Duality

V. Popkov

 
 
"The main and the first of sciences - is the science of arising "odd" and "even" notions in the meaning, they have with respect to the nature of things".
Plato, Afterlaw

1. Duality. Subject of analysis.

Strange as it may seem, but there is no philosophical dictionary, where you could find an article, devoted to the notion - "duality". This is more weird if we take into consideration the fact that these concepts are widely used in philosophy and various branches of special knowledge (in physics, mathematics, chemistry and others). However, till now there haven't been done any attempt to systematize, allowing for achievements of modern science, everything "really astonishing and divine for a thoughtful thinker - this is doubling of numerical values, characteristic of the whole nature, and vice versa, divarication - relation, observed in all kind of things" (Plato,1999).

The ultimate goal of such consideration, which, of course, deviates from this issue, is an attempt to create a coherent, logical and necessary system of general ideas, united by the concept of "duality", in whose terms each element of our experience could be interpreted.

Such investigation, to our mind, should first of all distinguish duality as a subject of special study, formulate main principles, revealing essence of duality and the character of its motion, and secondly, mark outlines of universal theory of duality, show its methodological power and heuristic value, comprised in a dual approach.

Duality is versatile and changeable, and this is very likely to be one of the difficulties in its description. Dualism, dyad, dichotomy, binary opposition, contrast, polarity - this is, perhaps, still incomplete list of notions in some way or other affecting duality essence. Not yet subtilizing various definitions and their relationships, "cohesion" with duality, let's point out that we could have declined from the old term - "duality" and create a new one, but we have chosen other way, as new meanings of old words do not remove their old meanings, but enrich them*. The other constant difficulty of reasoning with use of duality consists in necessity to "stretch" the idea and its verbal expression beyond the usual unary use. Thinking about something, we concentrate on something we think about, so we have to disregard everything irrelevant. But when we consider a separate fact, we always come across a latent premise - its coordination with surroundings, required for its existence. Therefore, modality of the thought in everyday life is continued by such phrases as "from one hand", "from another hand", and in philosophy this way of thinking was called by Hegel as doubling of phenomenon: "Thus we double the phenomenon, breaking it into two parts: internal and external, force and development, reason and consequence" (Hegel, 1998).

2. Duality. Views of ancients.

Idea of duality - is a very ancient idea, reached us through various myths of ancient peoples. Egyptian Heb and Nut, Sumerian Ki and An, Indian Prakriti and Purusha, Chinese Yin and Yang, Aztec's Ometikutl and Omesiguatl, Persian Ahriman and Ormyuzd - these are all names of gods or active dual principles, whose interaction, as ancients thought, set the world in motion. These dualities (for convenience we will call so the sides of duality) form an alliance, where the differences disappear, forming the whole, able to move and develop. Thus, Prakriti and Purusha unite in Pradkhan, Yang and Yin merge into Dao, Akhraman and Ormyuzd are combined in Mithra, Ometeotl forms a single whole for Ometikutl and Omesiguatl. A common idea, comprised in these ancient theories in different variants, comes to the following. The source of the world lies in manifestation of order inside shapeless foundation - chaos. As soon as the order is settled from the chaos, there appear various structures and pictures of the world. A real originating power, lying in the basis of various manifestations of the world, can be found in external duality motions. This duality is raised in the primary order as an act of pure discrimination where dual principles come from: heaven and earth, day and night, male and female. When dualities are united into an alliance, they possess vague (potential in the sense achievement possibility) energy. But separated as pure polarities, after submersion into shapeless foundation this vague energy transforms into active potential, able to make changes. Actually, this description (reproduced generally from Peat, 1987) already contains elements of modern theories (order, chaos, differentiation, flow, vibrations, connection and others), which, as we'll see, have direct relation to duality. We could only imagine sagacity and intuition of ancient thinkers, whose ideas reached us through the thick of milleniums to progress again on the way of different sciences integration on the basis of duality principle.

3. Duality. Ancient thought.

Cosmological ideas, created by ancient thinkers, continued the tradition to view the world as two principles. According to Anaximander and Heraclitus, form-creation proceeds in nature through rarifying and thickening of material substance - water and fire. Self-changing and self-organizing Empedokl's Spheros represents interaction of two "energetic" principles or forces, - Love (friendship, good) and Enmity (malice, hatred) and creates a huge vibration process, setting Spheros into motion. Love prevails, and homogenous unites with homogenous - Enmity retreats; if Enmity prevails, discord, disconnection of all grows, and Love steps back. Parmenid takes existence, that mean all being, in unity with non-existence or not being (that is neither defined, nor shaped). As the beginnings of all things there figure light and darkness (Fire and Night). These beginnings are counterbalanced, closed in harmony, whose presence is a manifestation of the fact, that there are things full of motion, changeability of life, and at the same time, stability, calmness of eternity, and death. In Plato's works, cosmos, built by demiurg, comprise beginnings, not subjected to him, - composition of homogeneous (identity) and heterogeneous (different), spontaneous and systematic, creating and destroying (Soroko, 1984). A common tissue of the world, which serves as a foundation for all arising, whose essence is a process, saving cohesion, is termed by Plato as "Godmother and Wet-nurse of all appearing". Plato speaks of it, as of a very vague notion, hard to understand (Plato, Timey). In this connection Whitehead (Whitehead, 1990) marks, that "space and time of modern mathematical physics, taken in abstraction from individual mathematical formulas, almost exactly coincide with Plato's "Godmother and Wet-nurse".

In established by Pythagoreans symbolics each of two pairs of conditions (limit - boundless, even - odd, entity - set, right - left, male - female, rest - motion, straight - curve, light - darkness, good - bad, square - oblong) fixes some certain side of life and death countervailing, one pole always prevailing over the other (life over death). Let's point out, following Soroco (Soroco, 1984), one of Pythagorean achievements - distinguishing of four basis elements or tetractid. Such elements are point, section, triangle, tetrahedron. In modern topological terminology they are called symplexes, by which dimensionality of space is expressed: zero-dimensional, one-dimensional, characterized by length, two-dimensional - by "flatness", three-dimensional - by "volume". The sum of "point units" (1,2,3,4), used to construct all mentioned basis elements, makes ten. Pythagoreans called this number as "the four". Let's mark asymmetry of basis elements: one of them - a point, do not exist in a real world, and, unlike others, is an abstraction. Citing B.Bellustin (Bellustin, 1923) Soroko writes, that a unity, according to Pythagoras, meant a spirit where the whole visible world came from. From a unity there comes deuce, a symbol of material atom. Absorbing the unity, deuce or material atom becomes three or mobile particle. This is a symbol of a live world. Live world plus unity makes four, which forms a whole, i.e. visible and invisible. As far as 10=1+2+3+4, it expresses "All". Let's note, that in a given interpretation each pair of basis elements have common nature, namely: unity and three mean movable live, fluid, have "stream nature", and two and four are nothing but stiffened, immovable, or have spatial nature. This is a principal remark, which, as we'll see later, always appear at a deeper consideration of duality as a phenomenon, and which have more complicated, four-element structure, instead of usual, two-side, used when we consider movements of opposites. Four-element basis ("fourfold") was also used by Aristotle in his doctrine about four reasons of opposition, four senses of forego relation, four ways of making conclusions, four kinds of quality, four meanings of essence and others (Aristotle. Metaphysics). Among basic categories, mentioned by Aristotle, four make self-closed, internally complete thesis: quantity, quality, relation, essence (Aristotle. Categories). Here, we can also distinguish the following pairs: quantity - relation, quality - essence, characterized by certain unity of their nature. Soroko quotes an extract, ascribed to an outstanding Pythagorean Filolai - "Harmony in general arises from contrasts. For Harmony is junction of diverse mixture and accord of discorded". An attentive eye could also see here pairs of connections inside contrasts: junction and accord, mixture and discord.

4. Duality. Hegel's dialectics.

Let's now pass from ancient myths and sciences immediately to one of the greatest thinkers - Hegel. Of course, his not less outstanding predecessors, such as Descartes, Spinoza, Fikhte, Shelling, Kant and others, could also tell much about duality, but this needs a special investigation. As for our goal, it consisted in showing the stability of a dual outlook which was stubbornly forcing its way, occupying minds of best scientists of all times and peoples. In particular, it was Kant, who first discovered pairs of categories in synthetic opinions. His works brought to clear realization of antynomy, raised by these very categorial pairs. So, since Kant's times it is customary to talk not about categories, but categorial pairs, i.e. material-ideal, cause-effect, shape-content, essence-phenomenon. In this sense we cannot even term as one word this new formation, consisting of two interconnected, dual to each other categories. However, there can be found modern manuals with one section devoted to the category of phenomenon, and the other - to the category of being.

Hegel - is one of those giants of thought, who still delight with power of "pure" solitary human mind. Having started with the simple statement (known before him, by the way), saying that "pure being is nothing", he built up a whole science "Science of logic" (Hegel,1998), as a universal method of cognition which was then applied by him with German pedantry and consistency, to construct philosophical theories of thinking ("Phenomenology of spirit"), nature, history.

