On the role of mathematics in the study of social objects

Shcherbakov Mikhail Gennad'evich Postgraduate student, Department of Business Law, Kazan (Volga Region) Federal University 428000, Russia, respublika Chuvashskaya, g. Cheboksary, ul. Bazarnaya, 7, of. 77 pravovednalog777@mail.ru

Mathematics is rightly called the queen of sciences. Mathematics, being the universal language of science, is the key to the knowledge of physical and social phenomena.

Galileo rightly noted that "the book of nature is written in the language of mathematics." ^{[3, p. 187]}

It is important to note that the task of mathematics is the theoretical description of natural phenomena, including through the formulation of mathematical theories and hypotheses (the world of ideas).

Abstraction, formalization and axiomatization allow mathematics to describe complex physical phenomena in a mathematical language that is particularly accurate.

Meanwhile, as a rule, in the humanities and social sciences, the mathematical method gives way to the direct analysis of phenomena, so the use of mathematical apparatus is reduced mainly through information and computer technologies.

To. Popper singled out the world: reality, mentality and universal ideas.

Currently, mathematics is a bridge between the physical and spiritual world.

G. Shteyngauz noted that "mathematics is an intermediary between spirit and matter" ^{[15, p. 187]}

In addition, mathematics, being an abstract science and possessing a special mathematical apparatus, deals with multidimensional phenomena.

For example, string theory (M-theory) deals with the microcosm of the world, the length of which does not exceed the Planck length (1,616 x 10-35 meters). The essence of the theory is that elementary particles are considered in the form of strings - vibrating strands of energy. ^{[14, p. 321]}

So, when a quantum string vibrates, it produces different properties of particles. M-theory states that all physical objects consist of vibrating strands of energy (Fig. 1).

Fig. 1 String theory

Moreover, M-theory considers eleven dimensions (ten spatial and one temporal).

It is impossible for a person living in three-dimensional space to imagine a ten-dimensional space.

For example, observing a projection on a plane (two-dimensional space), shown in Figure 2, it is difficult to imagine the real shape of a cube (three-dimensional space).

Fig. 2 Cube section by plane

Meanwhile, mathematics can successfully create models, for example, of a ten-dimensional space (Fig. 3).

Fig. 3 Multidimensional space

Thus, mathematics can describe objects having any dimensions.

It is important to note that at present, including due to the lack of necessary technologies, there is no evidence of the existence of energy strands predicted by M-theory.

Modern mathematical theories are so universal that they are suitable for describing not only physical, but also social processes.

N. Bourbaki noted that "in its axiomatic form, mathematics appears to be a cluster of abstract forms - mathematical structures, and it turns out (although it is essentially unknown why) that some aspects of experimental reality seem to fit into some of these forms as a result of predestination." ^{[1, p. 10]}.

It is important to note that in the digital world mathematics is a necessary and sufficient element of the development of civilization.

Ignashov S.V. notes that "mathematics serves as a civilizational engine for the development of culture, determining the pace and depth of the processes of rationalization and modernization of culture." ^{[4, p. 10]}

Meanwhile, mathematics expresses what is happening in quantitative relations and spatial forms, therefore it has a significant limitation in the description of social processes.

For example, it is impossible to express in mathematical language such subjective feelings as love, fear, hatred, joy, etc.

To. Jaspers notes that not only language skills or mathematical thinking are important for a person, but also readiness for spiritual comprehension. ^{[16, p. 123]}

Moreover, it is impossible to create a universal mathematical model describing the whole society.

P.S. Krasnoshchekov notes that "human society is much more complicated than any mathematical models. For example, a parameter characterizing an individual's mental makeup is not always a fatal constant for him. Its significance is largely determined by circumstances. One and the same individual in one situation can manifest himself as a pure individualist, and in another as a weak-willed conformist."^[7]

Meanwhile, in our opinion, it is necessary, on the one hand, to constantly improve the mathematical method, on the other hand, to expand the possibility of using the mathematical method in describing social processes.

Mathematical methods are the basis of modern modeling of social processes.

For example, P.S. Krasnoshchekov created a model of social behavior of people who act in a social group.

Moreover, in our opinion, topology and chaos theory can be used in the description of social processes.

The central institution of sociology is society, which, as a rule, is revealed through social connections.

