7.Modern Mathematics (794204)

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Modern Mathematics.

Mathematics is the science expressing and studying the relationships between quantities and magnitudes as represented by numbers and symbols. It is the oldest of the sciences. Contemporary maths is a mixture of much that is very old and still important (counting, the Pythagorean theorem) with new concepts such as sets, axiomatics, structure.

At present maths means the combination of pure and applied maths. Pure maths deals with logical structures, and applied maths is the study of real objects by means of math methods, that is, the study of structures modeling real phenomena.

Maths as a science, viewed as a whole, is a collection of branches. The largest branch is that, which builds on the ordinary integers, 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 subjects which follow the calculus in logical order are all developments of the real number system.

The creations of calculus, non-Euclidean geometries, set theory and cybernetics may be considered as revolutions in maths. (The calculus was the first achievement of modern mathematics). The second branch is geometry consisting of several geometries (Lobachevsky, Riemann).

Each branch of maths has the same logical structure: it begins with certain concepts and these concepts must verify explicitly stated axioms. From the concepts and axioms theorems are deduced. Hence, from the stand point of structure, the concepts, axioms and theorems are essential components of any compartment of maths.

A prominent feature of modern mathematics is a new point of view, a new conception of the nature of maths. From this point of view maths is a science in its own right, independent of the natural sciences while still giving aid to them. Another characteristic of modern maths are a higher level of abstraction and greater rigour. The modern form of axiomatic method and a very high level of abstraction characterize today’s maths.

New fields of mathematics such as Operation Research, Game Theory, Information Theory, Numerical Analysis, Probability and Statistics, Measure Theory, Set Theory, Mathematical Logic and there applications accelerated new mathematical inventions and theories. The inevitable result was the opening of new fields of maths (at present there exist more then 90!) and the increase of the number of the professional mathematicians.

The study of the algebra of real numbers has led to the development of what is now called modern algebra or abstract algebra. The concepts of modern algebra have been found to be extremely useful in other branches of maths, as well as in the physical and social sciences. A chemist may use modern algebra in a study of the structure of crystals, a physicist may use modern algebra in designing an electronic computer.

Mathematics has also prepared the ground for great scientific achievements. When Einstein started his study of what is now known as the general theory of relativity (a branch of physics which eventually led to the discovery of nuclear energy and hence to one of the greatest advances in the history of mankind), he found the tool for the description of his idea ready-made in the calculus. Besides, Einstein could not have developed his ideas without the applications of the absolute calculus and the non-Euclidean geometry of 4-dimension spaces, as expanded by Lobachevsky and Minkovsky.

There is no doubt that a variety of new mathematical techniques will have to be invented to solve innumerable scientific problems of the XXI century. Maths is one of the loveliest products of the human mind, one of the most dynamic forces in our culture.

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