21st March 2006
Conformality of 7 WH-questions to 7 Elementary Catastrophes
an exploration of potential psychosocial implications
- / -
Council of the Whys:
emergent wisdom through configuration of why-question dynamics
Applications of catastrophe theory
Cognitive feel for cognitive catastrophes: question conformality (Annex)
Correspondence of WH-questions to elementary catastrophes
Why-questions and the parabolic umbilic
Pointers to comprehension of multi-dimensionality of WH-questions:
Psychosocial implications of WH-questions as "catastrophes"
Skateboarding | Sexual attraction and
intercourse | Multiple intelligences | Psychosis
/ Neurosis | "Games people play" | Strategy games | Meditation | Symbolism
Psychosocial implications of WH-questions as "catastrophes"
Interrelating Cognitive Catastrophes in a "Grail-chalice" Proto-model
As a human response to the perception of a cognitively chaotic situation,
WH-questions (when, where, which, how, what, who/whose, why/wherefore) might be considered
to lend themselves to analysis with the tools of catastrophe
theory as developed by Rene Thom and others. Thom had developed
differential topology into a general theory of form and change of form as
a mathematical way of addressing the work on morphogenesis done
by C.H. Waddington in the 1950's. Thom's Classification Theorem culminates
a long line of work in singularity theory. The term "catastrophe theory"
was suggested by C. Zeeman (1977) to unify singularity theory, bifurcation
theory and their applications. The crucial theorems rigorously establishing
Thom's conjecture were proven by Bernard Malgrange (1966) and John N.
Mather (1968). Its essential concern is change and discontinuity in systems
(cf Robert Magnus, Mathematical
models and catastrophes). WH-questions may be considered as triggered
and formulated in response to discontinuity -- when habitual adaptive responses
to change are inadequate.
It is possible therefore that the set of WH-questions may in some way be
mapped onto elementary catastrophes. This is partially suggested by mathematical
techniques of conformal
mapping where, for example, the "cognitive flow field" around
one known shape (as with an elementary catastrophe) might be mapped onto
the flow field around a particular WH-question -- preserving the "angles".
Conformal mapping notably makes use of complex variables as combinations
of real and imaginary numbers. [applet]
Whilst the purpose here is to highlight the role of why-questions in the
set of WH-questions in the light of catastrophe theory, there is a certain
irony to the following description of Thom's own focus by Christer Persson
catastrophe theory: an introduction):
In science two main lines of questioning compete or co-operate; one asking "How?",
the other asking "Why?". In biology Thom had an irritating tendency
to counter each answer to a "Why?"-question with cascades of "How?"-questions,
the intent being to demonstrate the inadequateness or provisional character
of "guiding thought" in biology. When answered: "Because
messenger-RNA duplicates information from the DNA spiral and turns to ribosomes,
where proteins are synthesized...", he promptly
asked: When? How does it know when? How does it switch from one state to
another? Following what roads? Where is the map?
Given Rene Thom's interest in semantics and linguistics, the discontinuity
introduced into discourse by a question, and his predisposition to question
the assumptions of others, it might be asked whether he endeavoured -- perhaps
self-reflexively -- to relate elementary catastrophes to WH-questions in
some way that is not evident in the published literature.
This exploration develops aspects of earlier work on WH-questions (Functional
Complementarity of Higher Order Questions: psycho-social sustainability
modelled by coordinated movement, 2004;
with Questions of Higher Order: cognitive vigilance required for higher
degrees of twistedness, 2004). The dysfunctionality associated with WH-questions is explored separately (Question Avoidance, Evasion, Aversion and Phobia: why we are unable to escape from traps, 2006).
Catastrophe theory identifies degenerate critical points of the
potential function -- points where not just the first derivative, but one
or more higher derivatives of the potential function are also zero. Mathematically
these are called the germs (singularities or organizing centres) of the catastrophe
Thom listed all these germs and their unfoldings for cases involving up to
five parameters. He also proved that any family of potentials depending on
up to five parameters is structurally stable and equivalent around any point
to one of these canonical forms. Such equivalence and the properties of stability
and typicality arise from Thom's transversality and isotropy theorems and
from Mather's theorems on stable unfoldings [more].
When the degenerate points are not merely accidental, but are structurally
stable, they exist as organizing centres for particular geometric structures
of lower degeneracy [more]:
- For any system with four (or less) control factors and two (or less)
behaviour axes there are only seven elementary catastrophes possible
- Where the sum of the control and state dimensionalities equals eleven it
is possible to classify eleven families of catastrophes to some degree.
- Beyond this level of dimensionality even the categories of families of
catastrophes apparently become infinite and hence very difficult to classify.
For dimensionalities greater than five in the control space and two in
the state space the number of catastrophe forms is infinite.
In other words, given certain constraints, all discontinuous changes in
events can be described by one of seven elementary models.
When a system is therefore characterized, in spatial or temporal interpretations,
- Potential functions of one active (or "state", or "fast")
variable (or behavioural axis) and
- one input variable (control factor, or "slow variable"),
catastrophes take the form of a fold [more | applet]
- two inputs, catastrophes take the form of a cusp [more | applet]
- three inputs, catastrophes take the form of a swallowtail [more | applet]
- four inputs, catastrophes take the form of a butterfly (containing
"pocket" of compromise-- with a surface in 4D) [more | applet]
- Potential functions of two active (or "state",
variables (or behavioural axes) and
- three input variables (control factors, or "slow variables"),
catastrophes take the form of either:
- four inputs, catastrophes take the form of a parabolic
umbilic [more | applet]
The forms of the first four catastrophes have been clearly illustrated
in spatial (but not temporal) terms by folding paper by Leong Chen Chit
(Origami and Catastrophe
We can translate the first four manifolds of Catastrophe Theory into origami folds. The first one, the Fold manifold, is the equivalent, in flat origami, of the mountain/valley fold. It has no cusp point. The second catastrophe geometry, the Cusp manifold, is the equivalent, in flat folding, of the reverse fold; third, the Swallowtail manifold, is the equivalent of the double reverse fold; and fourth, the Butterfly manifold, the triangular sink fold.
Applications of catastrophe theory
After initial enthusiasm, Thom's approach has attracted criticism from
mathematicians with quantitative and predictive priorities, notably concerned
by "spurious quantization". However his considerable interest in
linguistic, semantic and psychosocial issues in the development of his general
theory continues to offer a qualitative approach that is appreciated in applications
of catastrophe theory in the social sciences [more | more].
Widespread use of catastrophe theory has been made for such modelling (cf
Brian R. Flay. Catastrophe
Theory in Social Psychology: some applications to attitudes and social behavior,
1978; Wolfgang Wildgen, Catastrophe
theoretical models in semantics, 2004). In The
Mathematics of Discontinuity, a balanced and extensive review of
the strengths and limitations of catastrophe theory in the light of such
criticism, is provided by J. Barkley Rosser, Jr (From Catastrophe to
Chaos: a general theory of economic discontinuities, 2000, Ch. 2). He
concludes that early criticism, now recognized as partly inappropriate, resulted
in the "baby being thrown out with the bathwater".
Such appreciation contrasts with that of the dismissive footnote of Philip
A. Schrodt (Patterns,
Rules and Learning: computational models of international behavior,
... chaos theory--despite its faddish character--has important
implications for international relations modeling that catastrophe theory--the
previous mathematical fad -- did not. The popular catastrophe theory
models required systems that were homeostatic and minimized a
quartic (cusp catastrophe) or hexadic (butterfly catastrophe) function and used
continuous time and contained two or more independent, real-valued
parameters. The "general" topological results of Rene Thom
basically applied only to mathematical abstractions, and only rarely to
empirically realizable systems. Chaos theory, in contrast, applies to models
that have been in common use for decades and have realistic features such
as quadratic feedback, discrete time, and interdependent parameters.
Schhrodt's position is consistent with advocates of complexity theory (as
compared to both catastrophe theory and chaos theory). This recognizes
that complex systems, considered in their totality, have more than one attractor
acting simultaneously and interdependently. The emphasis of catastrophe on
one form or another may therefore be considered a questionable form of reductionism.
Individual attractors can certainly be studied, but any assumption of their
independence is questionable, as strongly argued by Chris Lucas (Questioning
2006). For Lucas, WH-questions cannot in practice
be treated in isolation however distinct the sentences in which they are
From the perspective of complexity theory, in response to earlier drafts
of this paper, Lucas has proposed useful tabulations (as
below) of the relations between the set of WH-questions (for both questions
and answers) in terms of four methodological scopes.
(as developed by Chris Lucas, Questioning
Our Methodologies, 2006)
|When do I ask questions ?
|| Regularly ?
|Where do I ask questions ?
||One Place ?
||A Few Places ?
||Many Places ?
|Which systems are relevant to my questions ?
||One only ?
||A Few ?
||All of Them ?
|How do I ask questions ?
||In One Way ?
|| In Several Ways ?
||In Many Ways ?
||Every Way ?
|What questions do I ask ?
||Single Issue Ones ?
||Limited Issues Ones ?
||Multiple Issue Ones ?
||All Issue Ones ?
|Whom do I ask questions for ?
||Social Group ?
||All Stakeholders ?
|Why do I ask questions ?
||For Control ?
||For My Quality of Life ?
||For Group Quality of Life ?
||For Development ?
In general axiological terms
the first column, the scientific, is a systemic valuation mode -- it concentrates
on dualistic right/wrong approaches using Aristotelian or Boolean logic.
The second, personal, is an extrinsic valuation mode -- it concentrates
on acquisition or maximisation and is a fuzzy logic
approach, the third, humanistic, is an intrinsic valuation mode, it relates
to the whole and to matrix logic. The fourth, spiritual, is my holarchic
valuation mode, associated with integral logic.
It has been claimed that: "The politically-correct notion that 'What' and 'How' questions
belong to science, and that 'Why' questions belong to religion,
has been intellectually defunct for over a century" [more].
