(3) a. Mathematical Model of Internal Cohesion

(3) Mathematical Model of the 2nd Law

a. Mathematical Model of Internal Cohesion 

The core content explained by the 2nd law of political phenomena is the maintenance and cohesion of political organizations. To simplify the intent of the 2nd law, it can be stated as follows:

 

         [Ch.3.221] Political organizations are cohesive together because of (invasion) threats from outside the political entity, and this threat is a force that could deprive the political actors of the profits they pursue through cooperation (cohesion).

 

Let me denote the armament level of a political actor (organization)  as a 3x3 square matrix   \( [Ar]_{g} \) , and the degree of ideological diversity as a 3x3 square matrix  \( [Iv]_{g} \) . The armament level() refers to the tangible and intangible tools that a political actor(g) can use to protect and increase its survival and profit, such as weaponry systems including rifles, cannons, tanks, and warships, that can be utilized by a military. Ideological diversity( \( [Iv]_{g} \) ) refers to the extent to which different factions within a political actor(g) have varying beliefs and values. This can be measured by the amount of diverse information that is freely circulated and expressed within the political actor, such as religion, political ideologies, and information about other cultural regions. Therefore, the diversity of information about ideologies, religions, and other cultures that are disseminated within a political actor such as a state is considered as the degree of ideological diversity.

Let me define \( \vec{{H_S}_g} \) as the survival capacity of g that increases with enhanced cohesion force and  \( \vec{{C_T}_g} \)  as the survival capacity that decreases with an increase in invasion threats. Assuming that each value is appropriately quantified, the 2nd law of Political Phenomena can be expressed by the following formula.

         [Fmla.3.2.1]                   \( \begin{bmatrix}
\frac{d}{dt}\vec{H_{g}} \\
 \\
\frac{d}{dt}\vec{C_{g}} \end{bmatrix} = \begin{bmatrix}
[Ar]_{g} \frac{d}{dt}\vec{{H_S}_g} \\
 \\
[Iv]_{g} \frac{d}{dt}\vec{{C_T}_g} 
\end{bmatrix} \)

Let  \( \frac{d \vec{H_S} }{dt} \)  be denoted as "cohesion force ( \( \vec{S_g} \) )" and  \( \frac{d \vec{H_S} }{dt} \)   as "invasion threat( \( \vec{{T_{HR}}_g} \) )".

Cohesion force is the tendency of individuals to give up their partial interests in order to obtain the benefits of cooperation. It is manifested as obedience to power structures when the size of an organization is large.
Invasion threat refers to a threat that endangers the survival of a political actor, especially a nation- state. The most typical example is the threat of foreign invasion. Throughout human history, the external invasion and looting of a kingdom or a nation has been a repeated occurrence. There are no exceptions to this through the history of the East and West. Goguryeo constantly fought wars of aggression with China, and also fought against Baekje and Silla. Silla fought against the Tang Dynasty's invasion, and Goryeo was invaded by the Mongolian state of the Yuan Dynasty. During the Joseon Dynasty, Japan also invaded. China was repeatedly invaded and conquered by ethnic groups, and transformed into a state of that ethnic group. And when a crisis occurred, the people within the political structure worked together as one. If not, it led to destruction.

Now, if we substitute symbols for cohesion force and invasion threat, the equation above can be expressed as follows.

         [Fmla.3.2.2 ]                   \( \begin{bmatrix}
\frac{d}{dt}\vec{H_{g}} \\
 \\
\frac{d}{dt}\vec{C_{g}} \end{bmatrix} = \begin{bmatrix}
[Ar]_{g} \vec{S_g} \\
 \\
[Iv]_{g} \vec{{T_{HR}}_g} 
\end{bmatrix} \)