whence 
Philosophical Justification Of
The Golden Proportion Concept
Boris KosenokKuibyshev, 1990
CONTENTS:
1. History of beginnings of the golden section concept.
2. Natural sciences about the golden proportion.
3. The golden proportion concept at philosophical point of view.
4. Application of a philosophical interpretation towards economical and public processes.
“It’s quite possible that some mistakes have escaped my attention and still expect a penetrating view of a critically turned reader. I hope joy of discovery of the mistakes and feeling of intellectual superiority at the author at any rate will reward a lucky person for lost time and trouble which could be given to him by attentive reading of this book”
Lewis Carroll
The present work was written by me in 1990 as an essay on philosophy to pass candidate minimum, and certainly the elapsed time has changed my views at some truth that seemed stable then. But the conclusion I made then has stunned me by its obvious accuracy in today’s world and absolutely absurd then view that socialism in the USSR largely resembled feudalism of Ivan IV The Terrible instead of real communism (or its first stage, if you want – socialism). I think if you read this essay you will understand what I am trying to tell you. I want to let you know beforehand that first two chapters are brief expositions of the 9-12th sources of the used literature listed at end of the essay.
Besides that an article about the golden section has appeared on the Internet, for example at: http://www.n-t.org/tp/iz/zs.htm, http://www.n-t.org/tp/mr/szs.htm, http://www.list.ru/catalog/24889.htm, and that the ideas expressed in the 3rd and 4th chapters are only the author’s thoughts. I have exposed a text of the essay of 1990 to some amendments to correct some disharmonious expressions, in my opinion, and to design the editing at which I, no doubt, as the author have rights. The other things, also the references on the classics of Marxism-Leninism – unpopular here, but nevertheless studied in the West, are are left unchanged.
Philosophy today describes the real world very accurately but unfortunately, can not answer the practical questions which modern life puts before human society, its science, culture, and economics. How to make the development of society stable without revolutions and cataclysms? What is the perspective of society’s progress, political and economical structure? How to use the philosophical laws towards specific human activities? What is the quantitative correlation between the philosophical categories and laws? Where is, for example, a border between the transferring of quantity into quality?
How to pass from abstract reasoning and the description of surroundings to not only a notion of tendencies, but also to forecasting on numerals and the language of formulas – the most abstract language on which humanity describes and predicts all processes of its surroundings for a long time. Even the founders of dialectical materialism learnt in their deductions not only on abstract reasonings of Hegel and Feuerbach, but also on works in economics and natural sciences, were engaged in the “ignoble” analysis of capitalist economics based on statistical datas and economical works of other scientists.
The aim of this essay is to try to use philosophical categories and laws towards the mathematical expression of the golden section, which has a very strong influence on many sides of surroundings.
1. History of beginnings of the golden section concept.
The Golden proportion or golden section is the harmonic dividing of a segment “a” long into two parts in such way that its biggest part “x” is the mean proportional between the whole segment and its smaller part:
whence
The first references to the golden proportion were found in ancient literature in the 2nd book of Euclid’s “The Elements”, but the purpose of the golden section most probably was solved already by the Pythagoreans which was just a pearl of a Pythagorean doctrine the numeral harmony of the world. The whole ancient Greek science developed under the sign of the golden proportion. The Greeks were the first to determine that the proportions of a well-formed human body submitted to its laws and this is obvious from the example of antique statues (Apollo Belvedere, Venus de Milo). The Phrygian tombs and antique Parthenon, Dionysus theatre in Athens - – all of them are full of the golden proportion harmony. Interest to the golden proportion has increased with new force our days. The availability of the golden section is accentuated in a number works of music such as the compositions of Bach, Chopin, and Beethoven. Sergey Eisenstein used the golden section in the adjustment of episodes of his pictures. Academician G.V. Tzeretelli discovered that poetry harmony in the poem “The Knight In The Panther’s Skin” by Shota Rustavelli submitted to the golden section.
