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18th November 2003 | Draft
Hyperspace Clues to the Psychology of the Pattern that Connectsin the light of the 81 Tao Te Ching insights- / -This is part of a commentary on the Tao Te Ching Interpreted
Succinctly (original
order) and (alternative
order)
See also Commentary on Tao Te Ching Interpretation: and the possibility of higher order patterning Patterning possibilities are presented separately in detail in 9-fold Higher Order Patterning of Tao Te Ching Insights Context ContextThis is an exploration of the relevance to higher orderings of the insights of the Tao Te Ching of thinking with regard to what is termed hyperspace by mathematicians (understood here to include physicists). This is necessarily a presumptuous, and possibly foolish, undertaking -- since, for mathematicians, hyperspace requires consideration of at least 10 dimensions and the arduous mathematical training to understand the relevant equations. This exploration must therefore be more of an intuitive, speculative exercise in pointing to suggestive possibilities. There is some consolation in that one renowned physicist, Edward Witten, argues that physics is not about complex calculations: "The essence of it is that physics is about concepts, wanting to understand the concepts, the principles by which the world works" It is worth noting that several hundred international physics conferences have been convened to explore the consequences of the higher dimensionality associated with hyperspace. Field of consciousness and the Tao Te ChingThe individual, in framing and dealing with reality, can be understood as being at the centre of a field of consciousness offering a range of possibilities. This field can be understood as organized in many ways. The Tao Te Ching has long provided a much respected pattern of insights -- possibly to be understood as a distillation of awareness about awareness. For the purposes of this exercise, the focus here is on how the 81 insights of the Tao Te Ching might be understood as ordering the range of potential modes of awareness -- both explicit and implicit. In the accompanying exploration of the 9-fold Higher Order Patterning of Tao Te Ching Insights, much attention was given to their possible disposition in a 9x9 matrix -- as an array of insights (disposed in a peacock's tail in some cultural symbolism). Crudely this could then be seen as constituting a kind of setting for a children's game. As with hopscotch, for example, an individual might move from one cell to another -- with each cell being associated with a different perspective, insight or mode of awareness. And with each cell offering different kinds of connectivity to other cells. The challenge in that exploration was to find more powerfully integrative ways of ordering such an array -- hence the exploration of magic squares, and the possible relevance of mathematical objects of higher dimensionality, such as hypercubes. The emphasis however was on how any such order was to be comprehended. For mathematicians the exploration of hyperspace (according to the admirable description of Michio Kaku: Hyperspace, 1994) is based on the "field" theory originated by Faraday -- inspired by an agricultural metaphor. For him, a field occupies a region of three-dimensional space such that at any point in the space a collection of numbers can be assigned that describes the magnetic or electric force at that point. In its development by Georg Riemann (1854), a collection of numbers at every point could be introduced to indicate how much the space was bent or curved. On a two-dimensional surface, a collection of three numbers at every point completely described the bending of the surface -- whereas in four spatial dimensions a collection of 10 numbers was required at each point to describe its properties.
With this device Riemann could then describe N-dimensional space with a metric tensor that would then resemble a chess board that was NxN in size. In the quest to provide a unified description, the metric tensor could be expanded to N-dimensional space then portions of it -- in the form of rectangular pieces -- could be identified as corresponding to different forces embodied in the unified description. Whereas Maxwell's classical field equations for electricity and magnetism are 8 in number, these collapse into a single relativistic equation when time is treated as the fourth dimension -- because they then possess a higher symmetry. The development of theoretical physics over the past century has essentially been based on the search for the field equations of the forces of nature.
This approach was then extended by Kaluza, as indicated above, to provide a basis for unifying Einstein's metric with that of Maxwell. Further expanding the metric tensor in this way subsequently allowed all known forces (gravity, electromagnetism, weak and strong nuclear forces, and most fundamental particles) to be integrated into the unified description. Note that by slicing the metric tensor into its rectangular components, these are respectively descriptive of particular forces.
