*“Dogma is a direct prohibition to think”*

**Ludwig Feuerbach**

*“The death of dogma is the birth of morality”*

**Immanuel Kant**

*“Give me a place to stand and I will move the Earth”*

**Archimedes**

This work succinctly presents the vision of conditions of state origin on the basis of a mathematical model of interaction of structural points of Matrix of human existence (MHE) in the information-energy space (IES): “intellect”, “labor” and “good” earlier represented by authors. There were considered the features of the increment of extropy volumes and specificity of synergetic interaction of characteristics of MHE individual; product separation from the MHE product and the logical structuring of state origin stages.

The reasonability of using the term “INDIVIDUAL” in the analysis of pre-state period of human development: in everyday consciousness we often identify the concepts of “person”, “individual”, “personality”. The term “individual” is used to mean “a particular person”, a single representative of the human race, when it should be noted that this is not about all the mankind in general and not about any person of it.

Let’s simulate the operation of the structural points of MHE in pre-state period of society development.

**Description of Unmanifested and Manifested characteristics of MHE individual.**

Development of mathematical model of interaction of points of MHE in IES will be considered in accordance with the fact that Manifested quantitative characteristics of “intellect”, “labor” and “good” in the form of open in IES spheres *A, B, C* are generated from the point (expression (4)), we consider that intersection of these sets is not empty:

(7)

and also we think that reasoning about matrix points of one individual is transferred to the finite group of individuals for the main structural-logical elements of the system, in conditions of Pareto efficiency.

In what follows, a totality of Unmanifested characteristics of MHE for a finite set *L* of *i* individuals *(i = 1, 2, 3, …, n)* will be designated:

(8)

Then the totality of Manifested quantitative characteristics of MHE for a finite set *M of* *i *individuals *(i = 1, 2, 3, …, n)* will be designated as follows (picture 3.1):

(9)

*M _{1} … M_{i}* – Manifested characteristics “intellect”, “labor” and “good” of

**Picture 3.1** – The totality of individuals

Similarly, we will introduce the concept of crossing of Unmanifested characteristics of MHE for a finite set *H of* *i* individuals *(i = 1, 2, 3, …, n)*:

(10)

Crossing of Manifested quantitative characteristics of MHE for a finite set *D of* *i* individuals *(i = 1, 2, 3, …, n)* can be written:

(11)

Specificity of MHE is that all interactions between the structural points occur synergistically. There are two kinds of synergy between the structural points of MHE (Picture 3.2):

- SYNERGY OF THE FIRST KIND – synergy separately between any two of the three A, B, C volumes of Manifested quantitative characteristics of MHE (pairwise intersection of sets: );

- SYNERGY OF THE SECOND KIND –synergy between the three *A, B, C* volumes of Manifested quantitative characteristics of MHE (total intersection of set: ).

**Picture 3.2** – Synergy areas

In this connection, we introduce the following notions of synergy qualitative expression through extropy (Picture 3.2):

1) 1) EXTROPY OF THE FIRST KIND – extropy volume of synergy of pairwise intersection of Manifested quantitative characteristics of MHE (*A, B, C *sets):

– pictorial symbol (12)

It is defined by the functions of the extropies of respective MHE points of individual:

(13)

2) 2) EXTROPY OF THE SECOND KIND – extropy volume of synergy of all Manifested quantitative characteristics of MHE (*A, B, C *sets):

(14)

It is defined by the function of the extropies of respective MHE points of individual:

(15)

Within the author’s paradigm, the number of states of each of *A, B, C* sets separately will be less than the number of states of their pairwise intersections, and also less than the number of states of total intersection, in this case we have the next group of obvious inequalities:

(16)

where – is initial extropy (IES extropy); – is individual extropy; here and in what follows, using of symbol indicates that given correlations are true of the three extropies of the first kind ;

– is extropy of the second kind.

