Vinogradov Mathematical Encyclopedia. Mathematical encyclopedia. Old and new classifications of mathematics

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MATHEMATICS. Mathematics is usually defined by listing the names of some of its traditional branches. First of all, this is arithmetic, which deals with the study of numbers, the relationships between them and the rules for working with numbers. The facts of arithmetic are open to various concrete interpretations; for example, the ratio 2 + 3 = 4 + 1 corresponds to the statement that two and three books make as many books as four and one. Any relation like 2 + 3 = 4 + 1, i.e. the relationship between purely mathematical objects without reference to any interpretation from the physical world is called abstract. The abstract nature of mathematics allows it to be used in solving a wide variety of problems. For example, algebra, which deals with operations on numbers, allows you to solve problems that go beyond arithmetic. A more specific branch of mathematics is geometry, the main task of which is the study of the sizes and shapes of objects. The combination of algebraic methods with geometric ones leads, on the one hand, to trigonometry (originally devoted to the study of geometric triangles, and now covering a much wider range of issues), and on the other hand, to analytic geometry, in which geometric bodies and figures are studied by algebraic methods. There are several branches of higher algebra and geometry that have a higher degree of abstraction and do not deal with the study of ordinary numbers and ordinary geometric figures; the most abstract of geometric disciplines is called topology.

Mathematical analysis deals with the study of quantities that change in space or time, and relies on two basic concepts - function and limit, which are not found in more elementary sections of mathematics. Initially, mathematical analysis consisted of differential and integral calculus, but now includes other sections.

There are two main areas of mathematics - pure mathematics, in which the emphasis is on deductive reasoning, and applied mathematics. The term "applied mathematics" sometimes refers to those branches of mathematics that are created specifically to meet the needs and requirements of science, and sometimes to those sections of various sciences (physics, economics, etc.) that use mathematics as a means of solving their tasks. Many common misconceptions about mathematics arise from the confusion between these two interpretations of "applied mathematics". Arithmetic can exemplify applied mathematics in the first sense, and accounting in the second.

Contrary to popular belief, mathematics continues to develop rapidly. The Mathematical Review publishes annually ca. 8000 short summaries of articles containing the latest results - new mathematical facts, new proofs of old facts, and even information about completely new areas of mathematics. The current trend in mathematics education is to introduce students to modern, more abstract mathematical ideas at an earlier stage in the teaching of mathematics. see also MATHEMATICS HISTORY. Mathematics is one of the cornerstones of civilization, but very few people have an idea of the current state of affairs in this science.

Mathematics has undergone tremendous changes in the last hundred years, both in terms of subject matter and methods of study. In this article, we will try to give a general idea of the main stages in the evolution of modern mathematics, the main results of which can be considered, on the one hand, an increase in the gap between pure and applied mathematics, and on the other, a complete rethinking of traditional areas of mathematics.

DEVELOPMENT OF THE MATHEMATICAL METHOD

The birth of mathematics.

Around 2000 BC it was noticed that in a triangle with sides of 3, 4 and 5 units of length, one of the angles is equal to 90 ° (this observation made it easy to build a right angle for practical needs). Did you notice then the relation 5 2 = 3 2 + 4 2 ? We do not have any information regarding this. A few centuries later, a general rule was discovered: in any triangle ABC with a right angle at the top A and parties b = AC and c = AB, between which this angle is enclosed, and the side opposite to it a = BC the ratio a 2 = b 2 + c 2. It can be said that science begins when a mass of individual observations is explained by one general law; hence the discovery of the "Pythagorean theorem" can be seen as one of the first known examples of a truly scientific achievement.

But even more important for science in general and for mathematics in particular is the fact that, along with the formulation of a general law, attempts to prove it appear, i.e. show that it necessarily follows from other geometric properties. One of the Eastern "proofs" is especially graphic in its simplicity: four triangles equal to a given one are inscribed in a square BCDE as shown in the drawing. square area a 2 is divided into four equal triangles with a total area of 2 bc and square AFGH area ( b – c) 2 . In this way, a 2 = (b – c) 2 + 2bc = (b 2 + c 2 – 2bc) + 2bc = b 2 + c 2. It is instructive to go one step further and find out more precisely which "previous" properties are supposed to be known. The most obvious fact is that since the triangles BAC and BEF precisely, without gaps and overlapping, “fitted” along the sides BA and bf, which means that the two corners at the vertices B and WITH in a triangle ABS together form an angle of 90° and therefore the sum of all three of its angles is 90° + 90° = 180°. The above "proof" also uses the formula ( bc/2) for the area of a triangle ABC with a 90° angle at the top A. In fact, other assumptions were also used, but what has been said is enough so that we can clearly see the essential mechanism of mathematical proof - deductive reasoning, which allows using purely logical arguments (based on properly prepared material, in our example - splitting the square) to deduce from known results new properties, as a rule, do not follow directly from the available data.

Axioms and methods of proof.

One of the fundamental features of the mathematical method is the process of creating, with the help of carefully constructed purely logical arguments, a chain of statements in which each successive link is connected to the previous ones. The first fairly obvious consideration is that any chain must have a first link. This circumstance became obvious to the Greeks when they began to systematize the code of mathematical arguments in the 7th century. BC. It took the Greeks approx. 200 years old, and the surviving documents provide only a rough idea of how exactly they acted. We have accurate information only about the final result of the research - the famous Beginnings Euclid (c. 300 BC). Euclid begins by enumerating the initial positions, from which all the rest are deduced in a purely logical way. These provisions are called axioms or postulates (the terms are practically interchangeable); they express either very general and somewhat vague properties of objects of any kind, such as "the whole is greater than the part," or some specific mathematical properties, such as the fact that for any two points there is a single straight line connecting them. Nor do we have any information about whether the Greeks attached any deeper meaning or significance to the "truth" of the axioms, although there are some hints that the Greeks discussed them for some time before accepting certain axioms. In Euclid and his followers, the axioms are presented only as starting points for the construction of mathematics, without any comment on their nature.

As for the methods of proof, they, as a rule, were reduced to the direct use of previously proven theorems. Sometimes, however, the logic of reasoning turned out to be more complex. We will mention here Euclid's favorite method, which has become part of the everyday practice of mathematics - indirect proof, or proof by contradiction. As an elementary example of proof by contradiction, we will show that a chessboard from which two corner fields are cut out, located at opposite ends of the diagonal, cannot be covered with dominoes, each of which is equal to two fields. (It is assumed that each square of the chessboard must be covered only once.) Suppose that the opposite ("opposite") statement is true, i.e. that the board can be covered with dominoes. Each tile covers one black and one white square, so no matter where the dominoes are placed, they cover an equal number of black and white squares. However, because two corner squares have been removed, the chessboard (which originally had as many black squares as white squares) has two more squares of one color than squares of the other color. This means that our original assumption cannot be true, as it leads to a contradiction. And since contradictory propositions cannot both be false at the same time (if one of them is false, then the opposite is true), our original assumption must be true, because the contradictory assumption is false; therefore, a chessboard with two cut-out corner squares placed diagonally cannot be covered with dominoes. So, to prove a certain statement, we can assume that it is false, and deduce from this assumption a contradiction with some other statement, the truth of which is known.

