less we know, the more certain and precise we are in our
more we know, the more we realize the limitations of being certain and
is impossible to deal with fuzziness related to a higher level of
from the point of view of a lower level of consciousness
thing and no being can exist in itself or for itself but only in
relationship with other things and beings
CAN understand as much of the world as we have developed and
deeper our consciousness descends into the nature of existence, the
this nature reveals itself
In a broad
is the opposite of precision. Everything that cannot be defined
(that is, according to some broadly accepted criteria or norms of
and everything that has no clearly described boundaries in space or
is considered a bearer of fuzziness. In a narrow sense, fuzziness
to the definition of fuzzy sets as proposed by Zadeh in 1965: sets, the
belongingness to which is measured by a membership function whose
are between 1 (full belongingness) and 0 (non-belongingness).
characteristic of the images that raise and dissolve in our thoughts -
in our memories and reflections about the past and in our plans and
about the future. They have blurred boundaries and consist of fuzzy
'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
of the fuzziness in human evolution.
presence in our knowledge about the society and ourselves.
human dynamics at three major scales of their manifestations -
(intrapersonal dynamics), social (interpersonal dynamics) and
(universal dynamics) - results in the emergence of spinning webs and
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
with their infinite (in number and diversity) embodiments.
Paradox of Fuzziology
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
was incomparably deeper and broader than the transitory knowledge of
contemporaries, he used to say with a proverbial humility: "The only
I know for sure is how little I know". The awareness of "how little I
made Socrates capable to easily reveal the gaps in the 'precise' and
knowledge of his opponents. When the Athenians went to the famous
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
"That's why he is the wisest among you!" - was the answer of the Oracle.
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
complexity, we are fully aware about our limitations to be certain and
precise. It is then that we are ready to apply fuzziology
study fuzziness of human knowing - its sources, nature and dynamics -
in an endeavour to reduce or eliminate it but to understand and
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.
1973): As the complexity of a system increases, human ability to
precise and relevant (meaningful) statements about its behaviour
until a threshold is reached beyond which the precision and the
become mutually exclusive characteristics. It is then that the
statements are the only bearers of meaning.
by Zadeh for extending the applicability of his fuzzy sets theory and
logic to the analysis of complex systems .
Connectivity:No thing and no being can exist in itself or for
but only in dynamic relationship with other things and beings.
to the integrity of existence vitally supported by universal dynamics,
whose creative, sustaining or destructive powers are constantly
at different scales of the manifested world. It is through these
that everything that exists - from an elementary particle to a gigantic
galaxy - becomes connected in an all-embracing web of relationships.
Fractality (Mandelbrot, 1982): The geometry of nature is
and reveals itself as self-similar structures at different scales of
the basis of Mandelbrot' theory of fractals  and demonstrates the
self-organization works while unfolding the complex dynamics of nature.
Self-similarity is a kind of fuzzy repetition - each scale has common
with every other, and yet there are noticeable differences.
are inherent in the holistic unfolding of individual, social and
dynamics: the macrocosm is a fuzzy projection of the microcosm, the
world of individuals is a fuzzy projection of the inner world of their
experience, each level of development of consciousness has similarity
with the previous (less developed) and the next (more advanced) levels
and yet has its own distinguished characteristics - its own
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
the wholeness of existence, its manifested activities and its creative
potential are results of an all-embracing connectivity of everything
exists, that moves, changes and transforms in a gigantic self-organized
Web of Interdependent Dynamics. According to the Principle of
it is impossible to offer precise and yet meaningful explanations
to the overwhelming complexity of this web. Hence, any possible
that makes sense of the integrity of existential dynamics, their
actual or virtual appearance (as "manifested activities" or
to create") inevitably contains fuzziness.
wholeness of existence
immensity of its manifested
infinity of its potentiality
prevents fuzziology from looking for and from designing techniques to
control or eliminate the fuzziness of our knowledge of social
such techniques are hardly to be found. The fuzziness of social
has its deep roots in the very essence of existence - an essence whose
self-propelled unfolding makes universe "incomprehen-sible" for "our
and feeble minds" - expressions used by Einstein when describing his
"My religion consists of a humble admiration of the illimitable
spirit who reveals himself in the slight details we are able to
with our frail and feeble minds. That deeply emotional conviction of
presence of a superior reasoning power, which is revealed in the
Universe, forms my idea of God". (Quoted in the New York Times
on April 19, 1955)
and feeble minds"
are products of the unfolding of the mysterious essence of existence.
its fuzziness is not something 'over there' that can be onjectified,
defined and then studied and modelled; it is deeply inside each of us
therefore escapes the grasp of our reasoning. It can be felt,
realized in life. Being out of the realm of logical
formulations (no matter what kind of logic we decide to use, be it
deductive, abductive, binary, multi-valued or fuzzy), the journey into
the existential mystery needs 'preparation', in which the reasoning
of the human mind plays an important role - the role of a coordinator
the sense impressions, perceptions, sensations, feelings and emotions
a meaningful whole.
irreducible fuzziness at the essence of the existential dynamics. The
of this fuzziness activates the potential of fuzziology for construing
reality where the conscious revelation of our deep inner experience
the paramount role in making sense of existence, not the intellectual
about the outward, 'objective' world as perceived through our senses.
information from senses inevitably passes through mental and emotional
filters, consciously or unconsciously established in the process of
informed interactions. Some of these filters can irreversibly distort
sense information up to such a degree (in result of bias and prejudice,
brainwash and propaganda, attachments and delusions) that it entirely
to help people navigate the social complexity of their lives.
impossible to deal
with fuzziness related to a higher level of consciousness from the
of view of a lower level of consciousness.
