Every mathematical theory is a generalization. It makes it possible to solve a large number of different problems. The strength of mathematics lies in its abstractness from particular, specific properties of objects and phenomena. This is what makes it universal, suitable for solving specific problems in any field of nature and society.

Mathematics, like any science, has its own history of development. In its periodization, it is customary to distinguish four main periods:

1) the period of the birth of mathematics;
(2) The period of mathematics of constant values.]
3) The period of mathematics of variables;
(4) The period of variable-relations mathematics (modern mathematics).

The first period

The first period is characterized by the formation of the concepts of “number” and “figure”, the emergence of the rudiments of arithmetic and geometry, the development of arithmetic operations on natural numbers. However, at the turn of VI-V centuries B.C. mathematicians of antiquity, counting and measuring objects, were already distracted from their concrete qualitative nature.

During this period, the ancient Greek philosopher, religious and political figure Pythagoras of Samos (6th century B.C.) founded the religious-philosophical doctrine called Pythagoreanism.

Its representatives proceeded from the idea of number as the basis of all things. According to their view, the numerical relationship was the source of harmony of the Cosmos, whose structure was conceived as a physico-geometric-acoustic unity.

Second period

Since this period (6th to 5th century B.C.), it is possible to speak about the second period in the development of mathematics. The understanding of independent position of mathematics as a special science with its own subject (number and figure) has appeared. This period ends by the beginning of the 17th century.

Comparing the initial periods of development of mathematics, it can be noted that the first was characterized by empirical justification of the provisions of arithmetic and geometry, the second – by the penetration of abstract reasoning into mathematics. The surviving texts of Ancient Egypt, Ancient China, and Ancient India show that mathematics of this period had knowledge of a deductive character. For example, in ancient Egypt there was a known way of finding the volume of a truncated pyramid, and it could not be obtained empirically. Even then, some general techniques applied to homogeneous number problems had been developed.

The ancient Greek mathematicians of the Pythagorean school paid much attention to logical proof of mathematical constructions, making attempts to arrange the chain of mathematical proofs in a certain sequence. This method, called “deductive”, was developed by Euclid and Archimedes (c. 287-212 BC). It should be emphasized that their concept of proof does not differ significantly from the same concept today.

It is characteristic that mathematics of this period is built most often not only on the basis of the deductive method, but also on the axiomatic method. The axiomatic method is understood as such a construction of a certain scientific discipline, when a number of its provisions (axioms) are accepted without proof, and all other provisions (theorems) are strictly derived from the axioms according to the pre-fixed logical laws or rules.

A model of axiomatic construction of geometry and arithmetic was Euclid’s Elements (3rd century B.C.). Despite the fact that Euclid’s geometry was far from perfect, the merit of its author, who developed the axiomatic method and applied it to geometry and arithmetic, is undeniable.

Third period

The third period (the middle of the 17th-19th centuries) is connected with further widening of the range of quantitative relations and spatial forms studied. Mathematics is no longer limited to numbers and geometric figures, it reflects the ideas of continuity and motion. The notion of function comes to the fore.

The emergence of analytical geometry meant the discovery of a universal method of translating geometry questions into the language of algebra and analysis. For a certain period geometry was in a subordinate position. Arithmetic, algebra and analysis with the theory of functions began to be considered as parts of “pure” mathematics, the subject matter of which was understood as numbers, values and dependencies between them. Geometry came to be regarded as “applied” mathematics that applied the results of “pure” mathematics.

The emergence and development of differential calculus had the great advantage that differential expressions served from the very beginning as operative formulas for finding real equivalents.

Fourth Period.

The middle of the nineteenth century is generally considered the beginning of the fourth period, the period of modern mathematics. From this point on, mathematical theory became so abstract that it transcended the classical conception of mathematics, which viewed numbers and figures as its subject matter. New concepts and ideas led to the fact that the classical conception, although still officially recognized by most scientists, gradually became more and more in conflict with the actual state of science.