Speech therapy portal / Planning / Mathematical encyclopedia. Encyclopedia of mathematics See what "Mathematical encyclopedia" is in other dictionaries

Mathematical encyclopedia. Encyclopedia of mathematics See what "Mathematical encyclopedia" is in other dictionaries

04.01.2024

Mathematical encyclopedia. Encyclopedia of mathematics See what it is

Mathematical Encyclopedia - a reference publication 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 overview of the current state of the theory with maximum accessibility of presentation; These articles are generally accessible 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 mathematics teachers. Further, medium-sized articles on individual specific problems and methods of mathematics are provided; These articles are intended for a narrower readership and may therefore be less accessible. Finally, another type of article is brief references and definitions. At the end of the last volume of the Encyclopedia there will be a subject index, which will include not only the titles of all articles, but also many concepts, the definitions of which will be given within articles of the first two types, as well as the most important results mentioned in the articles. Most of the Encyclopedia articles are accompanied by a bibliography with serial numbers for each title, which makes it possible to cite them 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 previously (mainly these are articles in the Great Soviet Encyclopedia). The names of foreign (except ancient) scientists mentioned in articles are accompanied by Latin spelling (if there is no link 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 increasingly focus on the study of so-called algebraic structures. This had two advantages. On the one hand, the areas for which individual 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 was reflected in the appearance of two names: “classical algebra” and “modern algebra”. The latter is better 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., Ruzsa I., 1976

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

Mathematical Encyclopedia - a reference publication 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 overview of the current state of the theory with maximum accessibility of presentation; These articles are generally accessible 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 mathematics teachers. Further, medium-sized articles on individual specific problems and methods of mathematics are provided; These articles are intended for a narrower readership and may therefore be less accessible. Finally, another type of article is brief references and definitions. Some definitions are given within the first two types of articles. Most of the Encyclopedia articles are accompanied by a bibliography with serial numbers for each title, which makes it possible to cite them 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 previously (mainly these are articles in the Great Soviet Encyclopedia). The names of foreign (except ancient) scientists mentioned in articles are accompanied by Latin spelling (if there is no link 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 their most common form, or the word with the most important meaning is given first place. If the title of an article includes a proper name, it is placed in first place (the list of references for such articles, as a rule, contains a primary source explaining the name of the term). The titles of articles are given primarily in the singular.

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

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

Worked on volume 1

Mathematics Editorial Board of the Publishing House "Soviet Encyclopedia" - V. I. BITYUTSKOV (Editorial Head), M. I. VOITSEKHOVSKY (Scientific Editor), Yu. A. GORBKOV (Scientific Editor), A. B. IVANOV (Senior Scientific Editor), O A. IVANOVA (senior scientific editor), T. Y. POPOVA (scientific editor), S. A. RUKOVA (senior scientific editor), E. G. SOBOLEVSKAYA (editor), L. V. SOKOLOVA (junior editor), L. R. HABIB (junior editor).

Publishing house staff: E. P. RYABOVA (literary editors). E. I. ZHAROVA, A. M. MARTYNOV (bibliography). A. F. DALKOVSKAYA (transcription). N. A. FEDOROVA (acquisition department). 3. A. SUKHOVA (edition of illustrations). E. I. ALEXEEVA, N. Y. KRUZHALOVA (editor of the dictionary). M. V. AKIMOVA, A. F. PROSHKO (proofreader). G. V. SMIRNOVA (technical edition).

Cover by artist R.I. MALANICHEV.

Additional information about volume 1

Publishing house "Soviet Encyclopedia"