The golden proportion was ranked as the major aesthetic principle in the age of the Renaissance. Leonardo Da Vinci, Raffaello Sanzio, Mickelangelo, Titian and the other great painters of the Renaissance arranged their canvases deliberately by using the golden proportion. The Netherlands composer of the XV century, Jacob Obrecht, widely used the golden section in his music compositions that are still likened to a cathedral created by a genius architect.
2. Natural sciences about the golden proportion
In the XIX century not painters, but scientific experimenters who studied the regularity of phylotaxis (the position of leaves on a plant) turned to the golden proportion. It was found that flowers and seeds of a sun-flower, a camomile, the scales of fruits of a pineapple, coniferous cones, etc. are “packed” according to logarithmic spirals waving towards one another. Thus, the quantity of «right» spirals is always to “left” spirals as the adjacent Fibonacci numbers (13:8, 21:13, 34:21, 55:34) and limit of succession of which is the golden proportion.
Side by side with applied investigations, scientists are actively continuing to develop the number theory of Fibonacci and the golden section. Soviet mathematician U.Matiyasevich using Fibonacci numbers solved the 10th problem of Hilbert. Some refined methods in solving of a number of cybernetic problems (the theories of search, games, programming) using the Fibonacci numbers and the golden section have appeared. In the USA the mathematical Fibonacci association has issued a special magazine since 1963.
The discovery of generalized Fibonacci numbers and generalized golden sections by A.P. Stakhov and I.V. Vitenko became an outstanding event in 1964. And what is more, this discovery made on paper was confirmed by research. So, for example, invariants of known waves of electric activity in a man’s brain are equal to generalized golden sections. Well double alloys possess brightly expressed special characteristics (are thermal steady, are solid, are abrasive resistant, are steady against oxidation, etc.). If only specific weights of initial components are connected with each other by one of the generalized golden proportions. It has allowed the Byelorussian philosopher E.M. Soroko to suggest a hypothesis in his work “Structural harmony of systems” that generalized golden sections are numeral invariant of not only a human brain, but also of any self-organizing systems. This hypothesis (a law of harmony of systems) can have fundamental importance for new science studying processes in self-organized systems – a synergetics.
One of the ways of solving a problem of the safety of modern computers is the introduction of redundance. A scale of notation with irrational base and founded on Fibonacci numbers and the golden proportion has redundance that allows the creation of reliable computers with interference-free characteristics (by the way, the classical binary scale of notation is a particular case of this calculation).
3. The golden proportion concept at philosophic point of view.
The history of the golden section is the history of cognition of the world. Analyzing the foregoing one can wonder once again at the grandiosity of the world cognition process and at the discovery of its other new regularities. The golden section concept has covered all stages of cognition.
The first stage of cognition is the discovery of the golden section by ancient Pythagoreans. They transferred from the basic contemplation of reality to express it in the world of numbers, but concepts of causal and effect of the world were confused and a conjecture about the global significance of the golden section has remained just a guess for centuries. Nevertheless, people started to use the golden section in their works of art.
The practical needs of trade lead Fibonacci to discover his rows which were not connected to the golden section by anyone. Scientists opened the golden proportions in living and non-living substances, and already under this experience our contemporaries A.P. Stakhov and I.V. Vitenko made amazing discoveries of the generalized golden proportion and the generalized rows of Fibonacci. Their analysis leads investigators to results that stun by its simplicity and because of this are more considerable, “the golden section has redundance and stability which allow self-organized systems to be organized”
And it would be the right time to inscribe these magic numbers in temples to make bows and eulogize the Lord for the paradise, because even the most abstract science – mathematics can not begin to explain the essence of the golden section It is not able to explain to us why exactly these numbers, but not others, were put into a human brain, chemical compounds and works of great painters.In the fields where “narrow” sciences are weak and where it is impossible to explain various phenomena (chemical, biological, physical and social) with the help of any of sciences the philosophy – a science about the world – comes into its own. Though it is quite possible that the above-stated facts are not enough to raise a question about the belonging of the golden section characteristics to all reality of the present world. Nevertheless, let us try to do it and to explain the golden section concept from a philosophical point of view.What is the golden section in the eyes of philosophical science?
