The world of mathematics

Mathematics has always been held in respect. Being one of the most ancient sciences it is still ever young. Mathematics is a Greek word and it means “something that must be learned or understood,” “acquired knowledge”. It is supposed that mathematics had its birth among the ancient Greeks. The Greeks divided mathematics into arithmetic and geometry.

Mathematics has different interdependent branches. The largest branch is that which builds on the ordinary whole numbers, fractions and irrational numbers, or what, collectively, is called the real number system. Arithmetic, algebra, the study of functions, the calculus, differential equations, and various other subject which follow the calculus in logical order are all developments of the real number system. This part of mathematics is termed the mathematics of number. A second branch is geometry consisting of several geometries. Mathematics contains many more divisions. Each branch has the same logical structure: it begins with certain concepts. These concepts must verify explicitly stated axioms. From the concepts and axioms theorems are deduced.

XVII century introduced the idea of motion - variable quantity (I. Newton, G. Leibnitz). Arithmetic, i.e. theory of numbers, algebra, and geometry gave way to the notion of function, infinity, concept of limit, derivative, and integral. The so-called classical mathematics was created within the period of three centuries (from the XVII to the XIX century). From then on differential equations have been used to describe laws of nature. However, the so-called "classical mathematics" has retained its leading position and importance.

Much of most fruitful modern research work emerged as a result of classical mathematical analysis' deductions. The theory of both ordinary and partial differential equations, which is vital for the study of various quantities, is used in ever-growing application field of modern mathematics.

If before the XVIII century mathematicians used to be both philosophers and experimentalists, since the XVIII century mathematical research has become an independent profession. Professional mathematicians were now trained at universities. The scientific revolution had bequeathed to mathematics and major program of research in analyses and mechanics. The period from 1700 - 1800, the century of analyses, witnessed the consolidation of calculus and its extensive application to mechanics.

By the middle of the XIX century both fundamental human knowledge and accepted potential of mathematics led to a noticeable growth of limited number of people, engaged in active research. This process was determined by invention of typography and emergence of textbooks, which granted access to new achievements in the field of mathematics, by systematic university teaching of mathematics and, finally, by new prospects of expanding and deepening general human knowledge.

The XX century has seen a tremendous upgrowth of mathematics. Its field of application considerably expanded as early as in the beginning of the century, which led to further progress and development. Thus, mechanics and optics used to constitute the main branch of physics, closely connected with mathematical experiment. Nowadays, however, they have been supplemented by electrodynamics, the theory of magnetism and thermodynamics. Mathematics became especially important in terms of continuum mechanics study, namely gas dynamics and hydrodynamics, i.e. viscous and non-viscous. Most of the powerful abstract mathematical theories in use today originated in the XIX century. The growth of mathematics as a profession was accompanied by a sharpening division between mathematics and the physical sciences. One result of this separation has been that mathematics developed higher standards of rigour.

Since the second half of the XX century the number of professional mathematicians has sharply risen, and now amounts to hundreds of thousands. This is due to both the facts that computers are widely used nowadays and to mathematization of all the sciences as well as other domains of human activity.

Modern mathematics of the late XX century is characterized by a still wider use of mathematical procedures in various spheres of activities as well as by emergence of a number of new mathematical disciplines, such as informatics, mathematical economics, numerical analysis, games theory, digital mathematics, programming, harmonic analysis, Fourier analysis. The need to improve control over different systems (physical, economic, social etc,) following differential equations, led to working out the mathematical theory of optimal control. In its turn, the need to control conflicting processes led to the beginning and progress of differential game theory. Mathematical physics required working out generalized functions and complex variable functions theories. Functional analysis, a revolutionary branch of mathematics, as well as the theory of differential variable function spaces were created.

Vocabulary:

1. to expand – расширять(ся), увеличивать

2. to emerge – появляться, выяснять

3. to work out – разрабатывать, решать

4. motion - движение

5. variable quantity – переменная величина

6. equation – уравнение, равенство

7. fraction - дробь

8. to verify – проверять, подтверждать

9. to deduce – сделать вывод

10. rigour - точность