Encyclopedias, dictionaries, reference books

Scientific and editorial council of the publishing house

A. M. PROKHOROV (chairman), I. V. ABASHIDZE, P. A. AZIMOV, A. P. ALEXANDROV, V. A. AMBARTSUMYAN, I. I. ARTOBOLEVSKY, A. V. ARTSIKHOVSKY, M. S. ASIMOV , M. P. BAZHAN, Y. Y. BARABASH, N. V. BARANOV, N. N. BOGOLYUBOV, P. U. BROVKA, Y. V. BROMLEY, B. E. BYKHOVSKY, V. X. VASILENKO, L 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. ELUTIN, V. S. EMELYANOV, E. M. ZHUKOV, A. A. IMSHENETSKY, N. N. INOZEMTSEV, M A. I. KABACHNIK, S. V. KALESNIK, G. A. KARAVAEV, K. K. KARAKEEV, M. K. KARATAEV, B. M. KEDROV, G. V. KELDYSH, V. A. KIRILLIN, 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, Y. E. MAKSAREV, P. A. MARKOV, A. I. MARKUSHEVICH, Y. Y. MATULIS, G. I. NAAN, G. D. OBICHKIN, B. E. PATON, V. M. POLEVOY, M. A. PROKOFIEV, Y. 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. STUCALIN, A. A. SURKOV, M. L. TERENTYEV, S. A. TOKAREV, V. A. TRAPEZNIKOV, E. K. FEDOROV, M. B. KHRAPCHENKO, E. I. CHAZOV, V. N. CHERNIGOVSKY, Y. E. SHMUSHKIS, 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. ALEXANDROV, 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. MARZHANISHVILI, E. F. MISHCHENKO, S. P. NOVIKOV, E. G. POZNYAK , Y. V. PROKHOROV (deputy editor-in-chief), A. G. SVESHNIKOV, A. N. TIKHONOV, P. L. ULYANOV, A. I. SHIRSHOV, S. V. YABLONSKY

Mathematical Encyclopedia. Ed. board: I. M. Vinogradov (chief editor) [and others] T. 1 - M., “Soviet Encyclopedia”, 1977

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

Submitted for typesetting on June 9, 1976. Signed for printing on February 18, 1977. Printing text from matrices made at the First Model Printing House named after. 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 format 84xl08 1/14. Volume 36 physical. p.l. ; 60, 48 conventional p.l. text. 101, 82 academic. - ed. l. The price of the book is 7 rubles. 10 k.

Order of the Red Banner of Labor Moscow Printing House No. 1 "Soyuzpoligrafproma" 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.

20200 - 004 subscription © Publishing house "Soviet Encyclopedia", 1977 007(01) - 77

Mathematical Encyclopedia

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

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

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

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: Eye of the Theorem - Complex Function, 608 pp.
Volume 5: Random Variable - Cell, 623 pp.
Appendix to volume 5: index, list of noted typos.

Links

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

Categories:

Books in alphabetical order
Mathematical literature
Encyclopedias
Books from the publishing house "Soviet Encyclopedia"
Encyclopedias of the USSR

Wikimedia Foundation. 2010.

Mathematical chemistry
Mathematical foundations of quantum mechanics

See what "Mathematical Encyclopedia" is in other dictionaries:

Mathematical logic- (theoretical logic, symbolic logic) a branch of mathematics that studies proofs and questions of the foundations of mathematics. “The subject of modern mathematical logic is diverse.” According to the definition of P. S. Poretsky, “mathematical ... ... Wikipedia

Encyclopedia- (new Latin encyclopaedia (not earlier than the 16th century) from other Greek ἐγκύκλιος παιδεία “learning in a full circle”, κύκλος circle and παιδεία learning/paideia) brought into the system about ... Wikipedia

ENCYCLOPEDIA- (from the Greek enkyklios paideia training in the entire range of knowledge), scientific. or scientific popular reference publication containing systematized information. body of knowledge. The material in E. is arranged alphabetically or systematically. principle (by branches of knowledge).... ... Natural science. encyclopedic Dictionary

MATHEMATICAL LOGIC- one of the names of modern logic that came in the second. floor. 19 start 20th century to replace traditional logic. The term symbolic logic is also used as another name for the modern stage in the development of the science of logic. Definition… … Philosophical Encyclopedia

MATHEMATICAL INFINITY- common name for decomposition. implementations of the idea of infinity in mathematics. Although between the meanings of the concept M. b. and other meanings in which the term infinity is used, there is no hard limit (since all these concepts ultimately reflect very ... ... Philosophical Encyclopedia

MATHEMATICAL INDUCTION- complete mathematical induction (in mathematics is often called simply complete induction; in this case, this concept should be distinguished from the concept of complete induction considered in non-mathematical formal logic), - a method of proving general propositions in ... ... Philosophical Encyclopedia

MATHEMATICAL HYPOTHESIS- a presumable change in the form, type, character of the equation expressing the law of the studied area of phenomena, with the aim of extending it to a new, yet unstudied area as an inherent law. M. g. is widely used in modern times. theoretical... ... Philosophical Encyclopedia

