Cybernetics & Human Knowing vol 11, no1, 2004
Semiotic Seams: Fractal Dynamics of Reentry
Summary in quotes: Abstract:
This essay concerns fractal geometry as a bridge between the imaginary and the real, mind and matter, conscious and unconscious.
The logic rests upon Jung's theory of number as the most primitive archetype of order for linking observers with the observed. Whereas Jung focused upon natural numbers as the foundation for order that is already conscious, I offer fractal geometry, with its endlessly recursive iteration on the complex number plane, as the underpinning for a dynamic unconscious destined never to become fully conscious.
Everywhere in nature, fractal separatrices articulate a paradoxical zone of bounded infinity that both separates and connects nature's edges. By occupying the "space between" dimensions and levels of existence, fractal boundaries exemplify reentry dynamics of Varela's autonomous systems, as well as Hofstadter's ever-elusive "tangled hierarchy" where brain and mind are most entwined.
At this second-order, cybernetic frontier, the horizon of observers observing the observation process remains infinitely complex and ever receding from view. I suggest that the property of self-similarity, by which the pattern of the whole permeates fractal parts at different scales, represents the semiotic sign of identity in nature.
Swiss psychiatrist Carl Jung came to view number as the most primitive quality of existence. By crafting an archetypal theory, his theory of number doubled over as a theory of mind. Jung attributed to number the power to bring order into the chaos of appearances, referring to material existence not as objectively conceived, but rather as subjectively perceived by an observer. Here number links observers and observed, inner and outer worlds in a way parallel to second-order cybernetics. In the footsteps of Margaret Mead and Heinz von Foerster, this viewpoint is carried forth eloquently in this journal by the "Musings" of Ranulph Glanville, e.g., "the whole point of Second Order Cybernetics is that it asserts there is no observation without an observer. There is nothing spoken without a speaker, there is no action without an actor" (Glanville, 1998, p. 85).
Jung viewed number as the realm where mind and matter meet, sometimes referred to as the psychoid level of existence and at other times the Unus Mundus. Jung intuited that the realm of mathematical abstraction is discovered, in so far as it uncovers quantitative "facts" about the workings of the external world. At the same time, it is invented as an abstraction in the mind, indicating something qualitative about the subjective realm of meaning. For Jung, number serves as the most fundamental structure of perceived reality, the place where observers and observed merge at the level of synchronicity, symbol and meaning.
In building a bridge between mind and matter, Jung and his dedicated follower, Marie-Louise von Franz, were interested primarily in the counting numbers as symbols and founts of inexhaustible metaphor during the production of conscious experience. Whether in dream, mythology or art, the number one tends to symbolize undifferentiated unity; two signifies the first distinction or duality; three indicates dynamic change and movement away from the static opposition, and four suggests stable manifestation.
In this paper I move from the natural numbers that interested Jung to number as manifested naturally in what Benoit Mandelbrot (1977) calls the fractal geometry of nature. I argue for a link between imaginary numbers as iterated recursively on the complex plane and self-reflexive underpinnings of the dynamic unconscious.
... intrapsychic interpretation of fractal dynamics (MarksTarlow, 1995, 1999, 2002). As mathematically defined, fractals are partially infinite and partially finite. While their Euclidean dimension anchors and bounds them in the real world, they also partake of the infinite, which is necessary to calculate fractional dimensionality. As elaborated upon shortly, fractals occupy a twilight existence between "real" and idealized dimensions. Here the symmetry of identity - selfsimilarity - paradoxically exhibits parts as complex and infinitely detailed as their whole.
Fractal geometry represents the full mathematical fruition of this fourth stage of selfreflexive consciousness, where imaginary numbers are concretized in nature to simultaneously co-create world and self. Computer-generated fractals provide one of the most successful tools ever discovered/invented for simulating highly complex forms in nature. With fractals comes the cybersemiotic recognition of how the infinite becomes finitely embodied in nature. When the computer is used as a microscope on the complex plane to zoom in on ever-smaller scales of the Mandelbrot set (z~x2+ c), iteration exhibits unceasing complexity, shifting dynamically with the perspective of the observer.