An excellent example of proof by contradiction, which became one of the milestones in the development of ancient Greek mathematics, is the proof that is not a rational number, i.e. not representable as a fraction p/q, where p and q- whole numbers. If , then 2 = p 2 /q 2 , whence p 2 = 2q 2. Suppose there are two integers p and q, for which p 2 = 2q 2. In other words, we assume that there exists an integer whose square is twice the square of another integer. If any integers satisfy this condition, then one of them must be less than all the others. Let's focus on the smallest of these numbers. Let it be a number p. Since 2 q 2 is an even number and p 2 = 2q 2 , then the number p 2 must be even. Since the squares of all odd numbers are odd, and the square p 2 is even, so the number itself p must be even. In other words, the number p twice some integer r. Because p = 2r and p 2 = 2q 2 , we have: (2 r) 2 = 4r 2 = 2q 2 and q 2 = 2r 2. The last equality has the same form as the equality p 2 = 2q 2 , and we can, repeating the same reasoning, show that the number q is even and that there is such an integer s, what q = 2s. But then q 2 = (2s) 2 = 4s 2 , and since q 2 = 2r 2 , we conclude that 4 s 2 = 2r 2 or r 2 = 2s 2. So we get a second integer that satisfies the condition that its square is twice the square of another integer. But then p cannot be the smallest such number (because r = p/2), although initially we assumed that it is the smallest of such numbers. Therefore, our original assumption is false, since it leads to a contradiction, and therefore there are no such integers p and q, for which p 2 = 2q 2 (i.e. such that ). And this means that the number cannot be rational.

From Euclid to the beginning of the 19th century.

During this period, mathematics has changed significantly as a result of three innovations.

(1) In the course of the development of algebra, a method of symbolic notation was invented, which made it possible to represent increasingly complex relations between quantities in an abbreviated form. As an example of the inconvenience that would arise if there were no such "cursive writing", let's try to convey in words the ratio ( a + b) 2 = a 2 + 2ab + b 2: "The area of a square with a side equal to the sum of the sides of two given squares is equal to the sum of their areas together with twice the area of a rectangle whose sides are equal to the sides of the given squares."

(2) Creation in the first half of the 17th century. analytical geometry, which made it possible to reduce any problem of classical geometry to some algebraic problem.

(3) The creation and development between 1600 and 1800 of infinitesimal calculus, which made it possible to easily and systematically solve hundreds of problems related to the concepts of limit and continuity, only a very few of which were solved with great difficulty by ancient Greek mathematicians. These branches of mathematics are considered in more detail in the articles ALGEBRA; ANALYTIC GEOMETRY ; MATHEMATICAL ANALYSIS ; GEOMETRY REVIEW.

Starting from the 17th century. gradually clears up the question, which until now remained unresolved. What is mathematics? Before 1800 the answer was simple enough. At that time, there were no clear boundaries between the various sciences, mathematics was part of "natural philosophy" - the systematic study of nature by the methods proposed by the great reformers of the Renaissance and the early 17th century. - Galileo (1564–1642), F. Bacon (1561–1626) and R. Descartes (1596–1650). It was believed that mathematicians had their own field of study - numbers and geometric objects, and that mathematicians did not use the experimental method. However, Newton and his followers studied mechanics and astronomy using the axiomatic method, similar to the way Euclid's geometry was presented. More generally, it was recognized that any science in which the results of an experiment can be represented using numbers or systems of numbers becomes the area of application of mathematics (in physics, this idea was established only in the 19th century).

Areas of experimental science that have undergone mathematical processing are often referred to as "applied mathematics"; this is a very unfortunate name, since neither by classical nor by modern standards in these applications there are (in the strict sense) truly mathematical arguments, since non-mathematical objects are the subject of study in them. Once the experimental data have been translated into the language of numbers or equations (such a "translation" often requires great ingenuity on the part of an "applied" mathematician), the possibility of a wide application of mathematical theorems appears; the result is then translated back and compared with the observations. The fact that the term "mathematics" is applied to a process of this kind is one of the sources of endless misunderstandings. In the "classical" times that we are now talking about, this kind of misunderstanding did not exist, since the same people were both "applied" and "pure" mathematicians, dealing simultaneously with the problems of mathematical analysis or number theory, and problems of dynamics or optics. However, increased specialization and the tendency to separate "pure" and "applied" mathematicians significantly weakened the previously existing tradition of universality, and scientists who, like J. von Neumann (1903–1957), were able to conduct active scientific activities both in applied and in pure mathematics, have become the exception rather than the rule.

What is the nature of mathematical objects - numbers, points, lines, angles, surfaces, etc., whose existence we took for granted? What does the concept of "truth" mean in relation to such objects? Quite definite answers were given to these questions in the classical period. Of course, the scientists of that era clearly understood that in the world of our sensations there are no such things as Euclid's "infinitely extended straight line" or "point without dimensions", just as there are no "pure metals", "monochromatic light", "heat-insulated systems", etc. .d., which experimenters operate in their reasoning. All these concepts are “Platonic ideas”, i.e. a kind of generative models of empirical concepts, although of a radically different nature. Nevertheless, it was tacitly assumed that the physical "images" of ideas could be arbitrarily close to the ideas themselves. To the extent that anything can be said about the proximity of objects to ideas, "ideas" are said to be, so to speak, "limiting cases" of physical objects. From this point of view, Euclid's axioms and the theorems derived from them express the properties of "ideal" objects, which must correspond to predictable experimental facts. For example, the measurement by optical methods of the angles of a triangle formed by three points in space, in the "ideal case" should give a sum equal to 180 °. In other words, the axioms are placed on the same level as physical laws, and therefore their "truth" is perceived in the same way as the truth of physical laws; those. the logical consequences of the axioms are subject to verification by comparison with experimental data. Of course, agreement can be reached only within the limits of the error associated both with the "imperfect" nature of the measuring device and the "imperfect nature" of the object being measured. However, it is always assumed that if the laws are "true", then improvements in measurement processes can, in principle, make the measurement error as small as desired.

Throughout the 18th century there was more and more evidence that all the consequences derived from the basic axioms, especially in astronomy and mechanics, are consistent with experimental data. And since these consequences were obtained using the mathematical apparatus that existed at that time, the successes achieved contributed to strengthening the opinion about the truth of Euclid's axioms, which, as Plato said, "is clear to everyone" and is not subject to discussion.

Doubts and new hopes.

Non-Euclidean geometry.

Among the postulates given by Euclid, one was so non-obvious that even the first students of the great mathematician considered it a weak point in the system. Started. The axiom in question states that through a point lying outside a given line, only one line can be drawn parallel to the given line. Most geometers believed that the axiom of parallels could be proved using other axioms, and that Euclid formulated the assertion of parallels as a postulate simply because he failed to come up with such a proof. But, although the best mathematicians tried to solve the parallel problem, none of them succeeded in surpassing Euclid. Finally, in the second half of the 18th century. Attempts were made to prove Euclid's postulate of parallels by contradiction. It has been suggested that the parallel axiom is false. A priori, Euclid's postulate could turn out to be false in two cases: if it is impossible to draw a single parallel line through a point outside the given line; or if several parallel lines can be drawn through it. It turned out that the first a priori possibility is ruled out by other axioms. Having adopted a new axiom instead of the traditional axiom about parallels (that through a point outside a given line, several lines parallel to a given one can be drawn), mathematicians tried to derive from it a statement that contradicted other axioms, but failed: no matter how much they tried to extract consequences from the new "anti-Euclidean" or "non-Euclidean" axiom, the contradiction did not appear. Finally, independently of each other, N.I. Lobachevsky (1793–1856) and J. Bolyai (1802–1860) realized that Euclid’s postulate about parallels is unprovable, or, in other words, a contradiction will not appear in “non-Euclidean geometry”.