MATHEMATICAL SCHOOL IN POLITICAL ECONOMY- English mathematical school in political economy; German mathematische Schule in der politischen Okonomie. The direction in politics, economy, which arose in the second half of the 19th century, was given by representatives (L. Walras, V. Pareto, O. Jevons, etc.) ... ... Encyclopedia of Sociology

MATHEMATICAL SCHOOL IN SOCIOLOGY- English mathematical school in sociology; German mathematische Schule in der Soziologie. A trend in sociology that arose in the first half of the 20th century, the founders of sociology (A. Zipf, E. Dodd, etc.) believed that a sociologist’s theories reach the level of... ... Encyclopedia of Sociology

Mathematical model of buildings and structures- Mathematical (computer) model of buildings and structures - representation of buildings and structures in the form of a finite element diagram for carrying out numerical calculations when solving a set of problems arising during design, construction and... ... Encyclopedia of terms, definitions and explanations of building materials

Books

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

Download the book Mathematical Encyclopedia in 5 volumes absolutely free.

In order to download a book for free from file hosting services, click on the links immediately following the description of the free book.

Dear readers, if it didn’t work out for you

download Mathematical encyclopedia in 5 volumes

write about it in the comments and we will definitely help you.

We hope you liked the book and enjoyed reading it. As a thank you, you can leave a link to our website on the forum or blog :) The electronic book Mathematical Encyclopedia in 5 volumes is provided solely for review before purchasing a paper book and is not a competitor to printed publications.

The content of the article

MATHEMATICS. Mathematics is usually defined by listing the names of some of its traditional branches. First of all, it is arithmetic, which deals with the study of numbers, the relationships between them and the rules for operating numbers. The facts of arithmetic are susceptible of various specific interpretations; for example, the relation 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. a 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 to solve a wide variety of problems. For example, algebra, which deals with operations on numbers, can 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 analytical 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 is based on two basic concepts - function and limit, which are not found in the more elementary branches of mathematics. Initially, mathematical analysis consisted of differential and integral calculus, but now includes other sections.

There are two main branches of mathematics - pure mathematics, which emphasizes deductive reasoning, and applied mathematics. The term “applied mathematics” sometimes refers to those branches of mathematics that were created specifically to satisfy 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 confusing these two interpretations of "applied mathematics." Arithmetic may be an example of applied mathematics in the first sense, and accounting in the second.

Contrary to popular belief, mathematics continues to advance rapidly. The journal Mathematical Review publishes approx. 8,000 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 earlier in mathematics teaching. 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 enormous changes over the past hundred years, both in its subject matter and in its research methods. 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 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 90° (this observation made it easy to construct a right angle for practical needs). Did you then notice the ratio 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 right angle at apex A and the parties b = AC And c = AB, between which this angle is enclosed, and the opposite side a = B.C. the ratio is valid a 2 = b 2 + c 2. We can say that science begins when a mass of individual observations is explained by one general law; therefore, the discovery of the "Pythagorean theorem" can be considered 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 appear to prove it, i.e. show that it necessarily follows from other geometric properties. One of the eastern “proofs” is especially clear in its simplicity: four triangles equal to this one are inscribed in a square BCDE as shown in the drawing. Square area a 2 turns out to be divided into four equal triangles with a total area of 2 bc and square AFGH area ( b – c) 2 . Thus, 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 what “previous” properties are supposed to be known. The most obvious fact is that since triangles BAC And BEF exactly, without gaps or overlap, “fitted” along the sides B.A. And B.F., this means that the two vertex angles B And WITH in a triangle ABC together form an angle of 90° and therefore the sum of all three of its angles is equal to 90° + 90° = 180°. The above "proof" also uses the formula ( bc/2) for the area of a triangle ABC with an angle of 90° at the apex 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 - dividing a 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, using carefully constructed purely logical arguments, a chain of statements in which each subsequent link is connected to the previous ones. The first fairly obvious consideration is that in any chain there must be a first link. This circumstance became obvious to the Greeks when they began to systematize a body of mathematical arguments in the 7th century. BC. To implement this plan, the Greeks needed approx. 200 years ago, and the surviving documents provide only a rough idea of exactly how they operated. We have accurate information only about the final result of the research - the famous Beginnings Euclid (c. 300 BC). Euclid begins by listing the initial positions, from which all others are derived purely logically. 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, for example, “the whole is greater than the part,” or some specific mathematical properties, for example, that for any two points there is a unique straight line connecting them. We have no information as to whether the Greeks attached some 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, axioms are presented only as starting points for the construction of mathematics, without any commentary on their nature.