The question is whether it is fruitful to consider the magic square disposition of the 81 insights of the Tao Te Ching (in the accompanying paper) as in anyway corresponding to such a metric tensor. Each numbered insight would then hold an aspect of the information which -- with that associated with other numbers -- would define how much the "communication space" was bent or curved at that point. Recall that the geometry of such curvature in space-time had been determined by Riemann and Einstein to be indicative of the forces operating at that point. Experiencing the forces of unseen connectivity -- mathematically describedThe focus here is on comprehension by the individual in interacting with the contextual reality at any one moment. The question is whether there is any mathematically-based conceptual bridge that would clarify the relationship between "geometry" and "felt forces" in psychological and communication terms -- rather than in the material terms that are the focus of the metric tensor above. One insightful approach is that of Arthur Young who was inspired by his experience in inventing the Bell helicopter, because of the need for the operator to control movement in three dimensions. His theory of process is a formal analytical model based on number theory, geometry and topology -- which endeavoured to relate to psychologically-oriented modes of knowledge and insight. Young used this model to help comprehend and integrate a number of disciplines and areas of inquiry. His original study Geometry of Meaning (1976), derived from an ordering of 12 dimensionless physical constants, offers a useful basis for exploring a diversity of issues relating to learning/action cycles, dialogue, sustainable development and experience of past-present-future complexes. The mathematician who appears to have been most helpful in that respect is Ron Atkin (1972, 1974, 1976, 1977) -- whose ideas he has articulated more accessibly (Multidimensional man: Can man live in 3-dimensional space? 1981). Atkin proposed the use of simplicial complexes to analyze connectivity in social systems, like cities, committee structures, etc. Since then, Atkin's ideas have been developed further, resulting in a new combinatorial homotopy theory of simplicial complexes. In this setting, a graded group is associated to a simplicial complex, similar to the fundamental group of a topological space. However, the resulting theory is very different from classical combinatorial homotopy theory. Q-analysis is a combination of geometric and algebraic tools for studying relationships and connectivity among entities in a complex system. The research generalizes the idea of binary relation between two things, which underlies the highly successful theory of graphs and networks. Hypergraphs provide a first extension, allowing edges with more than two vertices. The methodology of q-analysis extends this by considering relational structure and multidimensional connectivity. Atkin was especially interested in traffic on hypergraphs. A review of the relevance of insights from q-analysis to an understanding of the psychology of operating in complex communication spaces is given separately in Comprehension: social organization determined by incommunicability of insights (also in Comprehension and Organization). Peter Jackson explores Atkin's ideas on cover set geometry to education (The Geometry of Intention: values in the creation of curriculae) Q-analysis has been used in the social sciences (Cullen, 1983; Macgill, 1985; Seidman, 1983), political science, industrial relations, community studies (Jacobson, 1998), planning (Johnson, 1981; Macgill, 1986), supply chain management (Rakotobe-Joel and Houshmand, 1999) and in organizational analysis [more]. It has been used to solve problems ranging from failure diagnosis in large-scale systems (Isida, 1985), traffic flows, organization of rule-based systems (Duckstein, 1988), multi-criteria decision-making (Chin, 1991). Q-analysis encourages inspection of data without distorting it -- contrasting with the conventional metric approach requiring manipulation of data involving some loss of information. Using q-analysis for organizational analysis, in the Management of Technology Group of the Simon Fraser University (UK), the focus has been on change decisions and management, which are often the marking points in the life of manufacturing organizations where such analysis has been explored as a change management tool that allows the analysis of the change process. The task involved the analysis of the relationship between various organizational forms in the studied artefacts and their respective characteristics in order to unearth the connectivity between various forms. The result of the analysis was then used to assess the change from one organizational form to another. Keys to success were: (1) confirmation of groupings, (2) verification of evolutionary pattern, (3) exploration of the relationships between organizational forms and characteristics sets [more]. This preoccupation with change processes is of course the core focus of the "sister" classic to the Tao Te Ching, namely the I Ching (or Book of Changes). Aron Katsenelinboigen (The Concept of Indeterminism and its Applications: economics, social systems, ethics, artificial intelligence, and aesthetics, 1997) says of Atkin:
Jacky Legrand (How far can q-analysis go into social systems understanding?) provides a detailed critical review of the applicability of q-analysis. She is concerned at the degree of "metaphorical discourse heavily flavoured by the methods of algebraic topology, abstract methodology, practical applications and their relationships" and the need to "separate the syntactic perspective from the semantic perspective". Her major conclusion is that "the gap between metaphorical discussion and woolliness is narrow. The understanding of some of Atkin's ideas has been too intuitive in the past. However the use of graphics as a language is a powerful thinking tool and Atkin has delivered a framework for thought". Possibility of nesting disparate systemic insight setsUsing, by analogy, the method indicated above of expanding the metric tensor, it is interesting to reflect on the possibility that the psychological "forces" in "communication space" of which an individual might be consciously aware -- or be forced to respond to -- could be represented as nested (and "integrated") in the following way.