With the advent of non-Euclidean geometry, several philosophical problems immediately arose. Since the claim to the a priori necessity of the axioms disappeared, the only way to test their "truth" remained - experimentally. But, as A. Poincaré (1854–1912) later noted, in the description of any phenomenon there are so many physical assumptions hidden that no experiment can provide convincing proof of the truth or falsity of a mathematical axiom. Moreover, even if we assume that our world is "non-Euclidean", does it follow that all Euclidean geometry is false? As far as is known, no mathematician has ever considered such a conjecture seriously. Intuition suggested that both Euclidean and non-Euclidean geometries are examples of full-fledged mathematics.

Mathematical monsters.

Unexpectedly, the same conclusions came from a completely different direction - objects were discovered that plunged the mathematicians of the 19th century. shocked and dubbed "math monsters". This discovery is directly related to very subtle questions of mathematical analysis that arose only in the middle of the 19th century. Difficulties arose when trying to find an exact mathematical analogue of the experimental concept of a curve. What was the essence of the concept of "continuous motion" (for example, the tip of a drawing pen moving across a sheet of paper) was subject to precise mathematical definition, and this goal was achieved when the concept of continuity acquired a rigorous mathematical meaning ( cm. also CURVE). Intuitively, it seemed that the “curve” at each of its points had, as it were, a direction, i.e. in the general case, in a neighborhood of each of its points, the curve behaves almost in the same way as a straight line. (On the other hand, it is easy to imagine that a curve has a finite number of corner points, "kinks", like a polygon.) This requirement could be formulated mathematically, namely, the existence of a tangent to the curve was assumed, and until the middle of the 19th century. it was believed that the "curve" had a tangent at almost all of its points, perhaps with the exception of some "special" points. Therefore, the discovery of "curves" that did not have a tangent at any point caused a real scandal ( cm. also FUNCTIONS THEORY). (The reader familiar with trigonometry and analytic geometry can easily verify that the curve given by the equation y = x sin(1/ x), does not have a tangent at the origin, but defining a curve that does not have a tangent at any of its points is much more difficult.)

Somewhat later, a much more "pathological" result was obtained: it was possible to construct an example of a curve that completely fills the square. Since then, hundreds of such “monsters” have been invented, contrary to “common sense”. It should be emphasized that the existence of such unusual mathematical objects follows from the basic axioms as strictly and logically flawless as the existence of a triangle or an ellipse. Since mathematical "monsters" cannot correspond to any experimental object, and the only possible conclusion is that the world of mathematical "ideas" is much richer and more unusual than one might expect, and very few of them have correspondences in the world of our sensations. But if mathematical "monsters" logically follow from the axioms, then can the axioms still be considered true?

New objects.

The above results were confirmed from another side: in mathematics, mainly in algebra, new mathematical objects began to appear one after another, which were generalizations of the concept of number. Ordinary integers are quite “intuitive” and it is not at all difficult to arrive at an experimental concept of a fraction (although it must be admitted that the operation of dividing a unit into several equal parts and choosing several of them is inherently different from the process of counting). After it became clear that a number cannot be represented as a fraction, the Greeks were forced to consider irrational numbers, the correct definition of which, using an infinite sequence of approximations by rational numbers, belongs to the highest achievements of the human mind, but hardly corresponds to anything real in our physical world (where any measurement is invariably subject to errors). Nevertheless, the introduction of irrational numbers took place more or less in the spirit of the "idealization" of physical concepts. But what about negative numbers, which slowly, meeting with great resistance, began to enter into scientific use in connection with the development of algebra? It can be stated with all certainty that there were no ready-made physical objects, starting from which we could develop the concept of a negative number using the process of direct abstraction, and in the teaching of an elementary algebra course we have to introduce many auxiliary and rather complex examples (oriented segments, temperatures, debts, etc.) to explain what negative numbers are. This position is very far from being "clear to everyone" as Plato demanded of the ideas underlying mathematics, and it is not uncommon to meet college graduates for whom the rule of signs is still a mystery (- a)(–b) = ab. see also NUMBER .

The situation is even worse with "imaginary", or "complex" numbers, since they include a "number" i, such that i 2 = -1, which is a clear violation of the sign rule. Nevertheless, mathematicians from the end of the 16th century. do not hesitate to perform calculations with complex numbers as if they "make sense", although 200 years ago they could not define these "objects" or interpret them using any auxiliary construction, as, for example, they were interpreted using directed segments negative numbers. (After 1800, several interpretations of complex numbers were proposed, the most famous being by means of vectors in the plane.)

modern axiomatics.

The revolution took place in the second half of the 19th century. And although it was not accompanied by the adoption of official statements, in reality it was about the proclamation of a kind of "declaration of independence." More specifically, about the proclamation of a de facto declaration of the independence of mathematics from the outside world.

From this point of view, mathematical "objects", if it makes sense to speak of their "existence" at all, are pure creations of the mind, and do they have any "correspondences" and whether they allow any "interpretation" in the physical world, for mathematics is unimportant (although the question itself is interesting).

"True" statements about such "objects" are all the same logical consequences from the axioms. But now the axioms must be regarded as completely arbitrary, and therefore there is no need for them to be "obvious" or deducible from everyday experience by means of "idealization". In practice, complete freedom is limited by various considerations. Of course, the "classical" objects and their axioms remain unchanged, but now they cannot be considered the only objects and axioms of mathematics, and the habit of throwing out or reworking the axioms so that it is possible to use them in various ways, as was done during the transition, has entered into everyday practice. from Euclidean to non-Euclidean geometry. (This is how numerous variants of "non-Euclidean" geometries other than Euclidean geometry and Lobachevsky-Bolyai geometry were obtained; for example, there are non-Euclidean geometries in which there are no parallel lines.)

I would like to emphasize one circumstance that follows from the new approach to mathematical "objects": all proofs must be based solely on axioms. If we recall the definition of a mathematical proof, then such a statement may seem like a repetition. However, this rule was rarely followed in classical mathematics due to the "intuitive" nature of its objects or axioms. Even in Beginnings Euclid, for all their seeming "strictness", many axioms are not formulated explicitly and many properties are either tacitly assumed or introduced without sufficient justification. In order to put Euclidean geometry on a solid foundation, a critical revision of its very principles was needed. Needless to say, the pedantic control over the smallest details of the proof is a consequence of the appearance of "monsters" that have taught modern mathematicians to be careful in their conclusions. The most innocuous and “self-evident” assertion about classical objects, such as the assertion that a curve connecting points located on opposite sides of a straight line, necessarily intersects this straight line, in modern mathematics requires a rigorous formal proof.

