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Mathematical LAB of Peace from Spherons’ Structural Harmony

Global Peace Mathematics

instead of military mathematics or societal mathematics of the spherons’ genetic structural harmony is a key scientific issue of the 21st century posed in the GHA for 13 years.

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Математика глобального мира

вместо военной математики или социетальная математика генетической структурной гармонии сферонов – это ключевой научный вопрос 21 века, поставленный в ГСГ за 13 лет.

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Mathematical LAB of Peace from Spherons’ Structural Harmony:

Thermodynamic / Stochastic Population Model
Based o­n the GHA International Statistical Research





Dr. Leo Semashko



In English: http://peacefromharmony.org/?cat=en_c&key=812

In Russian:


The internal harmony of the world is the o­nly true objective reality.

Henri Poincare

The significant problems we have cannot be solved at the same level of thinking

with which we created them….

We shall require a substantially new manner of thinking if humankind is to survive….

Peace cannot be achieved through violence, it can o­nly be attained

through understanding [science].

Albert Einstein


Mathematicians with physicists created in the 20th century "Absolute Evil - Nuclear Weapons"
(John Avery, 2017). Will mathematics be able together with sociologists and sociocybernetists

of a new, spheral, holistic manner of thinking to create in the 21st century

"Absolute Good of Global Peace Science" from structural harmony of spherons,

the task of which was set during 13 years in the GHA?



1. The Sociocybernetic ABC of Genetic Structural Harmony of Society


The life of any society in history requires four necessary and together sufficient static spheral/societal clusters/elements of conditions (s/clusters or s/elements), determining its constant integrity / holism:

1. Four resources/capitals: People, Information, Organization, Things (m/benefits and services) - PIOT resources (Comte, Park, Toffler, Parsons, Bourdieu, Braudel, Castells, Semashko and etc.)

2. Four processes of PIOT creating: Production, Distribution, Exchange, Consumption - PDEC processes (Smith, Ricardo, Marx, Keynes, Maturana, etc.)

3. Four spheres of PIOT production: Social, Informational, Organizational, Technical - SIOT spheres (sameauthors as in 1, 2 and many others).

4. Four classes/Spherons employed in the SIOT spheres by the PIOT production: Sociospheron, Infospheron, Orgspheron, Technospheron - SIOT classes/spherons (Parsons, Maturana, Giddens, Semashko, etc.).

The statics of 16 s/elements constitutes the basis of society in all its units and determines the dynamics of social and individual life throughout their history. 16 s/elements make up a deep, genetic structure or SOCIOCYBERNETIC GENOME (SOCIONOME) of humanity and any society (Semashko), which is reproduced in the whole infinite set of historical diversity of EVERY of these elements as well as in the national and civilizational diversity of their systemic cultural integrity.


Scheme. Sociocybernetic GENOME of Spheres and Spherons’ Structural Harmony

See the scheme in detail here: http://peacefromharmony.org/?cat=en_c&key=423


        Each of s/elements is EQUALLY necessary as a productive element, because the absence of ANY of them makes the life of society/human impossible. But o­nly o­ne of them - SPHERONS/PEOPLE is a living, defining and priority productive force, universal actor creating ALL elements, including o­neself (Maturana’s autopoiesis). However the Spherons productive power depends and is impossible without each of the other s/elements. EACH Spheron is the productive force, necessary for the life of every society actor, resource, capital, heart, energy, motor, etc. The Spherons discovery is confirmed by several examples of statistical researches in different countries, which can be reproduced in any country, in any of its institutions and at the level of humanity as a whole in any year of its historical time based o­n the population census (ibid.).

All s/elements can exist only in harmony (proportion, measure, consistency, coherence, symmetry, balance, etc.) among themselves. It is expressed by the fundamental Sociocybernetic / Tetrasociological Axiom of Structural Harmony (SASH): Harmonious (sustainable) existence of society in any place and time is determined by a measure in the proportion of four equally necessary and sufficient societal resources: People (P), Information (I), Organizations (O) and Things (T). The structural harmony of SOCIONOME constitutes a harmonious human/social nature (Kant, Poincare, Luhmann, Semashko, Soroko, Siebert, Baymuratov etc.).

The natural state of social structural harmony is spontaneous, nonlinear, stochastic and thermodynamic. It is accompanied in history by the sporadic disharmonies: violence, wars, other armed clashes, crises, increasing risks and challenges of various levels from global to local including the family and the individual.

The dynamics of the Spherons structural harmony, their direct and feedback connections constitute the subject of Tektology, Sociocybernetics, Tetrasociology, Organizational Governance science, System theory and Management (Bogdanov, Wiener, Bertalanffy, Ashby, Foerster, Beer, Pask, Geyer, Hornung, Scott, Semashko, etc.). Sociocybernetics of spheral elements, first of all Spherons of social production, determines the third-order Sociocybernetics, which organically includes the two preceding orders. The statics and dynamics of the spheral elements, as the alphabet of structural harmony, constitute the ABC of the third-order Sociocybernetics,which was born as a result of the synthesis of Tetrasociology and Sociocybernetics.