To. Max defined society as a special social organism, "the sum of the connections and relationships in which individuals are in each other." ^{[8, p. 13]} 

F. Tennis has developed a system about social relationships and social connections. According to F. Tennis, the basis of society is public relations. ^{[13, p. 45]}

M. Weber considers society as the interaction of people, which is the result of social action. ^{[2, p. 69]}

K. Marx noted that "man is not an abstract, huddled being somewhere outside the world. A person is a person's world, the state, society." ^{[8, p. 414]}

N.K. Mikhailovsky noted that "man is a particle of a higher whole - society, the criterion and goal of progress is man himself, man is not a means, but a result, a goal." ^{[10, p. 73]}

Zh. Mestre noted that "every nation has the government it deserves." ^{[9, p. 224]}

In other words, social ties are a system-forming element of the social system.

Thus, society is not just a collection of individuals, but mainly social ties that are the result of social relations. 

In other words, a person is permanently part of both a social group and the whole society (Fig. 4), therefore, his properties are continuously displayed in other elements of the social system.

Fig. 4 Society

In our opinion, the social relationship caused by the continuous mapping of the properties of one social object in another can be represented in the form of a topological variety.

Topology, being a branch of mathematics, studies a system of sets in their continuity. The founder of topology is A. Poincare, who connected mathematics with the qualitative characteristics of objects. ^[12]

In the mathematical aspect, social objects can be considered as mathematical sets.

What is a set? How do sets interact with each other? A set is a set of abstract objects (Fig. 5). For example, in the social aspect, an individual is a set. social group and society.

Fig. 5 The set

It is important to note that sets can be combined with each other. For example, elements of one set can be elements of another set (Fig. 6).

Fig. 6 Combination of sets

Thus, members of one social group can simultaneously be members of another social group, an individual can perform many social roles, and the social properties of an individual are the properties of a social group and the state.

It is important to note that in topology there is such a phenomenon as homeomorphism.

Thus, according to A. Poincare's hypothesis, every simply connected compact three-dimensional manifold without an edge is homeomorphic to a three-dimensional sphere (Fig. 7). ^[12]

In other words, homeomorphism is a constant continuous connection of objects that allows you to transfer the properties of one object to another.

Thus, homeomorphism is a property reflecting the continuous connection of objects.

Fig. 7 Homeomorphism of the sphere

The topological connection of objects can be expressed in mathematical language. For example: two objects and are called homeomorphic if there is a continuous one-to-one mapping, and the inverse mapping is also continuous.

Thus, the properties of one space are preserved when it is transformed into another space.

In the social aspect, the properties of a social group are reflected in the personality, and the properties of society in the social group, so it can be assumed that social objects are homeomorphic to each other.

In addition, in our opinion, we can try to mathematically describe some properties of society through chaos theory.

For example, in the mathematical aspect, society is a dissipative structure in which relations between people and social groups can be conditionally designated as trajectories.

What is a dissipative structure? According to I. Prigozhin, "dissipative structures are open systems that exchange energy (work results) or information with the external environment ^{[11, p. 187]}.

It is important to note that the "strange" attractor plays a special role in dissipative structures.

What is a "strange attractor"? According to mathematicians, a "strange attractor" is a nontrivial attracting closed invariant set lying in the phase space of the system inside the absorbing region, which includes all trajectories crossing the boundary of this region (Fig. 8).

Figure 8 Strange attractor

In other words, the "strange" attractor "attracts" all possible trajectories.

Meanwhile, in the social sphere, the role of a "strange" attractor associated with the regulation of relations (trajectories of movement) is performed by social norms.

For example, the rules of the road determine the order of movement of vehicles and pedestrians (Fig. 9).

Fig. 9 City

Moreover, for example, in the social sphere, morality plays a special role as social regulators.

 I. Kant noted that "act only according to such a maxim, guided by which at the same time you can wish it to become a universal law" ^{[5, 6, p. 321]}.

Thus, in the social sphere, social norms "attract" or "encourage" useful actions, as well as "repel" or "prohibit" illegal actions.

Another property of the "strange" attractor is that small changes in the initial conditions can lead to unpredictable consequences (the butterfly effect).

Meanwhile, for example, in the social sphere, the lack of formal equality of citizens (equality in initial conditions) can lead to a social crisis.

In this regard, it is necessary and sufficient to guarantee the legal equality of citizens.

In our opinion, the task of mathematics is to try to describe social processes in the language of mathematics, as well as to create mathematical models of social processes.

In conclusion, it should be noted that the mathematical method is a universal means of describing both the physical and social world. Moreover, in our opinion, mathematical modeling of social processes will become the main research tool in the future. Meanwhile, it must be remembered that the mathematical method has objective limitations associated with a high degree of uncertainty of social processes.

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