It may seem paradoxical to say that it is precisely because of its adherence to axioms that modern mathematics serves as a clear example of what any science should be. Nevertheless, this approach illustrates a characteristic feature of one of the most fundamental processes of scientific thinking - obtaining accurate information in a situation of incomplete knowledge. The scientific study of a certain class of objects suggests that the features that make it possible to distinguish one object from another are deliberately forgotten, and only the general features of the objects under consideration are preserved. What distinguishes mathematics from the general range of sciences is the strict adherence to this program in all its points. It is believed that mathematical objects are completely determined by the axioms used in the theory of these objects; or, in Poincaré's words, axioms serve as "definitions in disguise" of the objects to which they refer.

MODERN MATHEMATICS

Although the existence of any axioms is theoretically possible, only a small number of axioms have been proposed and studied so far. Usually, in the course of developing one or more theories, it is noticed that some schemes of proof are repeated in more or less similar conditions. After the properties used in the general schemes of proofs are discovered, they are formulated in the form of axioms, and the consequences of them are built into a general theory that is not directly related to the specific contexts from which the axioms were abstracted. The general theorems thus obtained are applicable to any mathematical situation in which there are systems of objects that satisfy the corresponding axioms. The repetition of the same proof schemes in different mathematical situations indicates that we are dealing with different concretizations of the same general theory. This means that after an appropriate interpretation, the axioms of this theory become theorems in every situation. Any property deduced from the axioms will hold true in all these situations, but there is no need for a separate proof for each case. In such cases, the mathematical situations are said to have the same mathematical "structure".

We use the concept of structure at every step in our daily lives. If the thermometer reads 10°C and the forecast office predicts a temperature increase of 5°C, we expect a temperature of 15°C without any calculations. If the book is opened to page 10 and we are asked to look 5 pages further, we do not hesitate to open it on the 15th page, without counting the intermediate pages. In both cases, we believe that the addition of numbers gives the correct result, regardless of their interpretation - in the form of temperature or page numbers. We don't need to learn one arithmetic for thermometers and another for page numbers (although we use a special arithmetic for clocks, in which 8 + 5 = 1, since clocks have a different structure than the pages of a book). The structures of interest to mathematicians are distinguished by a somewhat higher complexity, which is easy to see from the examples, the analysis of which is devoted to the next two sections of this article. One of them deals with the theory of groups and the mathematical concepts of structures and isomorphisms.

Group theory.

To better understand the process outlined above, let us take the liberty of looking into the laboratory of the modern mathematician and taking a closer look at one of his main tools - group theory ( cm. also ALGEBRA ABSTRACT). A group is a collection (or "set") of objects G, on which an operation is defined that associates any two objects or elements a, b from G, taken in the specified order (the first is the element a, the second is the element b), the third element c from G according to a strictly defined rule. For brevity, we denote this element a*b; the asterisk (*) means the operation of composition of two elements. This operation, which we will call group multiplication, must satisfy the following conditions:

(1) for any three elements a, b, c from G the associativity property is satisfied: a* (b*c) = (a*b) *c;

(2) in G there is such an element e, which for any element a from G there is a relation e*a = a*e = a; this element e is called the identity or neutral element of the group;

(3) for any element a from G there is such an element a¢, called inverse or symmetrical to element a, what a*aў = aў* a = e.

If these properties are taken as axioms, then the logical consequences of them (independent of any other axioms or theorems) together form what is commonly called group theory. Deriving these consequences once and for all proved to be very useful, since groups are widely used in all branches of mathematics. Of the thousands of possible examples of groups, we will choose only a few of the simplest ones.

(a) Fractions p/q, where p and q are arbitrary integers i1 (for q= 1 we get ordinary integers). Fractions p/q form a group with respect to group multiplication ( p/q) *(r/s) = (pr)/(qs). Properties (1), (2), (3) follow from the axioms of arithmetic. Really, [( p/q) *(r/s)] *(t/u) = (prt)/(qsu) = (p/q)*[(r/s)*(t/u)]. The identity element is the number 1 = 1/1, since (1/1)*( p/q) = (1H p)/(1H q) = p/q. Finally, the element inverse to the fraction p/q, is a fraction q/p, because ( p/q)*(q/p) = (pq)/(pq) = 1.

(b) Consider as G set of four integers 0, 1, 2, 3, and as a*b- remainder of the division a + b 4. The results of the operation thus introduced are presented in Table. 1 (element a*b stands at the intersection of the line a and column b). It is easy to check that properties (1)–(3) are satisfied, and the number 0 is the unit element.

(c) We choose as G set of numbers 1, 2, 3, 4, and as a*b- remainder of the division ab(ordinary product) by 5. As a result, we get the table. 2. It is easy to check that properties (1)–(3) are satisfied, and 1 is the identity element.

(d) Four objects, such as the four numbers 1, 2, 3, 4, can be arranged in a row in 24 ways. Each location can be visualized as a transformation that translates the "natural" location into a given one; for example, the location 4, 1, 2, 3 is obtained as a result of the transformation

S: 1 ® 4, 2 ® 1, 3 ® 2, 4 ® 3,

which can be written in a more convenient form

For any two such transformations S, T we will define S*T as a transformation that will result from sequential execution T, and then S. For example, if , then . With this definition, all 24 possible transformations form a group; its identity element is , and the element inverse to S, is obtained by replacing the arrows in the definition S to the opposite; for example, if , then .

It is easy to see that in the first three examples a*b = b*a; in such cases the group or group multiplication is said to be commutative. On the other hand, in the last example , and hence T*S differs from S*T.

The group from example (d) is a special case of the so-called. symmetric group, whose scope of applications includes, among other things, methods for solving algebraic equations and the behavior of lines in the spectra of atoms. The groups in examples (b) and (c) play an important role in number theory; in example (b) the number 4 can be replaced by any integer n, and numbers from 0 to 3 - numbers from 0 to n– 1 (when n= 12 we get the system of numbers that are on the clock faces, as we mentioned above); in example (c) the number 5 can be replaced by any prime number R, and numbers from 1 to 4 - numbers from 1 to p – 1.

Structures and isomorphism.

The previous examples show how varied the nature of the objects that make up a group can be. But in fact, in each case, everything comes down to the same scenario: of the properties of a set of objects, we consider only those that turn this set into a group (this is an example of incomplete knowledge!). In such cases, we say that we are considering a group structure given by our chosen group multiplication.

Another example of a structure is the so-called. order structure. A bunch of E endowed with an order structure, or ordered if between elements a è b belonging to E, some relation is given, which we denote R (a,b). (Such a relation should make sense for any pair of elements from E, but in general it is false for some pairs and true for others, for example, the relation 7

(1) R (a,a) is true for each a owned by E;

(2) out R (a,b) and R (b,a) follows that a = b;

(3) out R (a,b) and R (b,c) should R (a,c).

Let us give some examples from a huge number of various ordered sets.

(a) E consists of all integers, R (a,b) is the relation " a less or equal b».

(b) E consists of all integers >1, R (a,b) is the relation " a divides b or equal b».

As a last example of a structure, we mention the structure of a metric space; such a structure is given on the set E, if each pair of elements a and b belonging to E, you can match the number d (a,b) i 0 satisfying the following properties:

(1) d (a,b) = 0 if and only if a = b;

(2) d (b,a) = d (a,b);

(3) d (a,c) Ј d (a,b) + d (b,c) for any three given elements a, b, c from E.