II. A Brief History of Theoretical Integration of Tetrasociology and Sociocybernetics


The history of this constructive international scientific process began in 2002 with the publication of Leo Semashko's book "Tetrasociology: Responses to Challenges" (http://peacefromharmony.org/docs/2-1_eng.pdf) along with the forewords of the RC51 (Sociocybernetics) leaders Bernd Hornung (Germany) and Bernard Scott (England), dedicated to the XV World Congress of Sociology in Brisbane, Australia, July 7-13, 2002. This book was presented at 32 sessions of the Congress.

In 2016, this process found its key result in the creation of the international "Global Peace Science" (GPS) of 174 GHA coauthors from 34 countries, including the President of India, three Nobel Peace Prize laureates and dozens of outstanding scientists from all over the world


        In 2017, this process focused o­n creating a unique "Sociocybernetic Model of the Spherons’s Global Harmonious Governance" (SMSGHG) for the UN reform (http://www.peacefromharmony.org/?cat=en_c&key=769).

        In 2018, this process finds its logical continuation in two indissoluble directions: theoretical and empirical. For theoretical development, the GHA offers an international fundamental scientific work "Third-Order Sociocybernetics: Megascience for Genetic Structural Harmony of Spheres and Spherons of Social Production and Global Peace." This article of 20-30 pages is expected to be completed and published in October 2018 and submitted to the UN International Peace Conference in November this year, and in 2019 - at the RC51 next International Conference.

        The initial outline of this article "Spherons’ Sociocybernetic Megascience: Global Breakthrough of Politics, Business and Governance to Genetic Structural Harmony of Social Production and Universal Peace" in 7 pp. was published here:
http://peacefromharmony.org/?cat=en_c&key=810 Its center is the genetic structural harmony of spherons.

Classical cybernetics is also involved in harmonious research, as illustrated by the relevant mathematical studies in the Norbert Wiener Center for Harmonic Analysis and Applications: http://www.norbertwiener.umd.edu/. They are taken into account and included in our project. The Sociocybernetics of the structural harmony of spheres and spherons is much pushing the horizon of mathematical engineering in its new spheral technologies, revealing the brilliant philosophical definition of Henri Poincare: "The world inner harmony is the o­nly true objective reality" for all sciences, especially for mathematics.

        In the empirical direction, the GHA organizes "International Statistical Research of Spherons" (ISRS) in several countries (http://peacefromharmony.org/?cat=en_c&key=811). Based o­n their results, a special book is planned to be published: "Spherons: International Statistics" in October with its presentation at the UN Conference in November. This book will constitute the empirical basis of the above theoretical article.

         Within the framework of the integration of Tetrasociology and Sociocybernetics, the idea of ​​a thermodynamic / stochastic model of spheres and spherons of social production was logically born. The first attempt at its interpretation for understanding global peace was presented in the GHA collective article: Vera Afanasyeva, John S. Avery, Timi Ekimovich, Leo Semashko: "Thermodynamics and Negentropy of Spheral Classes Societal Harmony: Thermodynamics of Global Peace" (2017): http://peacefromharmony.org/?cat=en_c&key=756

(John Avery is a physicist, peacemaker, Nobel Peace Laureate: http://peacefromharmony.org/?cat=en_c&key=672).
        It is also formulated in the SMSGHG of the UN (2017. Its excerpts are published below).

        Below is a generalized attempt at a geometric formulation of the problem for the thermodynamic model of spherons.


Stochastic distribution of humankind’s Spherons, any country and region
between 4 spheres of social production

(Theory of Probability and Mathematical Statistics?)

Spherons are extremely large clusters / classes of the population, covering it as a whole, from birth to death,
employed in four spheres of social production.


III. Interpretation of the population distribution by spherons o­n a circular Scheme-1.
Short theses


1. Circle-1. It expresses the entire population from birth to death. Entrance to the circle 1 - birth, birth rate. The way out is death, mortality.

According to the discovery of Umberto Maturana, the whole life of any biological species is "AUTOPOIESIS", self-production. In biological species, it occurs according to the instinctive, innate laws of biological employment/life. At human, as a social being capable of living o­nly in society, it occurs not o­nly according to the laws of biological employment, but also according to the laws of social productive employment in 4 spheres of social production, which constitute "autopoiesis of society" or "social autopoiesis" or "sociopoiesis".

Since society is part of the biosphere, social autopoiesis is also a part of individual (biological) autopoiesis. But in another respect, individual autopoiesis of a person/people (food, sleep, sex, hygiene, protection from temperature fluctuations, communication, education, treatment, rehabilitation, leisure, recreation, sports, etc.), which is carried out o­nly in a social environment/conditions, is part of social life, social autopoiesis. Therefore, the relationship between autopoiesis and sociopoiesis as a part and the whole is different: from the bottom up, from the bio to the sociologist, o­ne: autopoiesis is the whole, and the sociopoiesis is part of it; but from top to bottom, from the socio to the bio, is another: sociopoiesis is the whole, and autopoiesis is a part of it. These opposing relations of bio-socio-genesis are inseparable, unified and constantly stochastically harmonized among themselves in dynamics, overcoming the death of any of the parties and providing them with a mutual life, being and reality in their deep inner harmony.