Russell (Russell, 1999) considered the Hegel's theory to be erroneous. Indeed, a great intention turned to complacent statement: "Everything real is reasonable, everything reasonable is real" and fixed, as the highest realization of an absolute idea: in nature - scanty knowledge of the beginning of XIX century, in history - excuse of violence, in law - predomination of State over person. In truth, "a mount gave birth to a mouth". Why it happened so and can Hegel's philosophy be considered erroneous only because it resulted in mentioned conclusions. We suppose, Hegel indeed was mistaken, but this mistake was also great, - this is still a stumbling block for philosophers. It is a formation problem (or, in Hegel's terminology, it could be treated as removing of contradictions between thesis and antythesis), which still keeps to be in the center of all philosophical schools' attention. As far as removal (synthesis) is, according to Hegel, embodied in absolute (absolute spirit, absolute idea) which is then materialized in a human spirit, nature, history and so on, there can appear arbitrary interpretations while passing from super-human to human, id est. synthesis in thinking (as well as in nature) is ambiguous. For example, from one and the same law of negation of negation, Hegel inferred stability and wisdom of Prussian State's system, and his passionate follower Marks - inevitability of a revolutionary crash of any state. But Hegel is great and incontestable, and nobody have refuted him yet in the moments important for aims of our research. Dual categorial pairs and formed by them "fourfold loops" of relations pierce through all sections of "Science of logic". Logical unity of the theory, based on duality - presence and interaction of two inseparably linked concepts, reflects, we think, an organic unity of all material.

Let's not go further into the thickets of Hegel's philosophy and dwell on one corollary, taken from "Science of logic", necessary to illustrate our subject. A preconceived expert could have nagged at casual nature of choosing this example to illustrate duality. The answer would be a modern one: the theory of Hegel's dialectics is self-similar (fractal); each section of "Logic" can give one and the same understanding of duality and its structure. Let's consider correlation "one and many" . Hegel writes (Hegel, 1998) - "one turns out to be absolutely incompatible with itself, push itself away from itself, and what it suppose it is - is many". We'll further cite our translation of an original Hegel's work (Hegel, 1975). "Alienating, considered in itself, take a part of negative relation (Verhalten) of many ones between themselves (gegeneinander), hence, their connection (Beziehung) corresponds them with each other (aufeinander),and as far as those, one corresponds with in its alienation, are the ones, it corresponds in them with itself".

To deeper understand the idea of a thesis, proposed by Hegel, let's remind that in German the words Verhalten and Beziehung, although being translated into Russian as synonyms meaning "relation, connection, behavior", in original text have indeed different meanings. If Verhalten - means relation, it is relation of "keeping, seizing" from somewhere inside (it becomes evident from the verb ending halten - seize, keep), stipulated by opposition of one to many as external to each other (this external, as if pushing one away from many, is indicated by prefix gegen in the word gegeneinander). From the other side, Beziehung - is also a connection, also a relation, but relation of drawing "outside" (verb zienen - pull, draw out), but which have the nature of internal, as arising from opposing one of many (prefix auf in the word aufeinander). Thus, in a dual complex "one-many" we have again four relations combined in pairs Beziehund-Verhalten, and aufeinander-gegeneiander. This structure is characteristic of all famous Hegel's formula, for instance, "reason is essential essence inside itself. Essence is essentially reason"; "essence appears, appearance is essential". It would have seemed to be a pun, if all Hegel's logic wasn't based on it. What's behind this? Are these Hegel's "loops" just speculative constructions, or they are based on some fundamental property of nature, and if they are, on which one? We answer positively and believe, that dual structure of Hegel's statements - is an attempt in zero approximation "to catch" in verbal expression a constantly changing, progressing world, which in every its moment is what it is, and at the same time is not, what it is. And what it is not, in a reversed way have an influence on what goes in the next moment of time and so on, and this turning from non-existence to existence - is something that is reflected in categorial pairs of Hegel. Unfortunately, these subtleties of philosophical sense of Hegel's expressions were completely lost (just like in other places of original "Logic") while translating into Russian, and this translation is being reproduced again and again with every new edition.

5. Duality. Evidences of XX century scientists.

Our review of philosophical understanding of duality would have been incomplete without mentioning, although shortly, ideas of some philosophers of XX century. Let's note at once, that the choice is stipulated by the frame of the review goals, which is to give just a general contour of the problem, and personal interests of the author. Whitehead - a great mathematician and original philosopher (Whitehead, 1990) concentrated on dynamic description of reality as a process. As a result it turned out that the notion "substance-quality" can be avoided. Such morphological description is substituted with the description of dynamic process. "Each real phenomenon behaves as a process; - it is formation". Whitehead distinguishes two types of formation: union and transition.

The first type of formation (union) is internally inherent in construction process of separately existing. The second type of formation (transition); due to this type of formation the process ceasing, in case of forming separately existing, constitute this existing as initial element of constituting other separately existing, which are shown up while the process is repeated. Union is directed towards its final cause representing its subjective goal; as for transition, it is a mechanism of working cause which is immortal past. According to Bogdanov (Bogdanov, 1990), preservation of the system "is the result of mobile equilibrium of the system with its environment, i.e. it is formed by two flows of activity - assimilation, absorbing activity from outside, and disassimilation, loss of activities, their transition into environment; it implies two rows, unbroken and parallel, of the process of progressive selection: positive and negative. In which direction do they regulate the progress? Obviously, in the direction of more stable relations, for less stable should be gradually rejected by negative selection (within gradually rejected relations can appear "points of growth" for more stable formations - V.P.), and more stable - be positively fixed. It can be seen now, that positive selection brings to the first type of formation which keeps on developing the chain of still new beings - the chain of animate, much dying at once, at times just marking the possibility of its birth. Negative selection is connected with formation of the second type - transition, and fix the born being in system of the world, constantly reproducing itself, and in this sense it is indeed a transition of constantly appearing and dying animate to immortal inanimate, forming the past.

Whitehead makes acute remarks considering existence of two types of formation, - "an implicit understanding of the two types of formation was inherent in works of Hume. Kant has almost made it explicit. This understanding was lost in evolutionary monism of Hegel and Hegelian schools".

Bohm (Bohm, 1987) states that "explicit order can be considered as a manifestation of unfolded deeper implicit order", and gives an example: a whirlpool in a river is an expression of the full stream of the river and its structure is constantly supported by dynamics of the river as a whole. We can say the whirlpool to be dynamic invariant of the river. "Reality, - Bohm believes, - shows itself through dual motion, in the sense, that a whole is enfolded inside individual, and in each field of space the nature of this reality can be found both by drawing outside in explicit form (which feeds the following implicit order) and by moving inside towards the very implicit order. In explicit order the bodies are external to each other and interact through local forces. On the contrary, in implicit order the structures enfold each other in such a way, that one structure can be both external and internal with respect to the other". Let's note, that Bohm, describing duality, also operates with "flow" model (the example of whirlpool in the river), when describes movements of dual reality in pairs of terms explicit-unfolded, implicit- enfolded.

6. Duality. Examples from special sciences.

Mathematics.

The simplest example of duality manifestation in mathematics is well-known from school course (Kiselev, 1958) existence of direct and inverse theorem. A theorem inverse to a given one, is the theorem, whose condition is the conclusion of the given theorem, and conclusion - is the condition of the given theorem. It's easy to find out, that if we proceed from the second theorem, which we call inverse, and consider it to be direct one, the theorem, called direct, would turn to be inverse. That's why they are often called not direct and inverse theorems, but mutually inverse theorems (Gradstein, 1973). Mutually inverse or dual to each other theorems, whose truth is proved, play a really big role in mathematical analyses. To analyze some certain objects it's often very important to know not only the properties of a given object, but also which properties can be considered as the main ones, fully enough defining the given object. These properties of objects (geometrical shapes, sets) are distinguished by proving inverse theorems. Thus, presence of duality helps to distinguish some essence of the object (main property, invariant) - the fact, most important for understanding the value of dual statements.

In projective geometry dual are the point and the straight line, lying through it; in this case they speak about a point, incident to a line and a line, incident to a point (Glagolev, 1963). Lines, planes, volumes and so on (proper elements of space) in projective geometry are closed on infinity by so called improper elements - correspondingly, infinitely remote point, infinitely remote line, infinitely remote plane and so on. Projective geometry bases upon absolute equality of rights within proper and improper elements. A line, dual to a segment, is that part of the line, which supplements the segment to the complete projective line, closed on infinity. In the same way a part of plane, bounded with a circle, is dual to the whole external part of the plane supplementing the circle to the complete projective plane. Duality principle in projective geometry states, that if some sentence considering points, lines and planes and relations of incidence between them is true, true would also be dual sentence, received from the given one if we change the words "line" and "point" (the words "plane" and "point" for projective geometry).

One of the examples of duality is the problem of linear programming (Karmanov, 1986). The direct task is to find maximum of objective function inside convex polyhedron, disposed in n-dimensional space. The polyhedron is set by linear form with "n"-limitations, laid upon m-dimensional vector-argument of the objective function. The inverse or dual task is to find minimum of inverse functional outside on the surface of polyhedron, in the space of m-dimensionality, all components of the "direct" vector transforming in accord with corresponding rules of matrices transformation into coefficients-constraints for inverse problem.

In the theory of categories (Bukur, 1972; Palenko, 1974) the category K* dual to the category K is determined as the category, having the same objects as K, but with morphisms (maps), operating in the inverse direction. As examples of categories can serve categories of sets, topologic spaces, groups an others. Each object A from K can be marked with "*" and called the object of category K*; each morphism f : AB is also marked with "*" and called a morphism f* : B*A*. The category (K*)* is equal to the category K. Each expression, formulated in the language of the categories theory, is connected with a dual expression, received by interpretation of the initial expression in dual category. The other way of receiving a dual expression consists in preserving of logical structure of the initial expression and inverse order of found in it "multipliers" (inverse arrows). Duality principle for the theory of categories states, that some expression is true in only case when in this theory the dual expression is also true. Duality principle helps to point out formal connections between notions and resultants, which in certain categories seem independent from each other. In particular, dual connection between visibility and accessibility of a system was found with the help of the categories theory. Visibility should be understood as possibility to define the system condition by efficiency. Accessibility - in defining the set of accessible conditions. The system looks like a pair of dual systems, accessibility of one being dual (in catedorial sense) to visibility of the other.