The benchmark of fractals is self-similarity, a newly discovered, self-reflexive symmetry in which parts of a fractal object resemble the whole. Sometimes this resemblance is exact, as in the case of linear fractals like the Koch snowflake; at other times it is approximate and statistical, as with the Mandelbrot set. In either case, rather than the whole being greater than the sum of its parts, the hallmark of organic systems as compared to purely mechanistic ones, instead we find the paradoxical embodiment of the whole detectable within its parts at multiple scales of observation.
Everywhere they arise, fractals occupy the boundary zone between dynamic, open processes in nature. This quality of betweenness is illuminated by a technical understanding of fractal dimensionality. Since imaginary numbers model hidden dimensionality, in the case of fractals, this consists of infinite expanses, or imaginary frontiers that lurk in the spaces between ordinary, Euclidean dimensions. Clouds are zero-dimensional points that occupy three-dimensional space, coastlines onedimensional lines that occupy two-dimensional planes, and monntains twodimensional surfaces draping a three-dimensional world. Quaternions are products of the hypercomplex plane consisting of one real and three imaginary axes. If imaginary numbers do relate to abstract processes in consciousness, and more specifically to the fuzzy zone between body and mind, then because they are three-dimensional shadows of four-dimensional space, quaternions may provide some clues as to the internal landscape of higher dimensional thought.
....fractal dimension represents a portmanteau that also indicates the quality of relations between observer and observed. Since fractal dimensionality signals what remains constant as we change scales, another paradox soon becomes evident: Fractal pattern simultaneously appears on all scales, at the same time that it demonstrates no characteristic scale - a quality called scale invariance.
....Although mountains and rivers appear to be stable things to our Western minds, they are actually continually moving processes that evolve dynamically on various time scales. The fiction of stability and thingness is reified by the English language with its predominance of nouns acting on and acted upon by verbs. The dynamism of fractals as processes-in-nature seems much more consistent with American Indian languages, where features such as lightning and coastlines are described using verbs.
As embodied in nature, fractals occupy the complex interface between chaotic forces, such as wind, water and heat comprising the weather. They represent the place where time gets etched into structure through process (see Kauffman, 1980). Fractal dynamics pervade our bodies (e.g., lannaccone & Khokha, 1996), where they comprise zones of openness, communication and transportation between various subsystems of the body, as well as between the body and the outside world (MarksTarlow, 2002). Blood circulates throughout the body in the fractal branching of arteries and veins. The lungs cycle oxygen in and carbon dioxide out through fractal bronchioles. Even the ion channels in our cells and the neural pathways in the brain, our main organ for perception and communication, are fractal. Skin pores, wrinkles and other markings on the sacks in which humans and other animals are enclosed are fractally distributed.
Self‑similar dynamics also pervade psychophysics, in that physical stimuli outside our bodies often follow power laws, which transmute into "just noticeable differences" in sensation and perception. In human physiology generally, as in nature broadly, fractals function in open systems as boundary keepers, both by separating and connecting various subsystems and levels of being.
.....The fractal self is multiply embedded within different scales of social observation (Marks‑Tarlow, 1999). Each of us has a proto‑self that is biologically driven and precedes consciousness. We have an intrapsychic self, revealed by dreams and unique patterns within our intrapsychic landscape. We possess an interpersonal self, brought into social existence through interaction with others. We have a cultural identity that enables us to share the language and customs of like others. And we possess a national identity that contrasts with people from other countries. Possibly, we possess a global identity opened up by high‑tech communication that allows instant access the world over.
...I have argued that self-similarity spans the full range of existence, from the most concrete, material levels to the most highly abstract and psychological ones. Viewed semiotically, I believe this newly discovered symmetry represents the sign for identity in nature, the pattern of patterning, by which essences actually precede evolution and biological reproduction. Due to self-similarity, wholes in nature can dynamically inform the shaping of self-similar parts.
...In a previous article for Cybernetics & Human Knowing (Marks-Tarlow, Robertson & Combs, 2002), I, with colleagues, speculate on the psychological significance of Varela's recursive dynamics, by examining the individuation process as lifelong, self-reflexive cycles of reentry.
