost art permitted its exponents to use over 100,000 filing locations (p. 120) and that these could be explored in any sequence. The importance of the art for orators, scholars, administrators, merchants, etc. is clear at a time when text reproduction was difficult and paper expensive.
What is even more disturbing is that with the Ramist educational reform in the 16th century, and increasing reliance on the printed medium, it is clear how the seeds were sown for the problems and dichotomy identified in the previous section. This reform explicitly rejected (for religious reasons) the use of memory triggering images, which seem to have been essential to the art, in favour of the present approach and its associated forms of classification. Memory is now considered as a "mechanical" facility only to be tested during examinations and otherwise to be enhanced by data banks processing information in serial order.
Whilst it would be foolish to deny the need for the reform, it seems clear that this cuts off some lines of investigation which could have proved fruitful (irrespective of the "nonsense" from which it is difficult to disentangle them). Despite the subsequent interest in memory of Bacon, Descartes and Leibniz, it does appear that insights were lost (or driven underground) with the rejection of the highly complex memory systems developed by Raymond Lull, Giordano Bruno and Robert Fludd. Although Yates acknowledges that, with the available information, their full scope eludes her, she makes it clear that they at least had more or less explicit concerns for:
8.6 Augmented comprehension: Having assembled much evidence, Yates bequeaths to others the problem of whether the Renaissance did in fact possess a secret memory technique for stimulating the human psyche to a wider range of creative achievement than ever before (p. 354). That the described techniques claim to provoke memory to retain the interrelationship between many elements in a whole pattern is clear. That this involved a preoccupation with complete and ordered sets in also clear, as is their relationship to number (in the light of von Franz's review of the same authors). Proportion, harmony and connection in the representation of such sets are considered vital to success in empowering this new comprehensive grasp which is consequently intimately related to artistic expression: poetry, painting, music, architecture, theatre . Current research on computer augmentation of intellect lacks this artistic dimension although it facilitates manipulation of categories (69).
The concern with personally meaningful vivid image ry is echoed in recent studies of symbols as signs charged with meaning (70), (71), (72). To exert their psycho-dynamic organizing effect such symbols presuppose homogeneity of signifiant and signifé (70, p. 20 - 21) Whether and how, such "charging" can be accomplishes is presumably the key to the question. Mircea Eliade has studied a primitive approach to this (73).
Contemporary interest is reflected in research on altered states of consciousness (74), and psychotropic research (75) although the process by which advertising and propaganda impart significance to isolated commercial or political signs is also of great importance. Other factors, such as iconicity, merit attention . But for the special significance of the configuration approach characteristic of the Renaissance representation of sets of categories-cum-symbols, insight can perhaps best be gained from contemporary use of the mandate as mentioned above (38). This preoccupation has been absent from western thinking until the work of Jung--it was with the rotoe of Lull, Bruno and company that development ceased. The technique must therefore have aimed to dissolve the dichotomy identified in the previous section, and to move beyond the neatly disciplined relationships of the concept triangle to a condition in which the "uninsulated" observer was impelled to move, change or create by exposure to symbols. That information no longer moves people to act, is a major preoccupation of those attempting to mobilize resources against world problems (76). The perspectives that Yates opens up suggests that the "quagmire" discussed above may conceal some valuable insights.
8.7 Convergence of concept triangle elements: a limiting condition: Given the preceding remarks, the question is whether anything useful can be obtained from the vast amount of material available on number symbolism in its different forms (see (9)). The answer would seem to be positive in the light of von Franz's approach. But there is an immediate problem of how to handle the subtle differences between authors and the way attribute sets are shuffled into new configurations. As noted by Varela above, the distinctions selected are as much a description of the author as of the subject matter. This question has been studied by W. T. Jones (17). It would seem that authors can get caught in a subtle trap which does not deny the significance of their insights but rather limits the sectors of society (or period) within which their interpretations can be fully communicated and for which they will be valid and socially significant.