**The particular case of functioning of mathematical model.** Let’s consider the particular case of functioning of proposed by the authors mathematical model of interaction of points of MHE in IES at no increase in “good” characteristics. After a small increment of Unmanifested characteristics of “intellect” and “labor” in the surroundings of the boundary spots of MHE points in IES (in subsequent moments of life):

(17)

there happens an increase in set of Manifested quantitative characteristics of MHE, i.e. an increase of spheres volume (areas of circles) of the form (3), and this by-turn leads to an increase of system extropy (5). Let’s see a mathematical description of this process. In this case, the system of vector functions (2) will be written in the form (consider the small increment at *h = m = 1*):

The increase in “good” will also provoke motivational skills (at least through the modification of the same instincts under the influence of individual experience), which again will have an impact (through “intellect”) on “labor” (at least to reduce the direct physical and temporal costs to produce the same amount of “good” (optimization of “labor”).

Thus, the increase of extropy will always be under fluctuations of all MHE points. Let’s introduce the sign: – the number of MHE states. The obvious is the fact that , because under such a complex system of synergetic interactions of points, and also taking into account the functioning of MHE in the open IES, changing MHE states according to the laws independent from consciousness (many of which have not been opened yet), lead to understanding that .

As part of the given mathematical model, this fact can be formulated so that any of the Manifested characteristics of MHE is not trivial, so the points of spheres are not generated to zero point:

It is necessary to take into account the fact that the paradigm will operate in the finite (boundary) conditions for each point. For example, excessive “good” will block “intellect” and its development, respectively, the ability to “labor” will also be significantly reduced. Accordingly, the abilities to “intellect” and “labor” can be blocked simultaneously, or vice versa. The result is that at the next stage of life development the ability to create “good” will tend to zero.

**Sequences of small increments of Unmanifested characteristics of MHE individual.** Let’s consider in the general form the sequences of small increments of Unmanifested characteristics of “intellect”, “labor” and “good” in the vicinity of the boundary spots of MHE points in IES:

It can be shown that the unlimited increase in one time interval of increments of Unmanifested characteristics of “intellect”, “labor” and “good” in the vicinity of the boundary spots of MHE points in IES will lead to a significant slowdown in growth of characteristics increments of at least one of the other points. On this basis, we have the following interaction of increments of Unmanifested characteristics of MHE in IES (on the example of increasing of “good” characteristics):

or blocking of both MHE points at the same time:

The most obvious illustration of a person’s development in favorable natural conditions (presence of ready nature “good”) does not develop MHE (Africa, Pacific Islands and others). And, on the contrary, we see the inverse extremum of the peoples of Far North, where the very existence of man is on the edge of possible, too small increase in “good” did not provide possibilities to develop “INTELLECT” (as human’s intelligence that is manifested in his cognitive and practical activities and different from the forms of sensory-emotional reflection of the world) and, accordingly, “LABOR” (as a form of activity aimed at the mastering and transformation in accordance with one’s own needs, interests and goals of the environment).

In this case, the extropy increment of the system will tend to zero (MHE does not develop):

**Conditions of mutual relations for Manifested quantitative characteristics of MHE**. Based on the synergistic aspects of the interaction of structural points of MHE in IES, we can formulate conditions of mutual relations for Manifested quantitative characteristics of MHE. Elements of spheres (circles) must satisfy the condition of complete mutual unabsorption and continuity, that is at the intersection the elements of A, B, C sets have at least one common point (Picture 3.3) and at least one point of each of the sets is not included in the intersection (Picture 3.4). Thus, we can highlight points of conjunction and points of non-conjunction on the spheres, as well as synergy areas.

*А, В, С* – Manifested characteristics of “intellect”, “labor”, “good”;

– point of spheres continuity

**Picture 3.3** – Point of spheres continuity

*А, В, С* – Manifested characteristics of “intellect”, “labor”, “good”;

– point of sphere *С*, unabsorbed by spheres *А* and *В*

**Picture 3.4** – Point of mutual unabsorption of spheres

Similar arguments can be given for MHE points *A* and *B*.

