The less we know, the more certain and precise we are in our
the more we know, the more we realize the limitations of
being certain and precise
It is impossible to deal with
fuzziness related to a higher level of consciousness from the point of
view of a lower level of consciousness
No thing and no being can exist in
itself or for itself but only in dynamic relationship with other things
We CAN understand as much of the
world as we have developed and realized within ourselves
The deeper our consciousness descends
into the nature of existence, the clearer this nature reveals itself
In a broad
sense, fuzziness is the opposite of precision. Everything that cannot
be defined precisely (that is, according to some broadly accepted
criteria or norms of precision) and everything that has no clearly
described boundaries in space or time is considered a bearer of
fuzziness. In a narrow sense, fuzziness relates to the definition of
fuzzy sets as proposed by Zadeh in 1965: sets, the belongingness to
which is measured by a membership function whose values are between 1
(full belongingness) and 0 (non-belongingness).
an essential characteristic of the images that raise and dissolve in
our thoughts - in our memories and reflections about the past and in
our plans and dreams about the future. They have blurred boundaries and
consist of fuzzy immaterial 'substance'. Having in mind how important
is to think in images for the development of our intelligence and
capacity to learn and know, to act and create, to evolve and transform,
one should not underestimate the role of the fuzziness in human
a substantial presence in our knowledge about the society and ourselves.
interplay of human dynamics at three major scales of their
manifestations - individual (intrapersonal dynamics), social
(interpersonal dynamics) and existential (universal dynamics) - results
in the emergence of spinning webs and 'whirlpools' of social
interactions, which constantly reproduce forces and energies to
strengthen or weaken the self-propelling capacity of these dynamics.
There are so many intricately interwoven factors and conditions engaged
in the realization of this capacity, that it is nonsensical to look for
or to apply precise descriptions and definitions when explaining or
dealing with their infinite (in number and diversity) embodiments.
Socrates Paradox of Fuzziology
Paradox: The less we know, the more certain and precise we are
in our explanations; the more we know, the more we realize the
limitations of being certain and precise.
Socrates' wisdom was incomparably deeper and broader than the
transitory knowledge of his contemporaries, he used to say with a
proverbial humility: "The only thing I know for sure is how little I
know". The awareness of "how little I know" made Socrates capable to
easily reveal the gaps in the 'precise' and 'certain' knowledge of his
opponents. When the Athenians went to the famous Delphic Oracle to ask
who is the wisest man in Athens, the answer of the Oracle was:
"Socrates". "But how he can be the wisest if he permanently tells us
that all he knows for sure is how little he knows" - responded the
crowd. "That's why he is the wisest among you!" - was the answer of the
acknowledgment of the fuzziness in our knowledge serves as a stimulus
for the lifelong search for truth and wisdom; and it is this search
that makes human life meaningful.
with social complexity, we are fully aware about our limitations to be
certain and precise. It is then that we are ready to apply fuzziology [2,3,4,5].
study fuzziness of human knowing - its sources, nature and dynamics -
not in an endeavour to reduce or eliminate it but to understand and
transcend its limitations so that,instead of an impediment, it serves
as a mighty stimulus for realization of human creativity.
principles and theorems are considered at the basis of fuzziology.
of Incompatibility (Zadeh, 1973): As the complexity of a system
increases, human ability to make precise and relevant (meaningful)
statements about its behaviour diminishes until a threshold is reached
beyond which the precision and the relevance become mutually exclusive
characteristics. It is then that the fuzzy statements are the only
bearers of meaning.
was used by Zadeh for extending the applicability of his fuzzy sets
theory and fuzzy logic to the analysis of complex systems .
of Connectivity:No thing and no being can exist in itself or for
itself but only in dynamic relationship with other things and beings.
relates to the integrity of existence vitally supported by universal
dynamics, whose creative, sustaining or destructive powers are
constantly demonstrated at different scales of the manifested world. It
is through these dynamics that everything that exists - from an
elementary particle to a gigantic galaxy - becomes connected in an
all-embracing web of relationships.
of Fractality (Mandelbrot, 1982): The geometry of nature is
fractal and reveals itself as self-similar structures at different
scales of manifestation.
is at the basis of Mandelbrot' theory of fractals  and demonstrates
the way self-organization works while unfolding the complex dynamics of
nature. Self-similarity is a kind of fuzzy repetition - each scale has
common features with every other, and yet there are noticeable
differences. Fractals are inherent in the holistic unfolding of
individual, social and existential dynamics: the macrocosm is a fuzzy
projection of the microcosm, the external world of individuals is a
fuzzy projection of the inner world of their experience, each level of
development of consciousness has similarity both with the previous
(less developed) and the next (more advanced) levels and yet has its
own distinguished characteristics - its own strength and weakness.
