Representation, Comprehension and Communication of Sets
the Role of Number (Notes)
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Notes to Parts 1, 2 and 3 Representation, Comprehension and Communication of Sets: the Role of Number (1978)
Notes to Part 1
 Obtaining a "good fit" is essentially a problem of design and indeed in his influential book on the subject, Christopher Alexander (ref. 2) devotes several chapters to the question. Deciding on the boundaries of a set and distinguishing its elements is a problem of design as Alexander would see it (as is the problem of elaborating a suitable representation, particularly when the relationships between the elements are taken into account). He notes:
"The ultimate object of design is form . . . every design problem begins with an effort to achieve fitness between two entities: the form in question and its context. The form is the solution to the problem; the context defines the problem. In other words, when we speak of design, the real object of discussion is not the form alone, but the ensemble comprising the form and its context. Good fit is a desired property of this ensemble which relates to some particular division of the ensemble into form and context." (p. 15-16)
"What does make design a problem in real world cases is that we are trying to make a diagram of forces whose field we do not understand. Understanding the field of the context and inventing a form to fit it are really two aspects of the same process. It is because the context is obscure that we cannot give a direct, fully coherent criterion for the fit we are trying to achieve; and it is also its obscurity which makes the task of shaping a well-fitting form at all problematic. . . I should like to recommend that we always expect to see the process of achieving good fit between two entities as a negative process of neutralizing the incongruities, or irritants, or forces, which cause misfit." (p. 21-24)
 It would be a simple matter to select, from papers of a wide range of disciplines or administrative activities, lists of "basic points" made (possibly with sub-point coding if any). Irrespective of content, the number of points should follow a pattern which could suggest interesting lines for future research. A rich source of popular material is The Book of Lists, edited by David Wallachinsky, et al. (New York, William Morrow, 1977) from information supplied for The People's Almanac. It contains 377 lists on all topics. Even if biased toward a particular format (of the Almanac) or to conform with the style of earlier lists, the results are still indicative. (1-10 items per list, 54.6%; 11-20, 35.0%; 21-30, 7.2%; 31-40, 1.3%; 41-50, 0.5%; 51-60, 0.5%; 61-70, 0.3%; 71-80, 0%; 81-90, 0%; 91-100, 0.3%;100+, 0.5%. With 10 items, 39.3%; 15, 8.0%; 20, 6.4%). A new edition is in production.
 For a comment on the general structural significance of the peaks in the curve, see ref (1), p. 604-607.
 Herbert Simon (ref. (5), p.39-40) notes that such constraints can now be less plausibly explained by a single parameter and that under certain circumstances the value falls from 7 to 2 (on which point see the peaks in the curve of Fig. l). It appears that it is short-term memory which can only handle information by chunks of 7. This constraint does not apply to long-term memory. However this does not change the fact that the sets under discussion usually contain about 7 chunks or less - possibly because access to such sets and their representations is necessarily via short-term memory.
 Alex Bavelas and Howard Permutter, classified work done at the Center for International Studies, MIT, quoted in "The relation of knowledge to action", by Max Millikan (see (40) p. 164).
 Antony Jay, in (8), identifies size limitations for organizations: "ten group" of 3-12 (work group, project group, task force); "camp" of 20-60 (work group plus those dependent upon their activity or servicing their requirements); "tribe" of 300-1000 (identity group, mutual recognition); "kingdom" of 5,000-60,000 (administrative, social, cultural or military coherence); "empire" of 100,000+. It would be interesting to explore the change in the nature of government once the number of ministries and cabinet ministers exceeds the critical number for small groups (see (7)) and the usual constraints on span of control.
 In the light of the NSF exercise, it will be interesting to note the organization of the results of the exercise launched in 1978 by the US Office of Technology Assessment "on the identification of major long-range problems and opportunities facing American society".