But, curiously, as exemplified in the anecdote concerning Thom (above), echoed
by Lucas (2006) and others [more],
why-questions indeed relate primarily to meaning, semantics, or values --
with science tending to reject or marginalize these as not being a valid
theme of research. The nature and extent of such question avoidance is discussed
Avoidance, Evasion, Aversion and Phobia: why we are unable to escape from
Philosophy makes a similar distinction, as noted by Lee Archie, et al (Reading
for Philosophical Inquiry A Brief Introduction to Philosophical Thinking):
Sometimes the distinction between science and philosophy is made by noting
that philosophy attempts to answer the question 'Why?,' and science
attempts to answer the question 'How?' ....
Is there a difference in the kinds of answers which would satisfy each
kind of question? Is the difference between why-questions and how-questions
the same as the difference between arguments and explanations?
The issue of the relationship between "how" and "why",
and their implications for governance, continue to be fundamental to the
debate between religion and science [more]. The challenge is highlighted
on a BBC "style
and usage" page on Why (2002):
It has long been clear that why questions are not as easily answered as
their how and what relatives. There is something in a why question that
ensures that, while they may be answered correctly, such correctness is
only in the mind of the perceiver....What then is different about why questions?
Well, why questions do not actually necessitate objectively true
answers. An infinite number of answers may be posed for any why question,
all of which may be true...
With such large emphasis on science in the modern
world, it is no surprise that science is often heralded as the answer to
all one's questions. Unfortunately, science has some major limitations
in this area. There is a large set of questions that science is necessarily
prevented from answering. And it is the set of why questions.
There is a simple reason for science's inability to answer why questions,
and that is that science assumes causality as its most fundamental
Why questions have a peculiar subtlety in their interpretation
that is unique to them. This is because one can mean two different things
by why and the difference between the two is often overlooked.
It arises from a tacit agreement to one of two juxtaposed world views.
Thom however defended his use of qualitative methods, arguing that science
constitutes a continuum between the poles of "acting effectively on reality" (with
quantitative tools) and "understanding reality" (with qualitative
approaches). The latter involved heuristic "classification of analogous
situations" by means of "geometrization that promoted a global
view while the inherent fragmentation of verbal conceptualization permits
only a limited grasp" (cf Rene Thom, Mathematical Models
of Morphogenesis, 1983).
Given the global condition at the beginning of the 21st century, it remains
unclear whether "chaos theory" or "complexity theory" will have
more to offer than "catastrophe
theory" in the face of the increasing number of catastrophes in an increasingly
chaotic society -- in what many see as a world that is increasingly complex
and incomprehensible -- and ungovernable, other than through processes of
fabricated threat and subterfuge (cf Promoting
a Singular Global Threat -- Terrorism: Strategy of choice for world governance,
2002). It is regrettable that the mathematical disciplines, with so much
to offer in reframing the situation, should be in thrall to such a degree
to the defence and security agendas exacerbating this condition -- or dedicated
to priorities in outer space more readily susceptible to mathematical solutions
When the Bombing Stops? Territorial conflict as a challenge to mathematicians,
In a highly problematic world situation
there is however merit in exploring the use of any approach -- however apparently
outmoded -- that may facilitate new thinking and the capacity to act on it.
Einstein noted that the thinking that had led humanity into its problems
would not be the thinking that would lead it out of them. He asserted that
moral questions -- namely including the why-questions -- were of utmost importance
for human existence and that in order for humanity to continue, it must create
a moral order. As argued by William L. Johnson et al. (Science
and Religion at a Crossroads: An Educational Perspective, Quodlibet
1, 6, 1999), "Science
must ask the 'why' questions as well as the 'how' questions. It cannot be
divorced from issues that take humanity quite beyond science itself".
John Archibald Wheeler evoked the possibility of a "meaning physics" in which
the "why' and "how" questions were resolved together in understanding
of the freedom and order of the development of the physics of the world
(Wheeler and Zurek 1983).
The challenge for institutionalized "science" and "scientists" in
a highly turbulent world is the risk of finding themselves perceived to be
trapped into responding only to "when", "where", "which" and "how" questions
-- as being the exemplification of "science". Their response to:
- what-questions, may then come to be characterized by the well-recognized problematic professional
and institutional dynamics and resistances associated with "scientific
revolutions" and paradigm shifts, including misinterpretation
and suppression of evidence. For scientists the determination of "what field" or "what speciality" is a typical preliminary to valid communication -- determinations typically subject to disruption by scientific revolutions that redefine boundaries between specialities.
- who-questions, whilst claiming impersonal objectivity,
may then be perceived as closely associated with the well-recognized, questionable, "unscientific",
professional issues of territory, groupthink,
science politics, mutual citation networks, creativity-inhibiting peer
review systems, and overriding patterns of personal and institutional ambition
(cf. Carl J. Sindermann. Winning the Games Scientists
Play, 2001). Of partcular interest is the predetermination of relevance on the basis of "who" is the source of information.
- why-questions, may only become evident in the consequences
unquestioning commitment to particular belief systems -- as is evident
with respect to faith-based science, to ethical issues relating to
responsibility of science, including complicity in development
of destructive technologies. The capacity
of conventional science to address why-questions is indeed a matter of
continuing debate (as any web search on "why
questions" science will
The challenge might be framed as follows:
- what is being done inappropriately is well-documented and widely recognized
(cf Encyclopedia of World
Problems and Human Potential)
- who is doing it is a matter of continuing debate in
the search for those to blame and those to emulate -- although one's own
complicity may go unrecognized
- why "who" is doing "what" is a question that is avoided
and framed as inappropriate -- especially with respect to one's own valued
or habitual initiatives
There is however a danger that the rigour of complexity theory may take
it beyond the point where it can be related to anything that can be grasped
with respect to practical policy initiatives -- a conceptual equivalent to
the application of the Peter
Principle. It is one thing to recognize the principle that 'The
first law of ecology is that everything is related to everything else' (Barry
it is quite another
to devise appropriate, communicable strategies in response to a particular
issue. The comprehensibility of an adequate explanation -- as an approximation in a world of compromise -- may be of greater value to sustainable social change than its diminished significance in a more fundamental framework. After all, even astrophysicists continue to use the geocentric phrase "the sun rises".
A generalization of catastrophe theory, avoiding controversial issues explored
by the Thom-Zeeman approach, has been produced by Vladimir
I. Arnol'd (Catastrophe
For Arnol'd: "Singularities, bifurcations, catastrophes are different terms
for describing the emergence of discrete structures from smooth, continuous
ones." His mathematical generalization of singularity
theory takes the focus
off the limited set of "elementary catastrophes" that are particularly susceptible
to visual representation (and real world examples) and stresses the much larger
range of singularities. However, in what follows, the concern is specifically
with discontinuities that are comprehensible and meaningful to the constrained
human mind as a description of behaviour -- rather than with singularities
that can only be represented mathematically. Of relevance to what follows,
however, is the focus in singularity theory on the failure
of manifold structure --
which might be understood in non-mathematical terms as the the kind of breakdown
of coherence and definition that evokes questions.
Cognitive feel for cognitive catastrophes: question conformality
This theme is developed in an Annex whose contents are:
Correspondence of WH-questions to elementary catastrophes
As described by Thom, the seven "elementary catastrophes" are
presented below (together with his "archetypal morphologies").
A possible cognitive correspondence to WH-questions or interrogatives has been tentatively
added (in italics). Most languages have seven interrogatives. The literature variously recognizes seven or eight WH-questions. For example the BBC recognizes eight [more], whereas as "interrogatives" only 7 are recognized in scriptural studies (cf The Christian and Missionary Alliance, Bible Quizzing Rule Book, 2004: "The seven permissible interrogatives are who (or a form of it), what, why, where, when, which, and how."). Note the assumptions in the table that: the who-question is considered to include "whose" and whom"; the why-question to include "wherefore" (even though the latter emphasizes purpose, whereas the former emphasizes cause); and that any "whether" question can be reduced to a which-question.
|WH-questions in relation
to the "elementary catastrophes"of catastrophe theory (adapted
from Rene Thom)
with addition of tentative cognitive correspondence to WH-questions
||Edge, end; refraction of sunlight
by raindrops to form a rainbow
||Fault; geological fault; transitions
from flight to fight, love to hate, and anxiety to calm in man and
||Slit, crack; behavior patterns
in some human nervous disorders; structural stability and buckling
||Pocket, shell; structural stability
||Arch; collapse of bridges; development
of sonar devices
(wave), Breaking down
||Needle, hair; flow of fluids
||Fountain, mushroom, mouth; atmospheric
fronts; problems in the field of linguistics; elastic stability
As noted by Alexander Woodcock and Monte Davis (Catastrophe Theory,
These stable unfoldings are called catastrophes because each of them has
regions where a dynamic system can jump suddenly from one state to another,
although the factors controlling the process change continuously. Each
of the seven catastrophes represents a pattern of behaviour determined
only by the number of control factors, not by their nature or by the interior
mechanisms that connect them to the system's behaviour. Therefore the elementary
catastrophes can be models for a wide variety of processes, even those
in which we know little about the quantitative laws involved. This is an
extraordinary idea: how is it possible that two processes can have features
in common even when they are on different physical scales, operate under
different quantitative laws and are affected by different sets of causes
The table above serves (tentatively) to distinguish the WH-questions in
terms of generic catastrophes:
- the first four questions deal primarily with operational tangibles --
through "when", "where", "which"
and "how" (corresponding, in mathematical terms, to systems with
only one behavioural variable and termed the cuspoid series). They might
notably be said to be typical of the "project logic" characteristic
of the majority of strategies in response to sustainable development. John
Ralston Saul (The Unconscious Civilization, 1995 ) has criticized
such "in-the-box" initiatives as not calling for imaginative
thought on the part of those managing their implementation
- the second three questions deal essentially with intangibles --
"what", "who" and "why" (corresponding to
systems having two behavioural output variables or order parameters, with
interaction between the order parameters in the second two cases). The
first two might be said to be typical of the politics, networking and spin
through which a plethora of projects are defined and funded in response
to strategies of sustainable development.