Aside from refining methods for sifting and testing complete sets, their relationship to one another must be clarified. This may be alluded to in terms of their relationship to sets of progressively greater "generality", here-and-now "concreteness" or "inclusiveness" [54, 55].
There is a qualitative convergence but its nature necessarily escapes verbal delimitation . It is a challenge to the comprehension of the observer and ultimately to the knower-known dichotomy. And, furthermore, whenever "fundamental" sets must be produced, they can only constitute aspects of a more fundamentally integrated understanding which must necessarily emerge progressively if future society is not to be deprived of all possibility of creative insight in this domain--to say nothing of any more mature insights on the part of the author. Clearly only the future can progressively identify and give content to more fundamental sets. Closure cannot be assumed --or, if comprehended, then not communicated. Closure in this domain cannot even be premature; it is impossible with maturing individuals in an evolving society (except for strictly limited purposes which carry the seeds of their own mortality). Any attempt at closure therefore merely sets the stage for production of "improved" versions, with all the resultant non-rational dynamics between the advocates of each and the hiatus as one version replaces another.
It is the argument of this paper that in such complete sets of a given number of elements, the latter are characterized by a qualitative pattern independent of the nature of the set elements. And as the set becomes more fundamental this qualitative characteristic predominates. For, as such sets become more fundamental or general, the characteristics of the elements (labelled by words) are increasingly affected (in their significance to the observer) by the semantic field associated with the numbers (whether explicit or implicit) characteristic of the representation of a given set. The label words therefore introduce increasing confusion, since the degree of precision they are expected to carry is severely eroded in comprehension by a cloud of polysemantic associations that are progressively more irrelevant to the elements distinguished. And, the more fundamental the set, the more probable it is that the numbers characteristic of the representation would more effectively label the set elements which in any case increasingly approximate the semantic fields of the numbers embodied in the representation . There is in fact a convergence or melding of the elements of the "concept triangle" (word, meaning, referent) with the observer, who is necessarily incorporated into the referent by the set, if it is fundamental. The 4 terms form a "concept tetrahedron" . Recent work has shown the link between the status of the observer and information systems viewed in the light of relativity theory (81).
This recalls Spencer Brown's point cited above that "We see now that the first distinction, the mark, and the observer are not only interchangeable, but, in the form, identical." (18, p. 76) This is clearly however a limiting condition and for less fundamental sets identity is necessarily not assumed, particularly since it is not experienced .
Even though the limiting condition may be ignored, the fundamental sets of interest here are sufficiently close to it, that any use of words must be viewed with great caution . What the words attempt to label is better coded by numbers with their qualitative associations. Differences in formulation of fundamental sets arise because each assumes it is containing fundamental elements but is effectively only containing those evident from an aspect of an even more fundamental domain (e.g. associated with a particular numeric quality). And each tends to focus on different aspects without being able to incorporate others even if the formulator is aware of them. At this level, however, there is a high degree of isomorphism between the numeric qualities characteristic of the sets centered on different aspects. This may be used to clarify the content of sets, and to identify more fundamental sets, without relying too heavily on the words used to carry meaning in any particular case.
9.1. Characteristics of multi-term systems: The remarks of the previous section provide a context within which efforts at establishing the characteristics of multi-term systems can be considered as defined in Annex 1 . This question cannot be explored here. It serves as an indication only therefore, that the results of J. G. Bennett's exercise are summarized in Annex 2. This suffers from the disadvantage of not establishing explicit links to the rich variety of cultural and mathematical material reviewed by von Franz in her study of the first four integers. Such material should be used to interpret and broaden the meanings, otherwise Bennett's (or any other) particular orientation is too easily assumed to exhaust the meaning associated with each system--thus subjecting the approach to the difficulties raised in the previous sections.
Bennett points out that "no one system taken alone can exemplify the organized complexity of real structures. We usually need to take more than one system into account in order to gain the insights needed for understanding any existing structure that we find. According to the aspect of structure that happens to be relevant to a given purpose, a system of one order may be more useful than another." (45, vol.3, p. 11-12).
