Patterns of N-foldness
Comparison of integrated multi-set concept schemes as forms of presentation
- / -
Paper prepared for a sub-project meeting (Geneva, June 1980) of the Forms of
Presentation group of the Goals, Processes and Indicators of Development (GPID)
of the United Nations University (UNU). Published in Patterns
of Conceptual Integration
Brussels, UIA, 1984, pp. 41-52. The examples to which this paper refers appear
in Annexes 0-20 (Examples
of Integrated, Multi-set Concept Schemes
) at the end of that volume.
[searchable PDF version
In order to provide information to facilitate investigations
suggested by the previous paper, examples of integrated multi-set
concept schemes of the most diverse nature are reviewed. As a cross-cultural, cross-ideological, interdisciplinary experiment, the
conceptual organization of the UNU/GPID project suggested comparison
with such examples. A method of comparison is tentatively suggested.
There are many forms through which new concepts and attitudes can be presented.
These are examined in a separate paper (1). In this paper the focus is on
how the pattern of concepts to be conveyed tends to be reflected in the organization
of the form - and how It may well be constrained in consequence. To what extent
is the "conceptual artist" constrained by the "medium of expression"?
Presenting GPID results
The purpose of this inquiry stems from concern about the manner in which
the UNU/GPID project will be able to portray the results of its own investigations
-- and especially the whole pattern of its final conclusions. It will of
course be possible to use film, theatre, poetry, posters, and many other methods
(1). But, because of their nature, these do not lend themselves to a systematic
precise elaboration of GPID concerns. They can only pick out portions
of the total pattern for special emphasis. And even if a series of them is
used, it Is not clear that any such "sequences" would protect the
essentially non-linear characteristics of the GPID pattern of conclusions.
When it is required to convey carefully a complex pattern, it Is of course
possible to do so using conventional text [even with appropriate illustrations).
This will probably be done in a general "GPID Rapport de Synthèse".
But the more carefully and precisely this Is done, the longer will be such
a document. And the longer it is, the more it will tend to defeat its purpose
as a medium of communication.
This problem is not new. For example, in the case of the famous Capacity
Study of the United Nations Development System (the "Jackson Report",
1969), the results were presented in two volumes, the first of which was a
short summary volume -- itself introduced by a summary. But in order to be
"readable", the underlying pattern of concepts had to be distorted
and abridged. This method is primarily useful for making a single point (e.g.
"something must be done") and not for presenting the pattern of
concepts which need to be taken into consideration in ensuring that appropriate
balance is maintained. Another widely-used approach is to produce a list of
points, as in the International Development Strategy of the Universal Declaration
of Human Rights. Such lists may be enriched by annotations. The lists are
structured somewhat like conference "resolutions" or "recommendations".
Whilst they are convenient as checklists, they imply by their structure a
simplistic linear "pattern" of relationships between the
paints in the list - even if some of the points have sub-points. But unfortunately
such lists, as they are used in practice, strongly de-emphasize any relationships
between the items (including the linear ones), even if they were recognized
when the list was elaborated.
How then is GPID to envisage the formulation of its results? Agreed there
will be books, collections of books, films, posters, etc. There may well
be many lists of GPID criteria, action points, values, priorities, etc., which
will partly determine the content of the former. But how is the underlying
"master" pattern to be expressed, orotected from "erosion",
and rendered widely communicable, given the constraints noted above? For whilst
parts of the pattern may be significant in themselves, the special significance
of GPID lies in its attempt to harmonize the parts, arising from the sub-projects,
into a coherent whole.
Investigation of precedents
If, as noted above, this is not a new problem, then there may be much to
be gained by reviewing a wide range of concept schemes to see whether there
are lessons to be drawn which could be of value to the GPID case. In selecting
such examples three major considerations need to be borne in mind:
examples are required which cover, in each case, more than one single
set of concepts [like the single set of 31 articles of the Universal Declaration
of Human Rights). The examples should have multiple sets of concepts
in which there is some relationship, or pattern of relationships, to be
maintained between such sets. This is necessary because GPID sub-projects
will each tend to generate concept sets in response to the few sets which
govern the whole project's conception.
- some of the examples should be from domains in which the prime emphasis
is not intellectual but rather expressive [as in the case of music,
art, dance, etc.). This consideration has already been accepted by the GPID
- some of the examples should be from the non-western cultures
which characterize many of the peoples to whose condition the GPID project
was designed to respond - especially in a "dialogue" situation.