Why-questions and the parabolic umbilic
It is the complexity of "why" (corresponding to the parabolic
umbilic) -- with 4 input (control) factors, rather than the 3 factors
in the case of "what" and "who" -- that might be understood
as characterizing the dynamics of the Council
of the Whys.
The six-dimensional parabolic umbilic is difficult to depict except in
very limited subsections (cf A N Godwin. Three
dimensional pictures for Thom's parabolic umbilic, Publications
Mathématiques de l'IHÉS, 40, 1971, pp. 117-138).
In seeking a correspondence between why-questions and Thom's "archetypal
morphologies" in the table, it is interesting that both a fountain and
a mushroom -- as traditional
cross-cultural symbols of creativity, fecundity and innovation -- are cited
as examples of the parabolic umbilic catastrophe (possibly to be understood
as a form of conformal mapping). In a discussion on innovation, human creativity
and complexity, Stephen J. Guastello (Chaos,
Catastrophe, and Human Affairs: applications of nonlinear dynamics to work,
organizations, and social evolution, 1995) explores the parabolic
umbilic in relation to "mushroom catastrophe dynamics". This was
found to explain the dynamics of creative problem solving in groups who were
working together in real time in an experimental situation.
The approach suggests the possibility of analyzing why-questions in terms
of "4 input control factors" and "2 behavioural output variables",
notably as to the manner in which these offer greater creative degrees of
freedom and potential for reframing. What indeed is the "shape" of
a "good question", or a pertinent question -- its "goodness
of fit"? How is "not the right question" recognizable? What
makes for a "burning question"? For example, how "problematic" (perhaps
in the light of complexity / chaos theory) does an experiential situation
have to become before "why" is a more appropriate question than
In particular such reframing may apply to:
- who: identity and "kinship" of both the "other" and,
self-reflexively, the questioner.
- what: categories through which reality, and responses to it, are articulated
The failure of Thom's catastrophe theory to provide descriptions of systems
with more than 5 significant variables can perhaps be related to the challenge
of determining the nature of higher order questions "beyond" the
limited set of WH-questions -- and the why-questions potentially describable
in terms of the parabolic umbilic.
Pointers to comprehension of multi-dimensionality of WH-questions
As noted above, the multi-dimensionality of the catastrophes makes them
difficult to depict except through particular subsections. These may even
conceal the challenge of comprehension, especially when the dimensions may
be interpreted either temporally or spatially. The few dynamic visualizations
tend to be of quantitative rather than qualitative value. The following
are therefore suggested as ways of coming to an understanding of the dynamics
of WH-questions, and especially why-questions, in the light of ways of engaging
with catastrophes variously understood.
A. Skateboarding pointers to comprehension of multi-dimensionality
It is curious, if not remarkable, that
many who are readily framed as least likely to be interested in comprehension
of WH-questions per
to be found exploring highly complex surfaces by skateboarding over
them to the limits of their kinetic skills. Skateboards first emerged, as
a historical coincidence, within the same decade that Rene Thom was
elaborating his theory. They are notably associated with a counter-culture
that contests a particular consumer-oriented development of public commercial
space and seeks to engage with that space in new ways (cf Ocean Howell. The
Poetics of Security: Skateboarding, Urban Design, and the New Public Space,
2001). Their significance has been compared to that of the Situationists.
The surfaces are constructed as skateparks by progressive local
communities and public institutions, or (possibly more questionably) as commercial
ventures. Some are of relatively simple configuration; others are highly
The dynamic of movement through such a space is central to the geometry
of skatepark design:
One of the most important elements of skate park design is flow. A skate
park's flow determines how a rider transitions from one ramp to the
next and how he/she moves from one area of the skate park to the next.
Proper flow allows for beginner, intermediate and advanced session areas,
each with several different ride lines. [more]
It would be worth exploring the jargon (cf Skateboard Science, Skateboarding
Glossary) and skills (note the many skateboarding
skateboarders) to determine the extent that they enact the behavioural
characteristics of the dynamic systems with fold and cusp catastrophes.
These are according to Zeeman (1977):
- Bimodality: if a system spends most of its time on either of
two widely separated conditions
- Inaccessibility: when intermediate values between the conditions are inaccessible.
- Sudden jumps: if the system jumps from one condition to another.
- Hysteresis: if there is a cycle of jumping back and forth due to
oscillations of the normal factor, but with the jumps not happening at
the same point.
- Divergence: resulting from increases in the splitting factor with
two parallel paths initially near one another moving apart if they end
up in different conditions after the splitting factor passes beyond the cusp
point (only in cusp catastrophes)
Given the kinetic gyrations of the most "aerobatic" skateboarders, it is
probable that they could well be exploring even more complex catastrophes.
In the light of the categorization of forms of intelligence by Howard
Gardner (Frames of Mind: the theory of multiple intelligences, 1985),
is the kinesthetic (bodily/kinetic) intelligence indeed more capable of
"comprehending" more complex catastrophes?
The skills of skateboarders certainly require a corresponding ability to
consciously address other WH-questions: when, where, which, how, what, and
who. With regard to "why", it might be argued that through the activity itself,
skateboarders are effectively enacting the "why" in the social-political
sense explored by Ocean Howell (The
Poetics of Security: Skateboarding, Urban Design, and the New Public Space,
2001). Skateboarders might be said to be "skating the why" just as dancers
of past cultures have enacted a reality that does not lend itself to verbal
B. Sexual attraction and intercourse as pointers to comprehension of multi-dimensionality
between sensory input regarding the human body and its movements, and their
semantic implications, as described above by Wolfgang Wildgen (Catastrophe
theoretical models in semantics, 2004; Morphogenesis
of limits: the relevance of dynamic
systems theory for cognitive linguistics, 2005) raise issues about
the nature of attractors and repellors in interpersonal encounter in relation
to the shapes of elementary catastrophes. Wildgen focuses only on a neutral
subset of the range of Thom's archetypal morphologies in his investigation.
The dynamics of arousal and intercourse have however been explored using
catastrophe theory by H M Hubey (Catastrophe
Theory and Human Sexual Response, 1991).
There is however the radical possibility that sexual attraction (or repulsion),
as a fundamental behavioural discontinuity, is intimately related to psychological
engagement with the spatio-temporal forms and patterns (as "catastrophes")
of interpersonal encounter. This is exemplified by the discontinuity of "falling
in love". To what extent is eye movement over the body of another associated
with a response to forms that lend themselves to mathematical description
as "catastrophes"? Human beauty has indeed been analyzed in terms
of geometric proportion. The shapes of the body that are a focus of attraction
can be closely related to the shapes descriptive of sections of the elementary
catastrophes -- even without taking into account dynamics associated with
them (cf Gurman
in the Nude).
Dali as a surrealist painter renowned for his attention to women, devoted
several of his last works to topology, inspired by Rene Thom. As reported by Thomas F. Banchoff (The Fourth Dimension and the Theology of Edwin Abbott Abbott):
Salvador Dali's painting Corpus Hypercubus of Christ crucified on a hypercube, symbolizes the infinite folded down into the finite for our benefit... We do not see things completely; we only see them in their illusions. Dali's final painting includes inflection points, and a swallowtail catastrophe, which forms the image of a chalice, once again combining mathematics and theology.
Dali's final works was entitled Topological
Contortion of a Female Figure (1983). It might be assumed that, as
an artist, Dali recognized the relationship of the attractive shapes of the
body of a woman to the forms of the elementary catastrophes. Of related interest
are three poems of Mary Jo Bang (The
Eye Like a Strange Balloon, 2004) on the topic of catastrophe theory.
As implied by T. van Gelder and R. Port (Beyond symbolic: towards
a kama-sutra of compositionality. In: Symbol processing and connectionist
network models in artificial intelligence and cognitive modeling : steps
towards principled integration, 1994), is there a cognitive dynamic to interpersonal
encounters of which the 64 positions of the Kama
Sutra could be considered an enactable
code for multi-dimensional understandings that cannot be verbally articulated?
(cf Boris Saulnier, Au-delà du
symbolique : la modélisation constructiviste et morphodynamique
des systèmes, et le défi de la compositionnalité,
2003). Such possibilities relate to the explorations of tantric
yoga. Intercourse might then be understood as "dancing with discontinuity" and
with the associated questions and answers -- expressed non-verbally through
to their consummation and "semantic" union (cf O E Rasmussen. The Dance of Meaning: the fundamentals of interpersonal reasoning and sense-making. European Chaos/ Complexity in Organisations Network (ECCON), 2005) .
Such considerations can only be suggestively reinforced through an applet
such as that of Lucien Dujardin (Catastrophe
Teacher: an introduction for experimentalists -- parabolic umbilic,
2005) through which 4 control parameters can be variously controlled.
C. Multiple intelligences as pointers to comprehension of multi-dimensionality
many approaches to knowledge organization -- as helpfully compared in tabular
form by Verna
Knowledge Evolution: Expanding Organizational Intelligence, 1997, Appendix).
Allee's comparison includes models by Peter
Senge, Erich Jantsch, Stafford
Kohlberg, Jean Piaget, Arthur
Young, Ken Wilber,
The comparison is made in terms of seven categories: data, information (procedural),
knowledge (functional), meaning (managing), philosophy (integrating), wisdom
(renewing), and union.
Both the multiplicity of models, and the distinction
of stages, phases or modes within models, is indicative of fundamental discontinuity
in knowledge space -- however this is understood. Allee notes in particular
the continuing influence of the 7-fold chakra system as one of the
oldest such organizing systems for domains of intelligence (cf Rolf von Eckartsberg, Maps
of the Mind: the cartography of consciousness, In: The
Metaphors of Consciousness,
1981). The situation might well
be described as a quagmire of competing alternatives with many of the models
implicitly endeavouring to subsume others. It is possible that these
discontinuities -- and the stability of the stages perceived -- could be
fruitfully explored in the light of the elementary catastrophe forms.
To clarify the relation to WH-questions, the focus here is placed on the
distinctions made by Howard
Gardner (Frames of Mind: the theory of multiple intelligences, 1985;
Intelligence Reframed: Multiple Intelligences for the
21st Century, 2000)
[more more more more] .
- linguistic intelligence: processing and using
oral and written language
intelligence: including rationality and technical understanding.