Let us give examples of metric spaces:

(a) the usual "three-dimensional" space, where d (a,b) is the usual (or "Euclidean") distance;

(b) the surface of a sphere, where d (a,b) is the length of the smallest arc of a circle connecting two points a and b on the sphere;

The exact definition of the concept of structure is quite difficult. Without going into details, we can say that on the set E a structure of a certain type is given if between the elements of the set E(and sometimes other objects, for example, numbers, which play an auxiliary role) relations are given that satisfy some fixed set of axioms that characterizes the structure of the type under consideration. Above we have given axioms of three types of structures. Of course, there are many other types of structures whose theories are fully developed.

Many abstract concepts are closely related to the concept of structure; Let us name only one of the most important - the concept of isomorphism. Recall the example of groups (b) and (c) from the previous section. It is easy to check that from Tab. 1 to table. 2 can be navigated using matching

0 ® 1, 1 ® 2, 2 ® 4, 3 ® 3.

In this case, we say that the given groups are isomorphic. In general, two groups G and Gў are isomorphic if between the elements of the group G and group elements G¢ it is possible to establish such a one-to-one correspondence a « a¢ what if c = a*b, then cў = aў* b¢ for relevant elements Gў. Any statement from group theory that is true for a group G, remains valid for the group G¢, and vice versa. Algebraically groups G and G¢ indistinguishable.

The reader will easily see that in exactly the same way one can define two isomorphic ordered sets or two isomorphic metric spaces. It can be shown that the concept of isomorphism extends to structures of any type.

CLASSIFICATION

Old and new classifications of mathematics.

The concept of structure and other concepts related to it have taken a central place in modern mathematics, both from a purely “technical” and from a philosophical and methodological point of view. General theorems of the main types of structures serve as extremely powerful tools of mathematical "technique". Whenever a mathematician succeeds in showing that the objects he studies satisfy the axioms of a certain type of structure, he thereby proves that all the theorems of the theory of structure of this type apply to the specific objects he studies (without these general theorems, he very likely missed would be out of sight of their specific variants or would be forced to burden their reasoning with unnecessary assumptions). Similarly, if two structures are proven to be isomorphic, then the number of theorems immediately doubles: each theorem proved for one of the structures immediately gives a corresponding theorem for the other. It is not surprising, therefore, that there are very complex and difficult theories, for example, the "class field theory" in number theory, the main purpose of which is to prove the isomorphism of structures.

From a philosophical point of view, the widespread use of structures and isomorphisms demonstrates the main feature of modern mathematics - the fact that the "nature" of mathematical "objects" does not really matter, only the relationships between objects are significant (a kind of the principle of incomplete knowledge).

Finally, it is impossible not to mention that the concept of structure made it possible to classify sections of mathematics in a new way. Until the middle of the 19th century. they differed according to the subject of the study. Arithmetic (or number theory) dealt with integers, geometry dealt with lines, angles, polygons, circles, areas, and so on. Algebra dealt almost exclusively with methods for solving numerical equations or systems of equations; analytic geometry developed methods for transforming geometric problems into equivalent algebraic problems. The range of interests of another important branch of mathematics, called "mathematical analysis", included mainly differential and integral calculus and their various applications to geometry, algebra, and even number theory. The number of these applications increased, and their importance also increased, which led to the division of mathematical analysis into subsections: the theory of functions, differential equations (ordinary and partial derivatives), differential geometry, calculus of variations, etc.

For many modern mathematicians, this approach recalls the history of the classification of animals by the first naturalists: once both the sea turtle and tuna were considered fish because they lived in water and had similar features. The modern approach has taught us to see not only what lies on the surface, but also to look deeper and try to recognize the fundamental structures that lie behind the deceptive appearance of mathematical objects. From this point of view, it is important to study the most important types of structures. It is unlikely that we have at our disposal a complete and definitive list of these types; some of them have been discovered in the last 20 years, and there is every reason to expect more discoveries in the future. However, we already have an idea of many basic "abstract" types of structures. (They are "abstract" in comparison with the "classical" objects of mathematics, although even those can hardly be called "concrete"; it is rather a matter of the degree of abstraction.)

Known structures can be classified according to the relationships they contain or according to their complexity. On the one hand, there is an extensive block of "algebraic" structures, a special case of which is, for example, a group structure; among other algebraic structures we name rings and fields ( cm. also ALGEBRA ABSTRACT). The branch of mathematics concerned with the study of algebraic structures has been called "modern algebra" or "abstract algebra" in contrast to conventional or classical algebra. A significant part of Euclidean geometry, non-Euclidean geometry and analytic geometry also became part of the new algebra.

There are two other blocks of structures at the same level of generality. One of them, called general topology, includes theories of types of structures, a special case of which is the structure of a metric space ( cm. TOPOLOGY; abstract spaces). The third block consists of theories of order structures and their extensions. The "expansion" of the structure consists in adding new ones to the existing axioms. For example, if we add the property of commutativity to the axioms of the group as the fourth axiom a*b = b*a, then we get the structure of a commutative (or abelian) group.

Of these three blocks, the last two until recently were in a relatively stable state, and the “modern algebra” block was growing rapidly, sometimes in unexpected directions (for example, an entire branch was developed, called “homological algebra”). Outside the so-called. "pure" types of structures lies another level - "mixed" structures, for example, algebraic and topological, together with new axioms linking them. Many such combinations have been studied, most of which fall into two broad blocks - "topological algebra" and "algebraic topology".

Taken together, these blocks make up a very solid "abstract" area of science in terms of volume. Many mathematicians hope to better understand classical theories and solve difficult problems with new tools. Indeed, with an appropriate level of abstraction and generalization, the problems of the ancients can appear in a new light, which will make it possible to find their solutions. Huge chunks of classical material came under the sway of the new mathematics and were transformed or merged with other theories. There remain vast areas in which modern methods have not penetrated so deeply. Examples are the theory of differential equations and a significant part of number theory. It is very likely that significant progress in these areas will be achieved after new types of structures are discovered and carefully studied.

PHILOSOPHICAL DIFFICULTIES

Even the ancient Greeks clearly understood that a mathematical theory should be free from contradictions. This means that it is impossible to deduce as a logical consequence from the axioms the statement R and its denial P. However, since it was believed that mathematical objects have correspondences in the real world, and axioms are "idealizations" of the laws of nature, no one had any doubts about the consistency of mathematics. In the transition from classical mathematics to modern mathematics, the problem of consistency acquired a different meaning. The freedom to choose the axioms of any mathematical theory must be obviously limited by the consistency condition, but is it possible to be sure that this condition will be satisfied?

We have already mentioned the concept of a set. This concept has always been used more or less explicitly in mathematics and logic. In the second half of the 19th century elementary rules for dealing with the concept of a set were partially systematized, in addition, some important results were obtained, which formed the content of the so-called. set theory ( cm. also SET THEORY), which has become, as it were, the substratum of all other mathematical theories. From antiquity to the 19th century. there were fears about infinite sets, for example, reflected in the famous paradoxes of Zeno of Elea (5th century BC). These fears were partly metaphysical, and partly due to the difficulties associated with the concept of measuring quantities (for example, length or time). It was only after the 19th century that these difficulties were eliminated. the basic concepts of mathematical analysis were strictly defined. By 1895, all fears were dispelled, and it seemed that mathematics rested on the unshakable foundation of set theory. But in the next decade, new arguments arose that seemed to show the inherent inconsistency of set theory (and all the rest of mathematics).