Johan Galtung, coordinator of the project, has, for example, called for daoist,
buddhist, and other influences on conceptualization within the project (3).
In addition to the above considerations, it would also be valuable to examine:
traditional concept schemes which have retained their significance
and communicability (above all at the village level], despite the passage
of considerable periods of time since they were elaborated.
- elaborate modern concept schemes which attempt to respond either
to the complexity of our condition using the full benefit of the range of
western disciplines, or to a complex body of data.
- concept schemes elaborated in the light of problems of comprehensibility
Clearly the schemes selected should preferably be as independent of one another
Ordering concept scheme material for comparison
Any such investigation can easily run the risk of being overly ambitious
and falling victim to the problem of "presentation"it is designed
to clarify. A filium ariadnis is required as a guide through the conceptual
labyrinths to be investigated.
An earlier paper considered the role of number in the formulation of complete
sets, and the associated problems of their representation, comprehension and
communication (4). Since tne concept schemes it is proposed to investigate
are supposedly made up of complete sets, it could well be that the number
of elements in any such set would provide a relatively simple and unambiguous...
way of ordering the material. The earlier paper in fact reinforced the arguments
of other authors in favour of such an approach. The question of whether the
GPID sets are necessarily "complete", and therefore whether more
can be learned from this Investigation of relevance to the actual integration^
of the GPID concept scheme, is dealt with in a separate paper (5).
Concept schemes selected
The information on each selected concept schemes is given in a specific Annex
to this paper, one scheme per annex. Details concerning the criteria whereby
material within each annex was selected, are given on an introductory page to
the set of annexes: Examples
of Integrated, Multi-set Concept Schemes. The provides a checklist to the
20 concept schemes.
Given the criteria listed above and the details at the beginning of each
annex, the individual concept schemes were selected for reasons summarized
on an introductory page to the set of Annexes. It should be stressed that
the exercise was undertaken (as a data gathering exercise) in an attempt to
bring together a rich collection of concept sets and without being able to
determine in advance what Kinds of conclusions could be drawn, or whether
all the concept schemes were equally appropriate in this context. The investigation
is conceived as open-ended with the possibility of adding other concept schemes
from other domains if this seems to be of value.
Preliminary comments on concept sets
In the earlier paper mentioned above (4), arguments were presented to indicate
that, depending on the number of elements in a given complete concept
set, different qualitative characteristics would tend to be associated with
that set or the relationship between its elements.
It is perhaps necessary to stress that this investigation is not concerned
with number symbolism which is a veritable quagmire, asdiscussed in the earlier
paper. Although as it was pointed out, phenomena characteristic of that historical
preoccupation are now recurring in the scholarly and administrative tendency
to enumerate lists of fundamental issues, principles, values, criteria, problems,
etc. [to be compared with the medieval predeliction for M virtues, sins, principles,
This Investigation is more concerned with how sets of a given complexity
[which necessarily increases with the number of elements) can be comprehended
and communicated. But it is especially concerned with how the different sets
within a scheme are interrelated, and how they can be comprehended and communicated.
If there is any substance to the basic argument, then it should be possible
to read "horizontally" through the annexes [examining only sets
of a given number of elements) and find some degree of equivalence between
them. One example is for a 5-set:
- Annex 15 (Systematics), indicates that 5-foldness
establishes the limits of a bounded significant entity, its connectedness
to its environment, and its potential in relation to it.
- Annex 4 (Tibetan Buddhism), indicates (in
point 5.10) that the Buddha took into account 5 conditions before taking physical
- Annex 17 (Synergetics), indicates (in point
5.2) that generalization is conceptually imaginable independent of frequency.
But frequency, as the fifth characteristic, is the unique special case variable.
Physical experience is dependent not only on the four generalizable characteristics,
but also on the fifth [which is associated with size).
- Annex 0 (UNU/GPID), indicates (in point 5.1)
that the project is characterized and defined by 5 features.
Despite the dissimilarity of the concept schemes, the notion of specificity
and delimitation is characteristic of all of them. If this is the case, why
is it not explicit in other annexes? There are several possible reasons:
- The scheme may not possess sets of a given number, or they may be implicit
rather than explicit. (The implication of this will be considered below)
- Sets of the given number may reflect "secondary" rather than
fundamental preoccupations, namely, they may be derived from more fundamental
sets which may [or may not) be characterized by the quality in question.