This deep essence was well expressed by Henri Poincare more than a century ago: "The inner harmony of the world is the o­nly true objective reality." This philosophy of harmony is the methodological foundation of the stochastic model of spheres and spherons of social production, their probabilities, thermodynamics, negentropy, etc.

2. Inscribed in the limit circle 4 circles denote 4 spherons, engaged/employed in labor activity in 4 spheres of production. Equal four circles express the equal necessity of each sphere and spheron, because the absence of any of them excludes the possibility of society's life. But the spherons are always unequal and different, both in terms of qualitative (professional) composition of employment, presented in the corresponding tables* (below), and in their number, presented in special statistics - in the statistics of spheral indices aggregating the indicators of traditional statistics. The statistics of the spherons in the first, rough approximation is presented in the relevant international statistics (see: Chapter 2 of the GPS).

Spherons have a thermodynamic, stochastic and network nature, which is naturally organized and ordered by their internal, genetic structural harmony as the "only true objective reality" (Poincaré) of the social world/noosphere (GPS).

3. SOCIOSPHERON (circle 2), the subject and product of which are people (P), occupies a priority place among the 4 equally necessary spherons. The priority of sociospheron is because the human resource (People) is the o­nly living, active and producing all resources, including itself, a resource. All people in their autopoiesis are part of the sociospheron, but o­nly when they are busy with themselves and are not engaged in the production of other PIOT resources. Here, the time factor, which determines the employment of each person, is the main factor in time and importance for the individual and society. Autopoiesis of a person takes all (or almost 100%) of his life in two periods: in childhood, up to labor/work employment and after it in old age.

The period of a person's labor employment is distributed among four spheres and spherons in accordance with his/her preferences, inclinations, vocational training and his/her main occupation in o­ne of spheres, in o­ne of the spherons. But even in the period of the individual's labor employment, a significant part of his life is kept constant behind the basic autopoiesis (above).

Any change in work employment of an individual between 4 spheres/spherons is carried out within and o­n the basis of his/her autopoiesis. Any break in employment, any rest and food in it, movement, etc. are automatically realized within and o­n the basis of autopoiesis.
Each individual in a period of employment in o­ne of the spheres, for example, a builder in the technosphere, can change the sphere and spherons several times every day. After a labor day, he can be busy for a while raising children (transition to sociospheron), then participate in a political meeting (transition to orgspheron), then devote some time to his favorite hobby - painting (transition to the infospheron), etc. All transitions are a stochastic process, determined by a variety of circumstances that are difficult to determine or cannot be accounted for. Therefore, they are thermodynamic, leading to the dissipation of energy, entropy of concentration of employment in spheres and spherons of employment.

As Heinz von Förster (2003) defined, "Autopoetic systems are thermodynamically open, but organizationally closed", o­n which the thermodynamic model of spherons is constructed. The measure of organizational closure/regulation of spheres/spherons determines the measure of their negentropy, thermodynamic efficiency and structural harmony. This is their main thermodynamic law, which determines the degree, dynamics and efficiency of population distribution among spherons and spheres of social production.

4. Sociospheron includes two permanent parts/subgroups: (A). People outside the labor employment (children, students, unemployed pensioners, the unemployed, sick, homeless, etc.) - the passive part of the sociospheron, limited o­nly by autopoiesis, and (B). People that are engaged work over people: teachers, medical staff, social workers, coaches, etc. - an active part of the sociospheron. The first group is the subject of employment of another group of sociospheron. Since the sociospheron prepares people for professional employment in four spheres, it is divided into four parts according to this feature.

5. Distribution and interdependence of 4 spherons. The sociospheron (in its active part), the infospheron, orgsferon, technospheron, represented by the circles in Scheme-1, prepared for employment in the relevant spheres, are distributed over them as thermodynamic spherons/clusters in accordance with stochastic laws.They are subject to mutual restrictions of spheres and spherons that provide them spontaneous harmony (coherence, balance, proportionality, measure, etc.).

6. If there is a thermodynamic/stochastic harmony of resources, processes and spheres of social production, then there is inevitably a similar harmony of the spherons employed in them as the main productive force in them.

7. The thermodynamic model of the stochastic population distribution by spherons is a necessary scientific step o­n the transition from their spontaneity and uncontrollability to their scientific awareness and governance/management at all levels of humanity.

Note. It is understood that the proposed theses-lemmas require corrections and additions in the process of further collective interdisciplinary scientific work, primarily in conjunction with mathematicians in the GHA Mathematical Laboratory. We also emphasize that for 40 years of working with the spheronsstructure, many of its variously valuable models and interpretations have been created (see below), of which the proposed o­ne is also far from ideal, with many shortcomings and omissions, the overcoming of which is the task of the proposed LAB.


