Mathematical LAB of Peace from Spherons’ Structural Harmony


Dr. Leo Semashko – TRANSCEND Media Service

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

“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:
  2. Four resources/capitals: People, Information, Organization, Things (m/benefits and services) – PIOT resources (Comte, Park, Toffler, Parsons, Bourdieu, Braudel, Castells, Semashko and etc.)
  3. Four processes of PIOT creating: Production, Distribution, Exchange, Consumption – PDEC processes (Smith, Ricardo, Marx, Keynes, Maturana, etc.)
  4. Four spheres of PIOT production: Social, Informational, Organizational, Technical – SIOT spheres (same authors as in 1, 2 and many others).
  5. 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:

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.

  1. 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” ( 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 (

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: 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: 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 ( 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):

(John Avery is a physicist, peacemaker, Nobel Peace Laureate:
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.

  1. 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).

  1. 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.

  1. 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.
  2. 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.).
  3. 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.
  4. 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 spherons structure, 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.

  1. 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:
    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.
  2. Semashko L. (1999) Macrosociological statistics. In the book: Semashko L. Sociology for Pragmatists, p.278-327:
    6. Semashko L. (2002) Tetrasociology: Responses to Challenges. SPb:
    7. Semashko L., co-authors (2003) New statistics and IT. In the book: Tetrasociology: from the sociological imagination … to harmony, p.39-40:
    8. Semashko L., co-authors (2008) Fractals of social harmony and the harmony index in the “golden tetrasociology”:
  3. Semashko L. (2009) Tetra-philosophy and tetrasociology: the science of social harmony. In the book: Harmonious civilization:
  4. Sergienko P. (2009) On the principles of harmonious formalization of the fractal matrices of tetrasociology:
  5. 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:
  6. Semashko L. (2016) Prospects and significance of global spheral statistics. In the book: Global Peace Science, p. 94-97:
  7. Semashko L. (2016) Global Peace Science. Actors and Genome. Ibid., P. 567-570.
  8. 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:
  9. Semashko L. (2009) Tetra-philosophy and tetrasociology: the science of social harmony. In the book: Harmonious civilization:
  10. Sergienko P. (2009) On the principles of harmonious formalization of the fractal matrices of tetrasociology:
  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:
  12. Semashko L. (2016) Prospects and significance of global spheral statistics. In the book: Global Peace Science, p. 94-97:
  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:
  1. 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.

  1. 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.

  1. Invitation to the Laboratory

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 ( 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:



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

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”:; Initiator, Manager, Coauthor and Editor in Chief of the book project “Global Peace Science” (GPS):

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

E-mail: leo.semashko [at]

Personal page:


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