First Impossibility Theorem
is impossible to eliminate fuzziness from any explanation that tends to
make sense of
The proof of
this theorem follows from the first two principles above. According to
the Principle of Connectivity, the wholeness of existence, its
manifested activities and its creative potential are results of an
all-embracing connectivity of everything that exists, that moves,
changes and transforms in a gigantic self-organized Web of
Interdependent Dynamics. According to the Principle of Incompatibility,
it is impossible to offer precise and yet meaningful explanations
related to the overwhelming complexity of this web. Hence, any possible
explanation that makes sense of the integrity of existential dynamics,
their unlimited actual or virtual appearance (as "manifested
activities" or "potentiality to create") inevitably contains fuzziness.
wholeness of existence
immensity of its manifested activities
infinity of its potentiality to create.
Impossibility Theorem prevents fuzziology from looking for and from
designing techniques to reduce, control or eliminate the fuzziness of
our knowledge of social complexity; such techniques are hardly to be
found. The fuzziness of social complexity has its deep roots in the
very essence of existence - an essence whose self-propelled unfolding
makes universe "incomprehen-sible" for "our frail and feeble minds" -
expressions used by Einstein when describing his religion. "My religion
consists of a humble admiration of the illimitable superior spirit who
reveals himself in the slight details we are able to perceive with our
frail and feeble minds. That deeply emotional conviction of the
presence of a superior reasoning power, which is revealed in the
incomprehensible Universe, forms my idea of God". (Quoted in the New
York Times obituary on April 19, 1955)
and feeble minds" are products of the unfolding of the mysterious
essence of existence. Therefore, its fuzziness is not something 'over
there' that can be onjectified, rationally defined and then studied and
modelled; it is deeply inside each of us and therefore escapes the
grasp of our reasoning. It can be felt, experienced and eventually realized
in life. Being out of the realm of logical formulations (no matter what
kind of logic we decide to use, be it inductive, deductive, abductive,
binary, multi-valued or fuzzy), the journey into the existential
mystery needs 'preparation', in which the reasoning power of the human
mind plays an important role - the role of a coordinator of the sense
impressions, perceptions, sensations, feelings and emotions into a
acknowledges the irreducible fuzziness at the essence of the
existential dynamics. The awareness of this fuzziness activates the
potential of fuzziology for construing reality where the conscious
revelation of our deep inner experience plays the paramount role in
making sense of existence, not the intellectual speculations about the
outward, 'objective' world as perceived through our senses. The
information from senses inevitably passes through mental and emotional
filters, consciously or unconsciously established in the process of
socially informed interactions. Some of these filters can irreversibly
distort the sense information up to such a degree (in result of bias
and prejudice, brainwash and propaganda, attachments and delusions)
that it entirely ceases to help people navigate the social complexity
of their lives.
Second Impossibility Theorem
impossible to deal with fuzziness related to a higher level of
consciousness from the point of view of a lower level of consciousness.
The proof of
this theorem follows from the Principle of Fractality when applied to
the unfolding of existential dynamics. From their manifestation at the
scale of non-animated nature, described by the ancient thinkers as
built by fire, light, air, water and soil, dynamics unfold to express
themselves at various scales (levels) of animated nature - at the
scales of plants, animals and humans. The unfolding of these dynamics
runs in parallel with a self-propelled expanding and growing of
complexity at each scale of manifestation. There is a stunning
diversity at the level of minerals, and also at the levels of plants
and animals. The complexity at each level of unfolding cannot be
reduced to the complexity of the previous level: animals' lives are of
a higher order of complexity than the lives of the plants, which are
much richer and diverse than the 'life' of minerals. When dynamics enter
the human scale, it is the human consciousness (as a holistic
experience, awareness and knowing of our own nature and the nature of
reality in which we exist and evolve) that expands and grows.
of knowing at each level of development of human consciousness can
hardly be grasped from a lower level of consciousness; what may appear
as a 'fuzzy mess' for an individual with a certain level of development
of her/his consciousness can be seen as saturated with meaning if this
individual puts in some effort and succeeds in developing a higher
level of awareness and intelligence, and in sharpening her/his capacity
to think, to feel and experience holistically, rather than solely from
a more narrowly established point of view ('worldview').
