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 [63].
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 [43]. 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 [30]. 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 [65], 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 [66], dance [67], 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 [68] (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 apparent
| Fig. 1: Schema of positional space relations. The relations of the outer ring contain identity as their inner relation. The relations of the inner ring contain delatation. Dilatation itself is a purely inner relation. (Source: W. von Engelhardt, "Sine und Begriff der Symmetrie".ln: Studium Generale 6 (1953) No. 9, S. 524 reproduced by permission of Springer Verlag and W. van Engelhardt) |
| Fig. 2: Schema used to interrelate information of different qualities of order in experimental music. (Reproduced with permission from: Henri Pousscur: Fragments théoretiques sur la musique experimentale. Bruxelles: Ed. de l'lnstitut de Sociologic de l'Universitc Librc de Bruxelles. ) |
| Fig. 3: Interrelationship of 4 activities at 3 levels in an organisation resulting in 12 sub-systems linked by input-out variables and by control variables. (Reproduced with permission from: Bernard Walliser: Systemes et Modeles; introduction critique a ['analyse des systemes, Editions du Seuil 1977, p. 115) |
| Fig. 4: Emergence and classification of tones governed by some integer ratios (deduced from Platonic texts) and their representation in concentric mandate form. (Reprinted by special arrangement with Shambhala Publications, Inc., 1123 Spruce Street, Boulder, Colorado 80302 U S A, from: "Myth of Invariance" by E. G. McClain, p. 168. Copyright 1978 by Shambhala Publications, Inc.) |
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).