The new paradoxes were very simple. The first of these - Russell's paradox - can be considered in a simple version, known as the "barber's paradox". In a certain town, a barber shaves all the inhabitants who do not shave themselves. Who shaves the barber himself? If a barber shaves himself, then he shaves not only those residents who do not shave themselves, but also one inhabitant who shaves himself; if he does not shave himself, then he does not shave all the inhabitants of the town who do not shave themselves. A paradox of this type arises whenever the concept of "the set of all sets" is considered. Although this mathematical object seems very natural, reasoning about it quickly leads to contradictions.

Berry's paradox is even more revealing. Consider the set of all Russian phrases containing no more than seventeen words; the number of words in the Russian language is finite, so the number of such phrases is also finite. We choose among them those that uniquely define some integer, for example: "The largest odd number less than ten." The number of such phrases is also finite; consequently, the set of integers they define is also finite. Denote a finite set of these numbers by D. It follows from the axioms of arithmetic that there are integers that do not belong to D, and that among these numbers there is the smallest number n. This number n is uniquely defined by the phrase: "The smallest integer that cannot be defined by a phrase consisting of no more than seventeen Russian words." But this phrase contains exactly seventeen words. Therefore, it determines the number n, which should belong D, and we arrive at a paradoxical contradiction.

Intuitionists and Formalists.

The shock caused by the paradoxes of set theory gave rise to a variety of reactions. Some mathematicians were quite determined and expressed the opinion that mathematics developed in the wrong direction from the very beginning and should be based on a completely different foundation. It is not possible to describe the point of view of such "intuitionists" (as they began to call themselves) with any precision, since they refused to reduce their views to a purely logical scheme. From the point of view of intuitionists, it is wrong to apply logical processes to objects that are not intuitively representable. The only intuitively clear objects are natural numbers 1, 2, 3,... and finite sets of natural numbers, "built" according to exactly given rules. But even to such objects, the intuitionists did not allow all the deductions of classical logic to be applied. For example, they did not recognize that for any statement R true either R, or not- R. With such limited means at their disposal, they easily avoided "paradoxes", but in doing so they threw overboard not only all of modern mathematics, but also a significant part of the results of classical mathematics, and for those that still remained, new, more complex proofs had to be found.

The overwhelming majority of modern mathematicians disagreed with the arguments of the intuitionists. Non-intuitionist mathematicians have noticed that the arguments used in paradoxes differ significantly from those used in ordinary mathematical work with set theory, and therefore such arguments should be ruled out as illegal without compromising existing mathematical theories. Another observation was that in the "naive" set theory that existed before the advent of "paradoxes", the meaning of the terms "set", "property", "relation" was not questioned - just as in classical geometry the "intuitive" nature of ordinary geometric concepts. Consequently, one can proceed in the same way as it was in geometry, namely, to discard all attempts to appeal to "intuition" and take as the starting point of set theory a system of precisely formulated axioms. However, it is not obvious how such words as "property" or "relation" can be deprived of their usual sense; yet it must be done if we wish to rule out such arguments as Berry's paradox. The method consists in refraining from using ordinary language in formulating axioms or theorems; only sentences constructed according to an explicit system of rigid rules are allowed as "properties" or "relations" in mathematics and enter into the formulation of axioms. This process is called the "formalization" of the mathematical language (to avoid misunderstandings arising from the ambiguities of ordinary language, it is recommended to go one step further and replace the words themselves with special characters in formalized sentences, for example, replace the connective "and" with the symbol &, the connective "or" - with the symbol Ъ, “exists” with the symbol $, etc.). Mathematicians who rejected the methods proposed by the intuitionists came to be called "formalists."

However, the original question was never answered. Is "axiomatic set theory" free from contradictions? New attempts to prove the consistency of "formalized" theories were made in the 1920s by D. Hilbert (1862-1943) and his school and were called "metamathematics". Essentially, metamathematics is a branch of "applied mathematics" where the objects to which mathematical reasoning is applied are the propositions of a formalized theory and their location within proofs. These sentences are to be regarded only as material combinations of symbols produced according to certain established rules, without any reference whatsoever to the possible "meaning" of these symbols (if there is one). A game of chess can serve as a good analogy: symbols correspond to pieces, sentences to different positions on the board, and inferences to rules for moving pieces. To establish the consistency of a formalized theory, it suffices to show that in this theory, no proof ends with the statement 0 No. 0. However, one can object to the use of mathematical arguments in the "metamathematical" proof of the consistency of a mathematical theory; if mathematics were inconsistent, then mathematical arguments would lose all force, and we would be in a situation of a vicious circle. To answer these objections, Hilbert allowed for use in metamathematics very limited mathematical reasoning of the type that intuitionists consider acceptable. However, K. Godel soon showed (1931) that the consistency of arithmetic cannot be proved by such limited means if it is really consistent (the scope of this article does not allow us to present the ingenious method by which this remarkable result was obtained, and the subsequent history of metamathematics).

Summarizing the current problematic situation from a formalist point of view, we must admit that it is far from over. The use of the concept of a set has been limited by reservations that have been deliberately introduced to avoid known paradoxes, and there are no guarantees that new paradoxes will not arise in an axiomatized set theory. Nevertheless, the limitations of axiomatic set theory did not prevent the birth of new viable theories.

MATH AND THE REAL WORLD

Despite claims of independence of mathematics, no one will deny that mathematics and the physical world are related to each other. Of course, the mathematical approach to solving the problems of classical physics remains valid. It is also true that in a very important area of mathematics, namely in the theory of differential equations, ordinary and partial derivatives, the process of mutual enrichment of physics and mathematics is quite fruitful.

Mathematics is useful in interpreting the phenomena of the microworld. However, the new "applications" of mathematics differ significantly from the classical ones. One of the most important tools of physics has become the theory of probability, which was previously used mainly in the theory of gambling and insurance. The mathematical objects that physicists associate with "atomic states" or "transitions" are highly abstract in nature and were introduced and studied by mathematicians long before the advent of quantum mechanics. It should be added that after the first successes, serious difficulties arose. This happened at a time when physicists were trying to apply mathematical ideas to the finer aspects of quantum theory; nevertheless, many physicists still look forward to new mathematical theories, believing that they will help them solve new problems.

Mathematics - science or art?

Even if we include probability theory or mathematical logic in "pure" mathematics, it turns out that at present other sciences use less than 50% of known mathematical results. What should we think of the remaining half? In other words, what are the motives behind those areas of mathematics that are not related to the solution of physical problems?

We have already mentioned the irrationality of a number as a typical representative of this kind of theorems. Another example is the theorem proved by J.-L. Lagrange (1736–1813). There is hardly a mathematician who would not call her "important" or "beautiful." Lagrange's theorem states that any integer greater than or equal to one can be represented as the sum of the squares of at most four numbers; for example, 23 = 3 2 + 3 2 + 2 2 + 1 2 . In the present state of affairs, it is inconceivable that this result could be useful in solving any experimental problem. It is true that physicists deal with integers much more frequently today than in the past, but the integers with which they operate are always limited (they rarely exceed a few hundred); therefore, a theorem like Lagrange's can only be "useful" if applied to integers that do not go beyond some boundary. But as soon as we restrict the formulation of Lagrange's theorem, it immediately ceases to be of interest to a mathematician, since the whole attractive power of this theorem lies in its applicability to all integers. (There are a great many propositions about integers that can be tested by computers for very large numbers; but, as long as no general proof is found, they remain hypothetical and of no interest to professional mathematicians.)