As for the methods of proof, they, as a rule, boiled down 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 a proof by contradiction, we will show that a chessboard from which two corner squares are cut out, located at opposite ends of the diagonal, cannot be covered with dominoes, each of which is equal to two squares. (It is assumed that each square of the chessboard should 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 how the dominoes are arranged, they cover an equal number of black and white squares. However, because the two corner squares are removed, the chessboard (which originally had as many black squares as white) has two more squares of one color than squares of the other color. This means that our initial assumption cannot be true, as it leads to a contradiction. And since propositions that contradict each other cannot be false at the same time (if one of them is false, then the opposite is true), our initial assumption must be true, because the assumption that contradicts it is false; therefore, a chessboard with two corner squares cut out diagonally cannot be covered with dominoes. So, in order 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 a 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, from where 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 is an integer whose square is twice the square of another integer. If any integers satisfy this condition, then one of them must be smaller 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, which means the number itself p must be even. In other words, the number p twice the size of 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. This gives us a second integer that satisfies the condition that its square is twice the square of the other integer. But then p cannot be the smallest such number (since r = p/2), although initially we assumed that it was the smallest of such numbers. Therefore, our initial 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 ). This means that the number cannot be rational.

From Euclid to the beginning of the 19th century.

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

(1) In the process of the development of algebra, a method of symbolic notation was invented that made it possible to represent in an abbreviated form increasingly complex relationships between quantities. As an example of the inconveniences that would arise if there were no such “cursive writing”, let’s try to convey in words the relationship ( 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 plus 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 in the period from 1600 to 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 discussed in more detail in the articles ALGEBRA; ANALYTIC GEOMETRY ; MATHEMATICAL ANALYSIS ; GEOMETRY REVIEW.

Since the 17th century. The question, which until now remained insoluble, is gradually becoming clearer. What is mathematics? Before 1800 the answer was quite simple. At that time, there were no clear boundaries between the various sciences; mathematics was part of “natural philosophy” - the systematic study of nature using the methods proposed by the great reformers of the Renaissance and 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 how geometry was presented by Euclid. 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 a field of application of mathematics (in physics, this idea was established only in the 19th century).

Fields of experimental science that have undergone mathematical treatment are often called "applied mathematics"; This is a very unfortunate name, since, neither by classical nor by modern standards, there are (in the strict sense) truly mathematical arguments in these applications, since the subject of study in them is non-mathematical objects. Once the experimental data are translated into the language of numbers or equations (such a “translation” often requires great resourcefulness on the part of the “applied” mathematician), it becomes possible to widely apply mathematical theorems; the result is then back-translated and compared with 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 talking about now, this kind of misunderstanding did not exist, since the same people were both “applied” and “pure” mathematicians, simultaneously working on problems of mathematical analysis or number theory, and problems of dynamics or optics. However, increased specialization and the tendency to separate “pure” and “applied” mathematics significantly weakened the previously existing tradition of universality, and scientists who, like J. von Neumann (1903–1957), were able to conduct active scientific work in both 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 “truth” mean in relation to such objects? Quite definite answers were given to these questions in the classical period. Of course, scientists of that era clearly understood that in the world of our sensations there are no such things as “an infinitely extended straight line” or “a dimensionless point” of Euclid, 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 physical “images” of ideas could be as close as desired to the ideas themselves. To the extent that anything can be said at all 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 to which predictable experimental facts must correspond. For example, measuring by optical methods 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, 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. logical consequences of the axioms are subject to verification by comparison with experimental data. Of course, agreement can be achieved only within the limits of error associated with both the “imperfect” nature of the measuring instrument and the “imperfect nature” of the measured object. 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 obtained 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 unobvious that even the first students of the great mathematician considered it a weak point in the system Began. The axiom in question states that through a point lying outside a given line, only one line can be drawn parallel to a given line. Most geometers believed that the parallel axiom could be proven by other axioms, and that Euclid formulated the parallel statement as a postulate simply because he was unable to come up with such a proof. But, although the best mathematicians tried to solve the problem of parallels, 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 a given line; or if several parallel ones can be drawn through it. It turned out that the first a priori possibility is excluded 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 draw consequences from the new the “anti-Euclidean” or “non-Euclidean” axiom, a contradiction never appeared. 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 axioms had disappeared, the only way left to test their “truth” was experimental. But, as A. Poincaré (1854–1912) later noted, in the description of any phenomenon there are so many physical assumptions hidden that not a single experiment can provide convincing evidence 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 seriously considered such a hypothesis. Intuition suggested that both Euclidean and non-Euclidean geometry are examples of full-fledged mathematics.