It is the ratio between some opposite properties of an object. Or in other words, it is the quantitative ratio between two opposites. Therefore, for better understanding of the golden proportion, we will look in detail at opposites as a philosophical concept and look at their ratios and properties.
The opposites are two sides of the same object or phenomenon that always contradict one another because of their absolute polarity. This contradiction means that none of the sides can be in harmony with one another, though they both belong to the same object or phenomenon. Thus, there is always a fight between two sides of an object or a phenomenon, but neither of the opposites can exist without one another, and each opposite is a reflection of the other.So for example, a kind man can not be kind if there is no malicious person or how will we know what a kind man must be? And a nice person can not be kind towards evil; in this case all his kindness is only a complicity to harm. So, evil is also in kindness.
All this directly proves that the unity of opposites is both reality of the opposites of existence and their fight.V.I. Lenin noticed that a condition of temporary balance is possible between opposites; it means that neither of the opposite sides has marked superiority at some stage of the development of a process. [1] As an example it may be reduced by any social and economic structure where there is a social agreement between opposite classes which takes into account the real the correlation of forces.Although these phenomena are different by nature, both of them are the achievement of certain compromise – relation between opposite sides. As the existence of opposites means fight seemingly temporary balance is the grouping of opposite forces constant in its progress. The changing of quantitative balance of opposite sides brings at a certain stage the impossibility of preserving previous relations by opposite sides (temporary balance). As a result, there is a leap – a qualitative change of relations, corresponding to the given quantitative correlation of forces of opposing sides of a phenomenon or a subject.Any subject or phenomenon defines quantity and quality. Quantity is inseparably linked with quality. Quality defines what the differences will be between a given phenomenon or subject and others. Quantity defines a subject or phenomenon in the aspect of the progressing stage. Unity of quantity and quality is a measure (limit) where a phenomenon or a subject stays itself.
But the setting of a new balance of forces brings the appearance of a new quality of phenomenon or a subject that denies present quality and does not change until quantitative changes cross again the limits where a subject stops being itself and gains new quality denying the previous one. Thus, the circle will close – it will be denying of denial. Nevertheless, what will happen to a subject if one of the opposites wins this never-ending fight? In nature, victory of one of the elements leads to a catastrophe and the disappearance of enemies of some animal species leads to the dying out of the species. A man is one of the few animal species that have enemies similar to themselves and this is a motive power of mankind’s progress on the present stage of human history. But what will happen to society when its internal enemies (as an opposite class, competitors or “enemies of a nation”) disappear?
Firstly, it will be the degradation of a conquered class, because there is nobody to fight with, so bygone organization and solidarity are not required any more.
Secondly, monopoly will progress and economics will fall into decay.
Thirdly, fear of the masses for freedom will disappear and “troubled times” will come. This is full of stagnation and crisis.
There are only two ways out of this: either the total disappearance or reappearance of new opposites in the classes, competitors, or “enemies of a nation”.
Therefore, for progress to be only positive, it is necessary to establish compromises between opposites, i.e. the establishing of correlations instead of the destroying of contradictions.
