Integrative Dimensions of Concept Sets
transformations with minimal distortion between implicitness
of set representation according to constraints on communicability
- / -
Paper prepared for Integrative Group B on the occasion of the 6th Network Meeting
(Tokyo, August 1981) of the Goals, Processesd and Indicators of Development (GPID)
project of the United Nations University. Published in Patterns of Conceptual
Integration. Brussels, Union of Internation al Associations, 1984, pp. 141-148) [searchable PDF version]
Summary: Explores the problem of minimizing loss of significance when
a concept set is considered too complex to be communicated comprehen- sibly
to a particular audience. In such a case the set must be transformed, possibly
by collapsing some distinctions, so as to reduce the number of conceptual
elements in the set by a factor of 2, 3 or more. If collapsed in an orderly
manner, maintaining integrative dimensions, some of the original distinctions
are present implicitly rather than explicitly.
2.Representation, of integrative dimensions
3.Representation of sets by regular polyhedra
4.Language for representation of concept factors
5. Progressive emergence and re-absorption of explicitness
6.Application to ordering a concept set (as a potential gestalt)
Sets of concepts, such as principles, problems, issues, propositions,
priorities, characteristics or properties of some entity, values, human
human rights, are fundamental to the organization of research, policy-making,
programme management, and education, for example. This paper is concerned with
the means whereby severe distortion can be minimized when a given concept set
has to be transferred from a
communication environment tolerant of high
connectivity [i.e. integration) between the elements of the set to one in which
such connectivity is neither acceptable nor perceptible without distortion.
Typically this is the problem of communication between the world of
and that of policy formulation or political decision-making. It is also characteristic of the different levels of the educational process and of communication
between different research disciplines, or even between different schools of
within the same discipline.
The question is therefore how best to "compress" a concept set
within a connectivity-insensitive environment.
The compression should ideally
be such that the set can be subsequently "expanded" to any
by a receiver whenever this is considered appropriate. The structure of the
compression should suggest possibilities of such an expansion and provoke
interesting questions in connection with it.
In any development process characterized by the co-presence of actors of
different levels of connectivity-sensitivity, the compression/expansion
is of considerable importance if distortions, confusion, and associated abuses
are not to be repeatedly introduced. To
the extent that the development process
is characterized by multi-level action, where the levels are not independent,
careful attention is required to communication between such levels.
Some aspects of these concerns have benn partially explored in earlier
2. Representation, of integrative dimensions
An "integrated" concept set must clearly have several
dimensions of commonality
linking the concepts. If there was only
one such dimension, the set could be
considered a string or unstructured list.
If there were two, it could be
considered as a 2-D matrix. Because of
the inadequacy of such forms in the
face of real-world processes, as discussed elsewhere (1), this paper
is primarily concerned with the case of three or more such integrating
Now in a connectivity-sensitive environment it is presumably possible to
each set element with a distinct and explicit
feature of the structure which is
used to map their interlinkage. For
those in that environment, the interconnections
between the set elements in terms of their commonalities will carry
rather than act as a barrier to it. But
in the case of the connectivity-
insensitive environment, these explicit interconnections must somehow be
up" and "packed into" whatever structure is used to represent
the set [a kind
of "reverse ontogenesis").
They have to become implicit,
or else they will act
as a barrier to the transfer of whatever significance can be communicated under
An incidental but very important aspect of this explicit/implicit
that clearly in the explicit form the concept elements are well-defined. Whereas
in the implicit form any such articulation is potential (or "embryonic") rather
than actual. There is an
"undefined" quality to the significance which can then
be communicated by the set representation.
This is discussed further in a later
The unpacking/packing stages, therefore need to be mapped onto a setof interrelated
structures (each corresponding to a different level of connectivity- sensitivity).
As a series, such a set would represent the steps by which "logical forms"
acquire a separate existence from their inter-embedded, "primordial"
origin in the process of conceptualization.
3. Representation of sets by regular polyhedra
Clearly the connectivity between the elements of a concept set could be represented
by a suitable n-dimensional matrix. This can be expressed geometrically as
a multidimensional polyhedron. Polyhedral structures of this Kind are significant
to such important areas as citation analysis (of research articles), linear
programming (for resource management), and Q-analysis (as applied to complex
organization communication patterns).
In none of these cases is the communicability or comprehensibility of the
set as a whole of special importance. Results are effectively achieved by
computer analysis of complex polyhedra essentially incomprehensible to the
human mind because the sets which they represent have few integrative dimensions.
If comprehensibility is considered a basic constraint on the types of polyhedra
which can be used to represent sets, the question is then what makes polyhedra
comprehensible as a whole. A partial response from educational research is
the occurrence of mutually reinforcing elements in the structural pattern.