Also (bearing in mind the limited value of label words for the system attributes identified in Annex 1): "The series of multi-term systems is a progression such that each system implies all the earlier ones and requires those that follow. We cannot understand the triad unless we already group the notions of universality and complementarity, and the dynamism of the triad is not realized without the activity of the tetrad. The later systems are not only more complex and more highly organized than the earlier ones; they embody an understanding of reality that is more comprehensive and practical. The progression is from abstractness to concreteness." (45, vol.3, p. 12).
But: "Not all structures exemplify all stages of the progression to the same degree. A given structure may exemplify one attribute strongly and others weakly.... One other general property of systems remains to be considered. This we shall refer to as term-adequacy. If the terms of a system cannot be clearly discerned in a given structure, the required characters will be lacking and the system in question is then inadequately represented." (45, vol. 3, p. 13). Namely the set is weak in that attribute.
In the light of this argument, attempts should be made to explore a 3-term set re-interpreted as a 4-term system or more, particularly in the case of fundamental sets. In Bennett's study of systematics , he finds that: "for purposes of practical utility, the systems fall naturally in groups of four. The first four from the monad (1-term) to the tetrad (4-term) help us to see how structures work. The systems from pentad (5-term) to octad (8-term) show why they work and how they enter into the pattern of reality. The third group from the ennead (9-term) to the dodecad (12-term) is mainly concerned with the harmony of structures: that is, the conditions that enable them to fulfill tinier destined purpose." (45, vol.3, p.12)
9.2. Clarification of specific sets: Two procedures are outlined (in Annex 3) for the clarification of material on complete sets. Both procedures ensure that any given set is embedded in a context. In the first case, this is in relation to alternative (or more superficial) possibilities. In the second, it is in relation to more fundamental possibilities.
By such procedures the set is being tested and refined in a manner which should establish the constraints on its meaningfulness and communicability to those who--in contrast to its vigorous advocates - may be sensitive to other aspects of the context in which it is embedded . The procedures necessarily highlight the extremely limited value of dependence on the univocal, unambiguous meaning of any words (in definitions) used to label such sets or their elements.
It should be stressed that, in contrast to the usual competitive preoccupation, the concern is not with establishing any particular set as the most valid. Rather it is to give some understanding of the probability that any such set will be advocated, perceived as valid, or widely comprehended and communicated. At the same time it supplies a context for elucidating the meaning underlying whatever marks (words, numbers, codes, etc.) are used to identify a set and its elements.
10.1 The above sections have identified: the constraints on set formulation imposed by number: the importance for comprehensibility of representation in 3 dimensions; the impact of particular number choices on the consciousness of those exposed to such sets; the problems of comprehension and the role of memory; and the properties exemplified by sets of a given number of terms. These are brought into focus by the problems of representing and comprehending multi-term sets. The problems have been strongly emphasized. Even a brief perusal of Annex 2 makes it clear that a verbal explanation in linear text form dos not come near capturing the gestalt quality of most of the systems identified. Just as when the elements of a set are listed, the sequential presentation introduces the time dimension to an extent determined by the number of terms. Von Franz notes: "Detailed investigation revealed, however, that number, understood as a psycho-physical motion-pattern, is intimately connected with the problem of time" (9, p. 235). The linear scanning required is not consistent with holistic comprehension of the single underlying concept. The manner in which the elements stand as "un-time-bound" aspects to the set as a whole is lost .