 An intergovernmental meeting may give rise to a many-pointed declaration as the basis for a programme of action. This is then progressively condensed into a programme grouped under a number of headings within the number constraint noted. (Consider the evolution of the UN Environment Programme from 1972, for example.) Where an action programme does not emerge, the number of points remains unconstrained by the limit, particularly in legalistic declarations of principles such as the Universal Declaration of Human Rights (31 articles). But even here, such a declaration would be unacceptable if it had 131 articles, so anew constraint may be in operation.
 Magoroh Maruyama has consistently argued that the hierarchical orientation is only one of four culturally determined epistemological standpoints and is characteristic of the following cultures: European (and American), Islamic, Hindu, Japanese, Yamato, Kwekiutl, for example (see (11) and (12)).
 "It appears that the attention paid hitherto in exact science to increasing precision of analysis into smaller and smaller parts needs now to be supplemented by a method capable of representing the processes of complex systems composed of many parts. But there is no sign as yet of a simple comprehensive method of describing the changing form or structure of a complex of relationships." (ref. (37), p237)
 Problems also arise when creation of the set is expected to improve the status and prestige of the producer at the expense of others --who may have produced their own or may thereby be challenged into doing so. Such dynamics cannot be discussed rationally in the same arena as for the content.
 An interesting example is the single sheet chart of the biochemical metabolic pathways in living systems: see (15).
 "The neophyte can ... grasp this unstable universe of powers which are both within and without. For him the symbol is like a magical and irresistible admission into this formless and tumultuous tangle of forces. With the symbol he grasps, dominates and dissolves it. Through the symbol he gives form to the infinite possibilities lying in the depths of his subconscious, to inexpressed fears, to primordial impulses, to age-old passions." (See (38), p. 22.)
 Jones discusses seven pre-logical axes of bias and their application to scholarly debates in the arts and in the sciences. (17)
 "The main difficulty in translating from the written to the verbal form comes from the fact that in mathematical writing we are free to mark the two dimensions of the plane, whereas in speech we can mark only the one dimension of time" (ref. (21), p. 92). And in conventional text, where subscripts and superscripts are not permitted, writing becomes as restricted as speech.
 "Any aggregate that is neither completely ordered nor completely disordered must have hierarchical aspects, but the perception of the levels of the hierarchy requires the recognition of a two-dimensional surface to define each three-dimensional unit in accordance with Euler's Law" (ref. (10), p. 81).
 Of special interest in the 2-dimensional case, is the situation when line coding is not permitted and ways have to be found to fit shapes together. The book by Critchlow (22) explores the variety of regular patterns which result. These patterns can be important when any attempt is made to represent sets and their subsets by nested areas.
 "If a fourth spatial dimension cannot be visualized, it is probably because geometry is concerned with relations that can use perceptual and physical space as a convenient image up to the third dimension, but no further. Beyond that limit, geometrical calculations - just as any other multidimensional calculations, such as factor analysis in psychology - must be content with Fragmentary visualisation, if any. This also means probably putting up with pieces of understanding rather than obtaining a true grasp of the whole." (ref. (21), p. 292.) Note that in ref. (39) it is argued that higher dimensions can be suitably visualized.
 "When man employs nature's basic designing tools, he needs only generalized angles and special-case frequencies to describe any and all omnidirectional patterning experience subjectively conceived or objectively realized. For how many cycles of relative-experience timing shall we go in each angular direction before we change the angle of direction of any unique system-describing operation?" ((1), p. 248-9).
Notes to Part 2
 It seems to be time to recognize the extraordinary resistance of each social science profession to the application of the insights of its own discipline to itself as a social group, and to integrate this into the research process. There is a real blindspot, as has been noted with respect to one discipline at least (but not necessarily by many of its practitioners): "But sociologists have been reluctant to test empirically the relevance of many hypotheses... for the development of knowledge in sociology. Studies on the impact of the social organisation of the discipline, the prevailing climate of opinion, and the social background and personal values of researchers have been out of fashion. . ." (p. 45) and "sociologists are notorious for studying everything except their own discipline and its institutional patterns" (p. 55) from the introduction to The Sociology of Knowledge, edited by J. E. Curtis and J. W. Petras. Duckworth, 1970.