A focus on topics that are far from immediate applications is not unusual for scientists working in any field, be it astronomy or biology. However, while the experimental result can be refined and improved, the mathematical proof is always final. That is why it is difficult to resist the temptation to treat mathematics, or at least that part of it that has nothing to do with "reality", as an art. Mathematical problems are not imposed from without, and, if we take the modern point of view, we are completely free in the choice of material. When evaluating some mathematical work, mathematicians do not have "objective" criteria, and they are forced to rely on their own "taste". Tastes vary greatly depending on time, country, traditions and individuals. There are fashions and "schools" in modern mathematics. At the present time there are three such "schools" which for convenience we shall call "classicism", "modernism" and "abstractionism". To better understand the differences between them, let's analyze the various criteria that mathematicians use when evaluating a theorem or a group of theorems.

(1) According to the general opinion, a "beautiful" mathematical result should be non-trivial, i.e. must not be an obvious consequence of axioms or previously proven theorems; some new idea must be used in the proof, or old ideas must be ingeniously applied. In other words, for a mathematician, it is not the result itself that is important, but the process of overcoming the difficulties that he encountered in obtaining it.

(2) Any mathematical problem has its own history, so to speak, "pedigree", which follows the same general pattern in which the history of any science develops: after the first successes, a certain time may pass before the answer to the question posed is found. When a decision is received, the story does not end there, for the well-known processes of expansion and generalization begin. For example, the Lagrange theorem mentioned above leads to the question of representing any integer as a sum of cubes, powers of 4, 5, etc. This is how the “Waring problem” arises, which has not yet received a final solution. Also, if we are lucky, the problem we have solved will turn out to be related to one or more fundamental structures, and this, in turn, will lead to new problems related to these structures. Even if the original theory eventually "dies," it tends to leave behind numerous living shoots. Modern mathematicians are faced with such an immense scattering of problems that, even if all connection with experimental science were interrupted, their solution would take several more centuries.

(3) Every mathematician will agree that when a new problem is presented to him, it is his duty to solve it by any means possible. When the problem concerns classical mathematical objects (classicists rarely deal with other types of objects), classicists try to solve it using only classical means, while other mathematicians introduce more "abstract" structures in order to use general theorems related to task. This difference in approach is not new. Starting from the 19th century. mathematicians are divided into "tacticians" who seek to find a purely forceful solution to the problem, and into "strategists" who are prone to detours that make it possible to crush the enemy with small forces.

(4) An essential element of the "beauty" of a theorem is its simplicity. Of course, the search for simplicity is inherent in all scientific thought. But experimenters are ready to put up with "ugly solutions" if only the problem is solved. Similarly, in mathematics, classicists and abstractionists are not very concerned about the appearance of "pathological" results. On the other hand, modernists go so far as to see the appearance of “pathologies” in theory as a symptom of the imperfection of fundamental concepts.

Mathematical encyclopedia - a reference book on all branches of mathematics. The Encyclopedia is based on review articles devoted to the most important areas of mathematics. The main requirement for articles of this type is the possible completeness of the review of the current state of the theory with the maximum accessibility of the presentation; these articles are generally available to senior mathematics students, graduate students and specialists in related fields of mathematics, and in certain cases to specialists in other fields of knowledge who use mathematical methods in their work, engineers and teachers of mathematics. Further, medium-sized articles on individual specific problems and methods of mathematics are provided; these articles are intended for a narrower circle of readers, so the presentation in them may be less accessible. Finally, there is one more type of articles - brief references-definitions. At the end of the last volume of the Encyclopedia a subject index will be placed, which will include not only the titles of all articles, but also many concepts, the definitions of which will be given inside the articles of the first two types, as well as the most important results mentioned in the articles. Most of the articles of the Encyclopedia are accompanied by a list of references with serial numbers for each title, which makes it possible to cite in the texts of the articles. At the end of the articles (as a rule) the author or source is indicated if the article has already been published earlier (mostly these are articles of the Great Soviet Encyclopedia). The names of foreign (except ancient) scientists mentioned in the articles are accompanied by Latin spelling (if there is no reference to the list of references).

Download and read Mathematical Encyclopedia, Volume 3, Vinogradov I.M., 1982

Download and read Mathematical Encyclopedia, Volume 2, Vinogradov I.M., 1979

Download and read Mathematical Encyclopedia, Volume 1, Vinogradov I.M., 1977

Algebra was originally a branch of mathematics concerned with solving equations. Unlike geometry, the axiomatic construction of algebra did not exist until the middle of the 19th century, when a fundamentally new view of the subject and nature of algebra appeared. Research began to focus more and more on the study of so-called algebraic structures. This had two benefits. On the one hand, the areas for which certain theorems are valid were clarified, on the other hand, it became possible to use the same proofs in completely different areas. This division of algebra lasted until the middle of the 20th century and found its expression in the fact that two names appeared: “classical algebra” and “modern algebra”. The latter is more characterized by another name: "abstract algebra". The fact is that this section - for the first time in mathematics - was characterized by complete abstraction.

Download and read Small Mathematical Encyclopedia, Fried E., Pastor I., Reiman I., Reves P., Ruja I., 1976

"Probability and Mathematical Statistics" - a reference book on the theory of probability, mathematical statistics and their applications in various fields of science and technology. The encyclopedia has two parts: the main part contains review articles, articles devoted to individual specific problems and methods, brief references giving definitions of basic concepts, the most important theorems and formulas. A significant place is given to applied issues - information theory, queuing theory, reliability theory, experiment planning and related areas - physics, geophysics, genetics, demography, and certain sections of technology. Most of the articles are accompanied by a bibliography of the most important papers on this issue. The titles of the articles are also given in English translation. The second part - "Reader on Probability Theory and Mathematical Statistics" contains articles written for Russian encyclopedias of the past, as well as encyclopedic materials published earlier in other works. The encyclopedia is accompanied by an extensive list of journals, periodicals and ongoing publications covering problems of probability theory and mathematical statistics.
The material included in the Encyclopedia is necessary for students, graduate students and researchers in the field of mathematics and other sciences who use probabilistic methods in their research and practical work.

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Mathematical Encyclopedia

Mathematical Encyclopedia- Soviet encyclopedic publication in five volumes devoted to mathematical topics. Released in -1985 by the publishing house "Soviet Encyclopedia". Editor-in-Chief: Academician I. M. Vinogradov.

This is a fundamental illustrated edition on all major branches of mathematics. The book contains extensive material on the topic, biographies of famous mathematicians, drawings, graphs, charts and diagrams.

Total volume: about 3000 pages. Distribution of articles by volumes:

Volume 1: Abacus - Huygens principle, 576 pp.
Volume 2: D'Alembert Operator - Co-op Game, 552 pp.
Volume 3: Coordinates - Monomial, 592 pp.
Volume 4: The Eye of the Theorem - Complex Function, 608 pp.
Volume 5: Random Variable - Cell, 623 pp.
Appendix to volume 5: subject index, list of typographical errors noticed.

