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21st March 2006 | Draft

Conformality of 7 WH-questions to 7 Elementary Catastrophes

an exploration of potential psychosocial implications

- / -


Annex to
Council of the Whys: emergent wisdom through configuration of why-question dynamics

Introduction
Catastrophe theory
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:

Skateboarding | Sexual attraction and intercourse | Multiple intelligences | Psychosis / Neurosis | "Games people play" | Strategy games | Meditation | Symbolism

Psychosocial implications of WH-questions as "catastrophes": when, where, which, how
Psychosocial implications of WH-questions as "catastrophes": what, who, why
Interrelating Cognitive Catastrophes in a "Grail-chalice" Proto-model (Annex)
Conclusion
References

Introduction

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 René 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 (Elementary 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 René 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; Engaging 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

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 geometries [more]. 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 [more].
  • 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, by:

  • 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 a "pocket" of compromise-- with a surface in 4D) [more | applet]
  • Potential functions of two active (or "state", or "fast") variables (or behavioural axes) and

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 & Catastrophe Theory):

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, 2004)

... 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 René 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 Our Methodologies, 2006). For Lucas, WH-questions cannot in practice be treated in isolation however distinct the sentences in which they are embedded.

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.

WH-Question Methodologies
(as developed by Chris Lucas, Questioning Our Methodologies, 2006)
Question Type Scientific Scope
("material")
Personal Scope
("living")
Humanistic Scope
("community")
Spiritual Scope
("holistic")
When do I ask questions ? Once ? Occasionally ? Regularly ? Constantly ?
Where do I ask questions ? One Place ? A Few Places ? Many Places ? Everywhere ?
Which systems are relevant to my questions ? One only ? A Few ? Many ? 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 ? Myself ? Social Group ? Humanity ? All Stakeholders ?
Why do I ask questions ? For Control ? For My Quality of Life ? For Group Quality of Life ? For Development ?
Attractor Type Point Cyclic Strange Transient
Valuation Type Systemic Extrinsic Intrinsic Holarchic
Logic Type Boolean Fuzzy Matrix Integral
Thought Mode Reactive Uniordinal Multiordinal Synergic

For Lucas:

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 elsewhere (Question Avoidance, Evasion, Aversion and Phobia: why we are unable to escape from traps, 2006).

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 premise....

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 René 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 (cf And When the Bombing Stops? Territorial conflict as a challenge to mathematicians, 2000).

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 Journal, 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 of "unscientific", unquestioning commitment to particular belief systems -- as is evident with respect to faith-based science, to ethical issues relating to the social 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 make apparent).

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 Commoner). But 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. Arnold (Catastrophe Theory, 1998). For Arnold: "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 René Thom)
with addition of tentative cognitive correspondence to WH-questions

Singularities
"catastrophes"

Organizing
centres

Physical examples
(substantives)

Dynamics

 

Archetypal morphologies
WH-questions
Destructive
Constructive
Question
Property

Fold

V=x3

Edge, end; refraction of sunlight by raindrops to form a rainbow

Being

Ending

Beginning

When
Time

Cusp

V=x4

Fault; geological fault; transitions from flight to fight, love to hate, and anxiety to calm in man and animals

Becoming

 

Capturing, Separating, Breaking

Engender, Uniting, Becoming

Where Location

Swallowtail

V=x5

Slit, crack; behavior patterns in some human nervous disorders; structural stability and buckling

Agitate

Rejecting, Tearing, Splitting

Crossing, Knitting

Which
(Whether)
Distinction

Butterfly

V=x6

Pocket, shell; structural stability and buckling

Give

Sending, Scaling, Exfoliating

Receiving, Giving

How Dynamic

Hyperbolic umbilic

V=x3+y3

Arch; collapse of bridges; development of sonar devices

Cresting wave

Collapsing, Breaking (wave), Breaking down

Covering

What Typology, Taxonomy
Nomenclature

Elliptic umbilic

V=x3-3xy2

Needle, hair; flow of fluids

Penetrate

Piercing

Filling, Annihilating

Who
Whom
Whose
Identity, Nomenklatura,
Authenticity

Parabolic umbilic

V=x2y+y4

Fountain, mushroom, mouth; atmospheric fronts; problems in the field of linguistics; elastic stability

Eject

Lancing, Pinching

Linking, Opening

Why
Wherefore
Reason,
Symbolism,
Auspiciousness
Value
Aesthetic

As noted by Alexander Woodcock and Monte Davis (Catastrophe Theory, 1976):

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,