 Jay Kelley remarks on an associated phenomenon: "When an investigator acquires data and facts, he is improving order within his own sphere. The entropy of the experimenter and his data pad and records is improving, but the moment the observer separates himself from his data, he no longer can claim the fun possession of value of the information; the information is continually devalued as the observer accumulates other knowledge and as time passes. These observations lead to deeper questions of the nature of order and its human implications ((41), p. 179). For him: "Value implies accessibility to information, which reflects how it is ordered or its entropy."
 "It seems to be quite evident that oneness stands out as the origin of the structure from whence feasible patterns can emerge as rigidly hierarchical, associative, or sequential. Of these the hierarchical patterns appear to have lasting qualities while associative and sequential features may confer richness and flexibility... Thus, whether negotiating a computer or a sociological system the human conceives patterns from his singular frame of reference and must see and interpret the learned pattern from this state of oneness. Language and other standard ordered patterns tend somewhat to alleviate the plausible dilemma of a human having to interpret for himself from oneness to many independent patterns."((41),p. 195)
 "The theme of this book is that a universe comes into being when a space is severed or taken apart. The skin of a living organism cuts off an outside from an inside. So does the circumference of a circle in a plane. By tracing the way we represent such a severance, we can begin to reconstruct, with an accuracy and coverage that appear almost uncanny, the basic forms underlying linguistic, mathematical, physical, and biological science, and can begin to see how the familiar laws of our own experience follow inexorably from the original act of severance. The act is itself already remembered, even if unconsciously, as our first attempt to distinguish different things in a world where, in the first place, the boundaries can be drawn anywhere we please. At this stage the universe cannot be distinguished from how we act upon it, and the world may seem like shifting sand beneath our feet." ((18), p. v)
 He argues in favour of the fundamental validity of the ancient philosophical intuition that the "dynamical situations governing the evolution of natural phenomena are basically the same as those governing the evolution of man and societies, profoundly justifying the use of anthropomorphic words in physics. Inasmuch as we use the word "conflict" to express a well-defined geometrical situation in a dynamical system, there is no objection to using the word to describe quickly and qualitatively a given dynamical situation. When we geometrize also the words "information", "message", and "plan", as our models are trying to do, any objection to the use of these terms is removed." ((32), p. 323)
 Rene Thom himself develops a set of "archetypal morphologies" ((32), p. 307)
 Note Marcel Granet's extensive study ((44), p. 127 - 248) of the use and significance of number in Chinese thought as a means of classifying and expressing qualitative distinctions.
 The rules given ((45), vol. 2, p. 7) are in effect incorporated into the system definition given in Annex 1. He also makes the following points. A class is an externally determined set of members and a system is an internally connected set of terms. When the internal connections are disregarded, the set degenerates from being a system to being a class. No actual class is wholly free from inner connections so that classes are abstractions whereas systems are concrete (although to different degrees).
 "it is also possible to have an 'ordered' class or series, such as the first ten numbers. This is not a true system, for it does not take any account of the mutual relevance of the terms except their order. Nevertheless, since the ordinal numbers are in certain respects intermediate between classes and systems, we cannot regard the distinction between class and system as wholly free from ambiguity." ((45), vol. 2, p.4)
 "In the realm of ideas, man can count up to two and sometimes, in specially favourable circumstances, as far as three. He has no notion at all of what would be required for entertaining richer combinations. This limitation applies not only to man's thought but also to his feelings and to his instinctive processes. His judgements of feeling reduce always to the choice between like and dislike, attraction and repulsion, interest and boredom. His instinctive reactions have the same dualism of pleasure and pain, of activity and repose, of stimulus and inhibition." ((45), vol. 1, p. 21)
 Varela ((42), p. 21) notes that to introduce more than two values in a calculus or a logical system has been a current field of investigation since Lukasiewicz (52). Such additional values are usually interpreted in terms of probability or necessity. Günther (53) has been alone in pointing out another possible interpretation of many-valued logics, namely as a basis for cybernetic ontology, that is for systems capable of self-reference.