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General and special reference books and encyclopedias in mathematics on the World of Mathematical Equations portal, where you can download the encyclopedia in electronic form.

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Mathematical encyclopedia (set of 5 books), . The Mathematical Encyclopedia is a convenient reference book on all branches of mathematics. The Encyclopedia is based on articles devoted to the most important areas of mathematics. Location principle...

Mathematical encyclopedia - a reference book on all branches of mathematics. The Encyclopedia is based on review articles devoted to the most important areas of mathematics. The main requirement for articles of this type is the possible completeness of the review of the current state of the theory with the maximum accessibility of the presentation; these articles are generally available to senior mathematics students, graduate students and specialists in related fields of mathematics, and in certain cases - to specialists in other fields of knowledge using mathematical methods in their work, engineers and teachers of mathematics. Further, medium-sized articles on individual specific problems and methods of mathematics are provided; these articles are intended for a narrower circle of readers, so the presentation in them may be less accessible. Finally, there is one more type of articles - brief references-definitions. Some definitions are given inside the first two types of articles. Most of the articles of the Encyclopedia are accompanied by a list of references with serial numbers for each title, which makes it possible to cite in the texts of the articles. At the end of the articles (as a rule) the author or source is indicated if the article has already been published earlier (mostly these are articles of the Great Soviet Encyclopedia). The names of foreign (except ancient) scientists mentioned in the articles are accompanied by Latin spelling (if there is no reference to the list of references).

The principle of arrangement of articles in the Encyclopedia is alphabetical. If the title of the article is a term that has a synonym, then the latter is given after the main one. In many cases, article titles consist of two or more words. In these cases, the terms are given either in the most common form, or the main word in meaning is placed in the first place. If the title of an article includes a proper name, it is placed first (in the list of references for such articles, as a rule, there is a primary source explaining the name of the term). The titles of the articles are given mostly in the singular.

The Encyclopedia widely uses a system of links to other articles, where the reader will find additional information to the topic under consideration. The definition does not refer to the term appearing in the title of the article.

In order to save space in the articles, the usual abbreviations of some words for encyclopedias are adopted.

Worked on volume 1

Mathematics Editorial Board of the Soviet Encyclopedia Publishing House - V. I. BITYUTSKOV (Head of the Editorial Board), M. I. VOITSEHOVSKY (Scientific Editor), Yu. A. GORBKOV (Scientific Editor), A. B. IVANOV (Senior Scientific Editor), O A. IVANOVA (senior scientific editor), T. Yu. L. R. KHABIB (Associate Editor).

Employees of the publishing house: E. P. RYABOVA (literary editorial board). E. I. ZHAROVA, A. M. MARTYNOV (bibliography). A. F. DALKOVSKY (transcription). N. A. FEDOROV (Procurement Department). 3. A. SUKHOVA (Editorial illustrations). E. I. ALEKSEEVA, N. YU. KRUZHALOV (redaction dictionary). M. V. AKIMOVA, A. F. PROSHKO (proofreading). G. V. SMIRNOV (technical edition).

Cover by artist R. I. MALANICHEV.

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A. M. PROKHOROV (Chairman), I. V. ABASHIDZE, P. A. AZIMOV, A. P. ALEKSANDROV, V. A. AMBARTZUMYAN, I. I. ARTOBOLEVSKY, A. V. ARTSIKHOVSKY, M. S. ASIMOV , M. P. Bazhan, Yu. Ya. Barabash, N. V. Baranov, N. N. Bogolyubov, P. U. Brovka, Yu. M. Volodarsky, V. V. Volsky, B. M. Vul, B. G. Gafurov, S. R. Gershberg, M. S. Gilyarov, V. P. Glushko, V. M. Glushkov, G. N GOLIKOV, D. B. GULIEV, A. A. GUSEV (Deputy Chairman), V. P. ELYUTIN, V. S. EMELYANOV, E. M. ZHUKOV, A. A. IMSHENETSKY, N. N. INOZEMTSEV, M I. Kabachnik, S. V. Kalesnik, G. A. Karavaev, K. K. Karakeev, M. K. Karataev, B. M. Kedrov, G. V. Keldysh, V. A. Kirillin, and I. L KNUNYANTS, S. M. KOVALEV (First Deputy Chairman), F. V. KONSTANTINOV, V. N. KUDRYAVTSEV, M. I. KUZNETSOV (Deputy Chairman), B. V. KUKARKIN, V. G. KULIKOV, I. A. Kutuzov, P. P. Lobanov, G. M. Loza, Yu. E. Maksarev, P. A. Markov, A. I. Markushevich, Yu. Yu. Obichkin, B. E. Paton, V. M. Polevo J, M. A. Prokofiev, Yu. V. Prokhorov, N. F. Rostovtsev, A. M. Rumyantsev, B. A. Rybakov, V. P. Samson, M. I. Sladkovsky, V. I. Smirnov, D. N. SOLOVIEV (deputy chairman), V. G. SOLODOVNIKOV, V. N. STOLETOV, B. I. STUKALIN, A. A. SURKOV, M. L. TERENTIEV, S. A. TOKAREV, V. A. Trapeznikov, E. K. Fedorov, M. B. Khrapchenko, E. I. Chazov, V. N. Chernigovskii, Ya. E. Shmushkis, and S. I. Yutkevich Secretary of the Council L. V. KIRILLOVA.

Moscow 1977

Mathematical encyclopedia. Volume 1 (A - D)

Editor-in-Chief I. M. VINOGRADOV

Editorial team

S. I. ADYAN, P. S. ALEKSANDROV, N. S. BAKHVALOV, V. I. BITYUTSKOV (deputy editor-in-chief), A. V. BITSADZE, L. N. BOLSHEV, A. A. GONCHAR, N. V. Efimov, V. A. Ilyin, A. A. Karatsuba, L. D. Kudryavtsev, B. M. Levitan, K. K. Mardzhanishvili, E. F. Mishchenko, S. P. Novikov, and E. G. Poznyak , Yu. V. PROKHOROV (deputy editor-in-chief), A. G. SVESHNIKOV, A. N. TIKHONOV, P. L. ULYANOV, A. I. SHIRSHOV, S. V. YABLONSKY

Mathematical Encyclopedia. Ed. collegium: I. M. Vinogradov (head of editor) [and others] T. 1 - M., "Soviet Encyclopedia", 1977

(Encyclopedias. Dictionaries. Reference books), vol. 1. A - G. 1977. 1152 stb. from ill.

Handed over to the set 9. 06. 1976. Signed for printing 18. 02. 1977. Printing of text from matrices made in the First Exemplary Printing House. A. A. Zhdanova. Order of the Red Banner of Labor, publishing house "Soviet Encyclopedia". 109817. Moscow, Zh - 28, Pokrovsky Boulevard, 8. T - 02616 Circulation 150,000 copies. Order No. 418. Printing paper No. 1. Paper size 84xl08 1/14. Volume 36 physical p. l. ; 60, 48 conv. p. l. text. 101, 82 accounts - ed. l. The price of the book is 7 rubles. 10 k.

Order of the Red Banner of Labor Moscow Printing House No. 1 "Soyuzpoligrafprom" under the State Committee of the Council of Ministers of the USSR for Publishing, Printing and Book Trade, Moscow, I - 85, Prospekt Mira, 105. Order No. 865.