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) [19] against such a representation may in certain cases constitute an ultimate constraint on clearly distinguishing the elements in a set.
7. Three-dimensional constructs
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 [20]. 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 [21] .
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:
"The aesthetic element is present in all visual accounts attempted by human beings. In scientific diagrams it makes for such necessary qualities as order, clarity, correspondence of meaning and form, dynamic expression of forces, etc. The value of visual representation is no longer contested by anybody. What we need to acknowledge is that perceptual and pictorial shapes are not only translations of thought products but the very fresh and blood of thinking itself. . ." (21, p. 134).
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 [22], 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 [23]. 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 [4]. 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 [24] 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 [25].
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:
"We therefore endeavor in the program outlined here to free our intuition from three-dimensional experience and to use much more general, richer, dynamical concepts, which will in fact be independent of the configuration spaces. In particular, the dimension of the space and the number of degrees of freedom of the local system are quite arbitrary--in fact the universal model of the process is embedded in an infinite-dimensional space." (32, p. 6).
He does however support the geometric representation argued above:
"I should like to have convinced my readers that geometrical models are of some value in almost every domain of human thought. Mathematicians will deplore abandoning familiar precise quantitative models in favor of the necessarily more vague qualitative models of functional topology; but they should be reassured that quantitative models still have a good future, even though they are satisfactory only for systems depending on a few parameters." (32, p. 324).
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.