Use of the term "communication space" here is misleading -- especially when it describes the space which may have the range of psychological dimensions with which psychologists, philosophers and meditators endeavour to come to terms -- or the space in which the many aphorisms reflecting the wisdom of a culture are recognized. In addition the experiential nature of that "space" changes with the dimensionality accorded to it. It is tempting to see the lower dimensional portions of the above as descriptive of the more "obvious" forces experienced in daily life, whereas the higher dimensional portions correspond to various emergent levels of meta-reflection on that experience -- and possibly as a result of extensive life experience. Confucius is reported to have declared that only those above 60 years could hope to understand the I Ching ! Possible psychological implications of magic square orderingIn the light of the exploration in the separate paper 9-fold Higher Order Patterning of Tao Te Ching Insights: Possibilities in the mathematics of magic squares, cubes and hypercubes the question is what psychological significance is to be attached to the "magic" dimensions of any such metric? How is that "magic" experienced? For a mathematician, their symmetry effects are appreciated as beauty. Such aesthetic beauty and elegance effectively function as a kind of attractor -- the more "perfect" the magic object, the greater the sense of symmetry and the more powerful the attraction. The difficulty with such mathematical objects is that, aside from their elegant symmetry, they appear to lack the kind of differentiated content that would enhance the richness of their significance from a psychological perspective. They lack engaging "content" and might well be described as sterile crossword puzzles -- interesting exercises for the mind that do not touch other psychological dimensions. The content of the 81 insights of the Tao Te Ching are in stark contrast to this. Each insight has a quality that may be variously interpreted as description, injunction, orientation, inspiration, etc. It is perhaps here that the notion of aesthetic associations -- as embodied in and triggered by music, song, poetry and drama -- might be seen as represented in some way by the "magic" mathematical properties of such ordering. This is perhaps most obvious in the mathematics of the particular harmonies of plucked strings (echoes of Pythagoras!). Are particular insights linked in a privileged associative manner that is in some way consonant with such properties? In this sense could the "magic" of a poem be described in terms of the "magic" properties of some suitable mathematical object -- remembering that the Tao Te Ching is also considered to be a poem? Is q-analysis of value in exploring the degrees of association in terms of connectivity? Michio Kaku (p. 130) makes the point that: " In some sense, the equations of physics are like the poems of nature. They are short and are organized according to some principle, and the most beautiful of them convey the hidden symmetries of nature". More intriguing is the possibility that the perception of the poem as "magical" may be dependent on the dimensionality of the "communication space" within which it is experienced -- such that the "magic" properties emerge. There is also the question of the ordering of the insights. What are the mathematical properties of the object used to map the 81 insights? Does identifying richer "magical" mathematical mappings ensure a more "magical" aesthetic appreciation of the poem? How do the various mathematical properties (perfect, semi-perfect, etc) affect the aesthetic properties? What is the psychological implication of using a magic hypercube instead of a magic square to order and interrelate the 81 insights of the Tao Te Ching? The important point to be made here is, as in mathematics, that "aesthetic" may have powerful additional implications. For mathematics the stress is on the theoretical implications of the beauties of symmetry as an indicator of "correctness" and "power of explanation". In the psychological case it is associated with integrative, memorable insights -- binding together patterns of insights in resonant forms that provide the foundation for the emergence and comprehension of higher forms of order, and the potential for identification with it in some enhancing manner. "Aesthetic" configures insights in such a way as to sustain the emergence of a higher, subtler form of psychological identity. Helpful research in this direction is that of George Gadanidis and Cornelia Hoogland (Mathematics as story, 2002) who contend that mathematics is an aesthetic and a storied experience. They explore the interplay between what is an ‘aesthetic mathematics experience’ and a ‘good mathematics story’ using a mathematical applet, namely a Colour Calculator -- that resembles the magic squares discussed in the accompanying paper. Navigating the psychological forces of "communication space"The chess board is often used as a metaphorical descriptor of the metric tensor and of magic squares. In playing chess or go, there is a real sense of "forces" of different quality and potential "in play" across the board -- as in any drama. In response to the play of forces in a complex social world, some are recognized as "skilled operators". Curiously