Interweaving fundamental patterning approaches to transformation

In Quest of a Dynamic Pattern of Transformations (Part #5)

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Mathematics: It is to be expected that mathematics, as the formal science of relationships, would be of most relevance to any understanding and representation of patterns of transformation. The difficulty for the purpose of this exercise is the extent to which such insight is "buried" in formalism through which people can engage with the experience of transformation processes only with difficulty. Mathematics as a discipline is only incidentally and indirectly interested in the experience of mathematics as it might serve as a carrier for the experiences of transformation, despite efforts to promote related understanding (cf. Jacques Hadamard, The Psychology of Invention in the Mathematical Field, 1949; Philip J. Davis and Reuben Hersh, The Mathematical Experience, 1981; George Lakoff and Rafael Núñez, Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2000).

More focused efforts to explore possibilities of this kind tend to be treated as marginal, if not suspect. They might include:

• geometry / topology, as vehicles of meaning, as exemplified by the work of:
  • R. Buckminster Fuller (Synergetics: Explorations in the Geometry of Thinking, 1975) discussed separately (Geometry of Thinking for Sustainable Global Governance: cognitive Implication of synergetics, 2009)
  • Arthur M. Young (The Geometry of Meaning, 1976), as developer of the first Bell helicopter he endeavoured to derive insights from the dynamics of piloting it with a view to developing a "psychopter" (cf. Engendering a Psychopter through Biomimicry and Technomimicry: insights from the process of helicopter development, 2011)
  • Ron Atkin (Multidimensional Man; can man live in 3-dimensional space ? 1981), as summarized separately (Comprehension: Social organization determined by incommunicability of insights, 1995).
  • Rene Thom (Structural Stability and Morphogenesis: an outline of a general theory of models, 1972), who notably took account of the challenges of semiotics (Esquisse d'une Sémiophysique: physique aristotélicienne et théorie des catastrophes, 1989).
  • Christopher Alexander (The Nature of Order: an essay on the art of building and the nature of the universe, 2003-4), as mentioned above, who has interpreted this overview in terms of the dynamics of complexity theory (New Concepts in Complexity Theory: an overview of the four books of the Nature of Order with emphasis on the scientific problems which are raised, 2003; Harmony-Seeking Computations: a science of non-classical dynamics based on the progressive evolution of the larger whole, International Journal for Unconventional Computing (IJUC), 2009). He emphasizes the importance of geometric adaptation in order to enable comprehension of a higher order, as discussed separately (Harmony-Comprehension and Wholeness-Engendering: eliciting psychosocial transformational principles from design, 2010)
  • Steven M. Rosen (Topologies of the Flesh: a multidimensional exploration of the lifeworld, 2006; Dimensions of Apeiron: a topological phenomenology of space, time, and individuation, 2004) who focuses on the engagement with paradox through the topology of the Klein bottle.
  • Douglas Hofstadter (I Am a Strange Loop, 2007) who has adapted a self-reflexive approach to paradox in terms of the Möbius strip, with implications discussed separately (Sustaining a Community of Strange Loops: comprehension and engagement through aesthetic ring transformation, 2010)
• symmetry of unusually higher order, as discussed separately with respect to:
  • Monstrous moonshine mathematics and Monster Group symmetry (Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks, 2007; Dynamics of Symmetry Group Theorizing: comprehension of psycho-social implication, 2008)
  • Mandelbrot set and its fractal organization (Psycho-social Significance of the Mandelbrot Set: a sustainable boundary between chaos and order, 2005)
• probabilistic approach to truth
  • Vasily V. Nalimov (Realms of the Unconscious: the enchanted frontier, 1982) provides a remarkable aesthetic synthesis of biological, mathematical, and linguistic manifestations of probability -- as previously summarized (Probabilistic vision of the world, 1995).
• changing classificatory frameworks
  • Patrick A. Heelan (The Logic of Changing Classificatory Frameworks, 1974; Space-Perception and the Philosophy of Science, 1983)
• number
  • Marie-Louise von Franz (Number and Time; reflections leading towards a unification of psychology and physics, 1974), most notably her comment to the effect:

    When we take into account the individual characteristics of natural numbers, we can actually demonstrate that they produce the same ordering effects in the physical and psychic realms; they therefore appear to constitute the most basic constants of nature expressing unitary psycho-physical reality. Because of this I would conjecture that the task of future mathematicians will be to collect their characteristics and analyze. when possible, every number in its logical relationship to all others. This research should be undertaken in collaboration with physicists, musicians, and psychologists who are conversant with the empirical facts about the structural characteristics of numbers in different mediums.

  • the cognitive implication of the organization of sets with various numbers of elements may also imply a dynamic between those elements (cf. Representation, Comprehension and Communication of Sets: the role of number, 1978; Patterns of N-foldness; comparison of integrated multi-set concept schemes as forms of presentation, 1980). Irrespective of the seemingly disparate nature of observed phenomena in different domains, if it proves appropriate in practice to the human brain to group the processes of each domain into the same number of clusters, then this establishes a degree of correspondence between those domains -- a pattern of connectivity calling for consideration as an artefact of human consciousness (cf. Theories of Correspondences -- and potential equivalences between them in correlative thinking, 2007).

As emphasized, the approaches cited above tend to be framed statically -- implying a dynamic, if at all. This is epitomized by names attributed to the necessarily dynamic elementary catastrophes (fold catastrophe, cusp catastrophe, swallowtail catastrophe, butterfly catastrophe). It is of course with the dynamics of breaking waves that surfers develop their expertise. Although seemingly "ludicrous", it is such cognitive capacity that is required of governance in navigating a crisis -- namely a dynamic process, not a static condition. Less evident is the nature of any psychological engagement with transformative processes described in this way.

Especially interesting is the case of sacred geometry, given the significance attributed to it in terms of spiritual experience and theology. In the latter case dynamics might be understood as more explicitly carried by associated sacred music and dance. Given the continuing incidence of conflicts sustained by faith-based governance, there is a case for exploring the dynamic implied by mathematical theology, as discussed separately (Mathematical Theology: Future Science of Confidence in Belief -- self-reflexive global reframing to enable faith-based governance, 2011).

As many have remarked, mathematicians would be much challenged to represent and solve in mathematical terms the aerobatics enacted by those with developed kinesthetic intelligence. Expressed otherwise and more provocatively, can mathematicians -- as mathematicians -- "do" mathematics? Or otherwise, what is it that is "done" in aerobatics that is considered irrelevant by mathematicians -- as partially highlighted by the above-mentioned efforts of Arthur Young? How does this relate to the arguments (above) of Deacon?

General systems theory: As noted above, the possibility of this paper was partially inspired by the original articulation of this theory, most notably in General Systems: Yearbook of the Society for General Systems Research. That inspiration is less evident in its subsequent embodiment in the International Society for the Systems Sciences. As mentioned above, the dynamics of "transformation" processes are effectively implicit in the dynamics of such systems, as distinctly explored by cybernetics notably through the International Society for Cybernetics and Systems Research. Although traditionally indifferent to cognitive and existential issues, these have however emerged more recently (cf. Cybernetics and Human Knowing: a Journal of Second Order Cybernetics, Autopoiesis and Cyber-Semiotics; European Society for the Study of Cognitive Systems). The early assertion of John Casti that "behaviorist/cognitive debates are vacuous at the system-theoretic level" seems to have been indicative of future preoccupations (Connectivity, Complexity, and Catastrophe in Large-Scale Systems, 1979, p. 86). Ironically, "vacuous" may turn out to be an essential prerequisite for further understanding -- notably in the light of the arguments for an equivalent to "zero" of Terrence Deacon (above).