 Matila C. Ghyka however draws attention (64) to the Hamiltonian Principle of Least Action as fundamental to further reflection on these matters. He and Bennett ((45), vol. 1) also refer to the implications of transfinite numbers in which the whole can be seen as reflected within the part.
 "Far from restricting our efforts to put questions to nature in the form of experiments, the notion of complementarily simply characterizes the answers we can receive by such inquiry, whenever the interaction between the measuring instrument and the objects forms an integral part of the phenomena." (Niels Bohr, in Essays 1958-1962; on atomic physics and human knowledge. Wiley, 1963).
 Subidvision of a set (by the act of distinguishing elements) has been used rather than articulation, although the latter is preferable. It implies a respect for the functional relationships between the system elements (and an expression of them), whereas the former is solely concerned with their classical logical relationships.
 See (9), p. 144. Jungian psychology regards such gods as archetypal figures representing energies locked within the individual human psyche.
 There is a difference between archetype and archetypal image. The latter is always variable, but behind these variants stands a constant, non-perceptual pattern. According to Jung a conscious and invariable definition of its meaning is not possible.
 "Unfortunately, my abstract model tends to fade out when I get a circuit that is a little bit too complex. I can't remember what is happening in one place long enough to see what is going to happen somewhere else. My model evaporates... In all fields there are such abstractions. We haven't yet made any use of the computer's ability to 'firm up' these abstractions. I think that really big gains in substantive scientific areas are going to come when somebody invents new abstractions which can only be represented in computer graphical form." (61)
 It is Ghyka (64) who has traced the Pythagorean developments, recording the modern mathematician's tendency to dissociate himself from that perspective. However Sallantin notes: "D'ailleurs est pythagoricien quiconque percoit un lien naturel entre le nombre Un et l'idée d'Unité, entre le nombre Deux et l'idée de Dualité" (48). He demonstrates that conventional arithmetic is in effect one of four types of arithmetic; the others have increasing degrees of indeterminacy and are more suited to handling problems in biology and physics. He proposes that one of them should be used as the basis for trialectic logic.
 It is interesting to compare Bennett's exercise (in Annex 2 with Neelameghan's (63) application of seminal mnemonic as a pattern for systems analysis, which makes an attempt to associate ideas with the numbers 1 to 7. Although quite independent, there would appear to be some similarity between them.
 "Our community life is perhaps so structured that the very moment we seek to grasp reality in all its concreteness we run after simulacra. The present set of texts takes as it hypothesis that illusion and simulation have assumed in the Twentieth Century a power hitherto without parallel. We have entered, perhaps, the age of the simulacrum." Special issue summary of Traverses (Paris, Centre national d'art et de culture), 10, fevrier 1978.
 "Topics are the 'things" or subject matter of dialectic which came to be known as topoi through the places in which they were stored" ((68), p. 46)
 Yates quotes a pre-Socratic text on memory, dated about 400 BC: "For things [do] thus: for courage [place it] on Mars and Achilles; for metal-working, on Vulcan; for cowardice, on Epeus" ((68), p. 44)
 Amongst others, Yates quotes Marsilio Ficino: "Aristotle and Simonides [the inventor of the memory technique] think it useful to observe a certain order in memorizing. And indeed an order contains proportion, harmony and connexion." ((68), p. 163)
 There is of course a paradox associated with any such ultimate set. The act of distinguishing it necessarily establishes at least two subsets, for it necessarily incorporates the distinguisher as Spencer Brown demonstrates (18).