some of the interactions are recognized as "doing a number" -- presumably deriving from numbered tactics in team sports. Thierry Gaudin (Les katas institutionnels. Transnational Associations, 30, 1977, 3,pp. 77-79) identified 21 tactical moves (katas) open to institutions. The term "field" has not only been applied to describing physical nature, but also to understanding of psychological and social natures. Gestalt psychologists (Köhler, 1942) have conceptualized a psychic energy field under stress and with forces tending toward sensory and cognitive unity and balanced simplicity. Influenced by Gestalt psychology, Kurt Lewin (1951) in his psychological field theory thought of psychic energy localized in systems of tension and forces. Needs generate the field within which our potential activities and goals become manifest. Following Lewin, Edward Tolman (1951) considered sensory and cognitive psychological elements as affected by need-push forces activated in an energy system. Sociologists also have used field in this meaning. B. F. Brown (1936), a student of Lewin's, considered social behavior a result of individual needs localized in energy systems of tension and forces [more]. Studies of the origins of chess-type games in different cultures, including 9x9 variants, emphasize the ways in which they reflect the psycho-social forces in play (see Pavle Bidev. Chess: a mathematical model of the cosmos, 1979; Ricardo Calvo. Ancient Gnosis and Chess Evolution, 1999 and Continued Extracts on Gnostic Elements in Chess, 1999). Ricardo Calvo concludes that "The movements of the pieces are based in mathematical considerations that are older than the game of chess itself" [more]. Of special intedrest are the cross-cultural anthropological studies of the phylogentic relations between some 40 chess-type games, their connection with divination and the development of the magnetic compass, and the relationship between the current 8x8 variant and the 9x9 variants -- such as the extant Xiangqi (Chinese) and Shogi (Japanese) (see Gerhard Josten. Chess: a living fossil, 2001). It has been argued -- notably in the light of the jumping Rook, together with the movements of all pieces of Chaturanga as seen in the numerical arrangement of a magic square of 8x8 (the so-called Safadi Board) -- that the chess movements were historically deduced from a "genetic code" of arithmetical operations. Knight's move: In chess, this is especially interesting given the potential significance of the moves of the knight -- as a "noble" rather than as a "commoner". The strangeness of the knight's move (a keima in the Japanese game of go), and its numerical symbolism, has traditionally been the focus of hypotheses connecting the origins and structure of chess with secret magical and religious rituals of ancient India. In their study of its significance, James E. Loder and W. Jim Neidhardt (The Knight's Move: the relational logic of the spirit in theology and science, 1992) focus on the expression of complementary thinking that facilitates positive interaction between science and Christian theology. A reviewer, Richard H Bube (The "Strange Loop" of Complementarity) notes: "The symbol of 'the Knight's move' refers to the unique move of the chess piece that is the only one not moving in a straight line, as an indicator of a leap of insight or a leap of faith. The book also draws heavily on the symbolism of the Moebius strip, the two-dimensional 'strange loop' twisted in the middle, which has a two-dimensional surface that can be totally traversed with continuous motion along the strip". The problem of the knight's tour [more] on traditional 64-board in chess was solved by Euler in 1759. Knight's tour and knight's path are special cases of Hamiltonian cycles and Hamiltonian paths in graph theory [In August 2003 it was announced that one of the classical unsolved problems of mathematics, concerning the existence of a path that could be traversed by a knight on an empty numbered 8 x 8 chessboard, had been proven to be without solution]. The move of the knight is used as a metaphor for the unexpected, and illogical, connections between ideas -- invisible to the "commoner". Sidney Cohen described LSD perception as a kind of knight's-move thinking which leaps over logical premises and formal syllogisms. "Knight's move thinking" is even considered a pathological condition of thought disorder denoting a lack of connection between ideas -- an illogicality of the loosening of associations (found in schizophrenia but to be contrasted with the flight of ideas which characterizes hypomania). Strategically it is appreciated as an out-flanking maneuver. [The knight is part of the emblem for the US Psyops as a traditional symbol of special operations signifying the ability to influence all types of warfare.] The knight's move can be used to illustrate how "innovation" can emerge from a point W (below). Whereas the "logical", "linear" moves from W are along any of the grey pathways (whether horizontally, vertically, or diagonally), the knight can move outside this logical framework, first to X, then to Y. In a sense the originality or novelty associated with X is "birthed" by the vertical and diagonal pathways from W. What is "birthed" is in a sense hidden from the linear outlook along the grey pathways from the W perspective.