Such essentially "memorable" features depend upon the the presence
of interlocking symmetries in the polyhedron.
Two additional constraints can be usefully imposed in this initial investigation.
Unless the polyhedron is convex it cannot easily be used to represent a closed,
self-contained set. Furthermore, unless internal connections are avoided
through the body of the polyhedron, the structure cannot be easily scanned
It is clear that with these constraints the most memorable polyhedra are
those in two interrelated series, the so-called Platonic and Archimedean polyhedra.
To these may be added at a later stage the semi-regular polyhedra with looser
symmetry and patterning constraints than the above.
4. Language for representation of concept factors
The term "concept factor" has been discussed in an earlier paper
( 2 ). It is used to denote the properties shared in different combinations
by each element of a concept set. Thus one element may be a combination of
4 factors: p, q, r, s.
The integration of the concept set may then be considered as based upon:
- the presence of factor p in some other elements,, introducing a pattern
- the factor p being itself part of a set of factors of a similar kind: p1p2p3...pn
The problem is to find a useful way of mapping concept factors onto selected
polyhedra. This calls for special attention to the manner whereby significance
can be carried by the features of the polyhedra. Aspects of this have been
extensively investigated by R Buckminster Fuller ( 3 ).
(a) Edge: An edge of a polyhedron can be usefully considered as a
vector [of which only a portion may contribute to the formation of the polyhedron
-- the remainder continuing beyond the constraints of the polyhedron). This
follows Buckminster Fuller's argument that "Lines are vector trajectories"
( 3, para 521.201). The dynamic quality of the vector suggests that it is
preferable to associate with it a concept factor understood in its dynamic
sense, namely as a process or as a verb (rather than as a noun, for example),
as has been argued elsewhere ( 1 ). Associating a non-dynamic sense with
the edge tends to render the concept prematurely explicit, precluding transitions
to more richly articulated representations. In the language of Q-analysis,
such an edge marks the presence of connectivity between vertices, with the
associated possibility of some form of "communication traffic".
(b) Vertex: The vertex of a polyhedron is defined by the intersection
of at least 3 edge/vectors. It requires at least 3 such vectors to give existence
to the vertex. This may be more clearly seen by comparing:
- points on a line: whose positions [e.g. in the case of a list of
concepts) remain unfixed in relation to one another except in terms of ordering
on the line. Conceptually they may be irregularly "bunched" together,
for example, although numbering them creates the erroneous impression that
they are equally distinct and separable (see Fig. 1)
- 2 intersecting vectors: which cannot be used to fix a point adequately,
as is evident from the need for "triangulation" in surveying and
navigation. Unfortunately the widespread use of 2-D matrices (e.g.
of concepts) creates the superficial impression that the relative positions
of the cells is fixed by the matrix context. The absence of any third vector
allows any such matrix a degree of freedom (and associated possibilities of
distorted interpretation) as represented by the following:
|Figs 1 and 2
The vertex, defined by 3 vectors, may usefully be associated with a well-defined
concept. It is then appropriate to associate it with a concept in a more
static sense, namely as a noun.
(c) Face: The face of a polyhedron is bounded by at least 3 edge/vectors,
with intersections between pairs of these vectors at at least 3 vertices.
Such a face can be used to represent a necessary degree of non-closure or
ambiguity in the concept with which it is associated. Thus in Q-analysis
each such face is a potential "hole" which engenders "traffic"
in the edges surrounding it, although it may be perceived as an underdefined
"object" around which traffic must pass.
"Generally speaking it seems to be confirmed that action [of whatever
kind) in the community can be seen as traffic in the abstract geometry and
that this traffic must naturally avoid the holes (because it is quite impossible
for any action to exist in a hole)" (4, p. 75).
This corresponds to points made by BucKminster Fuller: "Areas as (polyhedral)
"faces", are inherently empty of actions or events....Euler's topological
aspects have to be altered to read...."areas" = openings (i.e. where
there are no trajectories or crossings)" ( 3, paras 524.21-524.31).
"What Euler and all professional topologists and mathematicians called
"areas" are only windows in polyhedrally conceptual systems. You
look out the window at the nothingness of undimensional night..." ( 3,
It is therefore useful to associate a face with a concept in its underdefined,
controversial or problematic sense. Alternatively, the face may be associated
with a conceptual ("frontier") domain which, although as yet essentially
undefined, is nevertheless well-defined as an area of current investigation
by the concepts (as vectors and vertices) which surround it and through which
it must be approached.
The Polyhedra under consideration have faces bounded by 3, 4. 5, 6, 8, and
10 edges. Clearly the larger the number of edges, the richer and more difficult
to comprehend will be the underlying concept with which the face is associated
-- at least as it emerges within the concept set represented on the
polyhedron in question.