IV. The Final Fundamental Question for Mathematicians

        It is formulated as follows.
If the military science and mathematics develop almost 220 years, beginning with Baron von Bülow, can science and the mathematics of peace arise, at least today, through 220 years, when a universal qualitative model of the structural genetic harmony of spheres and spherons of social production that describes all states of society as a measure of this harmony was created? Obviously, the science of peace, like social science as a whole, in the unity of all its branches, is impossible without mathematics. Social science as a whole has not yet found its fundamental mathematics, and the o­ne that it uses sporadically is fragmentary and weak, leaving social science o­n 70% -95% by propaganda and zombification, lacking constructive social meaning and helpless to resist war.

        But which of the mathematical disciplines is the most integral, effective and adequate to become (or to create) social mathematics, including the peace mathematics? (Peace is not a physical or biological state, but o­nly a social state or a state of society as a state of harmony of its spheres and spherons.) From our point of view, the closest to the peace mathematics, capable of mathematizing the structural harmony of spheres and spherons are its following branches: probability theory, mathematical statistics, fractal theory (harmony of chaos), harmonic analysis, harmony mathematics ("golden proportions"), and system dynamics. Or are these disciplines limited in o­ne way or another, therefore, are not suitable for the role of social/peace mathematics and therefore a completely new mathematics is required as their synthesis? Naturally, o­nly the mathematicians are able to answer this fundamental scientific question of the 21st century.

        Within the framework of Tetrasociology (the theory of harmony of spheres and spherons for more than 40 years) and the GHA (the last 13 years), many attempts were made to find adequate social/peace mathematics. Let's list the most significant of them.

1. Semashko L., Kaganovich E. and Romanov V. (1983) The problem of the interrelation of spheral indicators. Published in the book: L. Semashko. Spheral Approach. SPg, 1992, p.227-232: http://peacefromharmony.org/docs/leo-semashko-spheral-approach-1993.pdf
2. Semashko L. (1983) Construction of a system of matrices of spheral indicators (ibid., Pp. 229-237)
3. Semashko L. (1983) Construction of a system of indicators and calculations in ASOIDO (ibid., Pp. 223-229)
4. Trifanov V., Bogatushin I. (1994) Development of mathematical models of spherical equilibrium. Manuscript, 42 pp.

5. Semashko L. (1999) Macrosociological statistics. In the book: Semashko L. Sociology for Pragmatists, p.278-327: http://peacefromharmony.org/docs/sociologiya-dlya-pragmatikov.pdf
6. Semashko L. (2002) Tetrasociology: Responses to Challenges. SPb: http://peacefromharmony.org/docs/2-1_eng.pdf
7. Semashko L., co-authors (2003) New statistics and IT. In the book: Tetrasociology: from the sociological imagination ... to harmony, p.39-40: http://peacefromharmony.org/docs/2-2_eng.pdf
8. Semashko L., co-authors (2008) Fractals of social harmony and the harmony index in the "golden tetrasociology": http://peacefromharmony.org/?cat=ru_c&key=370

9. Semashko L. (2009) Tetra-philosophy and tetrasociology: the science of social harmony. In the book: Harmonious civilization: http://peacefromharmony.org/?cat=en_c&key=379

10. Sergienko P. (2009) On the principles of harmonious formalization of the fractal matrices of tetrasociology: http://peacefromharmony.org/?cat=ru_c&key=373

11. Semashko L., co-authors (2012) The key mathematical problem of a harmonious civilization. In the book: The ABC of Harmony, p.54-55 and others: http://peacefromharmony.org/?cat=en_c&key=478

12. Semashko L. (2016) Prospects and significance of global spheral statistics. In the book: Global Peace Science, p. 94-97: http://peacefromharmony.org/docs/global-peace-science-2016.pdf

13. Semashko L. (2016) Global Peace Science. Actors and Genome. Ibid., P. 567-570.

14. Semashko L. (2018) Spherons’ Sociocybernetic MegaScience: Global Breakthrough of Policy, Business and Governance to Genetic Structural Harmony of Social Production and Universal Peace. Gene technology, engineering and therapy for pathological societies:


15. And others.

        Naturally, these mathematical attempts are weak and insufficient, they are relative, but they can push in the right direction and warn against the wrong direction.


V. Conclusion

In the end, we emphasize. Mathematicians and physicists created in the 20th century "Absolute evil - nuclear weapons" (John Avery, 2017). Will mathematicians, together with sociologists and sociocybernetics of the new formation, be able to create in the 21st century the "Absolute good of Global Peace Science" from the structural harmony of the spherons, the task of which has been set for 13 years in the GHA? This will be possible o­nly with the close cooperation of professional mathematicians in the special "Mathematical LAB of Spherons" aimed at creating an integral mathematics of modern social thinking for the structural harmony of spheres and spherons of social production and global peace. They will be able to create the peace mathematics, which prevents and eliminates all war. The work of such a LAB in Russia will allow preserving the priority of creating peace mathematics for Russia, as well as the priority of creating "Global Peace Science". Without such work, this fundamental priority of the 21st century will go to other countries, and Russia will again lose it in a key area. The competitive question of priority in creating the peace mathematics remains open for mathematicians of our century.

Invitation to the

The GHA is happy to invite professional mathematicians to the "International Mathematical Laboratory" to create the "Mathematics of Spherons, Harmony and Peace", the task of which was formulated in the GHA during 13 years and is briefly presented in the proposed article.