Mathematical "monsters".

Unexpectedly, the same conclusions were reached from a completely different direction - objects were discovered that shocked the mathematicians of the 19th century. shocked and dubbed “mathematical monsters”. This discovery is directly related to very subtle issues of mathematical analysis that arose only in the mid-19th century. Difficulties arose when trying to find an exact mathematical analogue to the experimental concept of a curve. What was the essence of the concept of "continuous motion" (for example, the point of a drawing pen moving on a sheet of paper) was subject to precise mathematical definition, and this goal was achieved when the concept of continuity acquired a strict mathematical meaning ( cm. Also CURVE). Intuitively it seemed that the “curve” at each of its points had a direction, i.e. in the general case, in the neighborhood of each of its points, a curve behaves almost the same as a straight line. (On the other hand, it is not difficult 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 mid-19th century. it was believed that the “curve” had a tangent at almost all 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 FUNCTION THEORY). (The reader familiar with trigonometry and analytical 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 a 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 ellipse. Because 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 only 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 one more side: in mathematics, mainly in algebra, one after another, new mathematical objects began to appear, which were generalizations of the concept of number. Ordinary integers are quite “intuitive”, and it is not at all difficult to come to the 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 different in nature from the process of counting). Once it was discovered that a number could not be represented as a fraction, the Greeks were forced to consider irrational numbers, the correct determination of which by means of 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 associated with errors). Nevertheless, the introduction of irrational numbers occurred more or less in the spirit of the “idealization” of physical concepts. What can we say about negative numbers, which slowly, encountering great resistance, began to enter 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, using the process of direct abstraction, could develop the concept of a negative number, and in teaching 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 situation is very far from a concept “clear to everyone,” as Plato demanded of the ideas underlying mathematics, and one often encounters 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 “made 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 using 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 de facto declaration of independence of mathematics from the outside world.

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

“True” statements about such “objects” are the same logical consequences of the axioms. But now the axioms should be regarded as completely arbitrary, and therefore there is no need for them to be “obvious” or deducible from everyday experience through “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 has become part of everyday practice so that it is possible to use them in different ways, as was done during the transition from Euclidean to non-Euclidean geometry. (It is in this way that numerous variants of “non-Euclidean” geometries have been obtained, different from Euclidean geometry and from Lobachevsky-Bolyai geometry; for example, there are non-Euclidean geometries in which there are no parallel lines.)

I would like to especially emphasize one circumstance that follows from the new approach to mathematical “objects”: all proofs must be based exclusively on axioms. If we remember the definition of a mathematical proof, then such a statement may seem repetitive. 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 apparent “rigor,” many axioms are not stated explicitly and many properties are either tacitly assumed or introduced without sufficient justification. To put Euclidean geometry on a solid basis, a critical revision of its very principles was required. It is hardly worth saying that pedantic control over the smallest details of a proof is a consequence of the appearance of “monsters” that taught modern mathematicians to be careful in their conclusions. The most harmless and “self-evident” statement about classical objects, for example, the statement that a curve connecting points located on opposite sides of a line necessarily intersects this line, requires strict formal proof in modern mathematics.

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 assumes that the features that make it possible to distinguish one object from another are deliberately consigned to oblivion, and only the general features of the objects under consideration are preserved. What sets mathematics apart from the general range of sciences is the strict adherence to this program in all its points. Mathematical objects are said to be completely determined by the axioms used in the theory of those objects; or, in Poincaré's words, axioms serve as “disguised definitions” 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, during the development of one or more theories, it is noticed that certain proof patterns are repeated under more or less similar conditions. Once the properties used in general proof schemes are discovered, they are formulated as axioms, and their consequences are built into a general theory that has no direct relation to the specific contexts from which the axioms were abstracted. The general theorems obtained in this way 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 specifications of the same general theory. This means that after appropriate interpretation, the axioms of this theory become theorems in every situation. Any property derived from the axioms will be valid in all these situations, but there is no need for a separate proof for each case. In such cases, mathematical situations are said to share the same mathematical “structure.”

