[An ASKLEPIA FOUNDATION Journal]
HAVE YOU BEEN TO THE PARADOX?
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logic: this or that; true or false; black or white) can be superceded by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
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
--James Hillman, HEALING FICTION
A FUZZY REVOLUTION?
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,
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
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
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.
BETWEEN DARKNESS AND LIGHT
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.
THE PROBABILITY PARADOX
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
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"
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
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,
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.
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