It would appear that the progressive formalization within the objectivity of cybernetics has effectively excluded consideration of the experiential subjectivity through which individuals control their relation with the world. Casti (1979) celebrates this transition in the dedication of his study (To Rudolf Kalman, who transformed systems theory from a mystical art into a mathematical discipline). As with mathematicians, it might be asked whether cyberneticians exhibit unusual skill in "doing" cybernetics -- in comparison with the control skills of those exhibiting kinesthetic intelligence, perhaps then to be understood as a "mystical art". Casti's arguments may well exemplify the manner in which cybernetics has been purged of existential significance.

Casti prefaces his own study with a quotation from Ludwig Boltzmann: There is much that is appropriate and correct in the writings of these philosophers. Their remarks, when they denounce other philosophers, are appropriate and correct. But when it comes to their own contributions, they are usually not so.

Pattern language: As noted above, a quite distinct approach is that of pattern language, as developed by Christopher Alexander and colleagues to provide a structured method of describing good design practices within a field of expertise. The original study describes and interrelates 253 patterns -- understood as together forming a language. Curiously, like all languages dealing with a complex activity, it has vocabulary, syntax, and grammar, but it is specifically noted as not applied to communication itself. Domains to which the approach has been applied include software engineering and, more generally, computer science, as well as interaction designs. Pedagogical patterns are used to document good practices in teaching. A set of some 136 patterns for using information and communication to promote sustainability, democracy and positive social change has been produced by Douglas Schuler (Liberating Voices: a pattern language for communication revolution, 2008).

Classical Chinese encodings: Perhaps most remarkable in terms of combining systematic formalization -- and a preoccupation with comprehensible relevance to individual and collective decision making -- are the several distinct encodings, highly esteemed over centuries within China. These most notably include the:

• I Ching (or Yi Jing), also known as the Book of Changes. This identifies 64 patterns (encoded in the form of hexagrams) and the relationships between them (Diagram of 384 Relationships between I Ching Hexagrams, 1983). The original commentary makes extensive use of metaphor to facilitate comprehension -- in contrast with conventional systems initiatives. This is a mode which can be adapted to current preoccupations (Transformation Metaphors -- derived experimentally from the Chinese Book of Changes (I Ching) for sustainable dialogue, vision, conferencing, policy, network, community and lifestyle, 1997)
  
  
• Tao Te Ching (9-fold Higher Order Patterning of Tao Te Ching Insights, 2003; Commentary on Tao Te Ching Interpretation -- and the possibility of higher order patterning, 2003). Navigational implications are explored in Hyperspace Clues to the Psychology of the Pattern that Connects, 2003).
  
  
• Taixuanjing (or T'ai Hsüan Ching or Canon of Supreme Mystery). It is also known in the West as The Alternative I Ching and The Elemental Changes. This identifies 81 situations and the relationships between them.

The relationship between these patterns is notably a current preoccupation in terms of the mathematics of magic squares (9-fold Magic Square Pattern of Tao Te Ching Insights: experimentally associated with the 81 insights of the T'ai Hsüan Ching, 2006; Reframing the Dynamics of Engaging with Otherness: triadic correspondences between Topology, Kama Sutra and I Ching, 2011). A remarkable effort has been made by Maurice Yolles and colleagues to relate the Chinese yin-yang insights to those of cybernetics (Toward a Formal Theory of Socioculture: a yin-yang information-based theory of social change. Kybernetes, 2008).

It is most curious that the sudoku puzzle -- intimately related to the mathematics of the magic square and its origins in these encodings -- should currently be such a recreational preoccupation worldwide (The Appeal of Sudoku, Psychology Today, 19 July 2009). Whilst there is no explicit link to such encodings, of concern here is the nature of the "satisfaction" in successfully completing such a puzzle and driving the continuing effort to do so. This is of relevance to recognition and appreciation of an underlying pattern of transformation. Of relevance to the argument of Deacon (above), regarding the role of absence and constraint, is recognition of constraint satisfaction.

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