**Speed of Manifestation**. The key issue of the analysis of synergetic interaction of structural MHE points in IES and, in fact, the main mechanism of MHE development is the SPEED OF MANIFESTATION of quantitative characteristics of MHE individual. It is caused by the vector gradient of extropy function (6) and (16). For further analysis of Manifestation process it is necessary to set the total extropy gradient of Manifested characteristics of MHE individual:

(17)

and also extropy (synergy) gradient of the second kind:

(18)

Then, in accordance with the expression (16), the quantity (modulus) of vector (18) will be bigger than the corresponding vector quantity (17):

(19)

Conclusion: the bigger the volume of the synergy of the second kind, the bigger the speed of MANIFESTATION.

**Increment of extropy of MHE individual.** General extropy of MHE system of interacting individuals. Increasing of extropy of the first and second kinds of synergetically interacting MHE points is caused by increment of quantities of volumes of Manifested characteristics of “intellect”, “labor” and “good” at every (q-m) stage of individual (group of individuals) life development. With *q = 1* we have initial extropy *(M)* of Unmanifested characteristics of MHE in IES of individual (group of individuals).

Taking into account that MHE points in IES interact synergistically, excluding dissipative extremums and catastrophic system resonances, the following increase of extropy of points of MHE individual is true:

(20)

where – are prolonged extropies of interaction system of MHE points for individual (in the following moments of life), , , – are *q-е* extropy increments.

On the basis of the foregoing, we can say that the general system extropy of interaction of society (group) of individuals is a kind of function g from general extropy, the first and the second kind of interaction of each interacting individuals:

(21)

where

Maximum increment of MHE extropy. Determination of the mechanism of finding the maximum increment of extropy of MHE individual is the most essential element of the analysis of human development and development of society as a whole. It should be emphasized that the extropy of MHE individual can have an unlimited increment and depends directly on the extropy increments of every Manifested characteristic of “intellect”, “labor” and “good” of individual.

Based on the given interaction conditions of MHE points in IES, it can be stated that the maximum extropy of MHE individual on q-m increment will be determined by the maximum q-m increment of extropy of the second kind individual, neglecting quantities of volumes of extropy of the first kind because of their smallness in relation to extropy of the second kind:

(22)

Analysis of increments of maximum general extropy of MHE system of interacting individuals, enables us to obtain a qualitatively new evaluation of community development of these individuals.

The mechanism of finding general extropy of MHE system of interacting individuals is as follows. Each individual of the entire finite group (society) of individuals has q-e increment of MHE extropy (true to each structural MHE point).

Maximum general extropy of MHE system of interacting individuals on q-m stage of the increment will be the maximum q-e increment of MHE extropy of a particular individual of all q-x increments of MHE extropies of the given group of individuals

:

(23)

Correlation (23) is applied for comparison of general MHE extropies of two systems of interacting individuals (Picture 3.5 a, b). We can conclude the following: q-e increment of general extropy of the first group of interacting individuals (Picture 3.6, a) will be bigger than q increment of general extropy of the second group of individuals (Picture 3.6, b), as within Pareto equality attitude to increments of extropy of individual to general extropy of the first group of individuals is bigger than the corresponding attitude of the second group:

(24)

**Picture 3.5**. – Detection of the general extropy of MHE system of interacting individuals

Let’s form HYPOTHESIS about equality of increment of general MHE extropy of individual and the quantity of gradient of MHE individual on q-m increment:

Comparing the derived expression with the expression (20), we will write:

Considering this iterative process in utmost crossing with we can obtain the following assessment:

Based on obtained assessment , we can determine the quantity of extropy volume of MHE individual, as well as to evaluate the nature of growth of extropy further and identify a number of psychological traits of individual (personality charisma, inclination for labor or intellectual capacity and others).

**Proportional correlations of MHE points of individual**. Considering further MHE behavior for any , submitted correlations show that the balanced development of MHE is only possible with a proportional increase in extropy of each of the A, B, C sets. That is, with any increase of Manifested quantitative characteristics of MHE the proportional correlation of the extropy increments of each of the MHE points should remain constant.