Psychological birth is preceded by the paradoxical union of opposites within the unconscious, as we begin our mental life with good/bad, you/me, inside/outside melded together. In order to make first distinctions, nascent consciousness must separate each pole from its opposite. Yet, complete psychological evolution requires that we bring together and balance all opposites. Inner development requires that we leave ourselves in order to get deeper inside. In fact, spiritual advancement can be characterized as the use of self-reflection to achieve increasing objectivity. Yet most spiritual paths aimed towards objectivity move in the opposite direction, e.g., using methods of meditation, to delve yet further inside one's inner life.
In sum, whereas Jung speculated about the archetypal significance of the natural numbers, in this essay I explore number as manifested naturally in the fractal geometry of nature. Whereas Jung speaks of number as an archetype of order which has become conscious, I present fractal geometry as deep order under chaos, which is yet to become conscious and impossible ever fully to do so. With fractals as a new metaphor of mind, we no longer need to deify or defile mechanism as metaphor. Instead we can use mechanism as a tool. As we continue to use the recursive formulas of chaos, complexity, fractal geometry and cellular automata to simulate the natural world, we see across multiple, nested levels of serial inclusiveness, how self-similar fractal dynamics recur as a meta-pattern.
Cycles of reentry continually oscillate between creating and erasing the scam where observer and observed, perceiver and perceived, inner and outer, self and other, intersect and self-cross paradoxically. At these semiotic seams, self and world appear mutually co-determining. Meanwhile, fractals represent another level of abstraction in human consciousness, suggestive of the dynamic underpinnings of the unconscious. In this elusive seam where the unconscious and conscious minds touch, we forever seek the means by which brains cobble minds that study brains. Here, the physiological act of making a distinction creates the world as we perceive it, while the world brings distinction to the consciousness of the perceiver.
Fractal Dynamics of the Psyche
The rise of cybernetics, the science of information, following World War II, brought a new metaphor to psychology - the notion of mind as mechanism. This metaphor inspired the cognitivist revolution, in which psychological activity was likened to information processing in machines. In the decades to follow, a wellspring of new knowledge and empirical methods followed and even continues today. While invaluable for its early insights, I believe the notion of mind as mechanism has run its course.
In this paper, I introduce a different guiding metaphor in order to conceptualize the psyche, one with particular significance to clinicians immersed in the complexity of human affairs. This new metaphor represents the pendulum swung full circle, from machine back to nature, where psychology started when it first diverged from philosophy during Renaissance times. Ironically a return to organic models occurs just as the computer plus related technology ascends takes an ever more central role in most our lives.
More than ever, the computer affords us rich tools for simulating nature's complexity. Among the most powerful of these is fractal geometry. Because fractals provide a lexicon for nature's outer complexity, it makes sense that this new geometry is equally as effective for describing the complex terrain characteristic of inner processes.
This paper introduces the significance of fractal geometry to the psyche. In the first section, I describe this new branch of mathematics plus how to render a fractal by computer. I then articulate the significance of fractals to the development of psychological identity. Next, I claim that self-similarity, the hallmark of fractals, is a useful lens for viewing personality organization and especially repetitive patterns of behavior. I also argue that related concepts of dimensionality and scaling help lend breadth to intraspychic analysis. I use the notion of fractal boundaries to illuminate paradoxes of subjectivity and interpersonal relationships. Finally, I assert that fractal boundaries are not just a source of endless confusion and deep psychopathology, but also a fount of novelty, creativity and endless mystery in us all.
Self-Similarity, Weather Storms and Brainstorms
To recognize the significance of fractals is to understand its hallmark self-similarity. Self-similarity is a newly discovered symmetry in nature by which parts of fractal objects relate to their wholes. That is, the overall pattern of a fractal is repeated at multiple size or time scales, from small to large scale. Sometimes this repetition is exact, as with a linear fractal. Most often, especially in natural fractals, self-similarity is approximate or statistical. This nonlinear property allows fractals as they appear in nature to embody irregularity, discontinuity, evolution and change.