        The expected ending result of the work of each mathematician of this LAB is a well-grounded mathematical model of the structural harmony of spherons in any mathematical discipline in an article up to 20 pages submitted before December 1, 2018 - this is the deadline. Articles in the doc. format are presented together with a short (up to 1 page) positive response of a professional mathematician.

        Each ending result is paid in the same way as for statisticians (http://peacefromharmony.org/?cat=en_c&key=801) in the amount of $3-5 thousand.

         Articles together with reviews are published in English and Russian in a special book: "Mathematics of Spherons, Harmony and Peace" at the end of 2018 or early 2019.

        Questions and articles sent to Leo Semashko by e-mail: leo.semashko@gmail.com


Excerpts from the GHA project (2017): "UN Harmony ... Sociocybernetic model of global governance at the UN level":


6. Tables of Spheres and the Model of Humanity’s Spherons:

the UNH SMSGHG object expression


        Indissolubility of the SIOT spheres and the Spherons involved in them is that the products "exits" of all spheres constitute their resources "inputs". It is expressed in the spheres statistical tables of their sector/branch structure in statics. These tables, presented in many of our works[78; 121; 125; 154; 165; 166; 168 and others], because they are very cumbersome, cannot be shown here. We confine ourselves to their simplest form. It is important that these tables are universal, include all, without exception, production functions and relevant professional groups of humanity’s spherons at all its levels.


TABLE-1. Statics of SOCIOSPHERE and employed in it SOCIOSPHERON, P1





P = P1 + P2 + P3 + P4


Employed in Sociosphere


P1, I1, O1, T1 for Sociosphere


People (P)

Teachers, etc.



People (P)

Doctors, etc.


Sport, Tourism

People (P)

Athletes,Coaches, etc.


Social Care

People (P)

Social workers


Other services (Church, Philanthropy, Trade Unions,… )

People (P)

Priests, Philanthropists, Trade unionists, Rescuers, etc.






Family, family employment

Family (P)

Parents, children, family members


Individual employment for o­neself = autopoiesis

Individual (P)



Notes to the table.

1. People/population (P) is the product of Sociosphere. People/population is produced for four spheres, therefore P = P1 + P2 + P3 + P4, where:

P1 – is employed in Sociosphere; P1 constitutes SPHERON-1 or SOCIOCLASS,

P2 – is employed in Infosphere; P2 constitutes SPHERON-2 or INFOCLASS;

P3 – is employed in Orgsphere; P3 constitutes SPHERON-3 or ORGCLASS:

P4 – is employed in Technosphere; P4 constitutes SPHERON-4 or TECHNOCLASS.

2. The employed in Sociosphere (P1), i.e. SPHERON-1 consists of two parts: the professionals working in social sector of this sphere (P1w) and all non-working (P1n) employed in the individual sector with self-production (pre-school children, students, housewives, non-working pensioners, disabled unemployed, etc.) and in the households. Therefore, the number of SPHERON-1 is the sum of employed in all sectors of this sphere: P1 = P1w + P1n.

3. To every SPHERONS a socially useful labor corresponds: P1w are employed by humanitarian labor, P2 – information labor, P3 – organizational labor, P4 – material labor.


TABLE-2. Statics of INFOSPHERE and employed in it INFOSPHERON, P2





I = I1 + I2 + I3 + I4


Employed in Infosphere

Resources P2, I2, O2, T2 for Infosphere


Information (I)

Cultural workers, etc.



Information (I)

Scientists, etc.



Information (I)

Artists, etc.


Religion (as information)

Information (I)

Priests, etc.


Media, publishing and advertising

Information (I)

Journalists, Publishers, Advertisers, etc.



Information (I)

Designers, etc.



Information (I)

Communicators, etc.


Information service

Information (I)

Programmers, etc.



Information (I)



Family, informational employment

Information (I)

Parents, children, family members


Individual informational employment for o­neself = autopoiesis

Information (I)

Individuals in their inf. employment for themselves


Notes to the table.

1. Information (I) is the product of Infosphere. Information is produced for four spheres, therefore: I = I1 + I2 + I3 + I4, where

I1 – Humanitarian information of SPHERON-1. I1 is required for it to produce people;

I2 – information of SPHERON-2. I2 is required for it to produce information;

I3 – organizational information of SPHERON-3. I3 is required for it to produce organizations;

I4 – technical information of SPHERON-4. I4 is needed for it to produce things, material wealth.

These clusters of information constitute the spheres of information or informational spheres of each SPHERON.

2. The number of SPHERON-2 is the sum of the employed in all sectors of this sphere.


TABLE-3. Statics of ORGSPHERE and employed in it ORGSPHERON, P3





O = O1 + O2 + O3 + O4


Employed in Orgsphere

Resources P3, I3, O3, T3 for Orgsphere


Organization (O)

Political figures, etc.



Organization (O)

Lawyers, etc.


Army and security

Organization (O)

Military, policemen, etc.


Tax Inspectorate

Organization (O)

Tax officials



Organization (O)

Managers, etc.


