The Asklepia Foundation


Chaos Theory and Fuzzy Philosophy

by Iona Miller, 1993

ABSTRACT: The notion of paradox comes from a consciousness conditioned to think in terms of opposites, dualistic paradox.  Self-referential paradox feeds back and annihilates itself.  Such bivalence (binary logic: this or that; true or false; black or white) can be superceded by multivalent consciousness which perceives in terms of degrees.  Multivalence more accurately reflects the complex dynamics of consciousness.  As in the case of fractal generation, solutions are not found in terms of this or that, but in terms of degrees of fractional transformation, relationships.  Fuzzy philosophy is based on acceptance of degrees of truth, the "grayness" of most propositions (truth values), the fractional solutions of fuzzy logic.  Human consciousness is a self-referential system which embodies this principle of a connection between logic and chaos, in holistic ("whole brain") awareness.

What we've been discussing is the beginnings of a whole new theory of dynamic logic, invented by Gary Mar and Patrick Grim in the department of philosophy of the State University of New York at Stony Book.  It provides a link between semantic paradoxes and chaos theory.

                                                             --Ian Stewart, SCIENTIFIC AMERICAN

The universe is deterministic but gray.  Chaos theory had already gotten the determination part right.  Fuzzy theory now confirmed that and should that all things were matters of degree too.

                                                              --Bart Kosco, FUZZY THINKING

Know Thyself is revelatory, non-linear, discontinuous; it is like a painting, a lyric poem, biography thoroughly gone into the imaginative act."

                                                               --James Hillman, HEALING FICTION


We can read the progressive theories of leading-edge science as modern myths.  They both create and reflect changes in collective consciousness and the global worldview.  These new myths permeate culture, fomenting change and opening new conceptual territory for exploration.

The "new myth" seems to be one of "guiding fictions," even "healing fictions."  Mythic consciousness and its practice, ritual life, requires a telos to create momentum--the dynamics of consciousness.  The fictional carrot dangles, ever-present before us.  As a culture, we are in the position of having to take ritual fictions (including scientific theories) seriously, while recognizing their status as fiction at the same time.  This means being in two ontological "places" at once, facilitating the development of new ways of thinking about the nature of knowledge, being, and reality.

The myths are bigger than any one of us.  Yet through Creative Mysticism we can enter the imagination, not as an exercise in "doing our own thing," but as a disintegration of the reference points that make it possible for the archetypal energies to live through us.  Thus radical deconstruction of ego leads beyond the paradoxes of yoked opposites.  Dogmas dissolve and merge on their own.  The surety of fact melds into psychic reality.  The new myth is not one of polarity, but plurality--panoply.

Given these choices, each of us is faced with a struggle to acquire an authentic sense of identity.  We have been saddled with a cultural mandate to "pick one" and sustain a strong ego identification at the expense of empathy, compassion, and perspective.  What for one is a liberating experience is terrifying for others, or leads to the despair of cultural disenfranchisement.  Our sense of identity impacts us individually, socially, and existentially.  Broadening those boundaries opens us to an expanded sense of self.

In my own search for identity, I first became aware of the notion of paradox as a teenager.  I was given a button with the cryptic message, "Have you been to the Paradox?"  Puzzling over its meaning, I finally found out it was a music club, which I subsequently patronized.  On the broader scale, I became aware of another club--a "Reality Club."  I continue to visit "the Paradox" as I define and refine myself, through opening not only to the opposites but to all degrees of interpenetration between.  At this "inn between," I've many times found the thread of my chaotic trajectory through life.

Traditionally logic and chaos have held sway in two separate camps.  But what of that boundary domain where the two meet and progressively meld into one another?  In this "twilight zone," two threads of the new science revolution have come together in a "sacred marriage" between logic and chaos.  Ian Stewart (1993) gives an example of dynamic logic:

"You take the train of thought involved in assessing the truth value of a set of self-referential statements and convert it into a dynamic process.  Then you can apply all the techniques of chaos theory to that process.  The escape-time plot is inspired by exactly the same method that creates all those wonderful multi-colored images associated with the Mandelbrot set: swirling spirals, sea horses, cacti, stars, and so on."

This statement could also be read as commentary on the self-referential consciousness process expressed as "Know Thyself."  This Apollonian dictum is the utterance of the ancient god of logic Himself.  The Jungian method of "Know Thyself" means a radical activation of imagination.  It calls to us, echoing through the aeons--urging us to turn our consciousness back on itself, assessing and experiencing the relative "truth" of what we find within.