- spatial / visual intelligence: processing of visual representations
- bodily-kinesthetic / motor intelligence: learning through
touch, muscles, skin, balance, etc
- auditory / musical
intelligence: deriving meaningful
patterns while listening
- interpersonal intelligence
- intrapersonal intelligence: providing understanding
of one's own needs and the capacity to satisfy those needs
intelligence: relating to and
making meaning of the physical world and environment
Gardner, and others, have also considered adding philosophical intelligence,
spiritual intelligence, existential intelligence, moral intelligence, and
others. It might be asked whether a particular intelligence is especially
empowered to enable a response to a particular question. How is the thinking
associated with a particular intelligence "formed" by such a question
and by the catastrophe(s) to which it is required to respond? If that is
the case, then the debate about "missing intelligences" in Gardner's
set highlights the concern as to the possibility of "missing questions" in
the WH set (cf Council
of the Whys: emergent wisdom through configuration of why-question dynamics).
In exploring the relation of such distinctions to WH-questions, there
are several approaches including:
- refining and constraining particular WH-questions so that they confirm less ambiguously to particular intelligences
- considering individual WH-questions to be more particularly, or primarily,
associated with one (or more) modes of intelligence, where other questions are
only of secondary significance
- clustering particular combinations of WH-questions which may then provide a better mapping to particular intelligences and to the associated catastrophe(s) to which they respond. This is illustrated by the following table.
|Table: Combinations of WH-questions, potentially characteristic of particular modes of intelligence
The elaboration of the above table derives in part from earlier work on such a tabulation (see Set of measure formulae as a template for WH-questions in: Functional
Complementarity of Higher Order Questions: psycho-social sustainability modelled
by coordinated movement, 2004) where the positions of "how" and "what" were interchanged. It was however triggered by the following comments of Peter Collins (personal communication) in relation to the four quadrant approach of Ken Wilber:
Using Wilberian terminology all phenomenal interactions can be explained in terms of the interaction of the four quadrants.... So we could then validly affirm that all of the WH-questions relate in varying manners to the interaction between the four quadrants. Clearly
for some of the WH-questions the interaction would be very limited. For
example "when" and "where" questions would largely relate (though not
exclusively) to just one quadrant.
However... "why" questions are potentially much more complex and would
entail a richer interaction of the four quadrants. For example if one
poses the question: Why did John Lennon die, one could attempt to answer
with an objective answer at an individual level e.g. because he was fatally
wounded by a gunshot. However one could also attempt to explain it at a
subjective individual level e.g. because his killer was motivated with
the desire to achieve some kind of distorted recognition through the act.
However one could also attempt to deal with the issue in (objective) collective
terms e.g. that the freedom to hold guns increases the risk of such killings
or perhaps in subjective (collective) terms that the cult of celebrity
leads many mentally unbalanced people to act irrationally.
Therefore though earlier WH-questions can be largely dealt with in terms
of one quadrant, more complex "why" questions entail a richer
interaction of all 4.
So the four control parameters [of a catastrophe] would be the inputs four quadrants. The two behavioural outputs [of a catastrophe] would then relate to both the analytic (real)
and holistic (imaginary) understanding emerging from this process. [emphasis added]
Clearly the most creative endeavour would arise from the successful spiritual integration of the quadrants providing an overall (holistic)
context for the creative interpretation of (analytic) phenomena.
The sequence of WH-questions of the above table has been ordered to define four (colored) "quadrants" which could prove to be consonant with those of Wilber:
- upper left: interior-individual (intentional) -- "I". Subjective-Individual. Defined above by 9 combinations of the higher-order WH-questions
- lower-left: interior-collective (cultural) -- "WE". Intersubjective-Collective. Defined above by 12 combinations of the 7 WH-questions
- upper-right: exterior-individual (behavioural) -- "IT". Objective-Individual. Defined above by 12 combinations of the 7 WH-questions
- lower-right: interior-collective (systems) -- "ITS". Interobjective-Collective. Defined above by 16 combinations of the lower-order WH-questions
The table as a whole might be considered as somewhat akin to the periodic
table of chemical elements in identifying "groups" as adapted elsewhere
Classification in an Integrative Matrix of Human Preoccupations,
1982) and currently used for subject access to online
interaction" of WH-questions, for which Collins calls, would then amount
to "molecules" -- memes -- combining elements from various parts
of that table.
The earlier approach (Set
of measure formulae as a template for WH-questions, 2004) suggests
the correspondence of WH-questions to physical properties that would
be significant in any further exploration of the correspondence to elementary
catastrophes. Such measure formulae also provide greater connectivity
to the physical dynamics of skateboarding.
D. Psychosis / Neurosis as pointers to comprehension of multi-dimensionality
theory offers a way of describing a variety of psychological systems that
have conditions that are prone to suddenly "flipping" between states,
instead of gradually moving between states in a linear manner.
There is continuing application of
catastrophe theory to stage-wise cognitive development and sudden attitude
change, including conversion. Work has been undertaken on the psychosocial
dynamics of conflict-cooperation (cf R.J. Rummel, A
Catastrophe Theory Model of the conflict Helix, with tests, 2002).
The relevance to psychopathology has been recognized (Major
Principles of Ontogeny and Phylogeny, Current Medical Diagnosis and Treatment
in Psychiatry, 2005), including study of anxiety (cf Ivan M.McNally, Contrasting
Concepts of Competitive State-Anxiety in Sport: multidimensional anxiety
and catastrophe theories. Athletic Insight, 4, 2, August
2002). Possible applications of catastrophe theory to psychoanalysis
have been explored, but with relatively little follow-up (cf R Galazer-Levy, Qualitative
change from quantitative change: mathematical catastrophe theory in relation
to psychoanalysis. J. Amer. Psychoanal. Assn., 26, 1978, pp. 921
- 935; J Sashin, Affect tolerance: a model of affect-response using catastrophe
theory. J. Sociobiol. Struct., 8: 1985, pp. 175-202; Michèle Porte, La
dynamique qualitative en psychanalyse, 1994). An annotated review of the
more general literature relating to
nonlinear dynamics in psychiatry has been developed
by David M. Kreindler and Charles J. Lumsden (Chaos
and Psyche, 2005) covering: metaphors and metatheories, neural
networks in psychiatry, psychotherapy
and phenomenology,disorder modeling (schizophrenia, mood and affect, personality),
Anthony Stadlen, in a
review of Cassandra's
Daughter: A History of Psychoanalysis in Europe and America (1999)
by Joseph Schwartz, notes:
Rene Thom's notion of science as "reducing the
arbitrariness of the metaphor" is pertinent to Dr Schwartz's
own view of science. And Christopher Zeeman's metaphoric geometry
of cusps and butterflies is less arbitrary, and reflects more accurately,
the experience of hysteresis, catastrophic change and transcendence than
such metaphors as "vicious circle", or "another
spiral of the dialectic",
or "'equidistant from id and superego", though these
are good enough for many purposes.
A study by J Bradmetz (A
Topological Model of Epistemic Intentionality, Axiomathes,
13, 2, 2002, pp. 127-146) focuses, beyond their linguistic and rhetorical
uses, on the mental epistemic verbs to know and to believe that
reveals a basic conceptual system for human intentionality and the theory
of representational mind. Cusp and butterfly models were used to
explore the formation of epistemic states. These treat to
believe as an intermediate state which lacks
stability and presents the delayed effect of hysteresis.
The preliminary nature of such applications does not preclude the possibility
of their relevance to the study of mental states and behaviours associated
with WH-questions. At issue however is the manner in which these mental states
are personally experienced rather than externally described by observers
(cf Denis Postle, Catastrophe
Theory: predict and avoid personal disasters, 1980).
Can the onset
and form of pathological states -- the pathways and sudden transitions
of attitude -- be experienced in ways that can be informed by the forms
of the elementary catastrophes? How might experience of these forms relate
to the experience of repetitive or other dysfunctional processes of formulating
WH-questions characteristic of such pathology -- in the "skatepark" of
the mind and/or emotions? (cf Doris Lessing, Prisons
We Choose to Live Inside, 1987)
E. "Games people play" as pointers to comprehension of multi-dimensionality
discussing the communication problems resulting from the discontinuities
between disciplines, Felix Geyer (The
Increasing Convergence of Social Science and Cybernetics, 1996)
One should not wonder about strange interpretations of concepts when one transfers research results from one field to another field. It is for this that the skill of communication comes into the game of science. One has to move to the language-and-cultural games people play, even when they are doing science. It is then effective to make uses of the knowledge and "know-how" of the human sciences. That includes the last level of complexity: ... on which the complexity can be measured by the availability of a language in which theories can be formulated and communicated between different disciplines to attack the problem in an heuristic way;
The Stanford Encyclopedia of Philosophy reports on the plausibility of the statement by H Gintis (Game Theory Evolving, 2000) that "game theory is a universal language for the unification of the behavioral sciences." K Binmore (Game Theory and the Social Contract, 1998) is reported as having modeled social history as a series of convergences on increasingly efficient equilibria in commonly encountered transaction games, interrupted by episodes in which some people try to shift to new equilibria by moving off stable equilibrium paths, resulting in periodic catastrophes.
On a more personal level, the process of transaction
analysis developed by Eric
Berne (Games People Play: the psychology of human relationships,
1964/1996) and promoted by the International
Transaction Analysis Association, suggests that such games and transactions
could be interpreted as stable patterns in a context of dynamic discontinuity
-- namely catastrophes (see also Claude
Steiner, Scripts People Live: transactional analysis of life scripts,
1975/1990; Larry McLauchlin, Advanced Language Patterns
Mastery). For transaction
analysis, the games are common counterproductive social interactions. Such
games are clustered as one of a variety of group-dynamic
games for which the same point could be made. This approach has been extended towards variants of "games played in groups" and "games organizations play" (cf Thierry Gaudin. Les Katas Institutionnels, 1977; Michael Maccoby, The Gamesman, 1976) [more]. The issue is the nature
of the WH-questions implicit in any such transactional game and whether
engagement in any game pattern is effectively the enactment of a question
that might be understood as taking the form of an elementary catastrophe.
