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MATHEMATICS HISTORY. The oldest mathematical activity was counting. An account was necessary to keep track of livestock and conduct trade. Some primitive tribes counted the number of objects, correlating them with various parts body, mainly the fingers and toes. A rock painting that has survived to this day from the Stone Age depicts the number 35 as a series of 35 finger sticks lined up in a row. The first significant advances in arithmetic were the conceptualization of number and the invention of the four basic operations: addition, subtraction, multiplication and division. The first achievements of geometry are associated with such simple concepts as straight lines and circles. Further development of mathematics began around 3000 BC. thanks to the Babylonians and Egyptians.

BABYLONIA AND EGYPT

Babylonia.

The source of our knowledge about the Babylonian civilization are well-preserved clay tablets covered with the so-called. cuneiform texts that date from 2000 BC. and up to 300 AD The mathematics on the cuneiform tablets was mainly related to farming. Arithmetic and simple algebra were used in exchanging money and paying for goods, calculating simple and compound interest, taxes and the share of the harvest handed over to the state, temple or landowner. Numerous arithmetic and geometric problems arose in connection with the construction of canals, granaries and other public works. A very important task of mathematics was the calculation of the calendar, since the calendar was used to determine the dates of agricultural work and religious holidays. The division of a circle into 360, and degrees and minutes into 60 parts, originates in Babylonian astronomy.

The Babylonians also created a number system that used base 10 for numbers from 1 to 59. The symbol for one was repeated the required number of times for numbers from 1 to 9. To represent numbers from 11 to 59, the Babylonians used a combination of the symbol for the number 10 and the symbol for one. To denote numbers starting from 60 and above, the Babylonians introduced a positional number system with a base of 60. A significant advance was the positional principle, according to which the same number sign (symbol) has different meanings depending on the place where it is located. An example is the meaning of six in the (modern) notation of the number 606. However, there was no zero in the ancient Babylonian number system, which is why the same set of symbols could mean both the number 65 (60 + 5) and the number 3605 (60 2 + 0 + 5). Ambiguities also arose in the interpretation of fractions. For example, the same symbols could mean the number 21, the fraction 21/60 and (20/60 + 1/60 2). Ambiguities were resolved depending on the specific context.

The Babylonians compiled tables of reciprocal numbers (which were used in division), tables of squares, and square roots, as well as tables of cubes and cube roots. They knew a good approximation of the number . Cuneiform texts dealing with solving algebraic and geometric problems indicate that they used the quadratic formula to solve quadratic equations and could solve some special types of problems involving up to ten equations in ten unknowns, as well as certain varieties of cubic and quartic equations. Only the tasks and the main steps of the procedures for solving them are depicted on clay tablets. Since geometric terminology was used to designate unknown quantities, the solution methods mainly consisted of geometric operations with lines and areas. As for algebraic problems, they were formulated and solved in verbal notation.

Around 700 BC The Babylonians began to use mathematics to study the movements of the Moon and planets. This allowed them to predict the positions of the planets, which was important for both astrology and astronomy.

In geometry, the Babylonians knew about such relationships, for example, as the proportionality of the corresponding sides of similar triangles. They knew the Pythagorean theorem and the fact that an angle inscribed in a semicircle is a right angle. They also had rules for calculating the areas of simple plane figures, including regular polygons, and the volumes of simple bodies. Number p The Babylonians considered it equal to 3.

Egypt.

Our knowledge of ancient Egyptian mathematics is based mainly on two papyri dating from about 1700 BC. The mathematical information presented in these papyri dates back to an even earlier period - c. 3500 BC The Egyptians used mathematics to calculate the weight of bodies, the area of ​​crops and the volume of granaries, the size of taxes and the number of stones required for the construction of certain structures. In the papyri one can also find problems related to determining the amount of grain needed to prepare a given number of glasses of beer, as well as more complex problems related to differences in types of grain; For these cases, conversion factors were calculated.