Physics.

An interesting example of duality can be consideration of gravity law. Newton's formulation of the law states, that gravitational force depends on mass of the bodies at final distance. It has the property of non-localization and long-range action (Feinman, 1968). The force, affecting a body, depends on remoteness of an other body in accord with the law of inverse square of interval. From the other hand, there exist an other interpretation of the gravity law, based on the concept of field. The theory of long-range action deals with the concept of force, which is a vector, while in the field description a number-potential is attached to each point in space, but it is not a vector any more, it is a scalar quantity. While passing from point to point the number changes, and force, affecting the body, acts in the direction of the quickest change of potential. Unlike long-range action, potential in the center of however small sphere is defined by potential in the vicinity of the point, we are interested in, and mass, comprised inside the sphere. This formulation is local in time and space, for it speaks about neighboring points. It is a dual form of describing usual Newton's formulation, which is continuous in time and discrete in space.

Fundamental Maxwell's equations, describing propagation of electromagnetic waves in space, can serve as an example of dual relations in electrodynamics. If there are no currants neither charges, equations come to take more symmetrical form. One pair of equations come from another by a simple replacement of electric-field vector E with vector of magnetic flux density B. Presence of currants and charges makes the picture a bit more complicated, but duality still shows up, having now some other appearance, connected with different symmetry of E and B vectors. Absence of magnetic charge results in vector B, unlike vector E, always having vortex nature, for it appears due to the change of currents and electric density. The value of such kind of vector is geometrically determined by a space element, in contrast to electric-field density vector, which can be compared with a directed segment. In this case they speak correspondingly about contra-variant and covariant vectors, whose transformation laws are mutually inverse when coordinates vary. Thus, physical interconditionality of the vectors E and B has a deep dual geometrical background, analogous to incidence of line and plane.

In physics such notions as intensive and extensive values are widely used. The first answer the question "how strong?", the latter - "how many?". The first have a character of force, the second are the amount of operations of the first. It is also spoken correspondingly about generalized forces and generalized speeds or flows.

In case of electric phenomena, for instance, density can be considered as a peculiar action, i.e. intensive value (generalized force), while currant intensity is quantity, i.e. extensive value (charge flux). In 1931 Lars Onzager found out that there exist remarkable correlation of mutuality (Prigozhin, 1986). For weak non-equilibrium electric phenomena true is the following: if force "one" (for example, gradient of electric field potential) affects flux "two" (charge flux), force "two" (gradient of charge distribution), in its turn, affects flux "one" (electric field flux). It happens in such a way, that both these influences are directed to the opposite side and aspire to bring the system back to equilibrium (in the given example it is a self-induction manifestation). Easy to see, that mutuality correlation is nothing else but manifestation of duality on a higher structural level. What is understood here under complication in structure of generalized forces and fluxes relations? First of all, it is redoubling of pair of "force-speed" relation and appearance of four notions (two generalized forces and two generalized speeds): secondly, their interconnection (cause "one" becomes effect "two" and vice versa), and finally, the dynamic character of transmitting energy, substance, information, pointing at the change, development of the system, its striving for a certain condition.

The first principle in thermodynamics connects total differential dU (function of system condition U) with partial differentials dW and dQ, where W and Q are, correspondingly, work and heat. Therefore, work and heat, taken separately, are "incomplete" characteristics of the system. Supplementing each other up to the "complete" characteristic (internal energy), both in physical sense (describing just one of two ways of energetic influence on the system) and in mathematical one (being partial differentials, i.e. not a function of system condition), thus, represent two dual structures within one "energetic" body, bound up with the first principle of thermodynamics. For equilibrium processes dW=Pdv and dQ=TdS, where P-pressure, V-volume, T-temperature, S-entropy. Extensive parameters V and S are called generalized coordinates, and intensive parameters P and T - generalized forces, "force" P being conjugated with coordinate V, and force T with coordinate S. There exist one-to-one correspondence between the pairs of parameters (P,V) and (T,S); the so called Jacobian of transition (Rumeur, 1977) from one pair to another equals to unit, what makes equal coordinate planes (P,V) and (T,S). Let's note, that there is asymmetry among the four parameters P, V, T, S; the first three parameters being really measurable, while entropy can not be measured. The given four is the parameters of dual notions of a higher level - four thermodynamic potentials: intrinsic energy U(V,S), free energy F(V,T), enthalpy H(P,S) and Gibbs potential G(P,T). Each potential is a function of two parameters, one appertaining to the pair (P,V), another - to (T,S). It can be shown that V, F, H, G are connected with each other by relations of duality.

However many manifestations duality has, the most staggering and, at the same time, fundamental property of matter is dualism of substance and waves. It is known, that under certain conditions light behaves as a Newton's particle, under other - as a wave. The same is true for an electron, proton and any material system. Dependently on a question we ask to a microparticle, it turns to us either with its wave or corpuscular nature. The fact, that on experiment stage there never appears logical contradiction between both manifestations, is provided by Heisenberg uncertainty relation, which states that it's impossible to localize the position of a particle together with exact knowledge of its impulse. The determinacy measure of simultaneous detection of a particle with its impulse measuring is Planck's quantum of action (DD~h, where h - Planck's constant) (Feinman, 1967).

Chemistry.

Chemical reactions strive to proceed in the direction of lessening free energy. Let's remind, that free energy - in a notion, pointing at maximum quantity of work, which can be received as a result of some chemical reaction provided heat contact with environment. We could draw mechanical analogy between behavior of two-loads system and chemical reaction (Etkins, 1987). So, a heavy load, while falling (moving in "natural" direction), can lift a lighter load (if, for example, the loads are interconnected by a thread, thrown over a block), provoking its movement in "unnatural" direction (against gravitation). Reaction is similar to the load, but the one, "falling" in the direction of reducing not potential, but free energy. A chemical reaction, going deeper down the free energy scale, can provoke the situation when the reaction, connected with it, proceeds in the opposite direction. In biochemistry such reactions are called conjugated (in out terminology - dual; V.P.)

Developing mechanical analogy Etkins proposed a visual picture of a live system. Live organisms and cells can be compared with an extremely complicated system of gear-wheels, catching on each other (by the way, pinion drives in mechanics - are also an example of dual systems, used for transmitting power - V.P.) If in some part of an organism a "heavy load" falls down the free energy scale, in its other part a "light load", on this account, can go up this scale, but on less magnitude. The mechanism of such biochemical "gear drive" is sure to be delicate and complicated. Although the organism as a whole moves along degradation way, i.e. way of free energy reducing, these processes are so complicated and interconnected, that incidentally there appear all most delicate phenomena conformable to life and consciousness. That why we have to feed ourselves: we assimilate matters, situated highly enough according to a free energy scale, and then, in an organism, they start to dissociate, causing rotation of internal "gear-wheels" in the organism.

Biology.

A fundamental example of duality is encoding of the genetic information. Ancestral information, comprised within DNA and RNA , is generally encoded by two classes of nitrogen bases - pyrines and pyramidines (Inge-Vechtomov, 1983). They are formed by five kinds of nucleotides: thymine, adenine, guanine, cytosine, uracile. The first four encode DNA, the four last - RNA. Two purine bases - adenine and guanine - make invariant part of both combinations, DNA and RNA. It is the presence of these two common bases, included one-by-one into the encoding four as "special" elements (the other three are pyramidines), that allows to carry out replication process. Double spiral is stabilized by hydrogen combinations between purines of one DNA chain and pyramidines of the other. Formation of fixed pairs adenine-thymine, guanine-cytozine, results in the fact that bases sequence of one chain defines its sequence in the other - in other words, chains of DNA double spiral are complimentary (Frank-Kamenetsky, 1988).

Discovery of DNA dual structure played in biology development the role, same as discovery of atomic nucleus played in physics, it brought to appearance of molecular biology, and in the last years - gene engineering.

Janus of biology - structure and function. According to Hamaley (Hamaley, 1999), development of structures is determined genetically, realization of functions - ecologically. Genetically inherited structures take a stabilizing role in evolution, consolidating the passed way, and ecologically dependant functions - the one swinging, providing search of movement direction today. Participation of both regulatory mechanisms serves to achieve a combination of directed (progressive) and adaptive components of evolution in point of transition from past to future. Influence of structure on functioning is evident; there is a movability range, amplitude of functional oscillations set to it; influence of functional oscillations on structure variations is less obvious, its mechanism - gradual accumulation of insignificantly small shears. These variations, though being very small, irreversibly accumulate. Biological time is changed by structures as by stakes of the way passed. Reconstitution of the structure on functional signs is almost impossible.

Vertical determinacy of structural progress and horizontal - of functional, give good reason to set a task of seeking the model reflecting four dimensional relation of current ecology and evolution. Probably, it can help to describe the cross of time and space in biological coordinate system. Structure reflects the way passed, functional quivering - choice of movement direction, search of next step. Ecological mobility of functions within the diapason, set by structural giant of the past, gives some basis to assume a vibration model of functional search of evolutionary path prolongation under conditions of a certain ecology.