 This relates to Jung's concept of "unus mundus" as an expression of the unity of existence founded: "on the assumption that the multiplicity of the empirical world rests on an underlying unity, and that not two or more fundamentally different worlds exist side by side or are mingled with one another. Rather, everything divided and different belongs to one and the same world, which is not the world of sense but a postulate whose probability is vouched for by the fact that until now no one has been able to discover a world in which the known laws of nature are invalid" (77).
 One is reminded of the possibility of a qualitative analogue to the "big bang" cosmological theory which postulates the universe as having been elaborated from a single homogeneous ball of proto-matter. That the analogue might operate on standing wave principles, also merits reflection (note ).
 Von Franz ((9), p. 77) notes the Chinese use of numbers as qualitative fields whose internal numerical structures "represent time phases of the fields dynamic internal structure." She quotes: "The ontological and logical ordering (of numbers) is translated into rhythmical and geometrical images. On account of their descriptive power, as exponents of concrete analysis, numbers are classificatory, and for that reason used to identify concrete sets. They can serve as rubrics, for they indicate the various types of organization which are imposed on things when they are manifest in their proper order in the cosmos." ((44), p. 123)
 In the light of the scheme presented in Annex 2, the 3-term "concept triangle" (see (59)) is preceded in the series by the traditional 2-term "knower-known". It may be followed by the 4-term "word-meaning-referent-observer" (and it is this which blurs into a single set at the limit condition). This series bears an interesting relationship to that derived from Galtung's "theory-fact-value" triangle as discussed in the conclusion. Note the terms change significance with addition of a term (see note ). Zeman (80) specifically proposes a "gnoseological triangle": objective reality, the observing subject (i.e. conscious man), and expression. This combined with the concept triangle, constitutes a tetrahedron (4-term).
 Except possibly through peak experiences (see (79)). Von Franz stresses Jung's view "that there is little or no hope of illuminating this undivided existence except through antinomies. But we do know for certain that the empirical world of appearances is in some way based on a transcendental background." ((9), p. 9). Historically this has been represented by symbols (p. 303).
 It is rather as though different witnesses to a crime were to attempt separately to describe the criminal by establishing an Identikit portrait (a definition) using the kit components (words). Not only do the portraits differ from one another, but possession of a portrait however good does not magically result in the capture of the person identified.
Notes to Part 3
 On this point, the relationship of time to the variety of standing wave configurations of sand particles vibrated on thin plates of metal merits attention (see ref. (78)).
 Rene Thom, on the first page of his study, makes the point that: "recognition of the same object in the infinite multiplicity of its manifestations is, in itself, a problem (the classical philosophical problem of concept) which, it seems to me, the Gestalt psychologists alone have posed in a geometric framework accessible to scientific investigation" ((32), p. 1).
Rudolf Arnheim in discussing the same question, notes that Gestalt psychologists recognize a tendency to "good form" or "well organized structure" (88). L. L. Whyte sees all mental processes such as memory, classification, choice, and will as "displaying a movement toward greater three-dimensional spatial order, symmetry, or form". And such morphic processes "are directly responsible both for the existence of forms, and of brain-minds themselves generating forms and being responsive to forms." ((85), p. xvi)
Jean Piaget also makes points which could be interpreted to be in support of this position: "As a result, spatial structures, from the biological point of view, bridge the gap between logico-mathematical structures, the nature of which is still unknown, and those structures which are either hereditary or, as is sometimes the case, acquired by learning" ((86), p. 309). Also: ". .. cognitive functions are an extension of organic regulations and constitute a differentiated organ for regulating exchanges with the external world. The organ in question is only partially differentiated at the level of innate knowledge, but it becomes increasingly differentiated with logico-mathematical structures and social exchanges or exchanges inherent in any kind of experiment." ((86), p. 369).
 I am indebeted to Colin Cherry (On Human Communication, 1968) for this insight (87).