Jung's solution to this injunction is summarized by Hillman:

We have already been given the clue in the instructor's manual as to how this third realm traditionally called 'soul' can be re-established--and by anyone.  Jung says he treated the figures whom he met "as though they were real people."  The key is that as though; the metaphorical, as-if reality, neither literally real (hallucinations or people in the street) nor irreal/unreal ('mere' fictions, projections which 'I' make up as parts of 'me', auto-suggestive illusions).  In an 'as-if' consciousness they are powers with voice, body, motion, and mind, fully felt but wholly imaginary.  This is psychic reality...

Hillman points out that we can remember, associating backward and downward into the forgotten and repressed.  And in psychic reality we find a multiplicity of answers to all major, archetypal, sorts of questions--relative answers.  Each archetypal perspective has its way of self-knowing.

In alchemy the multi-state paradigm is known as multiplicatio, which touches all points of the soul, all channels of images.  According to Hillman, it is "spirit's self-knowledge in the mirror of the soul, soul's recognition of its spirits."

There is no single way of knowing thyself, even though psychology has favored the method of introspection and insight.  According to Hillman, "Know Thyself terminates whenever it leaves linear time and becomes an act of imagination.  A partial insight, this song now, this one image; to see partly is the whole of it.  Self-understanding healed by active imagination."

Our unique method of knowing ourselves is through our own epistemological metaphors which reflect "how we know what we know" about ourselves.  They are the result of our personal experience of archetypal experiences--our direct experience of the nature of reality, expressed as image.  The non-linear co-relationship of simultaneous cause/effect means that archetypes become embodied as specific, yet dynamic, imagery.

The imperative to know ourselves challenges us to reflect on our identity, beliefs, assumed truths, interpretations, and even our archetypal experience and self-understanding.  This dynamic recycling of consciousness creates/reveals the dynamic flux of holistic feedback patterns.  It establishes a patterning matrix--a strange attractor--reflective of the organism's relationship to the whole, through a unique relationship of chaos and order.  The whole brain approach to existence is both/and chaotic and logical--logically chaotic, chaotically logical.  It is a holding of the tension of the opposites between the logical and natural mind.

Complex systems, such as human beings, manifest emergent properties which tend to be non-linear.  They are sufficiently context dependent that they become unpredictable under normal circumstances.  The reason has to do with the fact that sufficient context dependence leads to self-reference.  That is, insofar as one part effects another part, which in turn effects the first, the first can be seen to be effecting itself through the mediation of the second.  Multiply the parts and the system gets really out of control, at least as an object of modelling.

Godel proved why such self-referential systems are inherently unpredictable.  Any finitely axiomatized formal system rich enough in entailment to manifest self-reference displays true theorems not deducible from the axioms.  Since all mathematical modelling systems can be formalized as axiomatic systems, and since the project of such modelling is the essence of deterministic reductionism, it follows that systems capable of self-reference cannot be reduced to deterministic causality.  They can be modelled, but they become descriptive of organization, rather than predictive of future states (Naser, 1993, Bridge-L).

Like a fractal, the individual embodies the whole, to a greater or lesser degree, depending on the initial conditions of existence.  The physical body is conditioned by physical laws.  At the mesocosmic level, we either exist or we don't, we're alive or dead.  But the flow of our essence within the whole is not so black and white.  We remain ourselves, despite the loss of discrete body parts or faculties of perception.

At the quantum level, our atomic "matter" merges with the environment and the vacuum of non-existence.  The I-Not I dichotomy breaks down in pure consciousness.  Our physical constituents remain, even after death, with their own molecular and quantum perceptions, but the genetic information becomes obsolete.  We all decay according to the same biological process.

Free of the gravitational valley of corpo-reality, consciousness can soar unfettered.  When consciousness flows into an "escape-time plot," we experience the boundary-dissolving transcendence of cosmic consciousness.  An increased sense of freedom comes with liberation from the gravity of literalism.  Yet we are neither exclusively biological nor psychospiritual beings.  The nature of our existence is both/and--psychobiological.


To "fuzzy consciousness" nothing is absolute--everything is a matter of degree.  Diffuse awareness permits conceptual transcendence of all or nothing thinking, rigid definition of right or wrong.  Fuzzy thinking helps us consider the universe from several relative perspectives at once.  It frees us from having to "choose" identification with some polarities over others.  It is inclusive, rather than reductive.  It helps us juggle conflicting concepts, differing truths.