But the main area of ​​application of mathematics was astronomy, or rather calculations related to the calendar. The calendar was used to determine the dates of religious holidays and to predict the annual flooding of the Nile. However, the level of development of astronomy in Ancient Egypt much inferior to the level of its development in Babylon.

Ancient Egyptian writing was based on hieroglyphs. The number system of that period was also inferior to the Babylonian one. The Egyptians used non-positional decimal system, in which the numbers from 1 to 9 were indicated by the corresponding number of vertical bars, and individual symbols were introduced for successive powers of the number 10. By sequentially combining these symbols, any number could be written. With the advent of papyrus, the so-called hieratic cursive writing arose, which, in turn, contributed to the emergence of a new numerical system. For each of the numbers 1 through 9 and for each of the first nine multiples of 10, 100, etc. a special identification symbol was used. Fractions were written as a sum of fractions with a numerator equal to one. With such fractions, the Egyptians performed all four arithmetic operations, but the procedure for such calculations remained very cumbersome.

Geometry among the Egyptians came down to calculating the areas of rectangles, triangles, trapezoids, circles, as well as formulas for calculating the volumes of certain bodies. It must be said that the mathematics that the Egyptians used to build the pyramids was simple and primitive.

The tasks and solutions given in the papyri are formulated purely by prescription, without any explanation. The Egyptians dealt only with the simplest types of quadratic equations and arithmetic and geometric progressions, and therefore general rules, which they were able to deduce were also of the simplest type. Neither Babylonian nor Egyptian mathematicians had general methods; the entire body of mathematical knowledge was a collection of empirical formulas and rules.

Although the Mayans of Central America did not influence the development of mathematics, their achievements dating back to around the 4th century are noteworthy. The Mayans were apparently the first to use a special symbol to represent zero in their 20-digit system. They had two number systems: one used hieroglyphs, and the other, more common, used a dot for one, a horizontal line for the number 5, and a symbol for zero. Positional designations began with the number 20, and numbers were written vertically from top to bottom.

GREEK MATHEMATICS

Classical Greece.

From a 20th century point of view. The founders of mathematics were the Greeks of the classical period (6th–4th centuries BC). Mathematics, as it existed in the earlier period, was a set of empirical conclusions. On the contrary, in deductive reasoning a new statement is derived from accepted premises in a way that excludes the possibility of its rejection.

The Greeks' insistence on deductive proof was an extraordinary step. No other civilization has reached the idea of ​​arriving at conclusions solely on the basis of deductive reasoning, starting from explicitly stated axioms. We find one explanation for the Greeks' adherence to deductive methods in the structure of Greek society of the classical period. Mathematicians and philosophers (often these were the same people) belonged to the highest strata of society, where any practical activity was considered an unworthy occupation. Mathematicians preferred abstract reasoning about numbers and spatial relationships to solving practical problems. Mathematics was divided into arithmetic - the theoretical aspect and logistics - the computational aspect. Logistics was left to the freeborn of the lower classes and slaves.

The deductive character of Greek mathematics was fully formed by the time of Plato and Aristotle. The invention of deductive mathematics is generally attributed to Thales of Miletus (c. 640–546 BC), who, like many ancient Greek mathematicians of the classical period, was also a philosopher. It has been suggested that Thales used deduction to prove some results in geometry, although this is doubtful.

Another great Greek whose name is associated with the development of mathematics was Pythagoras (c. 585–500 BC). It is believed that he could have become acquainted with Babylonian and Egyptian mathematics during his long wanderings. Pythagoras founded a movement that flourished in ca. 550–300 BC The Pythagoreans created pure mathematics in the form of number theory and geometry. They represented whole numbers in the form of configurations of dots or pebbles, classifying these numbers in accordance with the shape of the resulting figures (“curly numbers”). The word "calculation" (calculation, calculation) originates from the Greek word meaning "pebble". Numbers 3, 6, 10, etc. The Pythagoreans called it triangular, since the corresponding number of pebbles can be arranged in the form of a triangle, the numbers 4, 9, 16, etc. – square, since the corresponding number of pebbles can be arranged in the form of a square, etc.