We use the idea of structure at every step in our daily lives. If the thermometer reads 10°C and the forecast office predicts a rise in temperature of 5°C, we without any calculation expect a temperature of 15°C. If a book is opened on 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 adding the numbers gives the correct result, regardless of their interpretation - as temperature or page numbers. We do not need to learn one arithmetic for thermometers and another for page numbers (although we do use a special arithmetic when dealing with clocks, in which 8 + 5 = 1, since clocks have a different structure than the pages of a book). The structures that interest mathematicians are somewhat more complex, which is easy to see from the examples that are discussed in the next two sections of this article. One of them will talk about group theory 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 a modern mathematician and taking a closer look at one of his main tools - group theory ( cm. Also ABSTRACT ALGEBRA). A group is a set (or “set”) of objects G, on which an operation is defined that matches any two objects or elements a, b from G, taken in the specified order (first is the element a, the second is the element b), third element c from G according to a strictly defined rule. For brevity, we denote this element a*b; The asterisk (*) denotes 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 holds: 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 called the singular or neutral element of a group;

(3) for any element a from G there is such an element aў, called reverse 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 turned out to be very useful, since groups are widely used in all branches of mathematics. From thousands of possible examples of groups, we will select only a few of the simplest ones.

(a) Fractions p/q, Where p And q– arbitrary integers i1 (with q= 1 we get ordinary integers). Fractions p/q form a group under 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 unit 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 a set of four integers 0, 1, 2, 3, and as a*b- remainder of the division a + b at 4. The results of the operation introduced in this way are presented in table. 1 (element a*b stands at the intersection of the line a and column b). It is easy to verify that properties (1)–(3) are satisfied, and the identity element is the number 0.

(c) Let's choose as G a 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 table. 2. It is easy to check that properties (1)–(3) are satisfied, and the identity element is 1.

(d) Four objects, such as the four numbers 1, 2, 3, 4, can be arranged in a row in 24 ways. Each arrangement can be visually represented as a transformation that transforms the “natural” arrangement into a given one; for example, the arrangement 4, 1, 2, 3 results from 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 determine S*T as a transformation that results from sequential execution T, and then S. For example, if , then . With this definition, all 24 possible transformations form a group; its unit element is , and the element inverse to S, 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 therefore T*S differs from S*T.

The group from example (d) is a special case of the so-called. symmetric group, whose applications include, 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 (with n= 12 we get a system of numbers that are on the clock dials, 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 form 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 (here is an example of incomplete knowledge!). In such cases we are said to be considering the group structure given by the group multiplication we have chosen.

Another example of a structure is the so-called. order structure. A bunch of E endowed with the structure of order, or ordered if between the elements a è b, belonging to E, a certain relation is given, which we denote R (a,b). (This relation must 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) true for everyone A, owned E;

(2) from R (a,b) And R (b,a) follows that a = b;

(3) from R (a,b) And R (b,c) should R (a,c).

Let us give several examples from a huge number of diverse ordered sets.

(A) E consists of all integers R (a,b) – relation “ A less or equal b».

(b) E consists of all integers >1, R (a,b) – relation “ A divides b or equal b».

As a final example of structure, let us mention the structure of metric space; such a structure is defined 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) ordinary "three-dimensional" space, where d (a,b) – ordinary (or “Euclidean”) distance;

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

The precise definition of the concept of structure is quite difficult. Without going into details, we can say that on many E a structure of a certain type is specified if between the elements of the set E(and sometimes other objects, for example, numbers that play an auxiliary role) relations are specified that satisfy a certain fixed set of axioms characterizing the structure of the type under consideration. Above we presented the 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) given in the previous section. It is easy to check that from the table. 1 to table 2 can be navigated using matching

0 ® 1, 1 ® 2, 2 ® 4, 3 ® 3.

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

The reader can 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 related concepts 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 manages to show 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 is studying (without these general theorems he would very likely have missed would lose sight of their specific options or would be forced to burden my reasoning with unnecessary assumptions). Similarly, if two structures are proven to be isomorphic, then the number of theorems immediately doubles: each theorem proven 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 goal 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 matter much, only the relationships between objects are significant (a kind of principle of incomplete knowledge).