Banking and Finance

Organization (O)

Bankers, Financiers, etc.


Non-governmental organizations

Organization (O)

Relevant workers



Organization (O)



Family, organizational employment

Organization (O)

Parents, children, family members


Individual, organizational employment for o­neself = autopoiesis

Organization (O)

Individuals in their org. employment for themselves


Notes to the table.

1. Organization (institutes, rules, order, norms) (O) is the product of Orgsphere. Organization is produced for four spheres, therefore: O = O1 + O2 + O3 + O4, where:

O1 – humanitarian organizations (schools, hospitals, etc.) of SPHERON-1. O1 is necessary for it to produce people;

O2 – informational organizations of SPHERON-2. O2 is required for it to produce information;

O3 – administrative organizations of SPHERON-3. O3 is required for it to produce organizations;

O4 – technical (economic and ecological) organizations of SPHERON-4. O4 is needed for it to produce things and nature protection.

These clusters of organizations constitute the spheres of organizations or organizational spheres of each SPHERON. Each cluster organizations combine 4 types of organizations: political, legal, financial, management. To every sphere of society and each SPHERON corresponds its own organizational area of politics, law, finance, and management. Each SPHERON has his own money as an organizational tool of its sphere: social, informational, administrative and economic money [170].

2. The number of SPHERON-3 is the sum of the employed in all sectors of this sphere.


TABLE-4. Statics of TECHNOECOSPHERE and employed in it TECNOSPHERON, P4





T = T1 + T2 + T3 + T4


Employed in Technoecosphere

Resources P4, I4, O4, T4 for Technoecosphere


Things (T)



Agriculture and forestry

Things (T)

Farmers, foresters



Things (T)




Things (T)



Trade, marketing

Things (T)

Traders, etc.


Housing and household services

Things (T)

Workers, repairmen, etc.


Fish industry

Things (T)



Conservation of nature

Things (T)

Environmentalists etc.



Things (T)



Family, material employment

Things (T)

Parents, children, family members


Individual, material employment for o­neself = autopoiesis

Things (T)

Individuals in their material employment for themselves


Notes to the table.

1. Things (T) are the product of Technosphere. Things are produced for four spheres, therefore: T = T1 + T2 + T3 + T4, where:

T1 – humanitarian things (buildings and equipment of schools, hospitals, etc., housing, food, etc.) for SPHERON-1. T1 is required for it to produce people;

T2 – informational things (buildings and equipment of institutions of science, culture, media, etc., books, televisions, computers, etc.) for SPHERON-2. T2 is required for it to produce information;

T3 – organizational things (administrative buildings and equipment of institutions of politics, law, finance and management, office equipment, etc.) for SPHERON-3. T3 is required for it to produce organizations;

T4 – technical things (industrial buildings and equipment of enterprises of the Technosphere, the means of material production) for SPHERON-4. T4 is required for it to produce things and for nature protection. These clusters make things realm of things, the economy or economic spheres (physical environment) each SPHERONS.

2. The number of SPHERON-4 is the sum of employed in all sectors of this sphere.

Basing o­n the proposed classification and definition of Spherons, covering the entire population of the planet, humanity appears as the unity of its four constant, harmonious Spherons (classes), which constitute its deep eternal societal structure of harmony and peace. This structure is the most complex object of human cognition; therefore it approached to its investigation in Sociocybernetics and Tetrasociology only at the beginning of the 21st century, although some of its properties and elements were discovered before (2.1. and 2.2.).

Proceeding from what has been said, the total number of humankind (population, people of the planet) is composed of the sum of the number of global Spherons:

PEOPLE (P) = P1 + P2 + P3 + P4.

We know today: “World population reached 7 billion o­n October 31, 2011 according to the United Nations Population Fund. According to current projections, the global population will reach eight billion by 2024, and will likely reach around nine billion by 2042.” [171].

The generalized representation of the humanity’s spherons is expressed by the following model.

Model-2: SPHERONS of Humanity

The total number of humankind (population, people of the planet) is composed of the sum of the number of global Spherons: PEOPLE (P) = P1 + P2 + P3 + P4. We know today: “World population reached 7 billion o­n October 31, 2011 according to the United Nations Population Fund. According to current projections, the global population will reach eight billion by 2024, and will likely reach around nine billion by 2042.” [171].

Today there are no statistics of Spherons. Therefore, the number of each Spheron of humanity today can be hypothetically determined according to the pattern of their statistical proportion found for o­ne country according to its population census in 2010. These calculations are presented in detail in Chapter 2 of "Global Peace Science" [78].

The number of SPHERONS of this country (Russia) in millions of people and in percentage of the total population (rounded) is equal to:

SOCIOSPHERON = 85 million or 59.4%

INFOSPHERON = 4 million or 2.8%

ORGSPHERON = 10.8 million or 7.5%

TECHNOSPHERON = 41 million or 28, 6%

Statistical error = 2.3 million or 1.6%

The size of humanity’s SPHERONS in 2011 ($ 7 billion) in this proportion is:

SOCIOSPHERON = 59.4% = 4.2 billion

INFOSPHERON = 2.8% = 0.2 billion

ORGSPHERON = 7.5% = 0.5 billion

TECHNOSPHERON = 28, 6% = 2 billion

Statistical error = 1.6% = 0.1 billion.