F. Strategy games as pointers to comprehension of multi-dimensionality
Games like chess
and go are frequently associated with comments concerning the distinct "energy" or
tensions characteristic of certain strategic conditions. A valuable description
of this subtle perception is provided (in translation) by Michel Bruneau
Chess Classification -- Chess
Theory) which explicitly acknowledges how difficult it is to explain
the meaning of "energy" in chess. The document distinguishes, and
comments on, 7 game conditions:
- "Quick divergency"
- "Slow divergency"
- "Damped divergency"
The document states that "chess energy" or "tension" is
the result of various imbalances appearing on the chessboard during the unfolding
of the game. Their brief description, in energy terms, of each condition
-- as a discontinuity -- suggests an intriguing resemblance to geometrical
descriptions of the 7 catastrophes. Their descriptions might be usefully
refined by a chess-playing mathematician familar with catastrophe theory.
Such descriptions might also be usefully confronted with analogous descriptions
by go-playing mathematicians (cf David H. Stern et al. Modelling
Uncertainty in the Game of Go; Bruno Bouzy and Tristan Cazenave, Computer
Go: an AI oriented survey, 2001).
The game of go (also known as igo, wei ch'i,
or baduk) uses
terms such as aji (literally "taste" or "flavor")
meaning latent energy --- a long-term potential for good or ill that persists,
ready for activation at the appropriate moment. There is the possibility
that such conditions are especially associated with particular strategic
questions -- as explored above. Go has been used as a research
domain for cognitive science cf Jay Madison Burmeister, Studies
in Human and Computer Go: Assessing the Game of Go as a Research Domain for
Cognitive Science, 2000). In a discussion of the neural and cognitive
aspects of go, Robert T. Myers and Sangit Chatterjee (Science,
Culture, and the Game of Go, Journal of Science and Culture)
symmetry, simplicity and the topological nature of the game of Go evoke
great fascination on the part of mathematicians, physicists and people
from every branch of science.... Typical professional Go players...
almost certainly do not approach the game from a mathematical vantage point.
Instead they appear to view Go as a highly pattern-oriented game, involving
deeper and deeper trees of strategies and sub-strategies.... This has given
rise to the intriguing notion that Go is in fact the classical AI problem
that chess turned out not to be, that solving Go will in fact require approaches
which successfully emulate fundamental processes of the human mind, and
the development of these approaches may both give us new insight in to
human thought processes and lead to the discovery of new algorithms applicable
to problems ranging far beyond Go itself.... One key element would appear
to be a pattern recognition and processing facility. Stronger amateurs
and pros, when confronted with a local board position, can immediately
point out the move considered to be "good shape" or "bad
The game has been understood in terms
of energy since its origins, as noted by Peter Shotwell (The
Game of Go: speculations on its origins and symbolism in Ancient China,
...it would seem likely that the players of
the first games of go would have been characterizing their play
as attempts to block and release qi by placing their stones down
on the board according to the tenants of feng shui (literally 'wind
the qi of "energy".
The ability to sense such energy patterns may be related to the phenomenon
of synaesthesia (cf
Richard E. Cytowic, Synesthesia:
phenomenology and neuropsychology -- a review of current knowledge,
1995). A number of synaesthetes have had significant chess-playing or mathematical
ability (cf Vladimir
Tammet). In this dynamic, patterned context, what might be" the shape
of a question" -- or of the 7 distinct WH-questions, given their strategic
importance in response to emerging "catastrophes"?
G. Meditation pointers to comprehension of multi-dimensionality
possibilities provide a connection to understandings from disciplines of
meditation with regard to "cognitive
catastrophes" as sudden spiritual insights or "peak experiences",
whether through the extensive contemporary explorations and syntheses of
Wilber and others -- or through the traditional insights associated with tantric
yoga (with respect
to the chakras)
or with comparable disciplines (cf Navigating
Alternative Conceptual Realities: clues to the dynamics of enacting new
paradigms through movement, 2002). The issue is whether different
stages or modes of meditation are especially associated with particular WH-questions
that experientially take forms describable, or recognizable, as "cognitive
catastrophes" (cf. Leslie L. Downing, A Catastrophe
Theory Model of Ideological Conversion and Commitment,
If the meditation process is described as a cognitive activity in a multi-dimensional
space, pathways defined by spiritual disciplines, as well as "mistakes" may,
for example, have some recognizable correspondence for practitioners to the
properties described (above) for fold and cusp catastrophes according to
Aspects of this possibility have been explored by S. David Stoney, Jr. (A Structure for Embodied Human Consciouness, Society for Neuroscience Abstracts, 24:1419, 1998) with respect to the cusp and butterfly catastrophes.
To the extent that the meditation is understood and experienced as an energy activity, as in the case of the tantric yoga focus on chakras, a particularly interesting issue is the relationship sought between understandings associated with the "simplest" and most "complex" chakras -- expressed in terms of the coiled kundalini. In terms of the multi-dimensional insights associated with catastrophe theory, this suggests the emergence of potentially higher forms of order than those associated with the elementary catastrophes. This could well be associated with yet higher-order questions (cf Engaging
with Questions of Higher Order: cognitive vigilance required for higher
degrees of twistedness, 2004).
H. Symbolism pointers to comprehension of multi-dimensionality
The common symbol indicative
of any form of WH-question is the question
mark. No distinction is made between types of question. It is interesting
to reflect on how cultures that may have some sensitivity to the intimate
relationship between a question and the associated nonlinear cognitive dynamics
might choose to represent distinct WH-questions -- or the associated "cognitive
catastrophe", however that is understood.
It is for this reason that it is worth noting the implications in the case
of Sanskrit as one of the languages considered sacred. The Vedas,
for example, are written in Sanskrit -- considered to be "the language
This derives in part from an understanding in the Hindu tradition that the
forms of the letters of such a language are themselves embodiments of different
patterns or movements of insight or energy -- cognitive, if not spiritual.
For example, R.K. Joshi (The Religious World of Letterforms Mediamatic, vol. 8, 1996, #4) indicates:
Various writing systems either pictographic, alphabetic or syllabic in nature, developed in different parts of the world over different time-frames, have been associated with the spoken or written expressions of linguistic thoughts....Yet many of these sign systems in the able hands of sensitive calligraphers or master writers reflected their inherent formal aesthetics through their well formulated structures, taking ordinary writing to the level of Art. At a still higher level, philosophies of writing have attributed certain sacred qualities to the written signs, even claiming spiritual experience through the ritualistic practice of writing. By taking the art of calligraphy to the sublime heights of meditation through the symbolic representation of deities in the form of letters (seed-syllables), written signs not only served to help acquire a knowledge base of the physical world around the human being, but also played an important role in their spiritual and metaphysical needs. Letterforms essentially aided communication with the unknown via the primal energy behind their worldly manifestations.
This understanding in relation to calligraphy, and associated meditation,
is common to a number of traditions, including the Coptic Gnostic, the Jewish
Kabbalistic, Tibetan, Zen, and Tantra (in its highest form) where they may
be used in the construction of integrative mandala-type images. It is suggestive
of their common role that "germ" and "seed" are terms
common both to the mathematics of the generation of the different forms of
catastrophe and to the focal denotations of the individual chakras and
their associated energy patterns (as bija) .
It is therefore tempting to consider
the question mark as a "generic" version of the cognitive shift
associated with the signs traditionally central to the 7 chakras (and a focus
of meditation on them) -- and considered here (as argued above) as having
a possible correspondence with the individual WH-questions. In the case of
chakra symbols, the so-called "petals" surrounding each might be
considered indicative of the dimensionality of the associated catastrophe.
There is the interesting possibility that other traditions of mathematics, as noted by ethnomathematicians, may have developed forms of representation more suited to "seeing" catastrophic forms and viable pathways of continuity through them. For example Bill Barton (Ethnomathematics and Philosophy, 1999):
For example, consider the traditional questions in the philosophy of mathematics: how do we come to know about mathematical objects? But if the way we talk about number or space uses action words, how are we to make sense of a question about mathematical objects? ... Those of us with conventional mathematical back-grounds tend to think in rectilinear grid systems - our graphs are drawn with axes as verticals and horizontals, our talk is full of 'ups and downs'. Many weavers of indigenous crafts, however, orient themselves to diagonal systems: weaving on the diagonal is an easier technique in many situations. What mathematical functions would interest us if our graphs were drawn using diagonal axes?
|Pattern of symbols in relation to WH-questions
(very tentative -- in process ***)
The cells of the above table, following earlier investigations (Functional
Complementarity of Higher Order Questions: psycho-social sustainability
modelled by coordinated movement, 2004), can be populated by
understandings of question combinations. Thus row-heading questions can
be combined with column headings (eg of the type Why/How) to
populate the lower-left quadrant -- or the reverse (eg How/Why)
to populate the upper-right quadrant. This suggests an inversion of the
dominant and subordinate role for the two original WH-questions in each
As a tentative illustration, a zodiacal symbol is used (to be understood
differently in each quadrant) -- given the possibility that these forms
offered traditional mnemonic reminders of the forms of the question in
each case (as argued with respect to the chakra letters). There
is an interesting possibility to be explored that the form of the constellations
were used by cultures of the past to hold such reminders of the "form" of
the challenging questions required for psycho-social sustainability (cf Edward
Matchett and George Trevelyan, Twelve Seats at the
Round Table, 1976). The challenge of encoding cultural insights
in this way to protect them against erosion of social memory has been
explored elsewhere (Minding
the Future: a thought experiment on presenting new information,
Learning and the Erosion of Collective Memory, 1980).
One approach to providing symbolic content to the 16 cells of the lower-right
quadrant is to use the codes of the Myers-Briggs
Type Indicator which is
derived from combinations of the the 4-fold set of psychological functions
identified by C G Jung. However since no mnemonic cultural symbols are used
to identify these combinations, one alternative is to use the tetragram coding
-- a reduction of the I
The octagram, doubling the tetragram, is used in the case of the Tao
Te Ching. Note the widespread use of n-grams,
including the quadrigram (Research
on N-Grams in Information Retrieval, 1997; Valeri F. Venda, et
Ergonomics: Theory, Laws, and Graphic Models, International
Journal of Cognitive Ergonomics 2000, Vol. 4, No. 4, pp. 331-349).