As Engels F. wrote, “Mutual penetration of opposites and transformation one to another, when they are reduced to extreme – is development in a contradictory way or the denying of denial.” [2]“… Naked denying”, Lenin V.I. wrote,” is not denying for nothing, not skeptic denial (hesitation, doubt are typical and essential for dialectics which undoubtedly contains an element of denial and besides, it is the most important element), but denying as connection point, development point with holding positive…”[3]
Thus, philosophy clearly and evidently has shown that a source, the working reasons of progress and the moving of the present world in total – are unity and the fight of opposites. The process of world progress happens by the transferring of quantitative changes into qualitative ones; the total trend or tendency of world progress is denying of denial. All happened in the world processes, all subjects or phenomena answer these philosophic laws. But at the same time, knowledge accumulated by mankind says that both in nature and human society there are stable formations that do not change both at the spatial and temporal point of view, also in these formations there are quantitative changes. All social and economic formations were there in a steady position for a long time.Geological epochs have lasted for millions and even billions of years on the Earth. Stability of surroundings had so strong an influence on humanity, that only persistent work by the best world scientists have dispersed the myth about the invariability and constancy of the world. Stability is a philosophical category opposite to variability and is as characteristic of the present world as variability.But stability is not peace. This is such a condition of a subject or phenomenon where quantitative changes do not lead to qualitative changes, i.e. opposite properties of a subject or a phenomenon are in dynamic balance. Dynamic balance is possible only when quantitative changes correspond to the quality of a subject or phenomenon, i.e. when they more fully reflect real arrangement or quantitative correlation of forces of opposite sides. But on the other hand internal quantitative changes bringing stable formation characterized by concrete quality loses its stability and transfers into another stable condition corresponding to given quantitative changes. In nature similar leaps happen rather slowly, because nature immediately reacts upon any changes, and the changing of epochs lasts for ages (sure, if there is no rapid increase of quantitative changes).In human society quantitative changes are accumulated to excess because of the lagging of cognition, and an explosive leap happens. From this point of view progress is a continuous chain of more or less stable formations with their internal quantitative changes which do not bring qualitative changes but forestalls them, and to leaps – changing of quality. The golden proportion has amazing properties such as redundancy and stability which allow it to conduct conformity between the golden proportions and steady formations. Then mathematical expression will gain philosophical meaning:So for example, the classic golden section :
section is the ratio of quantitative characteristic "x" of an opposite expressing nature of a subject or phenomenon to quantitative characteristic "(a-x)" of another opposite. This ratio is the ratio of the quantitative characteristic "a" of the whole subject or phenomenon to quantitative characteristic "x" of the opposite expressing nature of a subject.Otherwise, optimal “oppression” for stability of a subject is such “oppression” of an opposite by another one that it is equal to “oppression” of the whole subject or phenomenon by this “oppressing” opposite.
Otherwise, optimal “oppression” for stability of a subject is such “oppression” of an opposite by another one that it is equal to “oppression” of the whole subject or phenomenon by this “oppressing” opposite.
If to bring generalized golden sections to the same philosophic definition, then their mathematical expression:
(where S=1, 2, 3, … - is an order of the golden section)
will express the same as the classic one, but "s" in this case means the power to influence the whole subject or phenomenon onto an opposite defining nature of a subject or a phenomenon. This power defines the amount of inverse causations by which quantitative characteristics of a whole subject or phenomenon are connected with the opposite.Solving of the generalized golden sections equations:
|
S |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
… |
¥ |
|
Õ |
0.5 |
0.618 |
0.683 |
0.725 |
0.755 |
0.778 |
0.797 |
0.812 |
… |
1 |
Allows the drawing of a conclusion that increasing the quantity of inverse causations “a subject – an opposite” brings to influence increasing of the given opposite to the rest of the subject or a phenomenon. Paradoxically the given conclusion will seem not so defiant if one remembers that both classic and generalized golden sections belong to the stable condition of a subject or a phenomenon, then this conclusion is the indisputable rule of balance between the prevalent position of one of the opposites and the control of the whole subject, as the totality of properties, over this opposite.
4. Applying of the philosophic interpretation towards economical and public processes.
Application of the generalized golden sections in economics will deliberately promote the stable development of public production. It is possible to prove without the help of figures and statistics that the generalized golden sections underlie economic regularities in this case. Indirectly it is possible to determine this considering economics and finding connections between economics and the rest of public life.For example, the economic sphere of capitalist society is a contradiction not only between labour and capital, but also between market and state regulation of economics. At the primary stage of formation of bourgeois society a state did not interfere in the economic sphere being engaged only in internal politic and criminal cases, foreign relations, conquest of foreign markets and defence. Eventually when market economics malfunctioned, the ruling class had to give to a state functions in regulation of holy of holies at capitalist economics – a market. The practice of the development of capitalist economics made complex measures from the prohibition of trusts and the giving of state order to the creation of state enterprises.Considering contradictions between a market and a state, one can see when progressing a market power of state regulation of economics increased and became stronger.