- It may be difficult for a non-specialist to understand the significance
associated with the particular set in its own context.
The last point brings up the very interesting challenge which the variety
of material poses. A shift in perspective is required to avoid being
distracted by a stress on : topological features, gods, principles, concepts,
tones, strategies, physical movements, colours, etc. These must simply be
considered as vehicles for the expression and comprehension of an underlying
pattern. This point is specifically made in Annex
7 (point 0.12): "The hymns describe the numbers poetically, distinguish
"sets" by classes of gods and demons, and portray tonal and arithmetical
relations with graphic sexual and spatial metaphor."
Annex 7 is in fact a scheme which uses 5-sets
extensively. Its material does not clearly convey the same notion
as the examples above (although perhaps 5.1 may be significant to musicologists
in that sense). But the constraints governing the "birth" of a
tone in a yantra frame (see Annex 7.0.9), for
example, could well prove to be interpretable in the light of the conditions
governing the "birth of the Buddha" (Annex
4, 5.10) interpreted metaphorically.
There are many other examples of comparability between the annexes. The
problem is then to decide how the material can best be investigated further.
There is little advantage in simply pointing out examples such as those above,
for, as such, they only have curiosity value and do not lead us forward.
In a second part of this paper some of these equivalences will be examined
to discover where they lead us. But at this point we need a better "handle"
on the material than the many annex pages provide, even if specially ordered.
Pattern generation and comprehensibility
One way of presenting the range of schemes in the annexes, for a better overiew,
would be simply to produce a table in which the columns were used to indicate
the presence of sets of a given number of elements. The rows would then each
be used for a different concept scheme. But even this table would be too
unwieldy, because of the large number of sets in some of the schemes.
Anticipating somewhat the investigations of the second part of this paper,
two of the annexes in fact provide an interesting alternative. Both Annex
7 (Rg Veda tones) and Annex 17 (Synergetics)
lay considerable stress on the significance of powers of prime number integers
in basic pattern formation (e.g. Annex 7. 30.1,
17.0.16 and 17.0.17). Why should this be the case?
Taking 2 as the first "genuine" prime number, any sets which have
the number of their elements definable by 2 only can be matched to one another,
such that within a scheme the larger sets are effectively regular subdivisions
of the smaller. This can be made visually obvious (e.g. with concentric circles).
No new patterning effect enters, however large n becomes (see Annex
7.2.2). New patterning effects are only introduced when other prime numbers,
or their powers, are required to define the number of set elements.
As to why the stress is placed on integers, these are clearly important
to the definition of sets with an integral number of elements. A complete
set obviously does not contain 6.25 elements. Furthermore it is the powers
of prime numbers which ensure that the set subdivisions are "rational"
and, as such, more comprehensible to the human mind. The stress on
powers of the smallest integers also ensures patterns which are more
comprehensible and memorable than would otherwise be the case. These criteria
imply that the most "conceptually compact" sets will tend to emerge
in any scheme prior to the less comprehensible and less memorable sets.
Concept scheme factor table
In this light a more compact table can now be produced in which the columns
indicate the prime number factors of the sets characteristic of the
concept scheme indicated in the rows (see Table 1).
The powers of the factors required are indicated in the body of the table
in each case. Presenting the information in this way raises several interesting
It is probable that concept schemes with similar prime number factors
and powers will be more obviously comparable (equivalent, isomorphic), than
those without. It is not surprising therefore that a number of authors
have exolored the relationship between the Chinese Book of Changes (Annex
2 and the genetic code (Annex 5), for
- Concept schemes with few prime number factors will tend to differ significantly
from those with more. Thus those whose sets are primarily defined by 2 appear
to group their elements at a very "general" or abstract level.
Those including 5 as a factor introduce more specific notions (as indicated
earlier). Higher primes may well encompass even more concrete, "real-world"
aspects of phenomena.
- Whilst there may be some value in viewing the concept schemes with prime
factors as "richer", more "mature" and more "comprehensive",
it may be even more useful to consider concept schemes based on different
patterns as performing different functions in the psycho- social system.
What sorts of concepts tend to be carried by what patterns?
- The previous point raises the question of what the full range of patterns
is through which concepts could be usefully presented - what is the criterion
for "pattern stability", or "viability"?