We live in a symbiotic universe, part of the seamless webwork of existence, which is the root of deep ecology, the science of relationships.  We interact in a pluralistic social milieu, with seemingly irreducible diversity.  Within such multiplicity, common interpretation can only exist by coercion, and then conformity is only superficial.

Fuzzy philosophy is one way of affirming our oneness as humans while honoring and affirming our diversity.  Perhaps it is one way of embracing multiplicity which can help heal the global spiritual crisis.  This openness to many forms of validity is reflected in nature.  Within the laws of organic evolution, diversity is essential to the process of adaptive change.

Fuzzy consciousness cleaves neither to the Light nor the Darkness, neither this nor that, not even the Void (existence/non-existence).  The void is not empty once you're there; it is intense richness.  That which the outer personality calls nothing, the inner traveller calls All.

The "Fuzzy Principle" is described and experienced as vagueness, "shades of gray."  There are an infinite number of gray values on the continuum between 0 and 1, an infinite continuum of gray scores like fractal solutions which spin out an infinite number of virtually random decimals places.

In dynamic logic, statements can be more than true, false, or "indeterminate."  Indeterminacy defines a continuum of multiple values, partial truths, probabilities.   Our uncertain reasoning is "fuzzy."  According to Kosco, fuzzy logic "labels an idea or family of ideas--shades of gray, blurred boundary, gray area, balanced opposites, both true and false, contradiction, reasonable not logical--and those ideas are very old and have many ancestors."

Jung and the Jungians speak of balanced opposites as the coincidentia oppositorum.  They have raised the process of "holding the tension of the opposites" to a fine art.  This psychotherapeutic method has to do with keeping soul and spirit "distinct but conjoined," in a combination of solar and lunar consciousness.  It is the alchemical marriage of masculine and feminine principles, which are representative of all polarities.

Solar consciousness is bright, analytical, logical; lunar consciousness is dark, diffuse, relational.  Their alchemical combination is a sort of "illumined lunacy," according to Hillman.  Thus, the "fuzzy" domain is revealed as the realm of imagination between the spiritual heights and psychobiological depths--that limbo, or twilight zone, between heaven and hell.  This imaginal perspective is neither real nor unreal, in the conventional sense.

In the psychotherapeutic journey, both the unconscious and conscious are transformed through the imaginal morphological process.  The whole is a process of fuzzy transformations of imaginal forms, culminating in a state of consciousness which transcends them all.  An old alchemical text says, "Whence will come the Chamaeleon of our Chaos, in which all secrets are hid in their potential state."  One result of embracing the alchemical notion of coincidentia oppositorum is the logical consequence of the relativity of the God-concept.  It evokes the fundamental ambiguity of the divine nature of the self.

The coincidence of opposites is expressed in the image of a divine or royal marriage, which has universal cultural validity and redeeming effects.  It supercedes the constructive/destructive powers of the unconscious, allowing us to experience awareness of a fraction of the unconscious which gets us outside of ourselves, our times, and our cultural bias.  In alchemy this state is symbolized as an androgyne, that which is whole unto itself.

Mystical models of thing and no-thing, such as the Taoist symbol of yin-yang embody such polarity in ever-shifting proportions of dynamic interplay.  It reflects the interpenetration of existence with non-existence, of essence and Source.  The principles of coherence and correspondence unite them.  Zen Buddhism sees three truths with the same mind: things are real, unreal, and neither real nor unreal.  According to Buddha, "The no-mind not-thinks no-thoughts about no-things."

An epistemology is based on an inclusive "both/and" principle, which characterizes both the "structure of cognition" and "dimensions of variation of the human potential for spiritual realization" (Schuman, 1993, Bridge-L).  John Wiegley defines the paradox of oneness and duality:

The position [of zazen] expresses the oneness of duality: not two, and not one.  This is the most important teaching: not two, and not one.  Our body and mind are not two and not one.  If you think your body and mind are two, that is wrong; if you think that they are one, that is also wrong.  Our body and mind are both two 'and' one.  We usually think that if something is not one, it is more than one; if it is not singular, it is plural.  But in actual experience, our life is not only plural, but also singular.  Each one of us is both dependent and independent.