Dialectic logic.

Hegel showed, that all original notions the mind use, always contain categorial pairs. Having made this step, he got an opportunity to do something, no professional logician ever done before him: he gave the first classification of logical forms, marking their difference from grammatical forms.

A judgement is known to be represented by a "subject" and a "predicate", connected with the link-verb "to be". If the subject is something individual, and the predicate is something universal, we deal with logical form of judgement. So, a grammatical sentences like: "Ivan is a man", "Zhuchka is a dog" , represent judgements. A simple fact that a link-verb "to be" connects "a categorial pair" or a "dyad" serves as a logical basis to distinguish logical form.

The old formal logic was naively sure that judgements are divided into two classes: true and false. Common sense would argue if someone stated, that "individual is universal", or that "effect is cause". Which of the two judgements should be considered true? The answer sounds shortly: none. Hegel showed the judgement form not to be that logical form, where the truth is generally expressed. According to Hegel, logical form of the truth is a "conclusion". It consists of logical judgement forms, and logical judgement form (dyad) is preceded by and comprised of logical notion form.

Logical notion form itself consists of two different logical forms: the pairs individual-special, individual-universal, special-universal form "dyads", and their integrity, peculiar independence from logical form of dyad, requires its own name - "triads". These two logical forms, not yet distinguished as logical forms by Hegel (Kuznetsov P.G., 1988) form the basement of mathematics. It became possible to single them out only when N.Burbaki, in his analysis of mathematical structures, succeeded to distinguish two qualitatively different mathematical structures. Their outlined names roughly conform to "the law of composition" and "order relation". Their philosophic content lying beyond mathematics and belonging to logical forms, is designated with logical form of dyad and logical form of triad.

Mathematical form of recording logical form is formulas, where two expressions, different on writing, aggregate by a sign of equality (A=B), and both in the right and in the left part of the formula there can be found notions, related to one of dual pairs, included in triad and setting "order relation", "direction", "orientation". The logical form of "triad" is an outer-mathematical form of this "consequence". That's why the simplest logical form of conclusion should take this fact into account, so that instead of one form of record (A=B), we'll have four ones:

A = not B; not A = B; B = not B; A = not A

We have now minimum five formulas along with the initial record A=B.

We'll be able to better understand the depth of these expressions, which seem striking for the common sense, if dispose them in the following way:

A = not A A = not B
not A = B B = not B

This is just the same structure we've come across while considering relations of one and many through Hegel's notion of "repulsing". Where did it come from? We added to the initial record, and it was well-founded, proceeding from logic of movement, the particle "not" - because in the changing world A is B and A is not B.

Linguistics.

Written language can serve as a simple and visual example of duality. When You read these lines, your brain reads out the information contained in signs - letters, disposed in accord with certain rules within two-dimensional space, - on a sheet of paper. Order relation is set by the direction of reading out the letters - from left to right (although there exist languages, where the direction is opposite, for example - Ivrit), or directed vertically, for example - hieroglyphs. Besides, additional order relations are defined by morphology - the rules of forming words from letters. Thus, there is formed a frame ("a skeleton"), and now the matter depends on compositional relation - grammar - a code of rules. In accord with these rules, from separate words and punctuation marks, there are formed sentences, bearing information. The received system is highly flexible and able to adapt itself, since any written text gives us a big choice of formulations it could be continued with. But this possibilities are not boundless, there exist one other global dual system, formed by the whole environment of each phrase - it is a context. It is impossible to come over to account of something without breaking internal logic of the contents, otherwise a discourse, in every its part, would be wrong both morphologically and grammatically, it would loose coherency. Having lost this property a discourse becomes disconnected, not characteristic of a healthy person. So, the visible text on this page (material signs, made in paint, written on a plane of paper) is connected with an invisible, dual text (context), which can not be physically seen. This dual text is comprised within the same material structures as the direct one, but it can only be read by "immersion" of the direct text into the thought vehicle of a person. The vehicle should be developed enough to absorb this construction, - for example, a child, learning to read, to make words from letters, cannot apprehend a context during a certain period of time, while his brain is not trained enough.

7. Duality. Theoretical models.*

Tectological model.

"Complex" - Bogdanov wrote (Bogdanov, 1989), is contained in its environment both as casting material and moulding model, being determined by this environment in the first sense and partially determined "in the second". The pattern of moulding model and casting material perfectly illustrate a statistical model of duality. Bogdanov's complex is not just a collection of elements and their relations, it represents the process or continuous flow of generating elements - processes, encompassed by circles of construction and degradation. (Zeleny, 1999). Bogdanov's complex cannot be separated from its environment, for it would neither exist nor interact outside this environment; it is structurally connected with the environment, and thus creates its own environment, progressing jointly. Structural connection with the environment makes absolute separation of complexes impossible and establishes their common systematic convergence and divergence resulting in growth of complementary and supplementary ability of elements and their relation.

Bogdanov invented the concept of regulator (modern backward connection), a device, helping to keep processes on a certain level (for example, fly-wheel). He went further and invented the concept of biregulator - the system of mutually regulating one another without outer regulator; they interregulate. These are two aspects of entire equilibrium system. Biregulator is not just a simple backward connection, but a two-sided dynamic system. Not only thermostat regulates temperature, but temperature also regulates thermostat, it is carried out by way of mutual oscillation and hierarchy. Any complex should assimilate and disassimilate the required variation of the environment, it is structurally connected with. Every change and preservation to the account of assimilation is balanced with opposite changes of disassimilation processes. Thus, there is established a dynamic (mobile) equilibrium. According to Bogdanov, simple elements of a complex are bound into junctions, chains, linear chains being ingressively connected into non-linear and circular chains, which constantly and continuously form chains of a higher rank. Its ties: chains and networks (schemes) were the beginning of "mathematics of complication" proposed by him. Legible differentiation between organization and structure of the system is required for better understanding of a complex's dynamics. Organization is related to the network of generating elements processes (interaction rules), and structure is related to a certain time-space pattern. The basic pattern in the investigations on complicated systems stability is a relatively firm "skeleton", immersed into mobile, changeable medium. He gives various examples, when systems, having such a structure, are most capable of development and self-organization.

Informational-entropic model.

This model (Soroko, 1984) proceeds from the fact, that in structurally determined systems information together with entropy represent a remaining value or:

entropy (H) + information (I) = const

Informational entropy H in qualitative sense coincide with thermodynamic entropy S within a constant factor. The fact, that information and entropy are permutative, enumerative in common terms, allows to form an entire system, where extremities - are extreme values of information and entropy quantity. Every structure of an object, representing a system is in line with some via point of the scale - figuring point of the given structure. If we write the above-stated parity as follows (normalization per unit): H + I = 1, the figuring point will move within the interval [0,1]. Its movement to one side or another is in line with: 1) structure-genesis process, typical of self-organizing systems, whose behavior is based on self-regulation, backward connection; 2) process of degradation, disintegration of structures, typical of all changes, made in combination with the second principle of thermodynamics, i.e. characterized by entropy growth. Figuring point of the structure is a peculiar border between variety and monotony, plurality and unity, necessity and contingency, symmetry and asymmetry and so on (dependently on which structure-forming relation is taken as a ground).

Preservation of "entropy-information" relation for a certain object (it is adequate to established definite coordinate values of figuring point) can mean, that its structure has reached the stationary (stable) condition. This is the case of "unstable equilibrium" (Brilluen, 1980), or "flowing stability" (Bertalanffy, 1953).

Combinatorial-wave model.

"The probability approach - is one of three possible to define notions of information quantity: the other two - algorithmic and combinatorial ones (Kolmogorov, 1965). Contingency with its logical independence of events - the basis of the stochastic laws, and necessity with its logic, embodied in algorithms, in the laws of rigid determination, - two opposite, independent beginnings of count." (Soroko, 1984).

Combinatorial approach in definition of "information quantity" combines peculiarities of probability and algorithmic approaches. The basic material, helping its specificity to come out, - alphabets. An alphabet - is any basic system of signs, each being an independent unit, bearing minimum semantic load. Various combinations of these signs form "words", which form "sentences", just like it happens in natural human languages. "General principles of making complex systems from component elements (molecules from atoms, organisms from cells, biological populations from individuals and so on) coincide with the properties, according to which letters of alphabet make words, words - make phrases, phrases - complex report. To be short - reality is Text". (Sedov, 1978). But the world as a whole and in every its structure is, at the same time, Wave (Nalimov V., 1982) - structural resonance of a substratum, all things are formed of, id est. a kind of "interference" of oppositions within a unity. Dual sides are equal in their whole, and, consequently, any fulfills the whole "relation space" and interfere with the other, forming immovable "clusters" of the substratum. "Waving" is experienced by an extremely abstract "substance" - substratum of systems, frequent (probability) distribution of their structure. The model World - Text, World - Wave seems hard to understand, but cooperative phenomena resonance often occur in the nature (solitons, phenomena of superconductivity and superfluidity, Benar cells, chemical reactions of Belousov-Zhabotinsky and so on). A combination of combinatorial and continuous approaches is more visually realized in the following model.

Tensor model.

American electric engineer Gabriel Kron, in his research, devoted to the theory of generalized electrical machinery and to the method of piecewise solution of a required problem (diacoptics) (Kron, 1978), created and effectively applied in practice electrical (topological-wave in generalized form) models to solve technical, physical, statistical, economical and any other problems of major dimension. In his unique models (which are pretty hard to perceive, as they are formed of concepts from at least three fields of knowledge: techniques, mathematics, physics) he succeeded to describe real problems in a dual language, combining advantages of topological (structural) and functional (composition) approaches.

