Representation, Comprehension and Communication of Sets
the Role of Number (Notes)
- / -
Notes to Parts 1, 2 and 3 Representation, Comprehension and Communication of Sets: the Role of Number (1978)
Abstract | Part
1-3 | References | Annex
1 | Annex 2 | Annex
3 | Annex 4
Notes to Part 1
 Further attention should be given to 0-element sets
and their significance.
 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.
 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)
 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
 From which arises the whole problem of communication
with the non-scholar and between scholars of different disciplines.
 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),
 This point is discussed in further detail in a
 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.
 Note that this "basic distinction" constitutes
a 2-element set which is subject to many of the points made in this paper.
 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.)
 Although it is very seldom done, any conventional
hierarchical structure (e.g. an organization chart of a corporation) can
be curved into a circle with the superordinate element at the centre.
 Jones discusses seven pre-logical axes of bias
and their application to scholarly debates in the arts and in the sciences.
 "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.
 See ref. (22) and
 "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),
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.
 "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),
 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)
 In the light of the theme of this paper, it is
curious to note that Thom's catastrophe theory identifies only 7 distinct
forms of catastrophic discontinuity.
 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
 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)
 I am indebted to Ingetraut Dahlberg for drawing
my attention to refs. (62) and (63)
and the question of seminal mnemonics in general.
 For a historical review and bibliography, other
than that of von Franz (9), see Ghyka (64),
Butler (65), and Hopper (66).
 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.
 Although Bennett's analysis is used by him as
a basis for much wider investigation which is not a matter of concen here.
 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),
 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),
 The implications of the "imprinting"
process of learning should also be considered as well as the role of portraits
in political, religious and cultural personality cults.
 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.
 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
Notes to Part 3
 Systematics 1963 1970 (Institute for Comparative
Study of History, Philosophy and the Sciences, UK)
 Only by viewing an N-term set as an N-1 term and
an N+1 term system can its significance be established.
 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.
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.
 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.
 I am considerably indebted to Ira Einhorn for
drawing my attention to references: (42, 106
- 107, 112).
 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
| Annex 2
| Annex 4