Mystics live at "the Edge," with a foot in two worlds.  Such an image of "life at the dividing line" may point to the source of creativity and intelligence.  We might consider the bottomless fractal decomposition of what it means to live "between the cracks" (in the cracks between the cracks between the cracks).  Fractal descriptions capture the organic properties of living systems, but also function within hierarchy.  They define a linear trajectory--a linear index of levels, as they descend into their bottomless complexity.

We can make a distinction between "levels" of conceptual modelling, defining our linkage to the Divine sources.  This topology is the generic basis of "maps of consciousness," how we project toward that which we do not know.  Hierarchical/fractal modelling is not a contradiction, but may be a "fuzzy" term for a reiterating process of self-similar cascades which emerge in the reiteration and interpenetration of self and Self.  Forms are identical to the descending cascade of "levels of abstraction."

The division between parts and whole marks the dividing line between dependent and independent variables.  If logic flow is going down the hierarchy, every element below a higher element becomes a dependent variable, whose values change as they receive input from above.  And as logic flow ascends the hierarchy every element becomes independent, as it determines the composition of the forms above it.

Such relativistic models more accurately reflect reality.  They reflect the nature of consciousness more accurately.  Our individual boundaries are actually gradients.  In logic things can be 100% true or false; in nature they seldom are.  The facts usually are partially true or false.  To the extent science has measured facts and interpreted them in all or nothing terms, it has failed to describe experiential reality.  "Truth" doesn't always match the facts.


Have you been to the Paradox?  If you have embraced the worldview of quantum mechanics, "chances are" you have.  It is based on the a priori assumption of bivalent thinking, and is ultimately a linear mathematics attempting to describe a non-linear reality, of observers desperately trying to remain impossibly "objective."  Therefore its symbols don't necessarily equate with "the facts" of reality.  Multivalent experience requires participation.

Semantic paradoxes reflect classic conundrums, such as the paradox of the heap.  With each grain removed it is still a heap, until you remove one and have nothing.  The paradox revolves around when it ceases being a heap.  Fuzzy sets easily resolve the paradox of the heap, gliding smoothly across the truth continuum.  With each grain, it looses membership value.  This vagueness is best described as a "haze at the edges," though ambiguous.

With vagueness, more information does not lead to precision, just as more measurement in QM cannot resolve uncertainty.  Vagueness and precision are features of language (even mathematical language), not reality.  Ordinary, traditional logic is the structure of mathematics, it confirms or denies.  Yet, the nature of reality is that all that exists is continuous, even consciousness itself is a continuum.

Paradox has been considered fundamental to the nature of reality since the birth of quantum physics.  Assuming that all systems are ultimately quantum in nature, the primary paradox surrounds the act of measurement.  In the Copenhagen interpretation of QM, the mathematics describes the superposition of probability waves prior to observation as the pristine nature of reality.

This theory asserts that the act of observation collapses the probability wave into the reality we experience.  It is vital to remember that the superposition does not represent a set of alternatives--an either/or choice--but a genuinely overlapping combination of possible realities.  These realities not only co-exist, they overlap and interfere with each other by the wave interference phenomenon.

The Everett/DeWitt "many-worlds" interpretation of QM eliminates the measurement paradox described above.  It takes the mathematical description at face value.  Therefore, no special collapse into reality is needed at the moment of observation--the reality is already there.  Yet the superposition of other states is an inescapable component of QM.  This interpretation further assumes the reality, not mere potentiality, of all others worlds in superspace.

This theory implies that the world is continually splitting into countless copies of itself--stupendous numbers of branches, multiplying infinitely.  This sounds suspiciously like fractal generation, cascades of chaos.  Like the "butterfly effect" in chaos theory, every subatomic process has the power to multiply the world, maybe an enormous number of times. This ceaseless replication theoretically splits and multiplies our bodies, brains, and consciousness infintely.  What starts out at birth as one consciousness multiplies countless billionfolds by death.

The splitting process is totally unobservable because the separate worlds of superspace are completely discrete.  Rather than parallel universes, the mathematics of extra dimensions describes them as perpendicular to ours.  Local space is just one three-dimensional subspace from a superspace that contains an infinity of perpendicular directions.

The many-worlds theory lends a new perspective to the fundamental indeterminacy of QM.  The "missing information" which could lead to complete predictability is "hidden" from us in the other worlds to which we have no access.  Thus, superspace as a whole is completely deterministic; the random element comes from our sampling just a minute portion of the whole.  The widest view of superspace implies that every situation can be reached along some convoluted path of development, in at least one of these other worlds.