wisdom of Socrates could grasp was far beyond the understanding of his
contemporaries. And the enigmas of life which appeared fuzzy to
Socrates and kept the passion of his inquiry alive till the day he was
unjustly accused and killed, quite possibly never bothered the most of
the Athenians at that time.
inherent in the deepest spiritual wisdom of the ancient Vedas,
considered as the oldest written text on our planet (they came to us in
written form between 4000 to 6000 years ago) is almost ungraspable with
the level of consciousness of our generations.
message of the Second Impossibility Theorem is straightforward: the
threatening fuzziness of all those serious ecological and social
problems, which today's humanity creates, and by which more and more
people are tormented, can hardly be solved using the present egocentric
level of consciousness typical for us - the members of so-called
'developed' societies, driven competitively by an insatiable thirst for
money, power, superfluousness and pleasures.
to this theorem, it is essential to emphasize that the development of
consciousness is a process open for realization by each individual;
every human being has an enormous potential for growth in knowledge and
wisdom. This growth cannot be imposed from outside. Nobody can poure
wisdom into the brain of another person; the process of developing
consciousness needs continuous efforts to be exerted by those who
strive for wisdom. The external conditions can accelerate or decelerate
this process, and yet the major responsibility for realization of our
potential for growth in knowledge and wisdom is on our own shoulders.
The 'Possibility' Theorem
We CAN understand as much of the world as we have
developed and realized within ourselves.
The proof of
this theorem follows from the Principle of Fractality and from the
Second Impossibility Theorem.
of Fractality makes us understand why the macrocosm mirrors the
microcosm and the world outside reflects the world inside us. The inner
world is made not only of our senses, of our feelings and thoughts
shaped into images, ideas, emotions, aspirations, expectations, hopes,
dreams, but also of our deep spiritual attitudes and beliefs; through
all of them we perceive the world around us. The power of our
will is also in the inner world, together with our infinite potential
to create and realize ourselves in innumerable activities. We never
cease to modify the external world through actions emerging from the
inner world of each of us.
world also affects the world inside us. The lower the level of
consciousness, the stronger the influence of the external world, the
more silent the voice of the inner world and the weaker our spiritual
drives for self-realization. From the Second Impossibility Theorem
follows that when we grow in consciousness, we are able to see more of
its projections onto the world around us, to develop and realize
outwardly more of our inner potential to create. Then another type of
fuzziness, inaccessible from the previous levels of consciousness,
starts to irritate and challenge our minds and souls.
Transcending Fuzziness: Knowledge versus Wisdom
fuzziness of what we know about the unfoldment of life and existence
means to grasp how the dynamics of this unfoldment work.
dynamics have to be lived in order to be understood; they unfold
spontaneously; there is no possibility to rehearse them, to practise
them or to design special experiments for their study.
Existence is not
a product of human mind. Digging into the past, investigating the
present, planning about the future, writing papers and books about
reality are mind exercises - exercises in using different types of
logic, a kind of intellectual gymnastics at the surface of reality -
they have but a "frail and feeble" resemblance with life as a holistic
all its modification is only a means of expression, a play with words
regardless of their relationship to living experience and therefore
cannot be used as a criterion of reality.
Fuzzy logic is also a product of rational thinking and entirely
subjected to its 'IF...THEN' rules of inference. It works satisfactory
'up to a degree' when dealing with the fuzziness of human perceprtions
and words and has been used to put into computers memory as much as
possible of the experts' practical knowledge and competence, then using
them for the purpose of designing and controlling intelligent
engineering systems and robots. Instead of selecting either A or B
(where A and B are given decision options, alternatives, possible
actions, etc.), fuzzy logic selects both A and B with different degrees
of significance (preference, compatibility with chosen goals and
criteria, fitness, 'truth'). The use of different degrees of 'truth', in
parallel, creates a fuzzy framework, which is more adequate to the way
people express their perceptions in words and therefore more efficient
in using these perceptions in computer-controlled engineering and
robotic scenarios. Although fuzzy logic 'softens' the problem of choice
by replacing 'either...or' with 'as well as', it remains rigidly
attached to strictly pre-determined sets of alternatives (inputs,
outputs, goals, criteria) and lists of rules describing the mutual
relations between the alternatives. If the set of alternatives consists
only of A and B, with fuzzy logic we can never generate C; it is the
human operator who, based on his or her experience, can generate
(discover, create) new decisions.
