Structural Harmony of Systems

What is a philosophy? The word "philosophia" is Greek's word and means a "live to wisdom". In general sense it is determined as the doctrine of general principles of being and cognition, of the person and world relation; as the science of the general laws of nature, society and thinking development. The philosophy takes his origin in the earlier centers of human civilization so as India, China, Egypt but it reaches its classical form in the ancient Creek. Mainly the former philosophers of the ancient world strive to find a single source of diverse natural phenomena. According to their opinion, a Harmony was by this source. Heraklitus expressed this idea the most clear. He assumed that "harmonization law of opposite sides, which differ between one and another but belong to the same range and are submitted to the same logos".

It is clear that increasing the interest in problem of harmony and golden section in modern science has found its reflection in modern philosophy in form of new original philosophical concepts. The Byelorussian philosopher Eduard Soroko who advanced in the 80th the highly interesting concept of "structural harmony of systems" developed one of the similar concepts. This concept and the "Law of Structural Harmony of Systems" following from it rightfully can be considered as one of the greatest philosophical achievements of the 20th century.

The Byelorussian philosopher Eduard Soroko

Soroko's main idea is to consider real systems since "dialectical point of view". As is well known any natural object can be presented as the dialectical unity of the two opposite sides A and B. This dialectical connection may be expressed in the following form:

A + B = U (universum).(1)

The equality of (1) is the most general expression of the so-called conservation law.

Here A and B are distinctions inside of the unity, logically non-crossing classes or substratum states of any whole. There exists the only condition that A and B should be measured with the same measure and be by members of the ratio underlying inside the unity.

The examples of (1) may be probability and improbability of events, mass and energy, nucleus of atom and its envelope, substance and field, anode and cathode, animals and plants, spirit and material origin in the value system, profit and cost price, etc.

The expression of (1) may be reduced to the following normalized form:

`A +`B = 1,(2)

where `A and `B are the relative "weights" of the parts A and B forming some unity.

The partial case of (1) is the "law of information conservation":

I + H = log N,(3)

where I is the information quantity and H is the entropy of the system having N states.

The normalized form of law (3) is the following:

R +`H = 1,(4)

where is the relative redundancy, is the relative entropy.

Let's consider the process of system self-organization. This one is reduced to the passage of the system into some "harmonious" state called the state of the thermodynamic equilibrium. There exists some correlation or proportion between the sides A and B of the dialectical contradiction of (1) in the state of the thermodynamic equilibrium. This correlation has a strictly regular character and is a cause of the system stability. Soroko turns to the principle of multiple relations to find a character of connection between A and B in the state of the thermodynamic equilibrium. This principle is well known in chemistry as "Dalton's law" and in crystallography as the "law of rational parameters".

Soroko advances the hypothesis that the principle of multiple relations is the general principle of the Universe. That is why there exists in accordance with this principle the following correlation between the components R and R `H in the equality of (4):

log R = (s + 1) log`H(5)

or

log`H = (s + 1) log R.(6)

The expressions of (5), (6) may be represented in the exponential form:

R = (`H )s+1;(7)
`H = Rs+1,(8)

where the number s is called the range of multiplicity and takes the following values: 0, 1, 2, 3, ... .

Inserting the expressions of (7), (8) into the equality of (4) we get the following algebraic equations respectively:

(`H )s+1 +`H - 1 = 0;(9)
Rs+1 - R - 1 = 0.(10)

Marking in y the variables `H and R in the equations of (9), (10) we get the following algebraic equation:

ys+1 + y - 1 = 0.(11)

Let's introduce the new variable for the equation of (11). Inserting the expression of into (11) we get the following algebraic equation:

xs+1 - xs - 1 = 0.(12)

We can see that the latter equation coincides with the algebraic equation of the golden p-proprtion. The real root of the equation of (11) is inverse value to the golden p-proportion, i.e.

(13)

where ts is the root of the equation of (12).

In accordance with Soroko's concept, the roots of the equation of (11), which is equivalent to the equation of (13), expresses the law of the structural harmony of systems.

Summing up Soroko had formulated the following "Law of Structural Harmony of Systems":

"Generalized golden sections are invariants, which allow natural systems in process of their self-organization to find harmonious structure, stationary regime of their existence, structural and functional stability".

What peculiarity has "Soroko's Law"? Starting since Phyphagor the scientists were connected the concept of a Harmony with the only golden proportion "Soroko's Law" claimed that the harmonies state corresponding to the classical golden proportion is no only for the same system. "Soroko's Law" allows an infinite number of the "harmonies" states corresponding to the numbers ts or the inverse numbers (s = 1, 2, 3, ...), which are the real roots of the general algebraic equations of (11), (12).

The values of the "structural invariants" bs for the initial numbers s are given in the following table.

s1234567
bs0.61800.68230.72450.75490.77810.79650.8117

Let's consider now the application of Soroko's law for the thermodynamic and information systems. The state of the thermodynamic and information system is expressed with the help of entropy, which is a principal concept of thermodynamics and information theory. The expression of the entropy of the information source with the alphabet

A = {a1, a2, ..., aN}

has the following form:

(14)

where p1, p2, ... , pN are the probabilities of the letters a1, a2, ..., aN; N is the number of the letters.

As is well known the entropy of (14) reaches its maximum value

Hmax = log N(15)

for the case when the probabilities of letters are equal among themselves, i.e.

Using the concept of the relative entropy:

(16)

we can write the following evident equality:

(17)

In accordance with the "law of the structural harmony of systems" any system reaches its "harmonious" state in the case when its relative entropy of (16) satisfies to the equation of (9). It follows from this consideration the next expression for the entropy of the "harmonious" system:

(18)

It is clear that for the given parameter s the problem of obtaining the optimal set of the values pi (i = 1, 2, ... , n) giving the optimal value of the entropy has many solutions. However, the correlation of (18) plays a role of the "aim" function for the solution of different scientific and technical problems because it points to the "optimal" variant of the solution.

Soroko gives in his book "Structural harmony of systems" (1984) a number of interesting examples from different fields of science to support the law of the structural harmony of systems. For example let's consider such natural object as a "dry air", which is a basis of life at the Earth. There arises a question: is the structure of the "dry air" optimal? Soroko's theory gives a positive answer to the question. In fact, the chemical compound of the "dry air" is the following: nitrogen - 78.084%; oxygen - 20.948%; argon - 0.934%; carbon dioxide - 0.031%; neon - 0.002%; helium - 0.001%. If we calculate the entropy of the "dry air" using the formula of (14) and then its relative entropy taking into consideration that log N = log 6, then the value of the relative entropy of the "dry air" will be equal to 0.683. With high exactness this value corresponds to the invariant b2 = 0.0682. This means that in process of self-organization the "dry air" reached its optimal, harmonies structure. This example is highly typical and shows that Soroko's Law can be used at the present time for checking the biosphere state, in particular, the states of air and water basin.

It is clear that practical usage of the "Law of Structural Harmony of Systems" can give essential advantage for solution of many technological , economical, ecological and other problems, in particular, can promote to improvement of technology of structural and complicated products, to monitoring of biosphere etc.