In this case, it is possible to formulate the following important statement about the nature of proportional correlations of the interacting MHE points of individual (character of individual). For any natural value q, characterizing stages of extropy increment of MHE points of individual, there will be such positive constants (in general they may be functions of *q*), that for each of them there will be the finite limit of correlation of extropy increments of respective MHE points equal to these constants:

(25)

Each of the found constants is in your Pareto balance local area to avoid dissipative destruction of the system.

It is possible to have similar representation of the nature of proportional correlations of individuals society, provided the development of Manifested quantitative characteristics of MHE of each individual.

On the basis of these correlations, it is possible to formulate HYPOTHESIS about the possibility of creating a methodology with 3 test notes of MHE individual in general, as follows:

A – nature of the number of identification states;

B – characteristic of state changes of person’s motility at one period of time ;

C – cumulative number of states for a certain period of simultaneous interaction of characteristics *A* and *B*.

It is possible to formulate the following HYPOTHESIS: within the same initial genetic sets we see the difference of characters, due to the influence of proportional development of MHE (constants ).

Conscious reality separation. In the quality-quantitative analysis of the Matrix “work” we should single out both objective component of the interaction of structural MHE points in IES expressed in the growing extropy volume, and subjective components of manifested volumes *A, B, C*.

In each of the volumes *A, B, C*, expressing MANIFESTED reality, we single out volumes *A', B', C'* expressing CONSCIOUS reality.

As part of the given mathematical set model, *A', B', C'* are subsets of *A, B, C* sets (Picture 3.6):

characterized in IES by the extropy of higher level, defining function of extropy of Manifested MHE characteristics:

(26)

*А, В, С* – Manifested characteristics of “intellect”, “labor” and “good”;

*А', В', С'* – Conscious characteristics of “intellect”, “labor” and “good”

**Picture 3.6** – Separation of the Conscious from the Manifested

In accordance with the way how synergetic interaction of Manifested characteristics of MHE individual was expressed, we can highlight synergy areas of the first and second kind of Conscious MHE characteristics and also extropy of the first and second kind of Conscious MHE characteristics.

**Separation of the Appropriated. MHE product**. In the volume of CONSCIOUS reality *A', B', C'* there happens subjective APPROPRIATION of IES elements in the form of Matrix PRODUCT, expressed in volumes *A'', B'', C''* (Picture 3.7).

As part of the given mathematical model of the set *A'', B'', C''* are subset of the sets *A', B', C'*:

and also characterizing in IES by the extropy of higher level, defining function of extropy of Conscious MHE characteristics:

(27)

**Picture 3.7** – Separation of the (Appropriated) Conscious

Let’s consider that MHE points are in one of the possible (not forbidden), excluding dissipative and catastrophic resonances, states, enabling fluctuations of quantitative and qualitative MHE characteristics.

For example, a decrease in “good” and its main components (material and non-material) provokes motivational growth of mental abilities to adapt to new conditions, create new working skills and tools. Through these two components, a minimum is to return “good” in the initial state (in this case as a motivation we must take into account basic human instincts, directly related to the absence of good).

Further process analysis of Appropriation is characterized by specifying of gradient of general extropy of Appropriated characteristics of MHE individual:

(28)

and also by extropy (synergy) gradient of the second kind of Appropriated characteristics of MHE individual:

(29)

Thus, we can write the following inequalities, which should be satisfied by extropy:

(30)

where – is initial extropy of Unmanifested characteristics of MHE in IES (extropy of IES itself); – are general extropy, extropies of the first and second kind of Manifested characteristics of MHE individual respectively; – are general extropy, extropies of the first and second kind of Conscious characteristics of MHE individual respectively; – are general extropy, extropies of the first and second kind of Appropriated characteristics of MHE individual respectively.

Similarly to the expression (30), we can formulate a group of inequalities for general extropies of MHE systems of interacting individuals.