Originally undivided domination of market relations in economics, spontaneous development of economics brought to unstable structures. Market in economics is similar to entropy in nature – it creates destruction and it was necessary to ease the influence of market and free competition. It seems paradoxical but the first direct step to the suppression of market was making it more lively when the natural course of events brought market to monopolizing, creation of trusts. The first experiment was partly successful – it showed the state can control market, but at the same time enlivening of the market brought capitalist world to the verge of disaster in 1929-1932.
Then the ruling class had to allow a state to control economics again – creating state enterprises, controlling of finance, subsidizing unprofitable branches and social programs etc. In a state society instituted control over its predominant part – economics for creating a more stable structure. The economics is more developed, its influence is more on other spheres of human society’s activities, the more control and amount of reverse relations to restrain its mighty forces are necessary but the control must be not so general to not cramp space for productive forces. The generalized golden sections can help to establish this proportion.For example, for human society it will mean a balance between the influence of the ruling class over the oppressed class and influence of society over the ruling class. Or else, management of any class or strata to the whole of society must have reverse relations on which the whole of society will influence the ruling class or strata. But reliable reverse relation is necessary not only for the stable existence of society but also for passing by society cross leaps – transfers of quantitative changes into qualitative ones.
Let us consider the generalized golden section applied to well-known social and economic formation.
The social and economic formation of primitive society is notable for its social uniformity. There were no material contradictions and social differences because of rudimentary economics (if hunting and fishing can be called so) not allowing the creation of surplus products of work and also the object of appropriation. The absence of any contradictions allow such a society to manage without internal reverse relation and to exist steadily during tens of thousands of years.But absence of reverse relation brought primitive society to fall after agriculture and surplus products of labour had appeared.
It was changed by slave-owning systems with the frank exploitation of a slave by a master. In this social and economic formation two opposite classes appeared – masters and slaves, two opposites which lacerates social contradiction between the hard back-breaking work of a slave and the total appropriation of products of labour by his master. Reverse relation in such a society is out of the question. The idea of asking a slave if he likes orders from a master did not come into anybody’s head.If reverse relation is zero then mathematically the condition of this society can be expressed in such a way:
society slaveholders
------------------ = -------------------
slaveholders slave
Such expression means that the ruling class controls masses of slaves without correcting the real correlation between the forces of slaves and slaveholders. So ignoring the reaction arising in the slaves environment, of course, brings new conditions, new arrangements of forces in society, new quantitative changes need qualitative changes in society. And the ruling class does not want and also can not by nature put them into practice.As a result, huge slave-holding formations were destroyed under pressure of barbarians and slaves revolts.
Feudal formation appeared on the ruins of slave-holding states allowing a society to get reverse relation in the personal interest of a peasant in cultivating his own piece of land after serving feudal conscription. But at the same time it extended the influence of the ruling class of feudal lords to oppress the peasant class. It seems paradoxical, but this conclusion is confirmed not only by the mathematical solving of the golden proportion of the 1st order, but also by increasing the labour productivity of peasants in comparison with the labour of slaves. However, eventually the further increase of the labour productivity of peasants began to be held by feudal conscription and the increased power of feudal lords. Reverse relation of feudal society was, as it were, indirect. A feudalist didn’t take a direct part in the enterprise process, and further development brought to the ruling class had either to reorganize its relation – quality of its society, or perish. And happened it so, one part of the feudalists has managed to enter on a way of new economic relations in time, while another part has perished in the fire of revolution I.e. availability of single reverse relation between the ruling class and the whole of society makes the probability of reorganizational achievement of new quality possible, but at the same time the probability of the breaking of progress for holding power above society is possible too.The bourgeois society appeared as a result of peace transitions and revolutions acquired a mighty master in bourgeoisie, for henceforth the possibility to work for the rest of society began to depend only it on . At the same time bourgeois society set an example for double reverse relation:Firstly – is the personal interest of wageworkers in the rising of labour productivity.
Secondly – the relative political and personal freedom in democratic institutions.Certainly, the interest of workers in increasing labour productivity was under feudalism too, but not at work for a feudal lord because peasants had their own land. Under capitalism a worker works only for a master and receives wages and daily bread only from him. But the democratic institutions existed before but were intended only for the ruling class.That which should limit the powers that be, only makes it stronger. And it is excellently confirmed by the solving of the golden proportion of the 2nd order. In practice it is also possible to see how successfully the bourgeoisie use these two back relations in spite of contradictions between the social character of labour and the private way of appropriation of its products should blow bourgeois society apart.What is going to happen next?
