    
|
|
21st March 2006
| Draft
Conformality of 7 WH-questions to 7 Elementary Catastrophesan exploration of potential psychosocial implications- / - Annex to
Introduction
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 IntroductionAs 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):
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 theoryCatastrophe 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]:
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:
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):
Applications of catastrophe theoryAfter 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)
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.
For Lucas:
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):
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):
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:
The challenge might be framed as follows:
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 conformalityThis theme is developed in an Annex whose contents are:
Correspondence of WH-questions to elementary catastrophesAs 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.
As noted by Alexander Woodcock and Monte Davis (Catastrophe Theory, 1976):
The table above serves (tentatively) to distinguish the WH-questions in terms of generic catastrophes:
| |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||