In accordance with the presented reasoning we formulate the following HYPOTHESIS:

- Characteristics of individual Manifestations – part of the subconscious mind of individual;

- Characteristics of individual Consciousness – individual consciousness;

- Characteristics of individual Appropriation – separation of an individual from Nature.

Taking into account the given above expressions we can formulate cause-effect relation for each structural MHE point:

• Manifestation doesn’t mean Consciousness;

• Consciousness is not Appropriation, but Appropriation is both Manifestation and Consciousness;

• Appropriated part of IES can be both material and non-material, andis defined as MHE Product.

In what follows within the introduced mathematical model, for the comfortable schematic description of Appropriated characteristics of MHE individual, we will submit for consideration a certain set in the form of open sphere, circled near spheres of *i* individual (Picture 3.8):

(31)

Thus, in what follows we can submit for consideration the following sets of Appropriated quantitative characteristics of MHE individual:

(32)

where Р_{product} is MHE individual Product, which is a general component of Appropriated (useful and conscious) quantitative characteristics of “intellect”, “labor” and “good”.

– Appropriated MHE Product of *і* individual

**Picture 3.8** – Simplified vision of volumes of Appropriated MHE Product of *і* individual

**Goods separation from Product. Market assessment**. The product is consumed by the individual to so-called autonomy point (saturation). Surplus is formed from the part of the product that the individual does not consume because of the maximum saturation of it. The totality of surplus of one individual, and as a special case the need for it of another individual may be called T goods, which is MHE Product:

In what follows we will consider set T as an open sphere of the kind:

(33)

Interaction of individuals in relations needs-surplus (further supply and demand) lead to the market origin: GOODS GIVE RISE TO MARKET.

Nature of goods origin is that they are singled out by Matrix, part of the objective reality. One of the basic characteristics of the goods is their fragmentation, changing and accumulation. Thus, we can say that goods form needs-surplus relationship, provided that there are at least THREE states of needs-surplus: presence of the surplus only; presence of the needs only; presence of both surpluses and needs.

The exchange of goods (with the help of the market) suggests origination of the need for universal goods – MONEY.

With the market appearance a need for the socialization of the goods arises, for example, in its protectability from adverse environmental conditions.

Transformation process of MHE individual Product. Awareness of the individual (group of individuals) of a surplus of MHE product precedes certain state – initial Manifestation of MHE individual in IES (Picture 3.9), as well as the state which assumes the saturation of the products of MHE points interaction, and which can for convenience be called autonomy volume (Figure 3.10).

– volume of Appropriated Product of MHE individual “intellect”;

– volume of Appropriated Product of MHE individual “labor”;

– volume of Appropriated Product of MHE individual “good”;

– volume of individual autonomy

**Picture 3.9** – Initial Manifestation of individual

– volume of Appropriated Product of MHE individual “intellect”;

– volume of Appropriated Product of MHE individual “labor”;

– volume of Appropriated Product of MHE individual “good”;

– volume of individual autonomy;

– volume of possible personal consuming and distribution of individual

**Picture 3.10** – Achieving the autonomy volume by individual

The presence of surpluses of individual (groups of individuals) in such amounts when he is not able to distribute them, exceeding the volume , leads to manifestation of a new system state. It is appearance of individuals who can distribute the surplus so that the MHE extropy of communities producing surplus was proportional to supplemented corresponding sets *A'', B'', C''*.

On Picture 3.11, a in schematic form there is an exchange of undistributed surplus of MHE product of one individual (*М _{2}* has surplus of “intellect” characteristics) in the presence of needs for appropriate product of another individual (

*A''* –volume of Appropriated Product of MHE individual “intellect”; *B''* – volume of Appropriated Product of MHE individual “labor”; *C"* –volume of Appropriated Product of MHE individual “good”; – volume of individual autonomy; – volume of possible personal consuming and distribution of individual, *M _{1}* – individual who has not achieved the autonomy volume according to Appropriated characteristic of “good”,

**Picture 3.11, а** – Private case of appearance and redistribution of surpluses of MHE Product of one individual

A" – volume of Appropriated Product of MHE individual “intellect”; B" –volume of Appropriated Product of MHE individual “labor”; C" – volume of Appropriated Product of MHE individual “good”; – volume of individual autonomy; – volume of possible personal consuming and distribution of individual