In Russia it has happened as Marx foresaw, ”Sure, first it may happen only with the help of the despotic intervention into property rights and the bourgeous relations of production, i.e. with the help of measures which economically seem insufficient and insolvent but which outgrow themselves and are inevitable as a means of revolution in the whole method of production” [4]
Though sometime it would be possible for more peace to transfer to a new social and economic formation, it even is more probable that a military coup d`etat would occur. As one can see in the analysis of previous social and economic formation the further development must lead to the appearance of new reverse relations between society and a ruling clique and at the same time the reinforcement of its power.
If one remembers the tendency of progress of the present world i.e. about the denying of denial, then the first reverse relations existed during capitalism must become the same as the reverse relations during feudalism but in a higher stage of progress, i.e. a hired worker must be interested in the increasing of labour productivity because he works for himself. The second reverse relation must be changed in the same way. If during a capitalist democracy there was a parliament, then in the new society it must become public. And the third reverse relation, which was not during capitalism but rudiments of which existed at that time in a state, it is a public property managed by a group of highly qualified specialists It is very difficult to suppose on the present piece of historic development, what will be the next, the fourth reverse relation between a ruling clique and the whole of human society. But at the same time proceeding from the general tendency of the progress of the present world, i.e. from the law “denying of denial”, it is possible to suppose what will happen to the other three relations.Firstly – a worker will be interested in increasing labour productivity because he will be working for a society.Secondly – it will be a limited democracy, i.e. a person may not submit to decisions of the society if he does not agree with them.
Thirdly– everyone will be able to control the public processes.
Thus, consideration of social and economic formations with the help of the generalized golden proportions allows us to conclude that increasing the control on the part of society increases the power of the ruling class (the ruling group or administrative machinery) and the steadiness of society increases because of this, that raises the probability of change of leap stages smoother and without cataclysms
1. Engels F. “Feuerbach and the end of classical German philosophy”. Moscow, Political literature publishing house, 1989.
2. Engels F. “The dialectics of nature” – Marx K., Engels F. “Compositions”, volume 20, p.343.
3. Lenin V.I. “Balance of forces”. – Lenin V.I. “The Complete works”, volume 12, p.5
4. Lenin V.I. “Materialism and empiriocriticism”. – Lenin V.I. “The Complete works”, volume 18, chap. 1-2.
5. Engels F. “Anti-Duhring”. – Marx K., Engels F. “Compositions”, volume 20, part 1, chap. XII-XIII.
6. Marx K., Engels F. “The Communist Manifesto”. Moscow, Political literature publishing house, 1973.
7. Lenin V.I. “A synopsis of Hegel’s book “Science of logic”.- Lenin V.I. “The Complete works”, volume 29, p.207.
8. Engels F. “The origin of the family, private property and the state”. – Marx K., Engels F. “Compositions”, volume 21, chap. IX.
9. Stakhov A.P. “Introducing into the algorithmic theory of measuring”. – Soviet radio, 1977.
10. Stakhov A.P. “The algorithmic theory of measuring”. Moscow, Knowledge, 1979. Serial “Mathematics, cybernetics”.
11. Stakhov A.P. "Codes of the Gold Proportion". Moscow, Publishing house " Radio and communication". 1984.
12. Sockolov A. “Mysteries of the golden section”. – Engineering for youth, ¹5, 1978.
13. Soroko E.M. “Structural harmony of systems”. Minsk, Science and engineering, 1984.
[1] Lenin V.I. “The Complete works”, volume 12, p. 5
[2] Marx K., Engels F. , Ñî÷.,v.20, p.343.
[3] Lenin V.I. “The Complete works”, volume 29, p.207.
[4] Marx K., Engels F. “The Communist Manifesto”, p. 44
Ïåðåâîä âûïîëíåí ×èðêîâîé Í.Ô.
(Translation executed by Natalia Chirkova)