# Mathematics

The science of quantitative relations and spatial forms in the real world. Being inseparably connected with the needs of technology and natural science, the accumulation of quantitative relations and spatial forms studied in mathematics is continuously expanding; so this general definition of mathematics becomes ever richer in content.

The clear recognition of the independent position of mathematics as a separate science became possible only after the collection of a fairly large store of factual material, and arose first in Ancient Greece in the 6th–5th centuries B.C.. The development of mathematics up to that time is naturally referred to as the period of the origin of mathematics, and the 6th–5th centuries B.C. as the beginning of the period of elementary mathematics. During these first two periods mathematical investigation dealt almost exclusively with very restricted questions concerning fundamental ideas, already there in the very early stages of historical development in relation with very simple aspects of economic life. The first problems of mechanics and physics were already in this collection of fundamental mathematical ideas.

In the 17th century new questions in natural science and technology compelled mathematicians to concentrate their attention on the creation of methods to allow the mathematical study of motion, the processes of variation of quantities and the transformation of geometrical figures. With the use of variable quantities in analytic geometry and the creation of differential and integral calculus, the period of the mathematics of variable quantities began.

Further expansion of the circle of quantitative relations and spatial forms studied in mathematics led, at the beginning of the 19th century, to the need to consciously regard the very process of expansion as a topic of mathematical research by setting the problem of systematically studying, from a fairly general point of view, the possible types of quantitative relations and spatial forms. The creation of the "imaginary" geometry of Lobachevskii was the first significant step in this direction. The development of this type of research introduced such an important new feature into mathematics that mathematics in the 19th century and 20th century is naturally referred to as the special period of modern mathematics.

## 1. The origin of mathematics.

Consideration of the objects in the very early stages in the development of cultures leads to the creation of the simplest ideas of the arithmetic of natural numbers. Based on an elaborate system of verbal calculation, written systems of calculation arose and slowly the methods of doing the four arithmetic operations over the natural numbers were perfected. The demands of measurement (quantity of grain, length of a road, etc.) led to the emergence of names and notation for the simplest fractions and to the elaboration of methods for performing arithmetic operations on fractions. In this way material was accumulated which gradually added up to that most ancient mathematical science: arithmetic. The measurement of area and volume, the needs of building technology and, somewhat later, astronomy motivated the development of the rudiments of geometry. These processes occurred in many nations, largely independently and in parallel. Of special significance for the later development of science was the accumulation of arithmetic and geometric knowledge in Egypt and Babylon. In Babylon, on the basis of the techniques which were developed for arithmetic calculation, algebra appeared also in connection with the needs of astronomy, and the rudiments of trigonometry appeared.

## 2. The period of elementary mathematics.

Only after the accumulation of a larger amount of concrete material in the form of un-coordinated methods of arithmetic calculation and methods for determining area and volume, did mathematics arise as an independent science with a clear understanding of the originality of its method and the necessity for a systematic development of its basic concepts and assumptions in a fairly general form. In the application to arithmetic and algebra this process had already begun in Babylon. However, this new trend, comprising the systematic and logical succession of the construction of the foundations of mathematics, was fully defined in Ancient Greece. The system of exposition of elementary geometry created by the Ancient Greeks remained for more than two thousand years the standard for the deductive construction of a mathematical theory. From arithmetic gradually grew number theory. The systematic study of magnitudes and measurements was created. The process of forming (in connection with the problems of measurement of magnitude) the notion of a real number (see Number) turned out to be very protracted. The problem was that the idea of an irrational or a negative number was related to more complicated mathematical abstractions which, in contrast to the concepts of a natural number, a fraction or a geometric figure, have no fairly sound support in prescientific ordinary human experience. The creation of algebra as a literal calculus was completed only at the end of this period. The period of elementary mathematics ended (in Western Europe at the beginning of the 17th century), when the emphasis of mathematical interests shifted to the domain of mathematics of variable quantities.

