PREFACE
Different languages describing the same thing or scene or experience emphasize different aspects, constructing different realities accordingly. So a study of parallel descriptions will yield just that: the recognition that different cultures are able to describe the same thing in different ways. But the question is: what is the nature of this 'sameness'? The mandalas of Taoism and the archetypes of Jungian psychology, for instance, have a common ground rooted in the nature of pre-naming psyche, the underlayment before the advent of the discriminating mind.
But knowing this, that cross-cultural metaphors for aspects of reality are capable of pointing to the same thing, does not seem to have any earth shaking affect beyond the understanding and acceptance that certain principles of life and insights into its nature are universally understood. Moreover, this knowing does not, in general, seem to be able to push the mind that one step further: to the realization that perceptions are relative and subjective, that they are not absolute objective truths, that an examination of 'perception' itself may yield a new way of apprehending reality.
The mind is full to overflowing with images and metaphors, beliefs and moralities, points-of-view. Is there a fundamental self that once realized will allow us to get outside the prisons of our perceptual identifications? How much control and responsibility do we have over what we create, what we perceive, from moment to moment, in our local daily lives?
The mathematics of digitized bits approaching infinity in number, of differential units of some variable quantity, of instantaneous rates of change, of minimalized representations of shape and motion, manifests itself through a broad range of theoretical and practical models, inventions, and subsequent advantages, fostering rather dramatic social changes. Evolution: a new invention based, ultimately, on the calculus, like the computer, affords a plateau from which perspective-altering concepts, bearing on the nature of some aspect of reality, transcending the calculus, can be created, invented, discovered and studied.
As an intellectual construct, an elaborate and cohesive conceptual tool of nondescript parts standing for properties of change and motion, its nature lies independent, in the background of specific content or uniqueness of appearance, of a process or event or thing. Context determines what forms are used, what constants may come into play; the content is quantity dependent, the result of particular values, initial or boundary conditions, peculiar to a given situation. For instance: Regardless of the composition of a projectile, be it an arrow or a cannonball or a missile, for a given angle of elevation and initial velocity (initial or boundary conditions), and assuming gravity is the only force acting, the position of a projectile at any time (where it is on the parabola) may be determined from the parametric equations (the forms that come into play):
x = (v_{0} cos(a))t | y = -1/2 g(gravity)t^{2} + (v_{0} sin(a)) |
On another level, the calculus, the core of applied mathematics, has extended and expanded our capacities and understanding of the universe, and as a result, the very notion of what it means to be a human being has likewise been expanded, as the invention of language itself did many eons ago.
Mathematics that lays out the functional relationships among the parts of a linear dynamic, the mathematics of the Calculus, differential and integral equations, Fourier Series and special function substitutions, has therefore demonstrated its usefulness and applicability in the realm of quantification, the proof is all around us. Manipulating variants of time and space, mass and energy, through the medium of symbols, following certain basic rules and applying methods, both established and ingenious, for processing, has allowed us to not only map connections in a logical, cause-effect manner, but also to draw implications from resulting equivalences, for instance: E = Mc^{2}.
However, when it comes to dealing with a nonlinear dynamic, the mathematics of Newton and Leibniz must remain satisfied with a linear approximation, a linear model, unable to completely capture the nonlinearity, to keep it from spontaneously bubbling up or oozing across the edges of the frame. What is needed is a qualitative, or 21st century, approach, the mathematics of pattern, structure, arrangement; internal and holistic symmetries; complexities that arise and owe their being to function building or product formation, function seen as form.
Physicists have applied what is known as Group Theory to the study and modeling of particle-forces in nature, using the symmetry of different groups as equivalent to symmetries associated with laws of conservation found in nature. Group theory thus is not simply a classification system for mathematicians immersed in abstract ideas concerning number theory, crystal patterns, or multi-dimensional space, but has indeed proven itself to be a practical tool for the study and understanding of how this universe works. Wherever there are symmetry considerations, the application of group theory may shed light by precipitating the emergence of connections not hithertoo perceived or imagined.
Our perceptions of the world, our perceptual reality, necessarily involves considerations of symmetry, patterns of recognition and significance that vary widely from culture to culture, and from person to person, across both time and space. Is there an underlying sameness among apparently divergent perceptions? Or, is Reality nothing more than our perceptions, each world so generated not only a product of our minds, but a picture of some feature of reality in fact? How much of a hand do we have in creating Reality? What is real, and what is make-believe? Which is fundamental, and which, a derivative? Would knowing the difference make a difference?