It is obvious that similar extrapolation of proportions of o­ne country to the whole humanity is rough but the other does not exist today. The proportions of sizes of Spherons in each country and each region are very different. Their definition requires appropriate statistical studies as proposed in thismethodology, which was tested o­n o­ne country statistics. These computations are very labor-intensive work for all countries and they are available o­nly to the UNH.

The logic and o­ntology of SPHERONS are simple and indisputablealthough they are deep, do not lie o­n the surface and are not visible empirically, with the naked eye (see Appendix). If we recognize, o­n the o­ne hand, the Maturana’s "autopoiesis" (productive employment) of every person from birth to death, which can be carried out o­nly in the spheres of social production, and o­n the other hand, if we recognize the reality of four equal necessary and sufficient spheres of social production of PIOT resources, we are forced to recognize the reality of four equal necessary and sufficient Spherons. They unite ‘softly’ all people/individuals employed by autopoiesis in the relevant spheres of production, which cannot exist without Spherons as they are their main productive force.


8.1. Thermodynamic interpretation


SOCIONOME as a unity of four constant spheres of production and four constant in the structure of spherons occupied in them can be interpreted in dynamics as a four-chamber thermodynamic stochastic societal heart of humankind, which opposes to entropy of its energy and ensures the constant growth and strengthening of its negentropy. o­nly the continuous work of this heart ensures the life of humankind and its social "autopoiesis" as a whole throughout its history. The dual, biosocial nature of a human being requires interpreting functioning of the four-chambered productive heart of social harmony with its inputs and outputs of PIOT resources within the "autopoietic" heart of biological harmony with its birthrate entry and mortality output. Moreover, "Autopoietic systems are thermodynamically open but organizationally closed" [129].The result of this interpretation is a five-block sociocybernetic model of biosocial harmony as a source of survival and sustainable development of humankind/noosphere in the Earth's biosphere. Let's clarify some details and their some important characteristics.

If the spheres are the producing chambers of this heart, then the Spherons are their muscles, strength, motor and energy. Their "blood" is a continuous circulation through the channels of distribution and exchange of pulsating counter flows of life-sustaining PIOT resources, which they continuously reproduce every part of their life for the nutrition of all the Spherons, including themselves, but not o­nly themselves. Each of them cannot exist without feeding others. The destruction or absence of at least o­ne of the Spherons exsanguinates society and deprives it of life. Without each of the Spherons, society is dead. Therefore, they are equally necessary for society, regardless of their number. The reliability, stability and historical survival of this heart is ensured by its almost infinite division in levels from mega-, macro-, meso- to micro-levels in all human settlements to the family inclusively.

        Continuous pulsation of PIOT resources flows through inputs/outputs mechanisms of feedback and export/import channels of distribution and exchange can be expressed mathematically for all levels discretely of any dimension from a second to decades, centuries and millennia, depending o­n the practical and scientific goals of the managing subsystem - in our case – for the UNH SMSGHG. Based o­n any discrete dimension, the mathematical "Humankind Thermodynamic Spheral Model" (HTSM) is constructed with global inputs/outputs of people's birthrate/mortality (or migration/immigrationby regions) and production/consumption (or export/import) of other resources. Its first example is the "thermodynamic model of global peace" [168]. Similar approaches of thermodynamic modeling are demonstrated in other sources especially in the works of the Nobel laureate John Avery [172; 173; 174; 175].

Our HTSM rests o­n two interrelated nonlinear laws of stochastic distribution of PIOT resources between its chambers/spheres: 1. The law of proportions, measures and harmony (LPMH) of Spheron's integral employment and 2. The law of harmonious spheral limitation (LHSL) of PIOT resources. They exclude disharmony and entropy of any unbounded parts and attributes: the planet population, the richest people propertyand incomes, unlimited power, endless freedom, etc.

By analogy with this model, we can build a “Thermodynamic Model of Distribution and Exchange of Solar Energy” (TMDESE) between four natural spheres: Litho-, Hydro-, Atmo- and Bio- (including Noo/Anthropo/Socio) spheres for global harmonious governance of the Earth ecology and climate.

Another thermodynamic model. The spherical Sociocybernetic approach allows us to find a semblance of natural and societal spheres and express it in appropriate models not o­nly for society but also for nature, for harmonious governance of global energy o­n the Earth, for its harmonization in order to solve and prevent its environmental and climatic risks. For this purpose, we can build a “Thermodynamic Model of Distribution and Exchange of Solar Energy” (TMDESE) between the socio/noosphere and the four natural spheres. It is a model of the five-chamber spheral energy heart of the Earth, composed by the inseparable unity of its spheres: Litho-Hydro-Atmo-Bio-Noo/Socio spheres. This model is similar to SOCIONOME and TMDESE models (above), therefore it reflects the coherent part of the UNH SMSGHG object for global harmonious governance of the Earth's ecology and climate to harmonize them in the 21st century. [176]


8.2. Statistical interpretation


The dynamics of qualitative elements of SOCIONOMErequire their quantitative expression in adequate mathematical statistics. If the basis of the SOCIONOME is PIOT resources, then their quantitative, statistical measurements are necessary and sufficient for the quantitative measurement of all components of SOCIONOME.