Finally, one cannot fail to mention that the concept of structure has made it possible to classify branches of mathematics in a new way. Until the middle of the 19th century. they varied according to the subject of the study. Arithmetic (or number theory) dealt with integers, geometry dealt with straight lines, angles, polygons, circles, areas, etc. Algebra was concerned almost exclusively with methods for solving numerical equations or systems of equations; analytical geometry developed methods for converting 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 fragmentation of mathematical analysis into subsections: 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 early naturalists' classification of animals: once upon a time, both the sea turtle and the 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 new discoveries in the future. However, we already have an understanding of many of the basic "abstract" types of structures. (They are “abstract” compared to the “classical” objects of mathematics, although even those can hardly be called “concrete”; it is more a matter of the degree of abstraction.)

Known structures can be classified by the relationships they contain or by 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 ABSTRACT ALGEBRA). The branch of mathematics concerned with the study of algebraic structures is called "modern algebra" or "abstract algebra", in contrast to ordinary or classical algebra. A significant part of Euclidean geometry, non-Euclidean geometry and analytic geometry were also included in the new algebra.

At the same level of generality are two other blocks of structures. 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. “Expansion” of the structure consists of adding new axioms to existing ones. For example, if to the axioms of the group we add the property of commutativity 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 were in a relatively stable state until recently, and the “modern algebra” block was growing rapidly, sometimes in unexpected directions (for example, an entire branch called “homological algebra” developed). Outside the so-called “pure” types of structures lie at another level – “mixed” structures, for example algebraic and topological, together with new axioms connecting 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 substantial “abstract” field of science. Many mathematicians hope to use new tools to better understand classical theories and solve difficult problems. Indeed, with the 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. Vast 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 as deeply. Examples include the theory of differential equations and much of number theory. It is very likely that significant progress in these areas will be achieved once new types of structures are discovered and thoroughly studied.

PHILOSOPHICAL DIFFICULTIES

Even the ancient Greeks clearly understood that mathematical theory should be free from contradictions. This means that it is impossible to derive as a logical consequence from the axioms the statement R and his denial is not P. However, since mathematical objects were believed to have correspondences in the real world, and axioms were “idealizations” of the laws of nature, no one doubted the consistency of mathematics. During 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 condition of consistency, but can one be sure that this condition will be met?

We have already mentioned the concept of set. This concept has always been used more or less explicitly in mathematics and logic. In the second half of the 19th century. the elementary rules for handling the concept of set were partially systematized, in addition, some important results were obtained that formed the content of the so-called. set theory ( cm. Also SET THEORY), which became, as it were, the substrate of all other mathematical theories. From antiquity to the 19th century. there were concerns about infinite sets, for example, reflected in the famous paradoxes of Zeno of Eleatic (5th century BC). These concerns were partly metaphysical in nature, and partly caused by difficulties associated with the concept of measuring quantities (for example, length or time). It was possible to eliminate these difficulties only after the 19th century. 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 internal inconsistency of set theory (and 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 residents who do not shave themselves. Who shaves the barber himself? If the barber shaves himself, then he shaves not only those residents who do not shave themselves, but also one resident who shaves himself; if he himself does not shave, 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 telling. 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 finite. Let us 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; therefore, the set of integers determined by them is finite. Let us denote the finite set of these numbers by D. From the axioms of arithmetic it follows that there are integers that do not belong to D, and that among these numbers there is a 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 come to 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 had been developing 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 accuracy, 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 intuitively unrepresentable objects. The only intuitively clear objects are the natural numbers 1, 2, 3,... and finite sets of natural numbers, “constructed” according to precisely specified rules. But even to such objects, 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, they easily avoided “paradoxes,” but at the same time they threw overboard not only all modern mathematics, but also a significant part of the results of classical mathematics, and for those that remained, it was necessary to find new, more complex proofs.