## 3. The period of creation of the mathematics of variable quantities.

With the 17th century an essentially new period in the development of mathematics began. The circle of quantitative relations and spatial forms of mathematics studied now was no longer exhausted by numbers, quantities and geometric figures. On this basis there resulted the explicit introduction into mathematics of the ideas of motions and change. Already algebra contained the idea of dependence between variables in a latent form (the value of a sum depends on the values of the terms, etc.). However, in order to include quantitative relations in the process of variation it was necessary that the very dependence between the variables be made an independent object of study. Therefore, in the first scheme the notion of a function was put forward, which later played the same role of fundamental and independent object of study as the notion of quantity or number had played earlier. The study of variable quantities and functional dependence leads to the fundamental ideas of mathematical analysis, introducing explicitly into mathematics the idea of the infinite, the notions of a limit, a derivative, a differential, and an integral. Infinitesimal analysis was born; in the first place in the form of the differential calculus and integral calculus, allowing one to relate finite variations of variable quantities to their behaviour in an immediate vicinity of their individual values. The basic laws of mechanics and physics were described by differential equations, and the problem of investigating these equations comes to the foreground as one of the major problems of mathematics. The search for unknown functions defined by conditions of another kind (conditions of maxima or minima of certain related quantities) forms the topic of the calculus of variations (cf. Variational calculus). In this way, side-by-side with equations in which the unknowns are numbers, equations emerge in which functions are the unknowns and have to be determined.

The subject of geometry is also significantly expanded with the penetration into geometry of the ideas of motions and transformations of figures. Geometry begins to study motions and transformations for their own sake. For example, in projective geometry one of the basic objects of study is the set of projective transformations of the plane or space. However, the conscious development of these ideas dates only from the end of the 18th century and the beginning of the 19th century. Much earlier, with the advent of analytic geometry in the 17th century, the relation of geometry to the remainder of mathematics was essentially changed; a universal method was found for transferring questions of geometry into the language of algebra and analysis and for solving them neatly by algebraic and analytic methods. On the other hand, the broad possibility of sketching (illustrating) algebraic and analytic facts by geometric means was discovered, for example, in the graphical illustration of functional dependence.

## 4. Modern mathematics.

All the divisions of mathematical analysis created in the 17th century and 18th century continued to develop with great intensity in the 19th century and 20th century. The circle of applications to problems of science and technology was greatly expanded at this time. However, in addition to this quantitative growth, at the end of the 18th century and the beginning of the 19th century a number of essentially new features were observed in the development of mathematics.

The enormous amount of factual material which had been accumulated in the 17th century and 18th century led to the demand for a deep logical analysis and unification of it from a new point of view. In essence, the relationship between mathematics and natural science was no less close but was now increased in complexity. The majority of new theories arose not just as a result of the immediate needs of natural science and technology, but also from internal requirements of mathematics itself. Such, in essence, was the development of the theory of functions of a complex variable (cf. Functions of a complex variable, theory of), which occupied a central position in mathematical analysis at the beginning and middle of the 19th century. Another remarkable example of a theory arising as a result of the internal development of mathematics was Lobachevskii geometry.

In more immediate and continuous relation to the needs of mechanics and physics the formation of vector and tensor calculus arose. The translation of vector and tensor notions into infinite-dimensional quantities resulted in the framework of functional analysis and is closely connected with the requirements of modern physics.

In this way, as a result of both the internal requirements of mathematics and the new needs of natural science, the circle of quantitative relations and spatial forms studied in mathematics was greatly expanded: relations between elements of arbitrary groups, operations in function spaces, the whole diversity of forms of spaces of any number of dimensions, etc. are now parts of mathematics.

The essential novelty of this stage in the development of mathematics, beginning in the 19th century, is that questions concerning the necessary expansion of the circle of ideas in the study of quantitative relations and spatial forms themselves became the subject of conscious and active interest for mathematicians. If before, for example, the introduction of negative and complex numbers and the exact formulation of their rules of operation required protracted effort, now the development of mathematics required elaboration of methods for the deliberate and planned creation of new geometric and algebraic systems.