In addition to the classic, explicit, topological features (edges,
vertices, faces). Fuller considers a further four implicit features (3, para
1044.01) which may later prove of relevance to this approach. Of particular
interest are the axes of spin or symmetry and the associated concept of the
polyhedron centre. Both nay be used to represent aspects of the underlying
relationship of the observer to the set observed.
In this connection Fuller suggests a potentially significant generalization
of the classic Euler equation for any polyhedron:
no. of vertices + no. of faces = no. of edges + 2
"The observer and the observed are two ins with one relationship.
Euler said V + A = L + 2, but we may now say: The number of somethings + the
number of nothings = the number of [most economical) interrelationships +
2". ( 3. para 269.06)
5. Progressive emergence and re-absorption of explicitness
In considering the argument of the following sections, it is important to
trace the progressive emergence of factors into implicitness, through explicitness,
to re-expression (or re-absorption) as multiples. This is summarized in the
following table, based on the language introduced in the previous section.
[See Annex 19*******]
It is worth reflecting on the impossibility of providing a unique mapping
of 2 or 3-classes onto a polyhedron. The minimal polyhedron is the tetrahedron
which requires 6 edge/vectors to generate 4 vertices -- 4 being the minimum
number of factors that can therefore be mapped explicitly. In Fuller's terms
this is a characteristic of the minimum event. Within this tetrahedral mapping
the 2 and 3-factor features have already been reabsorbed as multiples [e.g.
3x2 opposing edges).
This suggests that it requires a minimum of 6 interrelated classes of information
to define a new identity which cannot be absorbed into one of those classes
or simply treated as their sum. Below 6 no new information is engendered
by interrelating the classes, even though such classes may establish dimensions
fundamental to the generation of some new conceptual structure. The establishment
is not "achieved" until at least 6 are interrelated.
Within this polyhedral mapping framework, the "new" information
engendered by the set as a whole may be associated with the definition of
the centre point of a polyhedron through the spherical disposition of its
parts equi-distant from that centre. This "virtual" point is not
a member of the set end is at another level in relationship to it. As such
it may also be considered an implicit mapping of the 1-class factor of the
The centre of the polyhedron may be usefully viewed as a kind of observational
origin through which the expression of the set is "born" in stages.
Fuller, for example, says that it is "the zero starting point for happenings
or nonhappenings: it is the empty theater and empty Universe intercoordinatingly
ready to accommodate any act any any audience." ( 3, para 503.031)
Larger factor numbers first appear implicitly through the centre. At this
stage it is as though their significance had not yet been disentangled from
the observing process -- there is both confusion as well as embryonic new
order. Only when the numbers are "objectified" and anchored in an
explicit new set of features (edges, vertices, faces) distant from the centre
is this confusion resolved. But once factors have been uniquely expressed
in this way, subseqeunt polyhedra of greater complexity re-absorb or re-express
these factors as multiples -- an integrated multiplicity that does not provoke
confusion in the observer.
Viewed in this way the series of polyhedra map not only the set but how the
set may be comprehended:
confusedly, because of the omni-directional relationships through the
- objectively, by sequential scanning of surface features with memorable
- objectively, but also subsuming an integrated multiplicity of sub-features
6. Application to ordering a concept set (as a potential
If research has generated a set of say 100 or more concepts (characteristics,
etc), careful consideration will presumably enable each of these to be attributed
to one or more "cover-concepts" (cf. the concept of a cover
set used in Q-analysis) . It is through such concept (categories, classes)
that the set of 100 will be comprehended and communicated, if this is possible.
Each such concept group may itself belong to one or more other concept groups
at a different level. The question is then how to organize and interrelate
these different groups so as to represent/describe the set as a whole and
to clarify the significance of each element, given the communicability constraints
The problem is to have the number of concept groups small enough to remain
comprehensible. As noted elsewhere ( 1 ), comprehensibility falls off rapidly
if the number exceeds about 6, unless small prime number factors can be used
to introduce pattern reinforcement.
The other requirement is of course to be able to use alternative mappings
permitting more or less explicitness according to user needs and tolerance
1. Anthony Judge. Representation, comprehension and communication of sets;
the role of number.1978 [text]
2. Anthony Judge. Concept factors in concept scheme integration. 1980 [text]
3. R Buckminster Fuller. Synergetics: explorations in the geometry of thinking.
Macmillan, 1975 [vol. I), 1979 (vol. II)
4. R H Atkin. Combinatorial Connectivities in Social Systems: an application
of simplicial complex structures to the study of large organizations. Basel,