Within the framework of the previous possibilities it is then useful
to consider the problems of "developing" a concept scheme from
a simpler pattern to a more complex one - by introducing sets based on a
larger prime as a factor.
- In the light of the previous point, special attention should perhaps be
given to the sets of a concept scheme based on higher primes - because, since
they are necessarily less memorable and less comprehensible, it is these that
are most vulnerable to "erosion" when the set is communicated through
simplifying channels, particularly those which emphasize "communicability"
in its popular sense.
These possibilities, if they prove to be well-founded, would suggest that
a fruitful line of investigation can be opened up. There is however an important
point to be clarified. The concept sets require for their definition not
only primes and their powers, but also multiples thereof (e.g. 22, 30, 52 as
in Annex 7 30.1) What are the constraints,
if any, on forming such multiples and on the powers used?
A further step can be envisaged which might clarify this difficulty. The
presentation of the numbers in the table bears an interesting resemblance
to that of the electron shell configuration in the periodic classification
of elements (Annex 14), The general implications
of the closest-packing problem in this classification do emerge from
the synergetics scheme (Annex 17 92.1). But,
it is three interrelated quantum numbers taking on integer values which determine
the energy levels of the electrons in an atom in terms of a radial and an
[Note : Such a relationship is not as far-fetched as might first appear,
Relatively litt le is known about the organization of concept sets in memory.
The synergetics omnidirectional "packing" approach has merit asapplied
to provides useful guidelines, then these would enable the patterns of a complete
set of concept schemes to be interrelated in a form of periodic classification.
It would then he probable that these would fall into groups with comparable
properties. The existence of as yet undiscovered concept scheme patterns
could also be predicted concepts. The related problem in the case of an atom
Is to "pack" electron Orbitals in a spherically symmetrical field
whose potential energy is dependent on a radius vector. The spherical symmetry
of the field permits separation of the Schrodinger wave-function equation
into radial and angular factors (compare Annex 17.0.12
and 17.5.2). The angular part is uniquely determined by the symmetry of the
field and not by the potential energy. An Individual's "field of attention"
could be described in similar terms in the light of Annex
17 - with more complex concept sets being represented by higher "quantum
numbers'. The most stable electron orbitals, which bring the electron closest
to the nucleus, are those with the smallest integer quantum numbers.
The energy sub-shells are made up of groups of 1. 3, 5, 7,...quantum cells
- each corresponding to two electron states, thus giving a maximum of 2, 6,
10, 14,....electrons respectively. Is there a useful "constraining"
relationship between the latter numbers and the prime number factors in the
table? The number 2 appears (as might be expected) to create a problem, from
which it could be inferred that the series 1,3,5,7,... are odd numbers
and not prime numbers. Atoms have however not yet been created to establish
whether the next number is 9 or 11. (G.S. Zhdanov, Crystal Physics. Edinburgh,
1965, p 31).
If the periodic classification provides useful guidlines, then these would
enable the patterns of a complete set of concept schemes to be interrelated
in a form of periodic classification. It would then be probable that these
would fall into groups with comparable proerties. the existence of as yet
undiscovered conceopt scheme patterns could also be predicted.
This whole approach would then have the merit of setting any particular scheme
in an appropriate context. It would help clarify how it Is likely to interact
with other schemes. But such a framework could then provide understanding
of how a new concept scheme should be "designed" in order for it
to be viable and have certain desired characteristics. This is its importance
But, additionally, it may prove to be the case that some characteristic
concept schemes are more "acceptable" and "implementable" than others - by
embodiment in organizations. Some may
be so "reactive" that they are
"adulterated" by association with other schemes - this
may be a desirable result. Others may
be "unreactive", retaining their
elemental characteristics - and this may prove undesirable. This possibility
of concept scheme "compounds" may be one of the factors hindering
of equivalence between the concept schemes in the annexes. We may be
comparing "compounds" (or "ores") which disguise the
characteristics of the
The previous section focuses on a means for codifying concept schemes, which
may fall within a general pattern. The disadvantage of this approach is that
any such "flat" tabular presentation [even of an essentially spherical
scheme) moves away from the more spherically-oriented presentations which
may well contain the key to greater comprehensibility and communicability
(4, and Annex 17 0.6). It is therefore Interesting
to note the basic argument of Annex 17 which
establishes the link between prime number factors and polyhedral spherical
approximations (Annex 18). Nested concept sets
may then be envisaged as "encoded" by nested polyhedra or their
topological features, for example (Annex 19).