10.2 It is for such reasons that Bennett, in his presentation of the systems in Annex 2, relies heavily on 2-dimensional diagrams with a high degree of symmetry. And indeed many complex structures are open to comprehension if they have a high degree of symmetry (82). The emergence of symmetry in science is also frequently considered an indicator of the adequacy of a description. As Rudolf Arnheim notes: "In a broader sense, symmetry is but a special case of fittingness, the mutual completion obtained by the matching of things that add up to a well-organized whole" (21, pp. 64-65). Symmetry has the special merit of enabling the mind to regenerate constantly those aspects of a pattern which fade from comprehension when they are not the focus of attention . It is in part for these reasons that asymmetric diagrams are seldom used for these purposes. Lack of symmetry limits the comprehensibility of conventional concept maps (83). Figs. 1-3 are thus interesting examples of "representational classification".
10.3 Given that symmetry is richer in 3 dimensions and that representation is then naturally more compact, the basic question still remains whether such packing of 3-dimensional structures should bear any isomorphic relationship to the manner in which concepts are "packed" in comprehension. Is it irrelevant that the geometry of such packing is fundamental to so many natural structures in the environment and to the design of artefacts? The argument may be made that concepts require an N-dimensional space as Rene Thom would seem to imply (see above). And yet he himself recognizes isomorphism between natural and social systems . And it is those very same natural systems requiring an "infinite dimensional space" which are so elegantly and symmetrically ordered (to one perception) in relatively simple 3-dimensional arrays (84, 131). Agreed, the N-dimensional space is required to order transformations and conflicts between such structures. But it would seem to be highly probable (particularly in the light of the ordering role of number) that there be a certain degree of isomorphism with "concept packing", at least in 3 dimensions and if only with regard to the iconicity of representations [3l, 64] (The very interesting question, of whether Thom's N-dimensional space can reflect the transformations and conflicts between such structures, namely the social dynamics of ideas and the organizations based upon them, is not an immediate concern here.)
10.4 Bennett, in presenting his schema (see Annex 2), makes use of several different 2 and 3-dimensional diagrams to symbolize a system of a given number of terms. He does this to bring out different qualitative aspects of the system in question. This suggests a much more general approach to the problem of representation using work in graph theory (see Annex 4)
10.5 Although the graph theory convention of points and lines may only be meaningfully representative to a segment of the population in western cultures , it is possible that symmetric patterns and solids are much more widely acceptable. Whatever the case, such structures may be used to order or classify the elements of a meaningful representation which could (and does traditionally) employ other forms and media, e.g. animation , dance , drama, ritual, music (90-95). Part of the general inability to perceive such underlying structures lies in the widespread "visual illiteracy" discussed by Arnheim (21, p. 294 315) - although "structural illiteracy" draws attention to an even more neglected aspects of it. (It is likely that there is a whole series of unrecognised nonfigurative classificatory "handicaps" equivalent to such "hidden" disabilities as dyslexia, discalcula, arhythmy, etc).
10.6 There is also good ground for arguing with Fuller (1) that ideal forms such as polyhedra conceal a basic design problem which must be solved to obtain a more complete representation in concrete reality. He does this by generating dynamically stable "tensegrity structures" each based directly on a given polyhedral form  (96). In this design problem and its solution may well lie the clue to the limited utility of ideal forms for representation, comprehension and (above all) effective implementation. For this reason, the author has explored the possibility of using tensegrity structures as a basis for new approaches to the representation of concept and problem complexes, and the creation of new kinds of organization (97, 102). Clearly this is relevant to the representation of the sets of interest here (98, 99,101).
10.7 The above procedures result in the generation of a multitude of
symbols which may be enrichened in various ways (e.g. colour coding, etc).
The question arises as to whether this multiplicity is not undermining
the original objective of representing and communicating the governing
central concepts--particularly since it is what already characterises the
representations of sets of various kinds. However, in remarking on the
But the point is that these divergent forms, and those arising from the procedures above, are generated by rules governed by numbers. The variations emerge from a general pattern or number field which we are slowly coming to understand (e.g. von Franz has a chapter on "Archetypes and numbers as 'fields' of unfolding rhythmical sequences" in which she grapples with the question).
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:
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:
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:
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:
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
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:
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:
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.
Abstract | Notes | References | Annex 1 | Annex 2 | Annex 3 | Annex 4
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