Geometrically n-grams are represented by star
polygons. Iin the I Ching hexagram a "moving
line" may occur where a solid line shifts
to a broken line (or vice versa) transforming one hexagram condition into
another. This convention is of potential interest for indicating
relationships between the cells in the lower-right quadrant as a result of "movement" in
the lines denoting the conditions -- thereby delineating pathways between
the cell conditions (cf Transformation
Metaphors (I Ching): dialogue, vision, conferencing, policy, network, community
and lifestyle, 1997),
With respect to the upper-left quadrant, potentially relevant is a modern
organization of 9 distinct modes of intelligence provided by the enneagram (cf
A G E Blake, The
Intelligent Enneagram, 1996). However, despite its reputed origins,
no distinct symbols are used to distinguish the nine individually within
the symbol that frames their relationship. There are however two distinct
sets of 9 symbols that merit consideration for that quadrant:
- A key Sanskrit work, Natya Shastra ("Scripture of Dance") deals with the different arts used to express feelings: primarily music, dance, literature and theater. It is the foundation of the fine arts in India, notably through distinguishing the rasas as forms of emotion susceptible to artistic expression: Adbhuta (Wonder), Hasya (Laughter), Shringara (Love), Shaanta (Peace), Bibhatsa (Disgust), Vira (Valour), Karuna (Pathos), Bhaya (Fear) and Raudra (Anger). (NB According to Bharata, there were 8 rasas and this was accepted until the first commentator on Natya Shastra began to speak of the rasas as nine in number). They were further articulated by Jagadguru Badshah (Book of Nine Rasas) who is renowned for having sought to bring cultural harmony, between the Shiyas and the Sunnis and between Hindus and Muslims through music.
With regard to rasa, Vilayanur S. Ramachandran (The
neurological basis of artistic universals,
Art and Cognition) argues that: "This Sanskrit word is difficult
to translate but it roughly means "Capturing
the very essence, the very spirit of something, in order to evoke a specific
mood or emotion in the viewer's brain".... I realized
that if you want to understand art you have to understand what rasa is
and how it is extracted by the neural circuitry in the brain".
- In Greek mythology, the Muses are nine archaic goddesses who embody the right evocation of myth, inspired through remembered and improvised song and traditional "music" and dances. They are: Calliope (epic poetry), Euterpe (music), Clio (history), Erato (lyrics/love poetry), Melpomene (tragedy), Polyhymnia (sacred poetry), Terpsichore (dancing), Thalia (comedy), and Urania (astronomy).
In each case the argument is that it is these symbols that are carriers of cultural insight into the nature and "form" of a particular cognitive discontinuity -- a particular dance between question and answer, between questioner and answerer. (Note that such an approach has been the subject of a variety of speculative experiments and proposals [more | more | more | more | more]). Nick Herbert (Quantum Reality: Beyond the New Physics, 1987) suggests that:
One of the greatest scientific achievements imaginable would be the discovery of an explicit relationship between the waveform alphabets of quantum theory and certain human states of consciousness.
Psychosocial implications of WH-questions as "catastrophes": when, where, which, how
The set of WH-questions can now be tentatively explored in terms of their
potential psychosocial implications -- assuming that the above argument
for the relationship with the elementary catastrophes has a degree of validity.
When-question: Here the focus is
clearly on timing and its unpredictability -- a one-dimensional concern.
This is the domain of a "fold
catastrophe". It can be understood as "the
start of something" or "the end of something", in other words
as a "limit", whether temporal or spatial. It
concerns the sudden transition between one condition and another. It has
no "cusp" point in contrast to the other catastrophes.
case of an ecosystem under stress, when its assimilative capacity is exceeded,
it enters a condition of "metastability" -- a region of potentially
rapid transition triggered by even a minor disturbance. Examples include:
- gradual increases in nutrients transforming an oligotrophic into a eutrophic
- overgrazing transforming a grassland into a desert scrub ecosystem,
- overfishing causing the sudden collapse of a fishery.
- goods and services
that are assumed in an economic model are no longer available; the assumptions
of the economic model are no longer valid.
In effect the system jumps from one stable equilibrium
to another without the intervention of a major external
disturbance. All catastrophes are generically described by fold catastrophes.
Much concern about the condition of modern society and
the planet may be described in terms of the triggering of such catastrophes --
hence the precautionary
principle. This is the challenge of "when" any system -- personal,
group, culture, ecology -- may "flip", possibly irreversibly, into
a new mode (cf Spontaneous
Initiation of Armageddon -- a heartfelt response to systemic negligence,
2004). That mode could of course be more desirable -- for some.
Where-question: Here the focus is
on location. As a "cusp
catastrophe", this is the spatial extension of the fold catastrophe
-- into four dimensions. The cusp is a pleat,
a fault; its temporal interpretation is "to separate, to unite, to
capture, to generate, to change". It is the point at which a curve crosses
itself. The catastrophe has a single cusp point -- the point of
coming together of two folds in a sharp spike like intersection. A cusp catastrophe
provides for a smooth, slow transition from one condition to another as well
as for a fast discontinuous transition. The notion of a "cusp of change" describes
a significant place of transition.
The cusp catastrophe has for example been used to model the dynamics
of brain processes resulting in occasional epileptic seizures [more] or the behaviour of a stressed dog, which may respond by becoming cowed or becoming angry.
Its relevance to policy analysis is highlighted by Lorraine Dodd and Mark
of intervention actions for HIV/AIDS, 2004) concerned with the basic
requirement for the analytical methods in determining the appropriate relationship
between social and political coherence and the experience of pandemic disease.
The main question is then, for analytical methods: what are the appropriate
modes of analysis to grasp these diverse data and make them tractable for
effective policy response? In a description of the challenge of understanding
the control space, the authors note:
Three-dimensional cusp models can be used to understand the control space provided that the system has what is known as a "gradient dynamic" -- that is, we are always trying to minimise some function such as cost or loss, and that there are two variables controlling this gradient. A vital element of a cusp catastrophe model is that it shows there is a range of values of the control parameters, where small continuous increases or decreases in them can result in large fluctuations in behaviour. For example, we could define the two control parameters as follows:
- a: system stress (level of uncertainty and trust in the situation
attributes plus number of conflicting cost assessments and conflicts
- b: need for intervention action (criticality depends on mis-match between
perceived current situation and desired situation).
Put simply the two control parameters represent, respectively, the system's unpredictability/complexity and the magnitude of the 'stakes' if intervention actions fail (or if no action is taken).
In effect such an approach addresses the where-questions: where to intervene and where to go in navigating the collective decision-space. Such concerns also apply in the study of individual human stress by Peter A. Levine (Accumulated Stress, Reserve Capacity, and Disease):
The underlying theme of this paper is that the accumulation of stress affects the reserve capacity of an organism, both in the maintenance of its functional integrity and in the resolution of subsequent exposures to stress... The basic physiologic relations of the autonomic, sympathetic and parasympathetic, can be represented by a simple mechanical analogy... which exhibits properties described by a relatively new branch of mathematical topology, catastrophe theory. The visualization gained by this re-presentation offers new insights into the nature and mechanisms by which stress accumulates. It also suggests "paradigms" by which stress, once it has already become internalized, may be successively resolved towards re-establishing a fuller adaptational range/reserve capacity.
There are several different kinds of cusp catastrophe of varying shapes and applications. The classic model is essentially a potential well -- a gravity well -- usefully understood as a 'strange attractor'. It is effectively a system that moves in a continuous loop within a defined space around one or more poles -- without making exactly the same loop twice. The irrecoverable state is then expressed as a loop initially traveling around two poles (violent and nonviolent), with the loop eventually settling around a single pole. More directly, it can also be expressed as a sort of 3-dimensional descent down a mountainside -- from one state to another -- with bumps along the way.
Clearly this could also be used to model a progressively converging
movement of the eyes in response to an attractive feature of a human body.
There are applications of such models to marketing and consumer catchment
areas -- including the challenge of switching between products. In the case
of larger social policy issues, it clearly has relevance to switching
between potentially dangerous attractors (unrestricted consumption of non-renewable
resources, gated communities) and alternative attractors (quality of life,
etc) -- or the reverse. The challenging where-question
remains that of determining the pathway -- whether into or out of any place
of power (or opportunity) or a "Garden of Eden".
Which-question: Here the focus is on
the modelling capacity of the swallowtail
catastrophe. In the
terms of Ole Elstrup Rasmussen (The
Discontinuity of Human Existence), this third order catastrophe
consists of a three-dimensional agent surface that determines a four-dimensional
surface of objective. This catastrophe has two cusp points, in contrast to
the single point of the cusp catastrophe. These bifurcation points of the
surface of objective projected on the agent surface, given certain conditions,
resemble the swallowtail. The control dimensions which constitute the agent
surface of the model are attention, positioning and intention. Rasmussen
uses it to model "human existence subsisting as development".
Stephen J. Guastello (Nonlinear
Dynamics in Psychology, 2001) points to the value of the swallowtail
model in relation to the which-question of decision-making:
The formation of roles would constitute fitness peaks, which denote relative fitness, local stability, and clusters of similar subspecies with regard to shared adaptive traits. The probability density function that is associated with the swallowtail catastrophe model ... describes the distribution of people into unstable and locally stable social roles. The swallowtail catastrophe structure contains a response surface of discontinuous events, or qualitatively different outcomes, such that there are two stable states, with a minor antimode between them, an unstable state, and a major antimode separating the unstable state from the two stable ones.