Kron constantly uses dual notions just as a method and convincingly shows effectiveness of such approach. Kron put a "live" electromagnetic structure, bound by a composition law, over a "dead" subject network, reflecting order relation. This unity of description let Kron not only propose a universal method of piecewise solution of complicated problems, but find out a remarkable self-organization property of systems, formed by "immersion" of structural "skeleton" (a notion with its triad structure) into the live substance of changeable environment, bounded with permeable "cover" (set by a composition law - "dyad"). Let's consider it in more details. Impendance, attributed to the elementary one-dimensional network - segment (one-dimensional simplex or, shortly, 1-network) is analogous to the notion of activity-resistance. One-dimensional simplexes can be combined into 2-networks (an analogue of usual circuitries on a plane) through matrix transformations, 2-networks serve as constructing material for associating them into spatial ones, 3-networks and so forth, up to multidimensional generalization (n-1)-network (where n - the number of linear graphs-impendances). It is obvious, that all networks of one dimensionality are dual (isomorphic) to each other, which is expressed in existence of orthogonal square matrix of transformation, C; in case of transition from the network of one dimension to the one of another dimension matrix of transformation C becomes rectangular. It is such topological structures, that Kron called a "dead" or subjected network.

The next step consists in "reviving the network", laying influence on it. In the elementary case of 2-networks there are laid currents and stress (I and e), where I and e - vectors with I and e components, enclosed to single impendances. As a result, in the network there appear responses E and i (also vectors of the same nature), connected with the laid currents and stresses by Om's "composition law"

e + E = Z(i + I),

where Z - a tensor quantity with components, attributed to separate (diagonal elements) and mutual (all other members) impendances. In case of transition from one network to another (analogue of progress), tensor (and, correspondingly, vectors E, i, I, e) transforms through matrix C, according to the law of "order"

Z*=CZCt-1 and e*=Ct-1e,

Where Z* and e* - are tensor and vector components in a new coordinate system (in a new network), Ct-1 - an inverse matrix to C.

At further generalization of the model two-dimensional networks can be replaced with multidimensional ones, and instead of four parameters there appear more and more complicated characteristics of electromagnetic field (electric and magnetic fields strength, charge density and so on), every direct p-network being connected with the dual to it (n-p) network; for example, in two-dimensional case a point network is dual to a contour one, and vice versa.

The idea of Kron's pioneer method consists in the fact, that having calculated an elementary electric machine, he succeeded to receive through tensor transformations a result for any other, however complicated, electric machine. It is very important, that Kron discovered a universal law: to receive correct transformation from one machine to another it is necessary to consider two networks - direct one and dual to it orthogonal network, for only in this case there is being retained the invariant - power, dissipated within the network.

And, at last, diacoptics - it is a method of tearing, utilizing a dual approach with respect to a transformation problem. This problem is very complicated: it is required to restore structural and functional peculiarities of every its part, being guided by functional signs arisen in points of discontinuity, and then receive the required solution by reverse transformation.

In his last years Kron (died in 1968) worked a lot at so called "polyhedral" automates. His works showed (Kron, 1960), that multidimensional polyhedron and dual to it orthogonal polyhedron, immersed in electromagnetic plasma, form non-mechanical oscillating structure, based upon a complex interaction (non-Riemannian dynamics) of generalized rotating electric machinery. Oscillating automate, as Kron showed, is capable of self-organization, self-adjustment to different parameters laid.

8. Duality. Non-linearity. Non-stability. Chaos. Fractals.

In this section we are going to confine ourselves to giving examples and comments, drawn from the work (Nicolis, 1990), undoubtedly making accent on the considered problem of duality.

Let's begin with the fact, that in natural sciences there are distinguished two kinds of systems: conservative, whose dynamics does not depend on time direction (invariant relatively to operation of time circulation, or substitution of t by -t in equations) and dissipate, showing irreversibility of time or dependence of their future condition on past, in the sense, that past in every moment of time is built in present condition and determines future. Strictly speaking, there are no conservative systems in the nature, - we can only speak about those models, suitable for usage under a certain condition, - when we can ignore the influence of environment or surroundings, or represent the problem in a quasi-conservative form (for example, a classical problem of a theoretical pendulum, which does not exist, can be given in a quasi-conservative system of real pendulum mechanism in a clock, receiving energy from outside). Quasi-conservative systems are widely used in technique, as they have the property of predictability and multiple identical reproduction, and do not depend on time. As for dissipate systems, typical of nature, live world, human world, they are much more complicated. A property of dissipation presupposes the presence of surroundings, environment, substance, i.e. in our context dissipation and flow, activity-resistance - are the same things.

Self-organization of a matter, resulting in hard behavior, with its all numerous examples (Nicolis, 1990), can be reduced to three major types, each corresponding to a certain kind of duality. They are: 1) bistability and hysteresis (existence of two, mutually supporting each other branches: examples are reaction of Belousov-Zhabotinsky, optical bistability in lasers and other); 2) oscillations, periodic and non-periodic, (two phases, one moving to another) and 3) space shapes (connected with a special manifestation of duality - fractals). Considering a possibility of describing a complex, always connected with a large-scale coherency and co-ordinated behavior, we should note, that an absolute order in the form of absolute absence of variability, represents an utmost case of coherency, when a subject behavior is so primitive, that we could hardly speak about complicated. From the other hand, strong variability, produced by casual noise and absence of correlation, attending such variability are the other, so unimpressive a form of organization. Complication of real system should be somewhere between these bounds, and, correspondingly, described in models, which consider structural and compositional relations. Later, we'll show, that these two sides mutually give rise and support to each other.

In general, the idea of system evolution being influenced by variation of some typical for such problem parameters, set by environment, can be written down in the following way:

where {Xi} - is a set of variable states; Y - a set of parameters, which can be changed by environment. For the stationary condition (even non-equilibrium) F({X},Y)=0. In linear systems (i.e. where X=kY) the result of combined action of two factors is their simple superposition. However, in non-linear systems a small growth of outer influence can result in very strong effect, with amplitude, incommensurable to the source effect. Non-linearity is always produced by a dual process: for example, in liquid medium mechanics speed of this or that characteristic variation begins to depend on the transition of this characteristic by the environment current, whose speed is one of the variables in the problem. Non-linearities, connected with regulation (remember biregulator) are of special interest. In this case, some matter X can abruptly accelerate or slow down self-production or production of an other component, closed, in its turn, on the first matter.

As we have already mentioned, dissipate systems contact with complicated, moreover, unpredictable environment. This environment continuously imparts to the system small (rarely considerable) quantities of substance, impulse or energy. From the other hand, inside the very system, each of its parts is also surrounded with personal environment and also can experience oscillations of internal state - fluctuation. This ubiquitous dual fact of continuous variation is mathematically expressed in instantaneous state of X(t) system not coinciding with stationary state Xs - it represents some neighbor state:

(1) X(t)=Xs+x(t),

where x(t) is called perturbation. How does the system react on such relations? The list of all possible variants is reduced to four cases:

1) X(t) state is locally unstable, but globally stable. It remains in some neighborhood of standard state at all times, exceeding the initial time. The system fluctuates around standard state.

2) X(t) state is locally and globally stable. Dissipate system eliminates perturbation and restores standard state. Xs state is called attractor (asymptotic stability).

3) X(t) state is locally and globally unstable. There is such initial perturbation, for which amplitude X(t) abruptly grows (usually according to exponential law) and the whole system "moves" into absolutely different state.

4) X(t) state is locally stable, but globally unstable. The system remains in some neighborhood of the standard state Xs, if perturbation does not exceed some threshold value and moves away from it in case of perturbations, exceeding threshold one.

Equation (1) can be rewritten as x(t)=X(t)-Xs, and we can suppose that x(t)=f(g), where l will be called an operating parameter.

Now, at small l, only one solution is possible, the one corresponding to the situation (2); this area is, like equations, characterized by a significant property of asymptotic stability - the system is able to cancel external perturbations and internal fluctuations.

While passing through extreme values of l=le parameter, the state becomes unstable, for fluctuations and external perturbations are not cancelled any more. Acting like intensifier, the system abruptly diverge from standard state and comes over to a new regime. Both these regimes interflow at l=le and l>le. This phenomenon is called bifurcation.