One profound insight to be derived from contemporary nonlinear science stems from the fact that human nature is embedded within nature at large, whose essence is chaotic and fundamentally unpredictable. Fundamental unpredictability means that the local, or minute-to-minute details of specific instances can never be precisely anticipated. Yet, beneath the surface of even the most turbulent chaos, usually lurks invisible, exquisite order in the form of fractal attractors. While self-similar patterns can be reliably detected at the global level, their local details remain uncertain and fundamentally unpredictable.
Fractals are a means by which time, or system dynamics, gets etched into form via self-similar, recursive loops that exist on multiple size scales. Fractals exist in the paradoxical space between dimensions, levels and forces of existence. They arise at the interface between processes, at boundary zones where they serve both to connect and separate multiple levels.
In the field of perception, many of our sensory systems, such as sight and hearing, follow psychophysical power laws. Power laws involve nonlinear, exponential relationships between variables, in this case between how energy is transduced from the material level of signals outside the body to the spiritual level of conscious perception.
Power laws are self-similar, because the same relationship holds between their variables no matter how they are scaled or rescaled. Newton’s universal law of gravitational attraction, F~ r -2, is another example of a power law. The same relationship holds between mass and gravitational attraction in Newton’s formula, whether manifest at the tiny scale of the wavelength of light or the cosmic scale of light-years.
Due to their nonlinear, exponential increases, power law dynamics in psychophysics ensure that at the lower end of the signal spectrum, tiny amounts of a signal can be detected, as little as a single photon for the eye or single decibel for the ear. Meanwhile at higher ends, because distinctions are made with far less precision, we become capable of perceiving the widest possible range of signals. There is a paradoxical element to power laws that the same relationship holds between variables at every scale means that they possess no characteristic scale. This hallmark of fractal dynamics is critical not just to psychophysics, where we enjoy the widest range possible of perception, but also to the psyche in general.
Power laws appear not just in psychophysics and Newton’s law, but also all over nature (see Schroeder, 1991). At times their appearance takes on a magical feel, because they connect seemingly unrelated things. For example, over a hundred years ago, the Italian economist Vilfredo Pareto recognized that a simple power law models the number of people whose personal incomes exceed particular values. More recently George Zipf recognized that a power law connects word rank and word frequency for many natural languages. Suppose we take any ordinary book and count all the words it contains. If we list the words first by rank order of word popularity and next by the actual number of many times each word appears, we find a power law relationship connecting the two. Power laws seem magical when they relate the apparently unconnected, e.g., qualities like rank order, to quantities like frequency.
Continuing with this survey of fractal dynamics related to human boundaries, along with Zipf’s law, self-similar dynamics are evident in language even more broadly, whose symbolic arena is one of the cornerstones of our humanness. Fractal dynamics afford language its remarkable flexibility the ability for a limited number of words and grammatical rules to enjoy unlimited combinations. At a purely formal level, language clearly consists of self-similar structures: words are embedded within words, phrases within phrases, thoughts within thoughts, etc. That our number system is fractal is so obvious as to seem almost trite. Yet, interestingly, only at the point when it became so, did the concept of infinity arise. That is, only when numbers served as place-holders, could they be recycled infinitely. This made way for continued novelty in the form of calculus and other mathematical advances.
Selves in the Paradoxical Space Between
The perspective of the self I offer is of an open, dynamical system that is fractally constellated. My view dovetails with Francisco Varela’s framework of autonomy in biological systems(e.g., Varela, 1979). Varela’s model involves endless feedback loops, which allow biological systems to re-enter themselves continuously. This results in paradoxical dynamics when biological systems are characterized in opposite terms, as being functionally closed yet structurally open. Selves follow the same pattern. They too are “closed” in that, when we are healthy, we retain a cohesive, coherent, ongoing sense of identity. Yet, selves are clearly open via interaction with the others, which constitutes the social mechanics of their negotiation.
Varela and the Uroboros
The psychological significance of reentry
Terry Marks-Tarlow, Robin Robertson, and Allan Combs
Cybernetics & Human Knowing vol 9 no2 pg 31
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