Vladimir Dimitrov

University of Western Sydney

Richmond, NSW 2753, Australia

Socrates Paradox of Fuzziology
Principles of Fuzziology
First Impossibility Theorem
Second Impossibility Theorem
The Possibility Theorem
Transcending Fuzziness: Knowledge versus Wisdom


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).

Fuzziness is 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 evolution.

Fuzziness has a substantial presence in our knowledge about the society and ourselves.

The constant 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.

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Socrates Paradox of Fuzziology

Socrates 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.

Although 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 Oracle.

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.

When dealing 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].

Fuzziology 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.

Several principles and theorems are considered at the basis of fuzziology.

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Principles of Fuzziology

Principle 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.

This principle was used by Zadeh for extending the applicability of his fuzzy sets theory and fuzzy logic to the analysis of complex systems [8].

Principle of Connectivity: No thing and no being can exist in itself or for itself but only in dynamic relationship with other things and beings.

This principle 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.

Principle of Fractality (Mandelbrot, 1982): The geometry of nature is fractal and reveals itself as self-similar structures at different scales of manifestation.

This principle is at the basis of Mandelbrot' theory of fractals [7] 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.

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First Impossibility Theorem

It 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.

The First 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)

Human "frail 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 meaningful whole.

Fuzziology 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.

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Second Impossibility Theorem

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.

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.

The fuzziness 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').

What the 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.

The fuzziness 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.

The practical 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.

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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.

The Principle 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.

The external 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.

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Transcending Fuzziness: Knowledge versus Wisdom

To transcend fuzziness of what we know about the unfoldment of life and existence means to grasp how the dynamics of this unfoldment work.

Condition 1. Existential 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 experience.

Condition 2. Logic in 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 logic needs 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 [8].

Example: 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).

Being beyond 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.

There are 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 existential dynamics.

(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 paradoxes dissolve.

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 creativity.


The content 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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1. Zadeh, L. 1965 Fuzzy Sets, Information and Control, 8, 1965, pp. 338-359

2. Dimitrov, V. 2000 Fuzziology in Search for Ways to Transcend Fuzziness, Internet paper http://www.uws.edu.au/vip/dimitrov/study-of-fuzziness.htm

3. Dimitrov, V. 2000 Introduction to Social Fuzziology, Internet paper

4. Dimitrov, V. 2000  Using Fuzziology when Collecting and Making Sense of Social Information, Internet paper

5. Dimitrov, V. 2000 The Conscious Resonance: Understanding Fuzziness of Knowing, Internet paper

6. Zadeh, L. 1973 A New Approach to the Analysis of Complex Systems, IEEE Trans. Syst., Man, Cybern., SMC-3, 1

7. Mandelbrot, B. 1982 The Fractal Geometry of Nature, Freeman Co., San Francisco

8. Dimitrov, V. and B. Hodge 2000 Why Fuzzy Logic Needs the Challenge of Social Complexity? With Fuzzy Logic in the New Millennium, Eds. V. Dimitrov and V. Korotkich, Publ. UWS, Richmond

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©V. Dimitrov, 2000

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