From simple geometric configurations some properties of integers arose. For example, the Pythagoreans discovered that the sum of two consecutive triangular numbers is always equal to some square number. They discovered that if (in modern notation) n 2 is a square number, then n 2 + 2n +1 = (n+ 1) 2 . A number equal to the sum of all its own divisors, except this number itself, was called perfect by the Pythagoreans. Examples of perfect numbers are integers such as 6, 28 and 496. The Pythagoreans called two numbers friendly if each number is equal to the sum of the divisors of the other; for example, 220 and 284 are friendly numbers (and here the number itself is excluded from its own divisors).

For the Pythagoreans, any number represented something more than a quantitative value. For example, the number 2, according to their view, meant difference and was therefore identified with opinion. Four represented justice since it was the first number equal to the product of two equal factors.

The Pythagoreans also discovered that the sum of certain pairs of square numbers is again a square number. For example, the sum of 9 and 16 is 25, and the sum of 25 and 144 is 169. Triples of numbers such as 3, 4 and 5 or 5, 12 and 13 are called Pythagorean numbers. They have a geometric interpretation: if two numbers from the three are equated to the lengths of the legs of a right triangle, then the third number will be equal to the length of its hypotenuse. This interpretation apparently led the Pythagoreans to realize a more general fact, now known as the Pythagorean theorem, according to which in any right triangle the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

Considering a right triangle with unit legs, the Pythagoreans discovered that the length of its hypotenuse was equal to , and this plunged them into confusion, for they tried in vain to represent a number as a ratio of two integers, which was extremely important for their philosophy. The Pythagoreans called quantities that cannot be represented as ratios of integers incommensurable; the modern term is “irrational numbers”. Around 300 BC Euclid proved that number is incommensurable. The Pythagoreans dealt with irrational numbers, representing all quantities in geometric images. If 1 is considered to be the length of some segments, then the difference between rational and irrational numbers is smoothed out. The product of numbers is the area of ​​a rectangle with sides of length and. Even today we sometimes talk about the number 25 as the square of 5, and the number 27 as the cube of 3.

The ancient Greeks solved equations with unknowns using geometric constructions. Special constructions were developed to perform addition, subtraction, multiplication and division of segments, extracting square roots from the lengths of segments; now this method is called geometric algebra.

Reducing problems to geometric form had a number of important consequences. In particular, numbers began to be considered separately from geometry, since it was possible to work with incommensurable relations only using geometric methods. Geometry became the basis of almost all rigorous mathematics at least until 1600. And even in the 18th century, when algebra and mathematical analysis were already sufficiently developed, rigorous mathematics was interpreted as geometry, and the word “geometer” was equivalent to the word “mathematician.”

It is to the Pythagoreans that we owe much of the mathematics that was then systematically presented and proven in Beginnings Euclid. There is reason to believe that it was they who discovered what is now known as theorems about triangles, parallel lines, polygons, circles, spheres and regular polyhedra.

One of the most prominent Pythagoreans was Plato (c. 427–347 BC). Plato was convinced that the physical world can only be understood through mathematics. It is believed that he is credited with inventing the analytical method of proof. (The analytical method begins with a statement that needs to be proven, and then consequences are successively deduced from it until some known fact; the proof is obtained using the reverse procedure.) It is generally accepted that the followers of Plato invented a method of proof called “proof by contradiction.” Aristotle, a student of Plato, occupies a prominent place in the history of mathematics. Aristotle laid the foundations of the science of logic and expressed a number of ideas regarding definitions, axioms, infinity and the possibility of geometric constructions.

The greatest of the Greek mathematicians of the classical period, second only to Archimedes in the importance of his results, was Eudoxus (c. 408–355 BC). It was he who introduced the concept of magnitude for such objects as line segments and angles. Having the concept of magnitude, Eudoxus logically and strictly substantiated the Pythagorean method of dealing with irrational numbers.