The extraordinary expansion of the subject of mathematics in the 19th century attracted vigorous attention to the question of its "foundations" , that is, a critical review of its initial conditions (axioms), the construction of rigorous systems of definitions and proofs and also a critical consideration of the logical methods used in these proofs. The standard requirement of logical rigour imposed on the practical work of mathematicians in the development of individual mathematical theories was added only at the end of the 19th century. A deep and careful analysis of the requirement of logical rigour in proofs, the construction of mathematical theories, questions of algorithmic solvability and unsolvability of mathematical problems, comprise the subject of mathematical logic.

At the beginning of the 19th century a new significant expansion of the domain of application of mathematical analysis was initiated. Up to this time the fundamental divisions of physics demanding the greatest mathematical apparatus were mechanics and optics. Now were added electrodynamics and the theories of magnetism and thermodynamics. The important division of the mechanics of continuous media underwent broad development. The mathematical needs of technology also grew rapidly. As the fundamental apparatus of the new areas of mechanics and mathematical physics, the theories of ordinary and partial differential equations and of the equations of mathematical physics were strongly developed (cf. Differential equation, ordinary; Differential equation, partial; Mathematical physics, equations of).

The theory of differential equations served as the starting point for the investigation of the topology of manifolds. Here were obtained the first "combinatorial" , "homological" and "homotopical" methods of algebraic topology. Another direction in topology arose, based on set theory and functional analysis, and has led to the systematic construction of the theory of general topological spaces (cf. Topological space).

An essential completion to the methods of differential equations in the study of the character and solution of technical problems are the methods of probability theory. If at the beginning of the 19th century the principal use of probabilistic methods was in ballistics and the theory of errors, then at the end of the 19th century and the beginning of the 20th century probability theory had many applications, due to the creation of the theory of random processes and the development of the apparatus of mathematical statistics.

The theory of numbers, a collection of diverse results and ideas, was developed in various directions in the 19th century as a structured theory (see Algebraic number theory; Analytic number theory; Diophantine approximations).

The emphasis of algebraic research shifted to new areas of algebra: the theory of groups, rings, fields, and general algebraic structures. On the interface between algebra and geometry grew the theory of continuous groups, the methods of which later permeated into all new areas of mathematics and natural science.

Elementary and projective geometry attracted the attention of mathematicians mainly from the point of the study of logical and axiomatic foundations. But the fundamental areas of geometry on which most of the significant scientific force was concentrated became differential geometry; algebraic geometry and Riemannian geometry.

As a result of the systematic construction of mathematical analysis based on the strictly arithmetic theory of irrational numbers and on set theory, the theory of functions of a real variable arose (cf. Functions of a real variable, theory of).

The practical application of the results of pure mathematical research required obtaining an answer to a given problem in numerical form. However, even after exhaustive theoretical analysis of a problem this often turns out to be a very difficult matter. The numerical methods of analysis and algebra that arose at the end of the 19th century and the beginning of the 20th century have grown with the manufacture and use of computers into an independent branch of mathematics: computational mathematics.

These fundamental distinctive features of modern mathematics and the basic directions of research within the divisions of mathematics are listed as at the beginning of the 20th century. To a large extent this division into branches has been preserved in spite of the tremendous growth of mathematics in the 20th century. However, the demand for the development of mathematics itself, the "mathematization" of various domains of science, the penetration of mathematical methods into many spheres of practical activity, and the rapid progress in computational techniques has led to a mixing of the basic efforts of mathematicians over the branches of mathematics and to the appearance of a whole series of new mathematical disciplines (see, for example, Automata, theory of; Information theory; Games, theory of; Operations research, and also Cybernetics; Mathematical economics). On the basis of problems in the theory of control systems (cf. Control system), combinatorial analysis, graph theory, coding theory, and discrete analysis arose. Questions on optimal (in some sense) control of physical or mechanical systems described by differential equations has led to the creation of the mathematical theory of optimal control (cf. Optimal control, mathematical theory of).

Studies in the domain of general control problems and related areas of mathematics, in conjunction with the progress in computational techniques, provided a basis for the automatization of new spheres of human activity.

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#### Comments

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Mathematics.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Mathematics&oldid=23895