In decisive moment (in l=le neighborhood) the system should make a critical choice. Only a case will decide through dynamics of fluctuations. Some special fluctuation wins and the system, having taken it as a starting point, turns to a peculiar historical object in the sense, that its further evolution will depend on this critical choice. It is evident, that a system choice, made to the benefit of one of the branches (b1 or b2) can again bring to the further process, analogous to the described one, and the system continues its progress. Let's point out, that the "won" state, for example b1 state, becomes the source of new fluctuations and external reactions. We could identify fluctuations and external influence with structural variations (both in micro and in macro-regions), and search of stable state with strive for a simple composition, possessing least energy. We could mark resemblance between these representations and concepts of mutation and selection in the theory of biological evolution. Fluctuations are the physical analogue of mutations, while stability search plays the role of natural selection. Even the diagram of bifurcation itself reminds with its structure phylogenetic trees, widely used in biology. Let's also point out one other very important circumstance - the connection between dynamic choice, made by the system as a result of bifurcation, and failure of symmetry. Solutions, arisen as a result of bifurcation, show in the form of another inner differentiation either between various parts of the system, or between the system and its neighborhood. The further discourse on this theme could have brought us to the notions of structurally-stable and structurally-unstable systems and terms of their realization, but this goes far beyond the frames of our investigation. Therefore, why not finish this section with one more fruitful attempt to combine dual notions inside one model, proposed by a French mathematician Mandelbrot (Mandelbrot, 1981; Paintgen H., 1993). Analyzing the stability state, we pointed out the presence of states, - attractors, the system aspire to return to, despite fluctuation and outer effects. It is known (Nicolis, 1990), that in mathematical models, set dimension, representing attractor, always smaller then the very phase space, and within three-dimensional space attractors, as a rule, are represented by points, closed curves or bounded two-dimensional surfaces. The question, which was, actually, positively answered by Mandelbrot, consists in the following: in three-dimensional phase space are there possible attractors of a transition type, lying between dual types: point-line, line-surface, surface-volume. Such objects, if they exist, would be neither points, nor curves, nor surfaces, not topological diversity at all. Mandelbrot gave them a special name - fractals. The simplest example of a fractal structure - is a set of Cantor (Cantor's "dust"). It appears while the following question is being answered - is there a set, intermediary between a set (like a set of natural numbers) and uninterrupted continuum (like the aggregate of points on a line). The answer would be positive. Let's consider the segment [0,1]. We'll divide it into three equal parts and throw out the middle interval. From each of two segments left, also divided into three equal parts, again throw out the middle interval and so forth. In a limit we'll receive uncountable set of isolated points. This set has no individual length (set of 0-dimension), for the length of its supplement to [0,1] segment equals to unity:

1/3 + 2/9 + 4/27 + 5/81 + = 1

But dimension (d) of this set does not equal to zero. Missing the proof, let's at once give the result dcantor 0,63. Thus, in this sense the Cantor's set is intermediary between point (d=0) and line (d=1), i.e. it is a fractal. With such set there is connected the appearance of the so called strange attractors, which are the natural models of chaotic behavior. The studying of strange attractors within chaotic dynamics makes us conclude, that in dissipate systems with many variables a remarkable manifestation of duality is possible: in such systems chaotic and regular behavior can exist not only separately, but simultaneously as well.

9. Duality and golden ratio.

The most simple example of "golden ratio" manifestation is the problem of a segment division in extreme and average relation. Let's take a segment AB, divided into two arbitrary parts by point C, so that AC is bigger then CB. It is required to find such position of point C, that the whole AB related to the bigger part As a result of simple transformations we can receive the equation X2 - x - 1 = 0, the required parity a=1,6180 being its positive root. (Shevelev, 1990). This remarkable number, called a "golden" proportion or ratio appears everywhere: in the nature, in human creations where the question is about most proportional, harmonic combination of parts, composing the whole. In some other interpretation golden ratio is the limit of series

where the whole numbers Zn are set by recurrent parity

Zn = Zn-1 + Zn+2

This line is called Fibonacci sequence and its first members look as follows:

1, 1, 2, 3, 5, 8, 13, 21, .

Golden proportion can be written as a / b = a + b / a, where a > b. Such record shows, that there are two values a and b, connected with a composition relation a + b and order relation a > b. It is required to find such a point inside a continuous network, which, without infringing order relation, would provide "equilibrium" of four parts, bound two and two by the order relation a + b > a, and a > b and the composition a + b / a and a / b. The number a = 1,6180 - is the only number, whose a-1 = a - 1 - 0,618, i.e. a - the only number "sewing together" two dual spaces: linear, set by a function y = x - 1 and hyperbolic y* = 1/x.

Fibonacci numbers figure in phyllotaxis laws of biology, known for a long time. According to these laws, in many cases of biological objects symmetric formations, in numeric performances of their configurations, there are realized pairs of Fibonacci numbers from dual to each other sequences (parastich and ortostih; Petuhov, 1981)

Zn-1 / Zn and Zn+2 / Zn

A very instructive example of connection between duality and golden ratio is given by the studying of the Carnot cycle. It is known, that heating and refrigerating machines are dual to each other, they pass one into another within circulation of the ideal Carnot cycle. If we make up our minds to find maximum efficiency value for refrigerating and heating machines simultaneously, we'll find out, that it equals a-1 = 0,6180. (Popkov, 2000).

10. Duality. Substance and consciousness.

Hitherto, we have considered duality from, let's say here, "divine" point of view, absolutely ignoring the fact, that we ourselves and our consciousness with its thoughts and feelings is a part of life stream, and therefore, we can assert, that duality is an integral part of this process. Appearance of quantum theory proved, that nature observation contains indissoluble connection between the spectator and the observed. An individualized condition of brain, thus, contains the objective, not individualized level. This dualism between objective and subjective points at a deep correspondence between substance and realization, that consciousness and substance can be viewed upon as two aspects of a single whole. An irremoved connection exists in a quantum theory between the spectator and the observed, as well as it exists between thoughts of those, who think. An act of thinking change the thinker. A thinker is a thought; the thought engenders the thinker, who, in its turn, again arises the thought. Kandel (Kandel, 1979) proposed a model of interacting neuronic networks, continuously dancing with each other, both being changed and established by the dance. Brain and consciousness are like a fountain, which lives until there exists a water stream, forming it. We can consider the following hypothesis. Inanimate substance, (not thinking, to be more exact), is considered as a "flexible", changeable frame, (skeleton), connected by order relations. The order relation in this space-topological cover is set by time, flowing from past to future. This reality is immersed into consciousness, exiting and influencing on it. "I am an object" as realization of "here and now" realizes its belonging to experience, constructed by internal attribution to the world of reality and the world of ideas. But, "I am an object", constructed in such a way, stays in the world of reality, and proves itself as an organism which requires inventing of ideas to fix its status in the world of reality. (Whitehead, 1990). A thought is not connected with order relation, it floats from past to present, and even future, unlike reality, is not an insuperable obstacle for it: due to the knowledge, a person can in many cases predict future with a certain accuracy.

The body and brain themselves are also material "topological" substance and immersed into consciousness; we feel first of all as consciousness, as "I", the whole Universe immersed into. "A Man - is a whole world of ideas, buried in night of "I" (Hegel, 1999). There are three groups of compositional relations between consciousness and reality: a) distinguishing oneself from oneself; b) distinguishing oneself from substance, which has no thinking; c) distinguishing oneself from thinking substance, i.e. from similar to oneself, other people. We have already mentioned, following Hegel, that a person doubles any phenomenon by breaking it into two parts. At the same time, only a human being doubles himself in such a way, that he is general for general (Hegel, 1999) For instance: an animal as such can never be shown, but we can always show a concrete kind of animal, id est. there does not exist an animal as such - it is general. But the property of being an animal, belonging to some kind as to general, - this is a property, arisen by a human brain; there is no general for an animal, only individual. So, if the world is immersed into consciousness, giving birth to a Man as general, mind, in its turn, arise general for world cognition, bisecting in into pairs of categories with a common name activity - resistance (force - manifestation, cause - effect and so on) and this world, transformed by a thought, also goes down into the consciousness.

"Circulation of "existence" and phenomenon - is a particular law of universum" and a Man is one of possible realizations of this law. Such circulation of existence and phenomenon principally change the view on a Man" (Lefevr, 1991) He gives results of an interesting experiment. A test group was given some tens of usual string beans and offered to sort out good and bad ones according to an arbitrary (inherent to each one) criteria. The result, average on the group, was rather unexpected: "good" beans related to "bad" ones in golden proportion (62% to 38%). We think, this is a manifestation of duality in the sense, mentioned above. Indeed, a person, under conditions of uncertain criteria, considers the pair good-bad as a whole, where, by virtue of moral purposes, good should be bigger then bad; - but how bigger - this is dictated by the "mechanism" of duality, laid in golden ratio.

Lefevr (1978) proposed a formalized algebraic way of analyzing interaction of the learning subject with reality, comprising other people as well. Using that approach, let's consider the so called general problem of philosophy about relation of substance and consciousness. There exist two groups of philosophical doctrines: idealistic (X1), materialistic (X2). In algebraic record, then, any object of reality T can be written in a man's consciousness as a pattern Q from various positions: Q1 = T + TX1 + TX2- this is dualism, no integral pattern. (TX1 should be read as T from X1 position). Q2 = (T + TX1 + TX2)X1 - the integrated pattern is created by means of idealistic approach. Q3 = (T + TX1 + TX2)X2- the integrated pattern is created form the position of materialism.

We suppose, a new approach is required, which could be written as follows:

Q4=(T+TX1+TX2)X1+(T+TX1+TX2)X2 integral pattern is created on the basis of new position of dual object nature synthesis. Having labeled TX1=T1* and TX2=T2*, we can write Q4 as Q4= T1*+ T2* X1+ T2*+ T1* X2 T1*+ T2* - is compositional relation; T2* X1+ T1* X2 -order relation.