The relationship between symmetrical forms and memorability has been explored
in an earlier paper (4). Of greater Importance however is the demonstration
of the existence of "transformation pathways" between certain polyhedra
thus maintaining structural continuity. This effectively amounts to a rational
change of "conceptual gear", in that no irrational discontinuity
is involved in the transition. Such transformations may be vital guidelines
to ways of "compacting" or "expanding" a concept set according
to the circumstances of communication. As an indication of how transitions
between concept sets might be effected,apreliminary version of a map is included
which shows some of these pathways on the basis of information in Annex
19 (see Fig.1).
It was not the intention of this paper to do more than present the
schemes collected and to indicate lines of Investigation which could be
opened up in relation to the presentation of GPID itself.
This paper has
stressed the structural aspects of pattern formation and has avoided the
question of content. Content, in the
light of this paper, is examined in
relation to the integration of the GPID concept schems in
a separate paper (5).
A special merit of the approach outlined lies in the concern with protecting
the GPID concept pattern as a whole. A great danger with many exciting "sensitizing"
forms of presentation is that they "excite" people's awareness very
successfully to levels at which there is no coherent pattern which can ensure
the permanent stabilization of their awareness at that level. Such temporary
increases in "attention potential" are associated wiht high "forgetabillty"
(witness the track record of various UN public information programmes )(7).Conceived
in this way the challenge is to find ways of stabilizing such patterns of
nested sets and of facilitating shifts of attention between them. This has
been compared to the oroblem of a "conceptual gearbox" in a separate
paper (8). One problem is to find ways of using "higher gears"
to mesh more effectively with the rate of social change. This is the question
of how to develop more operationally significant concept sets within the GPID
The more excitingly communicative, audio-visual forms are based on the
of minimum number of explicit messages per presentation, thus fragmenting
the whole pattern. It would therefor be
important to establish what forms
are useful to communicate what parts of that pattern. It is important also
for people to be ableto transfer, from the sets and
associated symbols of
a concept scheme with which they are familiar (and have faith in), to new
ones such as those of GPID, with a sense of continuity- otherwise the latter
will be perceived as of ersatz quality.
This is a reason for establishing
the relationship between a variety of concept schemes.
Whilst further invertigation is of course required, the method of
concept scheme material in terms of the number of set elements certainly
seems to show promise as a method of classification. Whilst conventional
classification focuses on documents,
this focuses on concept sets within_
a document. The presentation also
has advantages over linear text abstracts
It may also prove to be the case that this approach can provide a useful interdisciplinary,
or interparadigmatic,framework which respects
the concept sets of each paradigma and facilitates their comparison in an
explicit context. The GPID concept scheme will necessary be launched into
an environment inhabited by many other concept schemes with some of which
it will have to compete (for attention] as in any ecosystem. Its fate will
depend upon the ecological relationship it establishes with other schemes
(as predator? prey? symbiote? etc.... as indicated in Annex
16), or whether it can to some extent transform that ecology by revealing
principles which underly it.
1. Anthony Judge. Review of forms of presentation: paper for the Forms of
Presentation sub-project meeting (Geneva, June 1980)
2. Report of the Forms of Presentation sub-project meeting, Geneva, February
3. Johan Galtung. Towards synergy in networks of people with networks of
problems: a note on GPID methodology. (Paper for GPID Networks sub-project)
4. Anthony Judge. Representation, comprehension, and communication of sets)
the role of number. International Classification, 5, 1978, 3,
6, 1979, 1,6, 1979, 2 (in 3 parts) (also GPID Working paper)
5. Anthony Judge. Concept factors in concept scheme integration: GPID as
a case study (Paper prepared for the integrative Workshop meeting. Alfaz
del Pi. April 1980)
6. Martin Schönberger. Verborgener
Schlüssel zum Leben: Weltformel I-Gingim
genetischer Code. Frankfurt/Main, Fischer, 1977 (nr 480)
7. Anthony Judge. Mobilization of public opinion; yesterday's response to
today's problems. Transnational Associations, 31, 1979, 1/2,
pp. 8-14 [text]
8. Anthony Judge. The future of comprehension; conceptual birdcages and
functional basket-weaving, 1980