 It could be interesting to explore the possibilities of portraying each term in a multi-term system by a human or animal figure and animating their interaction on graphics devices to produce a cartoon effect, using a computer programme governed by the original structure. (Supposedly many folk tales are based on such structures)
 Rudolf Arnheirn notes ((88), p. 207-8) that: " . . . one must assume that structural characteristics of visual form are spontaneously related to similar characteristics in human behaviour. We have called this type of symbolism 'isomorphic' because this is the term used by gestalt psychologists to describe identity of structure in different media.... The gesture of a dancer. . . contain(s) structural features whose kinship with similarly structured mental features is immediately felt." Ritual dances are based on this insight and even have their modern advocates: Steiner's eurythmy. Gurdjieff's movements, Ichazo's Arica movements, and the like. The aim being to penetrate and express the more fundamental forms and to use them as a means of classifying experiences within a functional whole. It is no accident that Keith Critchlow in a book on design (22) incorporates Laban's use of the icosahedron for dance notation (89).
 It is interesting that in order to solve the problem Fuller has effectively had to confront the constraints of the basic duality with which our culture is faced as it is reflected in material forms. The "primitive" structuring effects of the duality have to be bypassed within a larger whole which depends on them for its integrity. This requires many more elements than the ideal forms, thus conforming to Bennett's insight that a higher number of terms is required to provide a better approximation to reality. (Although the higher number is effectively reduced by the encoding properties of the underlying polyhedron in each case).
 In terms of the status in society of fundamental sets, there would seem to be an amusing parallel between the role of temples to different deities in the Roman Empire and that of international agencies with respect to global society. Both the temples and the agencies each base their actions on welldefined sets of qualities.
 Possibly only by anthropomorphizing the representation of "world problems" which society faces will their nature and interplay be communicable to an adequate degree particularly in terms of how they are ordered or governed.
 Interpreting Bennett's scheme (Annex 2), It can be very tentatively suggested that sets of the following numbers of terms are required to encounter these current issues: mediation, relationships (3-term); retraining, resource renewal (6-term); organizational systems (7-term); worker individuality and human development (8-term);environmental processes (9-term); social innovation and creativity (10-term). Each stage requires more subtle skills in organization and governance in order to tolerate the additional freedom (i.e. reduction in imposed order) it implies and demands; in fact the challenge to policy at this time seems to lie with the I l-term approach of balancing order and disorder, rather than attempting to eliminate the latter (100). But understanding, if there is any, in terms of such multi-term sets seems to be only instinctive or intuitive, aided by frantic "rational" (2-term) attempts to order the component elements in isolation from each other, and a "fire-fighting" response to problems arising from their interactions - when they can no longer be ignored.
 Chinese philosophy, as exemplified by Lao Tzu and Chuang Tzu, is full of references to the attitude implied by the 12-term approach. This is also evident in the attitude advocated in Eastern martial arts, see Herrigel (103). It would be interesting to examine the Study of S. Boorman in this light (104). Clearly a strategy based on thinking in N+l terms is bound to out-manoeuver one based on only N terms, as well as appearing unpredictable and disorderly to the latter.
 Clearly Ashby's Law (105) concerning the necessary complexity for a control system also applies with regard to the complexity of representational device. However there is the paradox that representations which are as complex as that which they represent are of questionable value.
 Yates presentation (68) concerning rotoe suggests the possibility of an approach intermediate between conventionally static classification schemes and computer-based mathematical models (e.g. of social systems), namely a memorable pattern of classification possibilities implying the complete range of relationships between a set of categories.
 Don (107) discusses a model of the brain put forward by Powers (108) and based on ten hierarchical levels of control: musculo-skeletal intensity, sensation, configuration, transitions, sequence, relationships, control of patterned logical processes, principle, concepts. Again this bears comparison with a scheme such as Bennett's (Annex 2).