In the work (Lefevr, 1978) there is invented a useful notion "the shield of consciousness", - analogue of reflection (reaction) of consciousness on reality influence. The shield of consciousness works in accord with the following principle - "I think, that you think ", where under "you" there is understood either not thinking substance, and correction (self-concordance, adjustment) goes in percussive mode of interaction itself with itself (or, more precisely, initial representations with the ones, appearing anew); or, in a more complicated case, another thinking object is understood under "you". Bogdanov (1907) used a similar method, having introduced a substitution, which proceeds from acceptance of other people's mentality or sociality of knowledge. "When I substitute some certain statements of other people with some certain thoughts and feelings, only a certain content of the prepared is hypothetical here; it is often erroneous, I may not understand other people. But the substitution itself - not a hypothesis at all, but a constructive sign of knowledge Substitution does not represent going out the boundaries of possible experience. Its correctness is controlled by practice. Using substitution, Bogdanov (1907) looks into a very delicate question about dualism of consequent (causal) knowledge of reality and parallel (simultaneous) seizing it as a whole. This problem, in its more full aspect, has not been completely worked out till now, but in still larger measure attracts attention of explorers. Peat (1987) adduced a diagram, proposed by a well-known physicist V.Pauly (formulated, by the way, a principle of interdiction: two electrons can be in one quantum state, only having opposite spins).

Causality Space Simultaneousness
--|--
Time

All events flow and unfold themselves out in independent from time patterns quicker, then sequence of casual connections within linear time. He proposed a following approach: to consider substance order and thinking order as a whole, and in this case to determine: if these two orders are incompatible, the question is about dualism; or if they lie within the common spectrum. If thinking and matter can be understood as a manifestation of common order, it would be useful to consider them not as separate substances, but more likely as indissoluble manifestations of single individual whole. Peat does not explain, what he understands under the notion "order", but we can do this, if remember the Kron's model with its topological "skeleton", immersed into changeable plasma - "consciousness".

11. Duality. The theory outline.

Definitions.

Among rare sketches of already given facts and opinions there becomes more and more evident the outline of the theory, which have a good ground to pretend being universal, for, as well as tektology, is not attached to some certain science, but can draw new facts from all sciences to confirm necessity of its existence.

The foundation of a fruitful theory lies in the beginning with the postulates, which could be stuck to, while considering any problem, but before we try to do this, let's, first, get back to the problem of definitions and figure out the notion of duality (as far as we have decided not to introduce a new term), from kindred terms with prefixes "di", "du", "dia", "opposite" and others.

Dichotomy - is, according to the article in philosophical dictionary (Philosophical dictionary 2000), is carried in correspondence with a logical formula of the excluded third, and divides some set exactly into two classes of objects, while duality represents the whole, manifesting under various conditions this or that property, belonging to one of the branches of paired category.

Dyad (Greek) - a term to define the principle of uncertainty, set, unlike monad (a unity). Dyad is an opposition of monad. It is reckoned with the Pythagoren "four", which, as we have seen, can characterize duality.

Dualism - disability to connect two beginnings, more often material and ideal are implicated here. Thus, duality is considerably broader, then dualism, accepting the presence of the whole, expressed in dual measure of concept, from the very beginning (ad ovo). Avenarius realized, that there is a difference between dualism and duality, and expressed this with a categorical phrase "Diese Dualitat ist keine Dualismus" - This duality is not dualism. (citation - Bogdanov, 1907).

Binary opposition - a term (Soroko, 1984), closest to duality, for it points at dual existence, but the part of the term "opposition" distorts the meaning, brings it away from duality. To visualize let's give an example: a picture drifted on a tissue - can absolutely naturally be considered as a duality illustration, but we won't have the heart to call it binary opposition.

It turned out that most complicated is the interaction of duality with contrast and contradiction. The trend of contrast and contradiction is more completely worked out in Hegel's dialectics.

Duality, contrast, contradiction

As soon as the law of thinking takes a part of judgement, it differentiates the subject and the predicate, it contains differentiation. According to Hegel, difference progresses in a threefold form, with every step penetrating deeper and deeper. The first consists in external discrepancy - there is a difference. The second - is in internal or immanent difference, comprised in the fact that something differs from the other, which is its different. This form of differentiation is called contrast. And at last, the third form consists in differentiating itself from itself - this is contradiction. From the nature of contrast becomes obvious, that each of the two sides (something and its different and vice versa) is necessarily correspondent with the other one, connected with it, so that supposes and requires also non-existence of the other side, each side both supposes and negates the other side, consider it both positively and negatively, and therefore, form a whole contrast, and, which is the same, opposite to itself. In this opposition to itself is laid the essence of contradiction (Fisher, 1933).

There is no rise, change, life, progress without contradiction, this unity of opposite definitions in the essence of things. And where is the place for duality in this stream of life, what is its role in the structure of interdependent transitions and unity of contrasts?

Each of opposite sides of contradiction, comprising something, for example, - A, is called to life by its not-A, which is its different. Contradiction inside A is formed due to existence of contrasts between "A", becoming in reality, and its "not-A", counteracting this change in the real world, influencing "A". Let's label this its different by "A" with an asterisk or "A*". As far as for "A*" its different is "A", the sides of contrasts, correspondingly exchange sides; if in "A" essence the contradiction is formed by counteraction of "A*", opposite to it, this change is represented as "A*" becoming, that causes corresponding change (reaction) of "A" and so on, - thus, prolonging ad infinitum this pulsing of life. Thus, internal, immanent contradiction of some being turns out to be connected with external contrast of surrounding reality, the connection having a particular, dual character, described above.

It is obvious, that such understanding of reality manifestation through mutual transitions and interlacing, has an extremely important practical value for human activity. Planning our actions, preparing to introduce a new being into this world, we should clearly conceive, that its appearance will cause the reaction of response, counteracting changes. As far as the world where the becoming occurs, is infinitely multiform, manifestations of these responses and responses of a new being in correspondence with a dual character of interaction, are practically unpredictable. Something definite can be said only about very obvious consequences, but always with a certain share of mistake. That is why brain forces of a man should be forwarded not to counting the result, which often turns out to be mythical, but to forming, thinking out the measures, providing adequate reaction on the infinite chain of variations, progress, pulsing of life. Here becomes clear the whole senseless and perniciousness of the centralized decisions, strict plans and so on, but this is a theme for a separate discourse.

Duality of "denial of denial"

Dialectic "denial of denial" means motion, progress. This double denying contain the problem of becoming. How does a new appear and become in this complicated changing world? How is it connected with what exists, under which conditions a new receives the right to live and how inevitably it becomes old? Why progress has a cyclic, pulsing nature? These are the main questions, which are attempted to answer within the frame of the concept, or the so called "denial of denial" law . There are a lot of formulations and interpretations of this law. But, however different they would be in the form and completeness of the description, there can be distinguished two key moments in them, whose absence makes impossible to understand the "denial of denial " idea. The first of them pays attention to the process character of "denial of denial". It is stated, for example, that dialectic denial of denial comprise a triune process: destruction (overcoming) of past, cumulating (its partial saving, succession, translation) and constructing (forming, creating of new) (Introduction, 1989).

The other aspect is connected with acceptance of necessity to develop the process of progressing just of double denial. Dialectic denial or "removal" of contrasts provides saving of some residue, - a new being, which either strengthens itself through the following cycles of "removing", or, having no possibility to progress, - dies. But even its death does not pass without a trace, denial of death, denial of denial shows its creating role here as well. The fact of generating and death does not remain unnoticed, it leaves its changes in the world and creates premises for other beings appearance.

Knowledge of the dual mechanism of contrasts interaction can shed additional light on a delicate structure of mutual relations of contrasts in dialectic denial. Let's do it by formulating some basic statements.

Statement 1. An ability to progress (creativity) is characteristic of open systems, exchanging streams of energy, matter, information with surrounding reality. Such systems, as we have already noticed, are called non-equilibrium or systems of dynamic equilibrium. Their characteristic feature is constant destruction and restoration of equilibrium owing to the unity of the two processes - assimilation (absorption) and dissimilation (decompositionment and apportionment).

Statement 2. The more closed is the system, the less possibilities it has to progress. Limitation of energy, material, information streams results in system degradation and its further death. One of the hypotheses, explaining the reason, why everything live dies, is based on the fact that all cells gradually loose their ability to exchange, due to growth of errors while they replicate.

One of the scripts of degradation - is the system degeneration. Instead of spiral progress there takes place the movement on the circle, introducing nothing new, having no becoming. As an example many isolated systems can be taken, when we can ignore dissipation of energy. This assumption is laid in the foundation of many important applied calculation methods.

Statement 3. The process of "denial of denial" comprises four phases: first denial, progress, second denial or "denial-asterisk" and the next phase is progress again, but now it is "progress-asterisk". Progress-asterisk is removal of double denial influence.

Let's consider it in more details. We'll point out at once, that the first denial and progress, as well as the second denial and progress with "asterisks" represent common cycles, inseparable in time. Every generating of new - is a moment of progress and, at the same time, denial of old. But nevertheless, such partitioning is quite reasonable, as it brings us to studying one other very important problem, indissolubly connected with progress. The matter is about the problem of choice. Development is impossible without a choice between various ways of progressing. It is clear, that the choice is stipulates by the system openness, in a closed system the choice either limited or impossible: the more open is the system, the more possibilities of choice it has. The choice is carried out on the basis of internal criteria, characteristic of a given system. As a criteria there often works a principle of "minimax", i.e. receiving maximum results at minimum expenses. The significant are three circumstances. The first is, that the choice of one way of progressing excludes, or at least, restricts other variants. The second - the choice, though being carried out continuously, has a special importance in some node points, forming, according to Hegel, a "node line". In the theory of non-equilibrium systems these points, due to their importance, received a special name - "points of bifurcation" (Nicolis, 1990).

And finally, the richer is the possibility of choice and, correspondingly, conditions for the progress, the more indefinite is the possible result of this choice. This happens because all the rest, unclaimed ways of developing do not disappear, they form for a newly born being its different or the background, conditions, where the phase of the further progress is carried out, and the richer and diverse is the picture of their interaction, the less definite would be a result of the choice.