In the abstract of a later paper, notably summarizing his earlier work, (Leadership Emergence in Coordination-Intensive, Creative Problem Solving, and Production Groups, 2006) this is more specifically highlighted in response to the which-question decision processes through which a leader is identified::
In self-organizing processes, a system acquires a structure without any external intervention. This presentation describes the process by which an initially leaderless group differentiates into one containing leadership and secondary role structures and how the process was aptly described by the swallowtail catastrophe model. A subsequent study identified the control variables in the process of leadership emergence in creative problem solving and production-oriented groups which were found to be different in the two cases. The third study, which is considered in greater depth here, examined a different case of coordination-intensive groups. Coordination-intensive groups are particularly interesting because it is known that coordination can occur without talking and without leaders present, even though talking helps in some respects. Coordination is also interesting because it is a nonlinear dynamical process all by itself
The three control parameters of the swallowtail catastrophe were thought to be related to asking questions, making jokes, initiating a line of discussion, clarifying points made in a discussion, diffusing conflict, following others, and "gate keeping" (cf Stephen J. Guastello, et al, A Rugged Landscape Model for Self-Organization and Emergent Leadership in Creative Problem Solving and Production Groups, Nonlinear Dynamics, Psychology, and Life Sciences, 9, 3, July, 2005, pp. 297-333)/
A swallowtail catastrophe was determined to be the most appropriate means of dynamically modeling the macro features of a natural resource / environmental (pollution) system in relationship to the economic system. Under a given mode of specific change of economic activities, the related policy parameters were shown to have to satisfy some requirements to avoid a catastrophic shift (Liu-jun Chen and Fu-kang Fang, On the mechanism of catastrophic shift in natural resource and environmental system, The Fourth International Workshop on Meta-synthesis and Complex System, 2004, Beijing)
In the light of the work of Guastello, it might be asked whether
the swallowtail catastrophe is the most appropriate to model decision-making
in leading to emergence of alternative communities -- or lifestyles.
How-question: Here the focus is on the
effective navigation of a strategic space. The butterfly catastrophe can be considered as a metastable state that is transitional between the two main stable states -- namely the system can exist dynamically in that transitional condition for a time before entering one of the stable conditions. In the terms of Ole
Elstrup Rasmussen (The
Discontinuity of Human Existence), the fourth order butterfly
catastrophe is so termed, because, under certain circumstances, the projection
of the surface of objective onto the agent surface resembles one. It is the
first to exhibit a plane of bifurcation points. It is a swallowtail catastrophe
with an extra control dimension. Because of the added control dimension,
the agent surface becomes four-dimensional and the surface of objective five-dimensional.
This type of catastrophe is quite impossible to illustrate. He uses it to
model "human existence subsisting as correlation".
He suggests that the complicated movement modelled by this fourth order
catastrophe is the agent identifying the identity of objective. With respect
to the practicalities of the how-question, he helpfully argues:
Identifying is an infinite process. The concept, however, is a momentarily finite process which encompasses the categories that are identified. This means that none of the categories being identified disappear: they are embedded in the concept. Because of the distance, the categories still exist as different from each other... The categories exist in time and space, but, as they are also identified because of the correlation prerequisite, the concept encompasses identifying as identity... The most common example of identifying dissimilar categories is the process of buying and selling, that is, trade in the market place by simple exchange of commodities. Because of the exchange, the commodities are distributed in the network in a new manner. Things and information change position in the network following the rules that are determined by canalization.... Because of the exchange in the market place, a sack of potatoes and a chair, for example, change places in the network, but in the same process they are identified and systematized in the concept of commodity. The farmer exchanges his potatoes as a commodity with the carpenter's chair, which is also a commodity.
The how-question may be considered as intimately related to the following
- language: Enrique Bernádez (On the Study of Language with the Tools of Catastrophe Theory. Atlantis, 1995) indicates that a butterfly catastrophe is the kind of process found in all situations involving compromise, whether in political life or biological development "or in the opposition of two adjectives of the type black-grey-white and the like, with two extremes and an intermediate, usually badly defined 'compromise' state with rather fuzzy borders."
- motivation: S J Guastello (A butterfly catastrophe
model of motivation in organizations, Journal of Applied Psychology,
72, 1987, pp. 165-182) who studied the
efficiency of the butterfly catastrophe model for describing and predicting
performance changes in an educational setting.
- willingness to work: Daniel C. Kuang and Jenny C.
Kuang (Cusp, Quasi-Cusp
and Butterfly Catastrophe Models of Employee Strike Willingness, 2002
SIOP Conference) The butterfly catastrophe effectively unified the
complex and multidimensional strike willingness construct. Union and organizational
strategies in contract negotiation naturally emerged from the developed
models. Procedural justice is predicted to mediate strike willingness decisions.
- driving a vehicle: Furutani, Naomichi (A New Approach
to Traffic Behaviour: III. Steering Behaviour and the Butterfly Catastrophe.
Journal of Man-Machine Studies, 9, 1977, 2, pp. 233-254)
- dynamics of bilingualism: Ernest Querol (A
New Theoretical Model for the Study of Bilingual Contexts: the Catastrophe
the number and type of variables determining the use of a language since
discontinuities are the key factor in determining language shift for monolingual
speakers of either language (or a third) and for bilingual speakers having
a bias towards one language or the other. The processes of language shift
are described in terms of the stages of the butterfly catastrophe with
a clear identification of the control factors.
- competence development: Jytte Bang and Ole Elstrup
Development Learning by Problem Solving,
Cognitive Science Research, Lund University, 2000, No. 74) point out that
in order to explain different forms of problem-solving performance,
it is no longer necessary to assign different attributes to persons who
perform differently in complex situations:
The five-dimensional butterfly-catastrophe demands a four-dimensional
control space. The two remaining control parameters, feeling of availability
and achievement, are assumed to constitute the needed parameters. This
means that it is availability and achievement in connection with efficacy
and ruggedness which give rise to the third type of performance, muddling
through. In generating the butterfly catastrophe, one state variable (the
likelihood of performance) and four control parameters are needed (the
normal factor, efficacy; the splitting factor, ruggedness; the bias factor,
achievement; and the butterfly factor, availability).
for a task: Ole Elstrup Rasmussen and Jørgen
Aage Jensen (Preparations
for Modelling the Relationship between Competence and Qualifications.
Cognitive Science Research, Lund University,
2000, No. 77) were concerned to operationalize
the control parameters and behaviour forms in terms of the cusp and
In terms of behaviour, they infer from their study that "roaming"
could be seen as a fairly direct expression of the relationship, whilst another
interesting path is from "digging in" to "lingering" to "ensuring the
change of competence" that could make such a path possible. Comparing paths,
that from "roaming" to "exploring" would imply changes in both of the
control parameters; that involving a catastrophic
jump from "roaming" to "digging in", implying a constant ruggedness, but
a large change of efficacy.
- distinguishing performance features
indicating the different forms of performance
- distinguishing features that may indicate the conflicting control
parameters efficacy and ruggedness
- plotting the relation between these
groups of performance features
- competence reflected in the problem setting, manifesting
itself in the decisions made (as expressed
in the establishment of the unit of meaning), with development
of competence expressed in the change of the unit of meaning.
- enactment of qualifications, reflected in the problem solving,
manifesting itself in the choices made.
- emulsion inversion: A butterfly catastrophe model has
been applied to mimic dynamic inversion in emulsions. Psychosocial analogues
could be envisaged (including willingness to strike)
In the light of the above, the how-question in relation to (global) governance within the current mindset, might be more effectively addressed through the butterfly catastrophe model.
Psychosocial implications of WH-questions as "catastrophes": what, who, why
As illustrated by the challenges of operationalizing parameters for the
butterfly catastrophe, as well as the difficulty of comprehending (let alone
visualizing) the more complex umbilic catastrophes, there is almost no research
on the applicability of such models to the social sciences. One isolated
example is an analysis by
Puu (1979) of the structural change in regional trading systems
using the five dimensional hyperbolic and elliptic umbilic catastrophes.
The lack of such research inhibits reflection on their value in psychosocial
Such catastrophes are commonly observed in optics in the focal
surfaces created by light reflecting off a surface in three dimensions and
are intimately connected with the geometry of nearly spherical surfaces.
As catastrophes their geometric forms have in common, and are distinguished
from others, by possessing a point on a surface at which the curvature is
the same in any direction.
Seemingly more interesting than the "objective" optical examples are the
far more subjective aspects of the what-, who- and why-questions through
which humans define and relate to their environment -- as key processes in
the progressive "construction
of reality", whether as a social or individual enterprise.
What-question: Here the focus should
be on how this question might be clarified by the hyperbolic
umbilic catastrophe. The what-question, as framed above, is concerned
with distinguishing and identifying patterns, objects and entities in knowledge
space -- defining (or imposing) meaningful boundaries. This process is not
simply constrained by sensory input regarding entities distinguished as tangible "objects".
It may include intangibles such as conditions, patterns of behaviour or meaning.
What-questions call for the allocation of what is distinguished to some part
of an organizing framework, a typology or taxonomy, or some other construct
that provides a form of nomenclature. The what-question expects labels following
the detection of a degree of invariance. It is fundamental to information classification,
knowledge organization and its management -- irrespective of the degree of
illusion underlying the process.
In addition the what-question is concerned with moving beyond the constraints
of patterns and sets of categories considered inadequate to
interaction with emerging conditions. In this sense it is associated with
processes of creativity and innovation through which new objects and patterns
are designed -- whether tangible objects (art or technology) or intangibles
(including theories and aesthetic effects). This innovative aspect is significantly
dependent on thinking "out-of-the-box",
namely engaging in the kind of lateral
thinking required to respond creatively
to threats and opportunities. It is therefore to some degree associated with "new
thinking" and enabling "paradigm shifts".
Given this focus of the what-question, it might be asked what relation this
bears to the multiplicity of studies on light and radiation diffraction in
terms of the hyperbolic umbilic catastrophe. Yet, if the challenge of human
understanding of a confusing reality (with "the light of knowledge")
is compared with the considerable challenge of detecting patterns of order
in a crystal using x-ray analysis, then the possibility of a degree of isomorphism
between the pattern detection of the what-question and such light diffraction
studies merits exploration. Depending on how information is understood as
being distributed in knowledge space, the hyperbolic umbilic may prove to
be a valuable means of deriving significance from it. It may actually serve
to "throw light"
on the matter. There is a case for understanding the significance of "information
diffraction" patterns, and the emergence of disciplines as a consequence
of "knowledge diffraction" patterns.