Based o­n these measurements, statistics, developed about 40 years, is called “Tetranet Spheral Statistics of Harmony” (TSSH). It is built o­n alphanumeric indices of resource elements of PIOT, called spheral indices. It is presented in detail in many previous works; therefore here we confine ourselves to a brief description of o­nly its core - matrices of spheral indices or harmony spheral matrices.

The units of measurement of spheral indices, generalizing natural units of measurement, can be cost, time and energy units. We confine ourselves to the most developed today - the cost/money units.

We have established above that any society, including global humanity, in statics is expressed by the sum of all its resources PIOT: SOCIETY = P + I + O + T (1).

To produce each of the PIOT resources, an appropriate set of their parts is needed. Therefore, each PIOT is divided into four parts, necessary and sufficient as resources for the production of new PIOT. This regularity is expressed by a 4x4 matrix of indices, which is called the basic matrix of spheral indices or spheral statistics.

In the pure state, the 4-by-4 spheral matrix of PIOT resources looks like:

P1 + P2 + P3 + P4

I1 + I2 + I3 + I4

О1 + О2 + О3 + О4

T1 + T2 + T3 + T4 (2)

All matrixes that derive from a basic matrix are referred as spheral matrixes, which form spheral statistics. It is essentially different but connected with the traditional (branch or industrial) statistics. All branch indices are integrated in spheral indices, where they are systematized in harmonious spheral logic and Sociocybernetic system.

The spheral matrix columns represent inputs of the corresponding production spheres using PIOT as the resources, and the spheral matrix lines (rows) represent outputs of the corresponding manufacturing spheres that produce PIOT as the products.

The dual nature of the spheral resources (and the spheral indices) as input/resource and output/product of the SIOT spheres is expressed in double letter/numerical (alphanumeric) representation of these resources and corresponding indices.

On the foundation of the 4-by-4 basic matrix, the infinite hierarchical system of spheral matrixes may be created in the dimensions of 4-by-16, 4-by-64, 4-by-256, and so o­n. Their continuum is possible not o­nly in the positive but also in the negative, in fractional and in other infinities. However, at the first stage, we are bound by spheral matrixes o­nly in a positive infinity, that is, 4х4"n", where "n" is the row of natural numbers from +1 to infinity.

The entire infinite series of spheral matrices is constructed by a single algorithm:each line/row of any matrix turns into a column of the underlying matrix, and each index in the underlying matrix line appears as the sum of four indices that express a cluster of resources, necessary and sufficient for its production.

This generating algorithm expresses the fractal nature of social resources and their indices, which are connected with Benoit Mandelbrot’s fractal mathematics. "A fractal is a structure consisting of parts which are in some sense like the whole" [177]. Here, the main mathematical problem of harmonious statistical and fractal proportions / measures between the fundamental components of SOCIONOME, which are represented by matrices of sphere indicators, should be solved [178; 179]. They were used in some management projects of a large 5 million city in USSR in 1982-1988 [125], but for known reasons they were not recognized by the leadership.

The system of spheral/fractal matrices creates an Unified Logical Network Statistical Space of Humanity” (ULNSSH) that is able to integrate, compose and decompose by the PIOT indices all the diversity of national statistics, overcoming their inherent fragmentation, rupture and disharmony. Without such a single statistical system of indices, effective global harmonious risk governance is impossible in the UNH. Therefore, the mathematical statistics of similar matrices of spheral indices is a necessary scientific tool of the UNH SMSGHG. Its recursive, fractal use can be successfully integrated with other similar technologies, for example with recursive, fractal cybernetic processes in Stafford Beer's VSM Model [180].

The mathematics of the Sociocybernetic spheral governance may be o­ne of the fundamental scientific basis for the global harmonious digital economy, politics, education, artificial intelligence and management of the 21st century.


Leo M. Semashko, Ph.D. (Philosophy), Education: Moscow State University, Philosophical Faculty, “History of Foreign Philosophy” Chair; RANH Professor; Founding President, Global Harmony Association (GHA) since 2005; GHA Honorary President since 2016; State Councilor of St. Petersburg; Philosopher, Sociologist and Peacemaker from Harmony; Author of more than 400 scientific publications, including 18 books in 1-12 languages; Author of Tetrism/Harmonism as the unity of Tetraphilosophy and Tetrasociology – science of social harmony, global peace and harmonious civilization; Director: Tetrasociology Public Institute, Russia; Honorary Member of the RC51, Sociocybernetics, International Sociological Association; Director, GHA Web portal “Peace from Harmony”: http://peacefromharmony.org; Initiator, Manager, Coauthor and Editor in Chief of the book project “Global Peace Science” (GPS): http://peacefromharmony.org/docs/global-peace-science-2016.pdf.

Address: St-Petersburg, Russia; Phone: +7(812)-597-6571;

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