Fuzzy logicneeds a full description of the rules of relations
between the inputs and outputs that can occur in a considered
engineering context; when complexity increases, the list of rules
becomes extremely large and needs a great deal of expert information
(not easily available). Fuzzy logic resembles the way of thinking of
actors left with an agreed set of decision options, a list of rules of
behaviour and of instructions how to use them 'fuzzily' so that to
solve a specific problem; the actors are given no ideas how to go
beyond this set of options and to look for other solutions which might
be better. And even if they have information about something 'better',
they have not been taught how to free themselves from all those fuzzy
rules and instructions that keep the control system running. Fuzzy
logic, as any other type of logic, cannot transcend its own limits as a
tool of inference (based on certain premises) and thus, cannot be used
as a holistic criterion of reality .
If I have a headache, I use a medicine that aims to ease my headache,
regardless of whether that medicine might have a negative effect on
some other organs of my body, say my stomach or heart. This is an
example of using black-and-white logic. I am thinking only about my
headache and nothing else. With fuzzy logic I am looking for a medicine
that helps me decrease ('up to a satisfactory degree') my headache
while, at the same time, does not appear too harmful (that is, harmful
only 'up to a satisfactory degree') to the other organs of my body. In
both cases, the logical rule is quite simple: IF there is a pain, THEN
take an appropriate medicine. Let us imagine that I reject this rule
and instead of taking a medicine, I go for a long walk in the nearest
park, take a couple of deep breaths, or lay and consciously relax for a
while. This is a holistic approach - the approach of fuzziology
that is much broader than the use of an 'IF...THEN' rule; it is
compatible with the fuzziness of something that is essential for my
existence as a human being (like breathing, moving, relaxing).
the logic of rational thinking, the fuzziness is open for penetration
by our consciousness considered as a complex and co-evolving integrity
of four inseparable 'fractals': body, mind, soul and spirit. The
deeper our consciousness descends into the nature of existence, the
clearer this nature reveals itself. The opposite is true when using
the reasoning: the more we rationalise about reality, the more
paradoxical and incomprehensible it appears to our mind, the denser the
fuzziness of our knowing.
four essential differences between the approaches to fuzziness used by
knowledge (rational solutions, cogitation, use of quantitative or
qualitative methodologies) and wisdom (intuitive insights, inspiration,
use of concentration and meditation).
(1)Knowledge comes from without, wisdom wells up within.
Knowledge can be transferred, can be borrowed from books, can be
imparted and taught; wisdom is non-transferable, it is an individual's
insight into existence born while living and directly experiencing the
(2) While wisdom flourishes on fuzziness, knowledge
constantly tries to reduce or eliminate the fuzziness from the ways
chosen to lead towards selected goals and purposes, but the effect is
often the opposite. We may think we have 'cleared' the fuzziness from
the way to the goal A, but it suddenly becomes twice denser on the way
to the goal B. We may think that we have succeeded in eliminating the
fuzziness out of the ways leading both to A and B, but it
catastrophically explodes on the way to C.
(3)Knowledge is partial, it sets boundaries, hangs
labels, separates and generates precise definitions - definitions that
turn to be meaningless when seeking to describe complex phenomena and
processes. Wisdom is holistic. It accepts the unlimited, the timeless,
the infinite, and recognises that to stabilise a particular definition
of a complex dynamic pattern or process does not work. The words of
wisdom are always fuzzy, therefore they reach the hearts of many
different people and make sense for them in many situations.
(4) Knowledge prefers logical explanations to paradoxes,
while wisdom thrives on paradoxes and puts stress on the spirit of the
inquiry than on the search for intellectual solutions. Paradoxes cannot
be resolved intellectually - it is the spirit of the inquiry expressed
in the motivation, beliefs and aspirations of the researcher that make
In the light of
wisdom, fuzziology transforms the fuzziness inherent in human knowing
from an impediment of the process of learning to a powerful catalyst of
of this paper has been presented as an invited lecture at the 2001 Soft
Computing WSES Multiconference: Neural Networks and Applications, Fuzzy
Sets and Fuzzy Systems and Evolutionary Computations (11-15 February,
Puerto de la Cruz, Tenerife, Canary Islands). The author is grateful to
D. Levick for his help with editing the final text, and to B. Hodge and
R. Woog for their support and helpful discussions.
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