The proof of
follows from the Principle of Fractality when applied to the unfolding
of existential dynamics. From their manifestation at the scale of
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
The unfolding of these dynamics runs in parallel with a self-propelled
expanding and growing of complexity at each scale of manifestation.
is a stunning diversity at the level of minerals, and also at the
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
the human scale, it is the human consciousness (as a holistic
awareness and knowing of our own nature and the nature of reality in
we exist and evolve) that expands and grows.
at each level of development of human consciousness can hardly be
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
can be seen as saturated with meaning if this individual puts in some
and succeeds in developing a higher level of awareness and
and in sharpening her/his capacity to think, to feel and experience
rather than solely from a more narrowly established point of view
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
of his inquiry alive till the day he was unjustly accused and killed,
possibly never bothered the most of the Athenians at that time.
the deepest spiritual wisdom of the ancient Vedas, considered as the
written text on our planet (they came to us in written form between
to 6000 years ago) is almost ungraspable with the level of
of our generations.
the Second Impossibility Theorem is straightforward: the threatening
of all those serious ecological and social problems, which today's
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
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
be imposed from outside. Nobody can poure wisdom into the brain of
person; the process of developing consciousness needs continuous
to be exerted by those who strive for wisdom. The external conditions
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.
CAN understand as much of the world as we have developed and
The proof of
follows from the Principle of Fractality and from the Second
makes us understand why the macrocosm mirrors the microcosm and the
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
spiritual attitudes and beliefs; through all of them we perceive the
around us. The power of our will is also in the inner world,
with our infinite potential to create and realize ourselves in
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
world and the weaker our spiritual drives for self-realization. From
Second Impossibility Theorem follows that when we grow in
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.
another type of fuzziness, inaccessible from the previous levels of
starts to irritate and challenge our minds and souls.
Fuzziness: Knowledge versus Wisdom
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
there is no possibility to rehearse them, to practise them or to design
special experiments for their study.
Existence is not
of human mind. Digging into the past, investigating the present,
about the future, writing papers and books about reality are mind
- exercises in using different types of logic, a kind of intellectual
at the surface of reality - they have but a "frail and feeble"
with life as a holistic experience.
in all its modification is only a means of expression, a play with
regardless of their relationship to living experience and therefore
be used as a criterion of reality.
Fuzzy logic is
also a product of rational thinking and entirely subjected to its
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,
possible actions, etc.), fuzzy logic selects both A and B with
degrees of significance (preference, compatibility with chosen goals
criteria, fitness, 'truth'). The use of different degrees of 'truth',
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
scenarios. Although fuzzy logic 'softens' the problem of choice by
'either...or' with 'as well as', it remains rigidly attached to
pre-determined sets of alternatives (inputs, outputs, goals, criteria)
and lists of rules describing the mutual relations between the
If the set of alternatives consists only of A and B, with fuzzy logic
can never generate C; it is the human operator who, based on his or her
experience, can generate (discover, create) new decisions.
a full description of the rules of relations between the inputs and
that can occur in a considered engineering context; when complexity
the list of rules becomes extremely large and needs a great deal of
information (not easily available). Fuzzy logic resembles the way of
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
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.
even if they have information about something 'better', they have not
taught how to free themselves from all those fuzzy rules and
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
If I have a headache, I use a medicine that aims to ease my headache,
of whether that medicine might have a negative effect on some other
of my body, say my stomach or heart. This is an example of using
logic. I am thinking only about my headache and nothing else. With
logic I am looking for a medicine that helps me decrease ('up to a
degree') my headache while, at the same time, does not appear too
(that is, harmful only 'up to a satisfactory degree') to the other
of my body. In both cases, the logical rule is quite simple: IF there
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
for a while. This is a holistic approach - the approach of
that is much broader than the use of an 'IF...THEN' rule; it is
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
considered as a complex and co-evolving integrity of four
'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.
differences between the approaches to fuzziness used by knowledge
solutions, cogitation, use of quantitative or qualitative
and wisdom (intuitive insights, inspiration, use of concentration and
comes from without, wisdom wells up within. Knowledge can be
can be borrowed from books, can be imparted and taught; wisdom is
it is an individual's insight into existence born while living and
experiencing the existential dynamics.
wisdom flourishes on fuzziness, knowledge constantly tries to reduce or
eliminate the fuzziness from the ways chosen to lead towards selected
and purposes, but the effect is often the opposite. We may think we
'cleared' the fuzziness from the way to the goal A, but it suddenly
twice denser on the way to the goal B. We may think that we have
in eliminating the fuzziness out of the ways leading both to A and B,
it catastrophically explodes on the way to C.
is partial, it sets boundaries, hangs labels, separates and generates
definitions - definitions that turn to be meaningless when seeking to
complex phenomena and processes. Wisdom is holistic. It accepts the
the timeless, the infinite, and recognises that to stabilise a
definition of a complex dynamic pattern or process does not work. The
words of wisdom are always fuzzy, therefore they reach the hearts of
different people and make sense for them in many situations.
prefers logical explanations to paradoxes, while wisdom thrives on
and puts stress on the spirit of the inquiry than on the search for
solutions. Paradoxes cannot be resolved intellectually - it is the
of the inquiry expressed in the motivation, beliefs and aspirations of
the researcher that make paradoxes dissolve.
In the light of
fuzziology transforms the fuzziness inherent in human knowing from an
of the process of learning to a powerful catalyst of creativity.
of this paper
has been presented as an invited lecture at the 2001 Soft Computing
Multiconference: Neural Networks and Applications, Fuzzy Sets and Fuzzy
Systems and Evolutionary Computations (11-15 February, Puerto de la
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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