Fuzzy philosopher, Kosko rejects the notion of probability in favor of fuzzy logic.  Kosco notes,  "If fuzziness exists, the physical consequencesare universal, and the sociological consequence is startling:  Scientists, especially physicists, have overlooked an entire mode of reality."  We have overlooked this way of conceptualizing reality in favor of "crisp" logic.

Fuzzy logic is reasoning with fuzzy sets, degrees of embedding or enfoldment.  Logical truth differs from factual truth, being based on symbols and their formal relationships.  Probability is the fortress of a science based on bivalent thinking--it is an assumption, a worldview, a belief system.  It ranks or weighs future alternatives.  Kosco postulates a "probability instinct," in the Jungian sense, resulting from millennia of organizing our perceptions, memories, and expectations.

He postulates pure fuzz as a parent of probability, citing the fuzzy idea of containment--how much one thing contains another--how much one set contains another set.  The whole in the part is probability.  The subset theorem shows that the universe is deterministic but gray (this and not-this to some degree).  Fuzzy logic transcends paradox, eliminating probability, asserting that paradoxes of self-reference are half-truths, fuzzy contradictions.  The yin-yang equation holds where this equals not-this.  It is the midpoint of a reference "truth line" from zero to one.

Even in physics the truth of statements is a matter of degree.  Heisenberg's uncertainty principle shifted probability into an all-or-nothing bivalent truth, which requires rounding off descriptions, trading simplicity for accuracy.  In self-referencing paradoxes, rounding off leads to annihilation, infinite contradiction, like a Zen koan.

The ultimately linear nature of QM "causes" uncertainty in systems.  The nonlinear theories of quantum chaos offer a step toward the nonreality reality.  The measure of the whole in the part swallows up the old notion of "randomness" or the probability of a part.  The essence of fuzzy logic describes the whole in the part.

In Kosco's theory of fuzzy entropy, when A equals not-A fuzzy entropy is maximal.  He says, "At the midpoint you cannot tell black from white or white from black.  The midpoint is the black hole of set theory.  It is the gray hole of set theory...subsethood or degree of containment is the deepest and strangest idea in fuzzy logic and explains probability or "randomness" as the whole stuck in the part."

Lotfi Zadeh discovered fuzzy sets, nearly calling them "cloudy" instead of fuzzy.  Multivalence has also been termed "vagueness" in the past for obvious reasons.  Zadeh pointed out that, "for coping with the analysis of biological systems, and to deal effectively with such systems, which are generally orders of magnitude more complex than man-made systems, we need a radically different kind of mathematics, the mathematics of fuzzy or cloudy quantities which are not describable in terms of  probability distributions."

According to McNeill and Freiberger, "Zadeh realized that complex disciplines team concepts.  They include such notions as obscenity and insanity in law; arthritis, arteriosclerosis, and schizophrenia in medicine; recession, value, and utility in economics; grammaticality and meaning in linguistics; stability and adaptivity in systems theory; truth, morality, and causality in philosophy; and intelligence and creativity in psychology.  Fuzzy sets can describe them all."

Fuzzy thinking will allow us to revision our perspective on the relationship of man and God, man and the universe(s), life and death (not-life), morals and ethics.  It is part of our adaptive evolution, a new twist in worldviews, a new way to be conscious--logically.  Multivalued logic, though vague, reminds us of Hillman's diffuse awareness which he calls "anima consciousness."  Other synonyms are gray logic, cloudy logic, and continuous logic.  Diffuse logic takes the edge of extremism off left-hemisphere thinking.

Fuzzy philosophy reflects the fusion of cultural inputs in society and thought.  It is an androgynous consciousness, rooted in the interpenetration of masculine and feminine principles, melding the qualities of both hemispheres of brain functioning into a holistic continuum of perspectives.


Davies, Paul; OTHER WORLDS: A Portrait of Nature in Rebellion; Simon and Schuster, New York, 1980.

Hillman, James; HEALING FICTION, Station Hill, 1983.

Jung, C.G.; MYSTERIUM CONIUNCTIONIS, Princeton University Press, Princeton, 1963.

Kosco, Bart; FUZZY THINKING, Hyperion, New York, 1993.

McNeill, Daniel and Freiberger, Paul; FUZZY LOGIC; Simon and Schuster, New York, 1993.

Stewart, Ian; "A Partly True Story"; SCIENTIFIC AMERICAN, Feb. 1993, p. 110-111.

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