Metaphorically, the landscape of the psyche and its accompanying archetypes, constellations framing the content of consciousness, would seem to be apart from the rest of the universe in some absolute sense. And yet, perhaps there is a realm wherein the inner workings of the human psyche and that of the material universe become indistinguishable, insofar as they stem from the same numinous source.
The more deeply we delve into mathematics, the more do the concepts involved take on a metaphysical and non-contextual tone; they, in fact, become the context. Drawing on concepts from such areas as: Abstract Algebra, Linear Algebra, Point-set Topology, Algebraic Topology, Set Theory, Analysis, Chaos Theory, Fractal Geometry, Complexity, and nonlinear dynamics, I not so much 'construct a model' as discuss and explain, clearly, it is hoped, a way to orient oneself naturally, one's point of view, through an understanding of how things are related and connected. Conceptual pictures created (to be called 'models' for ease) are simultaneously Geometric and Algebraic, symbolic of a synthesis of these complementary reflections of the mind. This fusion acts as a dynamic model for the individuation process, and orients as the center of the conscious and unconscious aspects of the Self.
The first essay is terrain-like, going into some detail explaining the ideas involved. It centers around developing an appropriate geometric model, the solid sphere with extension into a countably infinite number of dimensions, together with its analogous algebraic counterpart, a subgroup of the full linear group - the orthogonal group of matrices modulo its algebraic center - this group is both infinite and simple.
The second is bio-environmental and contextual. The argument is constructed to support a phase-space condition referred to as Spontaneous Space. In a phase-space, the complete state of knowledge about a dynamic system at a single instant collapses to a point. Using fundamental ideas from structural mathematics to depict the action of the 'self' as it processes information in order to understand and map its immediate surroundings, and also by use of commonly known ideas found in various physical and metaphysical contexts, I attempt to substantiate, as the normal condition, the self's capacity to act and react spontaneously to life's vicissitudes through instinct, intelligence and assertive transcendence of ongoing, abstract restraints.
The third essay is a concentration, or emphasis, on 'the edge,' the edge where-on and -in and -of we live and have our being. This last also stresses the intrinsic nature of interwoven, gestalt-like processes. Even though traditional Gestalt Theory is limited to sensory perception, its structure can be seen working in other areas; in so doing it takes on an extension of its field, perfectly allowable from an algebraic point of view.
The second essay establishes the central generating feature of the overall model, having been described in all its gory detail in the first. From Modern Algebra we explore the concept of the Composition Series, a collection, or set, arranged in ordered hierarchical sequence, or nested layers, if you will, of 'simple' factor groups. The composition series is a particular construct of pure form capable of acting as a regulating and organizing system of symmetries within symmetries, each intimately related through progeny. This essay also identifies the geometric and algebraic complements of the model with Intuition and Rationality, Instinct and Intellect, respectively.
The algebraic model has no 'tangible' center; that is, its center takes on the nature and associated dimensionless properties of a singularity. That is to say, although the model itself, being algebraic, is coordinateable, its center is not. The ability to objectify reality, to 'know' that singularity, to measure it rationally as an empirical 'something,' is forever frustrated; ["The eye that cannot see itself."].
The geometric complement, a continuum equivalent on the level of the unconscious, does have a center. But this 'center' is diffused throughout, taking form as mandalic symbol, structured across space and time in a manner reflecting a composition series, and as such, has internally connected symmetries and dimensional meanings, each is representative of psychic functions and archetypal forces of the whole Self.
The possible arrangements of frames of thought-contents, defining and energizing structural mathematics, can be found embedded and mirrored in any system, nonlinear or idealized-linear, that concerns itself with the qualitative assessment of form, and to the relation -- part or set of parts to whole.
In particular, Jungian psychology's nested themes of synthesized dualities and mandala symbology, governing an individual's quest for individuation, self-realization and genuine freedom, lend themselves to a meaningful correspondence with the formalistic and intuitive relationships of shifting dimensional perspectives of dynamic, changing, transforming patterns of identity and orientation.
Pythagoras has been quoted as saying, "Limit gives form to the Limitless." 'Limit' also allows us to 'know,' be in contact with, and draw on the creative power of the 'Limitless.'
These essays were written based on the premise that there is only 'one eco-infrastructure,' and we, as witnesses and participants, directly affect what transpires, as we, in turn, are affected.