“Start doing the necessary, then the possible and suddenly

you see that you are already doing the impossible”

**Francis of Assisi**

Beginning: *Foundations of statehood*

Let’s see the mathematical model of interaction of matrix points of human existence (MHE) in IES (alpha) for the individual.

Originally MHE points, namely a set of “intellect”, “labor” and “good” of the individual are in IES, which lacks common categorical indicators, in this connection, the mathematical description of these objects, even very abstract, is almost impossible.

Presented MHE points may have common quantitative indicators (crossings) only after their Manifestation by the individual from IES. And as any individual (in particular, You and me) is a three-dimensional Euclidean space in which its elements (points) clearly have concept of measure, dimension, relations and others, then it is possible for it to construct the rules and laws of interaction of MHE points in IES. To do this, one must add the number of mathematical concepts and terms regarding the IES, as well as determine their structural interaction.

Let the parameters *u є U*, *v є V* and *w є W* – be Non-manifested quantitative characteristics of MHE individual in IES, where respectively *U* – are characteristics of “intellect”, *V* – «labor», *W* – «good» (may be generally functions of time: *u(t)*, *v(t)* and *w(t)*). Next we introduce the Cartesian coordinates *(x, y, z)* in IES that will set the three-dimensional Euclidean space *R ^{3}* with the introduced concept of space metric – function

(1)

Generally speaking, *x, y, z* – are the functions of the arguments *u, v, w* for the Euclidean space *R ^{3}*, the essence of which lies in the fact that to each characteristic of “intellect”, “labor” and “good’ they put point in correspondence

(2)

The system of functions (2) defines three sets a* є A*, b* є B* and c* є C*, points of Euclidean space *R ^{3}*, which are Manifested quantitative characteristics of MHE individual. Each of these sets is an open sphere in

(3)

where –* a _{0}, b_{0}, c_{0}* are the centers of spheres, and r

At the initial moment of life, a quantitative Manifestation of MHE characteristics for the individual is the point of the Euclidean space *R ^{3}*, which is determined by a system of vector functions (2), in which all three functions identically degenerate into one (Picture 2.1):

(4)

**Picture 2.1** – Quantitative Manifestation at the initial moment of life

In further life period the quantitative characteristics of MHE individual increase significantly, thus, the points generate open spheres, which points are determined by correlations (3) (Picture 2.2).

**Picture 2.2 (a)** – Quantitative Manifestation in further moments of life

**Picture 2.2 (b)** – Quantitative Manifestation in further moments of life

For the analysis of qualitative changes of the quantitative characteristics of MHE individual we can use the function of entropy *S* of the system (function characterizing the disorder of the system state). From the point of view of mathematical realization, it is practical to use opposite concept (function characterizing the order of system state) – extropy function . Hereafter, we will define extropy function as the maximum distance from the center to the boundary point of the sphere or the ball radius:

(5)

General system extropy will be defined with nonlinear function of three arguments, continuous on all definitional domain *D _{S} є R^{3}* of the following view:

(6)

Hereafter, for simplicity of perception and simplification of mathematical transformations, we’ll consider a system of sets (3) (open spheres) in the form of intercrossing circles with preservation of their qualities and structural interactions (Picture 2.3), what is quantitative Manifestation of matrix points of human existence.

**Picture 2.3** – Quantitative Manifestation of matrix points of human existence

The described mathematical model of Quantitative Manifestation of matrix points of human existence can be represented as a dimensionless structural-logic interaction of Manifested matrix points of human existence (Picture 2.4), as well as visually compare with previously submitted scheme of interaction of matrix of human existence in the information-energy space – time (Picture 2.5).

**Picture 2.4** – Dimensionless Manifestation of matrix points of human existence

**Picture 2.5** – Scheme of interaction of matrix points of human existence in the information-energy space – time

to be continued ...

**Author: Lebedenko S.S.
Co-author of mathematical model: Yurechko V.Z.**