**Picture 3.11, б** – Private case of appearance and exchange of surpluses of MHE Products of two individuals

Separation of subsequences of MHE Product. The obvious is the fact that as a result of synergetic interaction of structural MHE points every volume of Appropriated MHE characteristics contains parts of other Appropriated MHE characteristics. In other words, the sequence of set elements of any MHE point contains subsequences of the two other MHE points (Picture 3.12):

– subsequence of elements of Appropriated Product volume of “intellect” and “labor” of MHE individual;

– subsequence of elements of Appropriated Product volume of “intellect” and “good” of MHE individual;

– subsequence of elements of Appropriated Product volume of “labor” and “good” of MHE individual;

– subsequence of elements of Appropriated Product volume of “intellect”, “labor” and “good” of MHE individual, for whom the following conditions must be met:

(34)

A", B", C" – volumes of Appropriated Product of “intellect”, “labor” and “good” of MHE individual respectively; – volume of possible personal consuming and distribution of individual; *Рprod* – MHE individual Product; – goods with predominant “intellectual” component, – тgoods with predominant “labor” component, – тgoods with predominant “good” component (resource as a private case)

**Picture 3.12** – Schematic image of structural-logical connection of elements of MHE individual points in the process of separation of MHE Appropriated Product (goods)

Distribution of surplus through the market does not solve the problems of the protection of MHE individual from the external factors, as well as other problems related to the development and maintenance of long-term stable state of MHE in the boundary conditions – the conditions of maximum extropy increment of MHE individual and general extropy of MHE system of interacting individuals.

These tasks are solved by the socialization of the surplus and the creation of institutions ensuring the optimal functioning of MHE individual.

Thus, we come to the Manifestation and Consciousness (understanding) of the conditions of state origin:

1) state is the result of increasing of the general MHE extropy of society of individuals to the condition of the impossibility of self-distribution of surplus produced for personal consumption;

2) state is the result of the socialization of surplus of MHE individuals for the purpose of establishing institutions providing long-term optimal functioning of MHE.

**CONCLUSIONS****. **

1. Manifestation of MHE in IES by individual, the following Consciousness and Appropriation of the Manifested by increasing MHE extropy, produces goods;

2. The gradient of the synergy of the second kind of Appropriated MHE characteristics determines qualitative and quantitative parameters of both individual and communities of individuals, as well as the parameters of the goods produced by them;

3. The parameters of the goods produced by the society are functionally depend on the overall characteristics of MHE communities of individuals, in a private case – on the characteristics of MHE individual;

4. The development direction of the society (or community of individuals) functionally depends on the qualitative and quantitative characteristics of the goods produced by the community of individuals;

5. Goods, being both a surplus of MHE vital activity and a part of Manifested IES with increased extropy can be regarded as an initial component of any configuration of society, both existing and most efficiently possible from the point of view of the individual, including the state.

Research. On the basis of drawn up by the authors mathematical model of functioning of structural MHE points in IES, and obtained with its usage qualitative and quantitative evaluations of the nature of the development of social society of individuals and identifying of the main stages of state origin, the following studies were carried out:

- Substantiation of social, oriented on MHE control functions, independent from the state (functions system ) are logically consistent with the mathematical models presented in the works of George Akerlof, John Nash and Ronald Coase;

- Substantiation and sampling of MHE feedback, market, universal goods, state institutions and institutions of local systems control as well as global;

- Description of different variants of system reforms in Ukraine on the basis of mathematical model of the optimal functioning of MHE social groups of individuals (population);

- The mathematical substantiation of different variants of “soft” modifications of leading world democracies to the state of optimal functioning of both internal and external component;

- Determination of new optimal institution proportions of both public systems and public-social systems.*Авторы*

*Authors S.S. Lebedenko, V.Z. Yurechko
Graphic designer A.M. Slyahtitski*