The vast majority of modern mathematicians did not agree 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 jeopardizing existing mathematical theories. Another observation was that in "naive" set theory, which existed before the advent of "paradoxes", the meaning of the terms "set", "property", "relation" was not questioned - just as in classical geometry the "intuitive" was not questioned. the nature of ordinary geometric concepts. Consequently, one can act in the same way as it was in geometry, namely, discard all attempts to appeal to “intuition” and take a system of precisely formulated axioms as the starting point of set theory. It is not obvious, however, how words such as "property" or "relation" can be deprived of their ordinary meaning; yet this must be done if we wish to exclude such arguments as Berry's paradox. The method consists in refraining from using ordinary language in formulating axioms or theorems; only propositions constructed in accordance with 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 "formalization" of mathematical language (in order 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 symbols in formalized sentences, for example, replacing the connective "and" with the symbol &, the connective "or" - with the symbol b, “exists” with the symbol $, etc.). Mathematicians who rejected the methods proposed by intuitionists began 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 propositions of a formalized theory and their arrangement within proofs. These sentences are to be regarded merely as material combinations of symbols produced according to certain established rules, without any reference whatsoever to the possible "meaning" of these symbols (if any). A good analogy is the game of chess: symbols correspond to the pieces, sentences correspond to different positions on the board, and logical conclusions correspond to the rules for moving the pieces. To establish the consistency of a formalized theory, it is enough to show that in this theory not a single proof ends with the statement 0 No. 0. However, one can object to the use of mathematical arguments in a “meta-mathematical” proof of the consistency of a mathematical theory; if mathematics were inconsistent, then mathematical arguments would lose all force, and we would find ourselves in a vicious circle situation. To answer these objections, Hilbert allowed very limited mathematical reasoning of the type that intuitionists consider acceptable for use in metamathematics. However, K. Gödel soon showed (1931) that the consistency of arithmetic cannot be proven by such limited means if it is truly consistent (the scope of this article does not allow us to outline 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 set was limited by reservations that were specifically introduced to avoid known paradoxes, and there is no guarantee that new paradoxes will not arise in axiomatized set theory. Nevertheless, the limitations of axiomatic set theory did not prevent the birth of new viable theories.

MATHEMATICS AND THE REAL WORLD

Despite claims about the independence of mathematics, no one will deny that mathematics and the physical world are connected to each other. Of course, the mathematical approach to solving 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 microworld phenomena. 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 very 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 more subtle aspects of quantum theory; Nevertheless, many physicists still look with hope at new mathematical theories, believing that they will help them solve new problems.

Is mathematics a science or an art?

Even if we include probability theory or mathematical logic in “pure” mathematics, it turns out that less than 50% of the known mathematical results are currently used by other sciences. What should we think about the remaining half? In other words, what are the motives behind those areas of mathematics that are not related to solving physical problems?

We have already mentioned the irrationality of 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 it “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 current 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 often today than in the past, but the integers with which they operate are always limited (they rarely exceed a few hundred); therefore, a theorem such as Lagrange's can only be "useful" if it is applied to integers within some boundary. But as soon as we limit the formulation of Lagrange’s theorem, it immediately ceases to be interesting for a mathematician, since the entire attractive power of this theorem lies in its applicability to all integers. (There are a great many statements about integers that can be verified by computers for very large numbers; but since no general proof has been found, they remain hypothetical and of no interest to professional mathematicians.)

Focusing on topics far removed 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 conclusive. This is why it is difficult to resist the temptation to regard mathematics, or at least that part of it that has no relation to “reality,” as an art. Mathematical problems are not imposed from outside, and, if we take the modern point of view, we are completely free in our choice of material. When evaluating some mathematical works, mathematicians do not have “objective” criteria and are forced to rely on their own “taste.” Tastes vary greatly depending on time, country, traditions and individuals. In modern mathematics there are fashions and “schools”. Currently, there are three such “schools”, which for convenience we will call “classicism”, “modernism” and “abstractionism”. To better understand the differences between them, let's analyze the different criteria mathematicians use when evaluating a theorem or group of theorems.

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

(2) Any mathematical problem has its own history, a “pedigree”, so to speak, which follows the same general pattern according to 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 solution is obtained, the story does not end there, because 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, fourth, fifth powers, etc. This is how the “Waring problem” arises, which has not yet received a final solution. Moreover, if we are lucky, the problem we solve 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 usually leaves behind numerous living shoots. Modern mathematicians are faced with such a vast array of problems that, even if all communication with experimental science were interrupted, their solution would take several more centuries.

(3) Every mathematician will agree that when a new problem arises before him, it is his duty to solve it by any means possible. When a 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 relevant to task. This difference in approach is not new. Since the 19th century. mathematicians are divided into “tacticians” who strive to find a purely forceful solution to the problem, and “strategists” who are prone to roundabout maneuvers that make it possible to crush the enemy with small forces.

(4) An essential element of the “beauty” of the theorem is its simplicity. Of course, the search for simplicity is characteristic of all scientific thought. But experimenters are ready to put up with “ugly solutions” if only the problem is solved. Likewise, 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 in the appearance of “pathologies” of theory a symptom indicating the imperfection of the fundamental concepts.

Related materials:

Site Map