 Recent work needs to be related to that of Zipf (109),used by Kelley (41), for despite revision by Mandelbrot (110), it is strongly critized by Rapoport (111). There may be a link in this context between Zipf's Principle of Least Effort and the Hamiltonian Least Action Principle (see note ).
 Margalef (113) suggests that it is possible to measure the "maturity" of an eco-system as closely related in one respect to its diversity or complexity, and in another to the amount of information that can be maintained with a definite spending of potential energy. This is a question of patterning. A highly diversified community has the capacity for carrying a high amount of organisation and information, and requires relatively little energy to maintain it. Conversely, the lower the maturity of the system, the less the energy required to disrupt it. Anything that keeps an eco-system oscillating (or "spastic") retains it in a state of low maturity. (Hence the danger of simplistic reorganisation of organisational, conceptual or value systems.) A mature ecosystem has a maximum number of trophic levels of which, curiously in the light of this paper, the number rarely exceeds 7.
 From Yates presentation (68), one may suspect that Giordano Bruno's "seals" served this purpose in relation to his own texts. A similar role may be ascribed to the lapidiary seals collected by Rziha (114) as reported by Ghyka (64)
 Interesting examples, which have never been cross-linked, include Abellio (115), Buckminster Fuller (1), Haskell (116) Dodd (117), Lock Land (118), Langham (29), Young (25) and (26), Bennett (45). The Eastern equivalent which has attracted the most attention is the I Ching: see Needham (119), Blij (120), Gardner (121), Sung (122). The recently remarked link between the I Ching code and the genetic code raises many questions, see Schonberger in (121)
 Bennett notes ((45),vol.3,p.25) that: "Many of the difficulties in the interpretation of natural phenomena arise from treating qualities as if they remain the same in passing from one system to another." (e.g. from a 2-term system to a 3-term system, the added third term modifies the qualities originally expressed by the other two terms)
 Addition of "representation" as a fourth element is almost certainly insufficient simply as a passive pattern, at the best inviting to the attention. As with language in the West, it may simply classify experience without opening the observer to the action it suggests. Here lies a danger. Already with crude representations users of the flood of text information are overloaded to the point of blockage or effectively insulated from experience by suitable explanation and depiction. Some more iconic sophisticated representation may only reinforce the user's passivity, whereas appropriate representation may offer the user the visual configuration through which to act participatively and experientially (cf. the contrast between McLuhan's "hot" and "cool" media). "Activating potential" would thus seem to be a fifth element in the series and an appropriate constraint on representation. (I am indebted to Anthony G. E. Blake, for provoking these insights.).
 See (128) "Both geometry and topology deal with the notion of space, but geometry's preoccupation with shapes and measure is replaced in topology by more abstract, less restrictive ideas of the qualities of things.... (giving). . . a richer formalism to adapt as a tool for the contemplation of ideas. . ."
 The fruitful area identified is the use of a non-Boolean (non-distributive) lattice structure of complementary or dialectically developing languages (perspectives, categories) which reflects the logic of quantum mechanics (140,141). A developmental sequence may emerge either as the result of research or of comprehension (cf. programmed learning pathways) through stages which appear mutually incompatible for some period. From the diagrams used by Heelan and de Nicolas both sequence and complementarily can simultaneously be represented by developmental pathways of polyhedral form which, in their examples, privilege a single vertex (e.g. in a cubic structure) as the "least upper bound element". Richer possibilities, corresponding to non-dualistic complementarily of multi-term sets, could well become comprehensible in the light of the full range of polyhedral structures nesting polyhedral pathways to distinguish levels of co-existing incompatible perspectives (possibly linked by experiential or non-cumulative learning pathways, as might be represented by a circular chain of overlapping Venn circles) from levels at which complementarily is evident. Such polyhedral encirclement, of an unknown to be defined progressively without closure, could facilitate the relationships between viewpoints as discussed elsewhere (142).
Abstract | Part 1-3 | References | Annex 1 | Annex 2 | Annex 3 | Annex 4
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