The geometry of the hyperbola basic
to the catastrophe is interesting in relation to understanding of the challenging
relationship of subject-object, question-answer, etc. It offers two perspectives,
corresponding to the two focal points:
- centering "objective" perspective at one focal point, the "subjective" perspective is inaccessible; centering "subjective" perspective at one focal point, the "objective" focal point is inaccessible
- the hyperbola is the locus of all points whose distances, x and y, from two fixed points, A and B, is a constant difference, y - x = k
Different philosophies favour one or the other stance to different degrees,
typically disparaging the other as ill-founded or illusory. The perspectives
are oriented in completely opposite directions -- however complementary they
may appear to others. The challenge lies in the nature of the relationship,
defined by the geometry of the hyperbola, between features centred on one
focal point and features centered on the other. Whether "objective" reality
is considered primary and the "subjective" secondary (or "subjective" reality
is considered primary and the "objective" derivative) is of less
consequence than the relationship -- of reflection -- maintained between
the two. The hyperbola also provides interesting geometry through which to explore the implications of a "shadow", understood in Jungian analysis as the dark, hidden side of the human psyche. In effect, whichever curve of the hyperbola is privileged, it is "shadowed" by complementary perceptions associated with the other.
The what-question can then be variously answered depending on the
part of the geometry from which it is posed. Of great interest is the capacity
of the hyperbola (and the hyperboloid) to function as a template interrelating
polarized perspectives twisted in relation to each other (see below).
To the extent that a form of "binocular" conceptual vision is
possible within the "objective" reality, this hyperbolic model
might be used to explore classic cultural situations (cf C P Snow (The
Two Cultures and the Scientific Revolution, 1959; Samuel P. Huntington,
The Clash of Civilizations and the Remaking of World
Order, 1998). It
corresponds to the understanding of "seeing with two eyes" (cf
John Polkinghorne, Seeing
with Two Eyes: Mutual Harmony between Science and Religion, 2004;
John A T Robinson, Truth is Two-Eyed, 1979; Magoroh Maruyama, Peripheral
Vision: polyocular vision or subunderstanding?, 2004) [more].
The hyperbolic umbilic might therefore serve to map the basic "self-other"
conditions, associated with the what-question, through its potential to
reframe approaches -- through a degree of twistedness -- to the many
polarized situations (and the dynamics of polarization) now tretching
the fabric of society to tearing point.
Who-question: Here the focus should be
on how this question might be clarified by the elliptic
umbilic catastrophe. The who-question, as framed above, is concerned
with detecting and distinguishing identities in psychosocial space. Beyond
the preoccupations of the what-question, it is thus concerned with identifying
and relating to
"significant others" whether in kinship networks, communities or
as enemies from other groups. It is also concerned with the identity of the
questioner -- the "who am I" question so important to self-esteem,
religion and philosophy -- and the challenge to provide a credible answer to
the "who are you" question
asked by others.
In society, issues relating to the who-question are the focus of a variety
- marketing is concerned with the manner in which products and services
define and enhance the identity of the individual or group (as consumers
or voters), notably as fashion statements
- public relations is especially concerned to frame and enhance the identity
of an individual or a group -- the processes of "image management" vital
to the life and survival of celebrities and to sustaining their
self-image, or re-inventing it
- psychotherapies of every kind are concerned with assisting the individual
to better understand who they are, who they can become, and how they can
best relate to others
- those concerned with the complex processes in society that come into
play in response to those who distinguish themselves as heroic,
saintly or demonic exemplars -- or may be promoted as such
Given this focus of the who-question, it might be asked what relation
this bears to the multiplicity of studies on optical image analysis, light
scattering (including backscattering and twinkling), and the study of diffraction
patterns (in association with those modelled by the hyperbolic umbilic catastrophe)
The concept of psychosocial identity is a complex and subtle one -- often
especially for the entity concerned, whether individual or collective (cf
J. P. Cornelissen, On
the 'Organizational Identity' Metaphor, 2002). It
is perhaps not surprising that "light" is frequently used as a
metaphor to describe issues of identity:
- in the "light" of public opinion
- the "limelight", notably in performances
- media "spotlight"
- "light" as a metaphor in redemption in religion -- "enlightenment"
, "I am the light"
- "light of awareness" emerging through education and individuation
The range of light-based metaphors used in discussion of aspects of identity
and image, of the entity or of others perceiving it, again suggests
the possibility of a degree of isomorphism with the optical focus on light
(and blurring), including complex challenges of image analysis, which merit
exploration. Depending on how identity knowledge is understood as being distributed
in psychosocial space, the elliptic umbilic may indeed prove to
be a valuable means of deriving significance from it -- as a "container"
for quite distinct perspectives (and in striking contrast to the hyperbolic
model). It may actually serve to address a range of challenging issues of
identity -- whether individual or collective (cf Quotations
The secondary forms of the why-question, the whom-question and the whose-question,
raise interesting issues in this context, notably regarding the meaning of
possession and property, especially intellectual property and the notion
of home(land). What possibilities are there for reframing "who" in these
extremely sensitive conditions -- and how can the elliptic umbilic facilitate
The geometry of the ellipse basic
to the catastrophe is interesting in relation to understanding of the challenging
relationship of identity -- defined either in in individualistic terms or
in social terms. Like the hyperbola, it offers both these perspectives, corresponding
to the two focal points:
- centering "individualistic" identity at one focal point; centering "social" identity (within a community) at a second
- the eclipse is the locus of all points whose distances from the two fixed focal points have a constant sum
The ellipse is then the locus of combinations of individual-social identity
in relation to those two points -- a pattern of degrees of hybridization.
The ellipse may of course have a variety of forms (including circular)
depending on the dimensions of the major axis relative to the minor axis
-- suggestive of the relative importance attached to the individualistic
or social understanding of identity. The who-question can then be variously
answered depending on the part of the geometry from which it is posed. The
need to combine the perspectives, as with the requirement for two eyes to
achieve stereoscopic vision, is also indicative of the importance of both
focal points to the dynamic positioning of an identity within psychosocial
Through the elliptic umbilic model the who-question might be more
effectively addressed by its capacity to reframe identity as shifting continuously
around (and changing its basis between) two extremes -- individual-collective,
Why-question: Here the focus should be on how this question
might be clarified by the parabolic
umbilic catastrophe. The why-question, as discussed above, is concerned
with new understanding -- potentially outside a framework immediately meaningful
to the questioner. It relates not only to the "external" world of objects and
processes, however they have been defined, but also to the "internal" condition
of the questioner -- whether in existential, ethical, moral, philosophical
or spiritual terms. At its most fundamental, it may seek to focus on the question
of "why am I (alive)".
In society, issues relating to the why-question are the focus of a variety
of disciplines including:
- market research (notably through opinion surveys) seeks to determine why
particular products and services are purchased (or policies are supported)
- motivation research seeking
to understand, from a psychological perspective, the internal state or condition
that activates behaviour and gives it direction, desire or want
- vocational guidance research seeks, in part, to respond meaningfully to a person's
question as to why they should engage in society
- values research seeks to understand the values and principles that
determine individual and collective action
- experiential / lifestyle research seeks to understand why individuals
or groups engage in alternative lifestyles and are able to sustain them:
- philosophy is much preoccupied with the why-question and how it may be
- theology / religion
seeks to respond to the why-questions of generations of followers in the
light of understanding of the belief system
- ethical frameworks:
- mythology / creation myths:
- Joseph Campbell (Myths of Light: Eastern Metaphors of
- attractors **** scarcity ***
The geometry of the parabola
to the catastrophe is of value in understanding the challenging relationship
of the questioner to the space-time manifold in which he/she is located and/or
moving. It has three significant attributes:
- the focal point, that may be considered the locus of the questioner
- the vertex, positioned equidistantly between the focus and the directrix.
- the directrix, namely the line which, together with the focus, serves
to define a conic section as the locus of points whose distance from the
focus is proportional to the horizontal distance from the directrix (perhaps
to be understood as the axis of grounded reality).
Metaphorically, these attributes of the parabolic umbilic offer a means of ordering understanding
in response to the why-question from a wide variety of philosophical, ideological and spiritual standpoints -- especially
if consideration is given to the orientation of the parabola (for example
in relation to the symbolism of the holy chalice). The why-question can then be
variously answered depending on the emphasis placed on particular parts of
Interrelating cognitive catastrophes in a "Grail-chalice" proto-model
The possibility of interrelating a representation of the elementary catastrophes in a "Grail-chalice" proto-model is discussed in an Annex with the following contents:
The pattern of relationship between the WH-questions suggests that the essence
of human identity may be fruitfully understood as a play on a pattern of
questions. These may be variously interrelated or nested within one another
-- just as the fold catastrophe is a generic version of the others. In the
light of understandings of the set of chakras and their relationships,
these too (as psycho-energetic functions) may each be understood as individually
associated with WH-questions and "catastrophes" (whether as crises
or opportunities). Energetically the human body may therefore be understood
as an implicit set of questions -- perhaps to be played like a flute, such
as by "opening" and "closing" the chakras.
From a larger perspective, and in relation to any "transcendent" awareness
of a harmonious cosmic
the catastrophes associated with particular types of question may also be
understood as "mistakes" -- as discontinuities to be regretted. From a Taoist perspective, the need to answer a question may already imply that a mistake is being made.
The first four catastrophes, and the associated WH-questions, are reasonably
comprehensible precisely because they deal with tangibles. The other three,
the umbilics, are a fundamental challenge to comprehension because of their
degree of multidimensionality. However it would appear that, through the
potentially associated questions regarding existential intangibles, they
do offer pointers to lines of exploration.
In conclusion, it might be said that explorations such as the above, point
to the value of Thom's emphasis on more general qualitative and philosophical
concerns than those with which mathematicians are conventionally preoccupied.
Mathematics can do much to order the relationship between the questions that
humans face -- individually and collectively -- in responding to the challenges
of life in the 21st century.
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