The mathematical concern with the knight's tour might perhaps be usefully explored in relation to the cultural Grand Tour considered appropriate to the education of nobility of the 18th century -- through which they learned about the politics, culture, and art of neighboring lands. Psychologically it was an exploration of different realms of "communication space" -- distinct from those accessible through the logical framework of the point of departure. Other moves: One of the merits of board games, such as chess and go, is their capacity to give people a sense of the psychological significance of other moves. One of the merits of certain Eastern martial arts, such as aikido, is to extend this to more complex dimensions of communication space. As Clifford Pickover (The Zen of Magic Squares, Circles, and Stars, 2002) has noted, since the dawn of civilization humans have invoked such magical patterns to ward off evil and bring good fortune -- yet who would have guessed that in the twenty-first century, mathematicians would be studying magic squares so immense, and in so many dimensions, that the objects defy ordinary human contemplation and visualization? String theory and modular functionsThe scope of the unification in physics achieved by the above generalization of field theory proved inadequate (given the exclusion of certain fundamental particles) to the challenge of a complete "theory of everything" -- integrating space-time and matter. String theory, with its unusual geometry -- strings vibrating self-consistently in 10 or 26 dimensions -- proved to be the missing link. According to Michio Kaku (co-founder of string theory): "The distinguishing feature of a string is that it is one of the most compact ways of storing vast amounts of data in a way in which information can be replicated" (p. 156). And: "The symmetries of the subatomic realm are but the remnants of the symmetry of higher-dimensional space" (p. 159). As remnants they emerge from the curling up of that space -- as with such visible symmetries as rainbows and crystals. A major concern for physicists is why string theory is defined self-consistently in only 10 or 26 dimensions. The explanation is associated with the modular functions identified by Srinivasa Ramanujan (1887-1920) and named after him. As Michio Kaku explains:
Despite this, physicists remain mystified as to why such magic numbers emerge so definitively. The 10-dimensional theory of hyperspace remains untestable and Michio Kaku (p. 179) asks the question: "Is beauty, by itself, a physical principal that can be substituted for the lack of experimental verification?" But, relevant to the possible higher integration within the Tao Te Ching, Michio Kaku (p. 172) acknowledges that as mysterious as are the modular functions was the self-taught Ramanujan: "...the strangest man in all of mathematics, probably in the entire history of science. He has been compared to a bursting supernova, illuminating the darkest, most profound corners of mathematics...". The incredible theorems in number theory that he exuded -- "half a dozen new ones, almost every day" -- have aroused wonder at the unconventionality of his thinking processes. He has been described as intuition incarrnate (Robert Kanigel, 1991). He is estimated to have produced between three and four thousand theorems -- as many as two-thirds being new to mathematics [more]. For Jonathan Borwein (Ramanujan and Pi. Scientific American, February 1988, p. 112): "He seems to have functioned in a way unlike anybody else we know of. He had such a feel for things that they just flowed out of his brain. Perhaps he didn't see them in any way that's translatable." For Ramanujan, they emerged from his "dreams" inspired by the Hindu goddess Namakkal. It is curious that, of the mathematicians acknowledged to be the greatest of all time (Archimedes, Euler, Gauss, Jacobi, Newton and Ramanujan), both Ramanujan and Newton were inspired in ways which are considered so irrational that they are an embarrassment to their professional peers. How did Ramanujan's "dreams" work -- given that he believed them to be the inspiration of a goddess? Why was the, spiritually inspired, Newton's work on alchemy -- that he believed to be fundamental to his understanding -- considered irrelevant to his mathematics? Are such eccentricities to be equated with substance abuse as incidental to mathematics -- or do they have a role in "Knight's move thinking"? How does intuition work? Cosmology -- Big Bang to Big CrunchAs explained by Michio Kaku, introducing the higher dimensions of hyperspace may also be essential for prying loose the secrets of creation. For:
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