The work of Eudoxus made it possible to establish the deductive structure of mathematics on the basis of explicitly formulated axioms. He also took the first step in the creation of mathematical analysis, since it was he who invented the method of calculating areas and volumes, called the “exhaustion method.” This method consists of constructing inscribed and described flat figures or spatial bodies that fill (“exhaust”) the area or volume of the figure or body that is the subject of research. Eudoxus also owns the first astronomical theory that explains the observed movement of the planets. The theory proposed by Eudoxus was purely mathematical; it showed how combinations of rotating spheres with different radii and axes of rotation could explain the seemingly irregular movements of the Sun, Moon and planets.

Around 300 BC the results of many Greek mathematicians were combined into a single whole by Euclid, who wrote a mathematical masterpiece Beginnings. From a few shrewdly selected axioms, Euclid derived about 500 theorems, covering all the most important results of the classical period. Euclid began his work by defining such terms as straight line, angle and circle. He then stated ten self-evident truths, such as “the whole is greater than any of the parts.” And from these ten axioms, Euclid was able to derive all the theorems. Text for mathematicians Began Euclid served as a model of rigor for a long time, until in the 19th century. it was not found to have serious deficiencies, such as the unconscious use of assumptions that were not explicitly stated.

Apollonius (c. 262–200 BC) lived during the Alexandrian period, but his main work is in the spirit of the classical tradition. His proposed analysis of conic sections - circle, ellipse, parabola and hyperbola - was the culmination of the development of Greek geometry. Apollonius also became the founder of quantitative mathematical astronomy.

Alexandrian period.

During this period, which began around 300 BC, the nature of Greek mathematics changed. Alexandrian mathematics arose from the fusion of classical Greek mathematics with the mathematics of Babylonia and Egypt. In general, mathematicians of the Alexandrian period were more inclined to solve purely technical problems than to philosophy. The great Alexandrian mathematicians - Eratosthenes, Archimedes, Hipparchus, Ptolemy, Diophantus and Pappus - demonstrated the strength of the Greek genius in theoretical abstraction, but were equally willing to apply their talent to the solution of practical problems and purely quantitative problems.

Eratosthenes (c. 275–194 BC) found a simple method for accurately calculating the circumference of the Earth, and he also created a calendar in which every fourth year has one more day than the others. The astronomer Aristarchus (c. 310–230 BC) wrote an essay About the sizes and distances of the Sun and Moon, which contained one of the first attempts to determine these sizes and distances; Aristarchus' work was geometric in nature.

The greatest mathematician of antiquity was Archimedes (c. 287–212 BC). He is the author of the formulations of many theorems about the areas and volumes of complex figures and bodies, which he quite strictly proved by the method of exhaustion. Archimedes always sought to obtain exact solutions and found upper and lower bounds for irrational numbers. For example, working with the regular 96-gon, he flawlessly proved that the exact value of the number p is between 3 1/7 and 3 10/71. Archimedes also proved several theorems that contained new results in geometric algebra. He was responsible for the formulation of the problem of dissecting a ball by a plane so that the volumes of the segments are in a given ratio to each other. Archimedes solved this problem by finding the intersection of a parabola and an equilateral hyperbola.

Archimedes was the greatest mathematical physicist of antiquity. He used geometric considerations to prove theorems of mechanics. His essay About floating bodies laid the foundations of hydrostatics. According to legend, Archimedes discovered the law that bears his name, according to which a body immersed in water is subject to a buoyant force equal to the weight of the liquid displaced by it. While bathing, while in the bathroom, and unable to cope with the joy of discovery that gripped him, he ran out naked into the street shouting: “Eureka!” (“Opened!”)

In the time of Archimedes, they were no longer limited to geometric constructions that could only be done with a compass and a ruler. Archimedes used a spiral in his constructions, and Diocles (late 2nd century BC) solved the problem of doubling a cube using a curve he introduced, called the cissoid.