So, the first phase of progress - is the choice, denial from "its different", - of external contrast of all ways of development except for one. This is a becoming, arising of something new in conditions of uncertainty, in a hardly non-equilibrium field. This non-equilibrium is, at same time, cause and effect of appearance of something new. Then there comes the phase of progress, the new affirms itself, external to the becoming contrasts also undergoing the process of change, turning into its other "different"; and we have to make a choice again - the second denial, or denial of denial. Here, two results are possible, - either the next phase of progress (progress with asterisk in the changed conditions), or death (considered as a premise for generating some other new).

This, we think, is the "mechanism" of self-renewing the substance movement, based upon dialectic denial of denial. This is a unity of calm (laminar), to a great extent predictable and indefinite, vortical, hardly predictable stages of development. Existence of some uncertainty is an integral characteristic of a developing system, the reservoir, where new possibilities for progress are drawn from. The presence of uncertainty is, at the same time, payment for progressing and an installment into development fund.

Proceeding from the principle of the world unity we can suppose existence of the principle of uncertainty for developing macrosystems: "The more we localize the system movement, restricting possibilities of choice, the less possibility we leave to the system for self-progressing, self-regulation".

Striving for progress, we should submit the presence of uncertainty and give up the idea, that the process of development can be subject to the strict deterministic laws. There is required a dual to determinism probability description for learning such systems.

In the light of said above we should give a new glance on the problem of relations between spontaneous and conscious, casual and indispensable. Old, half-slighting attitude to left parts of these dichotomies (spontaneity, causality) as something second-rate, existing only because we haven't yet completely learnt the reality, should be replaced by respect and comprehension of the equivalent role, they take in the changing and progressing world along with indispensability and consciousness.

Dynamic principle of duality

To finish our short review of duality manifestations in various fields of human knowledge, let's try to formulate a general principle of duality and consequences from it. So, here is the principle of duality:

In a developing interdependent reality becoming of a new being results in such a change in its external surroundings, which cause internal contradiction of this being, resulting in the change of external contrast and appearance of internal contradictions in the beings, forming this environment. Thus, there is formed a qualitatively new environment, the initial being playing a role of external contrast, and the development cycle recommences.

Consequence 1. "Due to infinite diversity, surrounding the given being of the world, internal contradictions, displaying themselves in external contrasts, are represented as unpredictably appearing new beings, each able to serve as a point of the development process in accordance with the principle of duality".

Consequence 2. "The harder is the system on introducing new beings into a real world, the quicker and sharper become the contradictions, and if there is necessary mechanism of correction, contradictions grow and result in the system crisis".

Consequence 3. "Systems, representing multidimensional "conducting" networks, immersed into a changeable pulsing environment, possess the greatest adaptability and ability for self-regulation".

12. Duality. Search algorithm of a new.

In accordance with the logic of account, in this section we'll try to show some typical procedure, taking a part of search network or framework - means, which can be immersed into any environment for studying. The possibility to formulate such rules in steps (algorithm) is shown by repeating signs of duality manifestation in nature and human knowledge. Besides, above we have pointed at heuristics and methodological importance of the duality principle application in practical and scientific activity, and therefore, our analysis would have been incomplete, if we haven't tried to present a reproduced by any investigator mechanism of using basic results, stated in this work. Of course, this attempt is nothing but a rough reflection of more deep and delicate methods, we have already mentioned, and which progress and can be developed in the future. But, nevertheless, even in such simplified form, the given algorithm can play an important role. As a proof there can be used results of the work (Popkov, 2000), where the authors succeeded to get new results in the well-known and learnt long ago Carnot cycle by using dual approach.

For clear perception we have given the algorithm in an imperative style, which, of course, does not mean, that the research will go as fast, as it is shown in the procedure. Certainly, each step will require additional reflections, comparison, with the above given models as well, reading and studying of special literature and so on, - everything, that accompanies the work of investigator.

So, here is the algorithm of duality application while considering any process (for reality is always a process).

Step 1. Search for a system, dual to a given phenomenon. Remember, that it always exists - this is an objective property of reality. If it is impossible to find out a dual system immediately, bisect the phenomenon, break it into two parts. Such bifurcation is ambiguous, but it is always possible. The bifurcation logic is dictated by the subject and the goal of the research, as well as by opportunities to use the rules of the following steps.

Step 2. Outline the parts, corresponding with two characteristic branches of duality: order relation and composition relation. Remember, that the order relation (the structural part of duality) is characterized by a discrete description, the composition relation - by indispensable functional connection.

Step 3. Restrict in the beginning of the problem consideration with the stationary linear case.

Step 4. Find out linear approximation of composition relation, find four of interconnected in pairs parameters. The characteristic feature - one of them in one of the pairs should possess a special character.

Step 5. Search for the composition law among these four parameters (or four possibilities of description). Try to make a model, either qualitative or quantitative (in each case try to write down a mathematical dependence, such as X + Y* = F (X* + Y)

Step 6. Study carefully the order relation. Find ways to describe transition from one structure to another. The characteristic feature of such possibility is the presence of invariants.

Step 7. Search for invariants of the system. Find out those structure transformations, correspondent to invariants.

Step 8. While searching invariants, first of all, pay attention to various combinations of parameters from point 5.

Step 9. Combine both descriptions of points 4 and 6. One of the starting point to search such combination - a special character of one of the parameters, sometimes showing the possibility of transition to a dual description. As a result of applying the dual description, receive new results.

Step 10. Search for a dual "invisible" system. With this purpose make an operation, such as "circulation of arrows". Proceed from the fact, that in a simple approximation: f (a, b) f* (a*, b*), where the parameters, marked with the sign (*) are dual to the parameters without this sign and belong correspondingly to inverse and direct spaces. As a result of the operation on circulation receive a new result.

Step 3*. If it is impossible to outline a dual system in point 2, use the rule: "Structure (order) engenders function (composition); function engenders structure". For the rule could work there is required the point 4*.

Step 4*. Introduce heterogeneities (fluctuations) into the composition law, which means the transition to a more complicated non-linear model. Non-linearity will generate the structure, analyze its stability and symmetry (symmetry always points at existence of invariant properties).

Step 5*. Then move on in accordance with the algorithm logic, set by points 4-10. Namely, with the logic, for a simple analysis is no more possible; the number of parameters is already not four, but bigger, as usual it is divisible by two (4, 6, 8, ) The order law would be hard to distinguish, it will be "interlaced" into compositional relation and so on; in general, much here depends on the concrete analysis and the investigator's grounding, to give a proper, detailed description.

So, the stated is summarized. In the proposed algorithm there can be traced a notorious conflict of opposites "unity", but had we not made a principle step towards a dual description with an accompanying exposure of dual branches, we wouldn't have moved that far on the way of understanding the working mechanism of mutual transitions and dual parts interaction, especially its " invisible" part. This is the heart and fascination of the trend, we develop, which along with theoretical findings (as we could see, rather elegant and complicated ones) allows to receive non-trivial results in many simple cases, by considering simple relations of duality. In this case the principle of duality fully deserve to pretend being a universal organizational principle, consonant with A.Bogdanov's Tectology.

13. Duality. Instead of the conclusion.

 
"Fallacious dualism is always stipulated by an abstraction being taken as a final concrete fact. The Universe is dual, for it is, at the same time, both elapsing and eternal; each complete element of reality possesses both physical and spiritual sides; each actuality requires some abstract characteristics; each event combines a formal spontaneity and objective planning"
(Whitehead A. "Process and Reality")

It is said, that XXI century - is the age of Aquarius, which is characterized by spreading of intellectual energy, strengthening of en exploratory essence of a Man, overcoming of arisen prejudices, breaking of usual ideas and generating of new ones. From this point of view, the theory of duality, whose general features are outlined in the introduced work, fully correspond with the spirit of the new age. Let's mention at once, that this trend of seeking special remaining relations, having universal character, applied for analyzing the mass of phenomena, which seem absolutely heterogeneous, appeared long time ago, in the end of XIX century, at the same time with discoveries in the field of thermodynamics and electromagnetism. The XX century has intensified this orientation, which was favoured by new discoveries of universal character: creation of relativity theory and quantum mechanics, developing of genetics and cybernetics. All these helped to overcome a hypnotic influence of mechanicism, to find possibilities for developing new ideas in many fields of scientific thought and practical activity.

This trend is sure to be still intensified, moreover, in two directions. First, the very character of scientific research will change. The modern science is known to be divided into local, disciplinary "lands" - each, though studying one and the same - the Universe, has, nevertheless, absolutely different, special languages. Overcoming of such "caste nature", seeking common, universal principles, where compositions of special sciences could be threaded - this is a gratifying task, worthy any investigator. Secondly, even more important is the forming of a new style of thinking for new generations and not only investigators. Even in a high school students are overworked, many times studying various interpretations of facts, whose number continuously grows, exceeding now all reasonable limits. The same is with higher and special educational institutions. Instead of slightly associated facts, the attention should be concentrated on general principles of substance organization, - then facts of any science would be easily placed into the cells of the universal network of principles. The most significant breakthrough in this direction was made by A.Bogdanov in his Tectology, which also needs to be brought up to the recipes of practical use, suitable for any mind, trained in the slightest degree.

We hope, that further development of dual approach would help to solve this two-uniform problem. The algorithm of seeking new, based on the general principle of duality, just represents a small, but, we believe, very timely step in the required direction.

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