During the Alexandrian period, arithmetic and algebra were treated independently of geometry. The Greeks of the classical period had a logically substantiated theory of integers, but the Alexandrian Greeks, having adopted Babylonian and Egyptian arithmetic and algebra, largely lost their already developed ideas about mathematical rigor. Lived between 100 BC and 100 AD Heron of Alexandria transformed much of the geometric algebra of the Greeks into frankly lax computational procedures. However, in proving new theorems of Euclidean geometry, he was still guided by the standards of logical rigor of the classical period.

The first fairly voluminous book in which arithmetic was presented independently of geometry was Introduction to Arithmetic Nicomacheus (c. 100 AD). In the history of arithmetic, its role is comparable to that of Began Euclid in the history of geometry. It served as the standard textbook for more than 1,000 years because it taught the teachings of whole numbers (prime, composite, coprime, and proportions) in a clear, concise, and comprehensive manner. Repeating many Pythagorean statements, Introduction Nicomachus, however, went further, since Nicomachus also saw more general relationships, although he cited them without proof.

A significant milestone in the algebra of the Alexandrian Greeks was the work of Diophantus (c. 250). One of his main achievements is associated with the introduction of symbolism into algebra. In his works, Diophantus did not propose general methods; he dealt with specific positive rational numbers, and not with their letter designations. He laid the foundations of the so-called. Diophantine analysis – study of uncertain equations.

The highest achievement of Alexandrian mathematicians was the creation of quantitative astronomy. We owe the invention of trigonometry to Hipparchus (c. 161–126 BC). His method was based on a theorem stating that in similar triangles the ratio of the lengths of any two sides of one of them is equal to the ratio of the lengths of two corresponding sides of the other. In particular, the ratio of the length of the leg lying opposite the acute angle A in a right triangle, to the length of the hypotenuse must be the same for all right triangles having the same acute angle A. This ratio is known as the sine of the angle A. The ratios of the lengths of the other sides of a right triangle are called cosine and tangent of the angle A. Hipparchus invented a method for calculating such ratios and compiled their tables. With these tables and easily measurable distances on the surface of the Earth, he was able to calculate the length of its great circle and the distance to the Moon. According to his calculations, the radius of the Moon was one third of the Earth's radius; According to modern data, the ratio of the radii of the Moon and the Earth is 27/1000. Hipparchus determined the length of the solar year with an error of only 6 1/2 minutes; It is believed that it was he who introduced latitude and longitude.

Greek trigonometry and its applications to astronomy reached its peak in Almagest Egyptian Claudius Ptolemy (died 168 AD). IN Almagest the theory of the movement of celestial bodies was presented, which prevailed until the 16th century, when it was replaced by the theory of Copernicus. Ptolemy sought to build the simplest mathematical model, realizing that his theory was just a convenient mathematical description astronomical phenomena, consistent with observations. Copernicus's theory prevailed precisely because it was simpler as a model.

Decline of Greece.

After the conquest of Egypt by the Romans in 31 BC. the great Greek Alexandrian civilization fell into decay. Cicero proudly argued that, unlike the Greeks, the Romans were not dreamers, and therefore applied their mathematical knowledge in practice, deriving real benefit from it. However, the contribution of the Romans to the development of mathematics itself was insignificant. The Roman number system was based on cumbersome notations for numbers. Its main feature was the additive principle. Even the subtractive principle, for example writing the number 9 as IX, came into widespread use only after the invention of typesetting in the 15th century. Roman number notation was used in some European schools until about 1600, and in accounting a century later.

INDIA AND ARAB

The successors of the Greeks in the history of mathematics were the Indians. Indian mathematicians did not engage in proofs, but they introduced original concepts and a number of effective methods. It was they who first introduced zero both as a cardinal number and as a symbol of the absence of units in the corresponding digit. Mahavira (850 AD) established rules for operations with zero, believing, however, that dividing a number by zero leaves the number unchanged. The correct answer for the case of dividing a number by zero was given by Bhaskara (b. 1114), and he also owned the rules for operating with irrational numbers. The Indians introduced the concept negative numbers(to indicate debts). We find their earliest use in Brahmagupta (c. 630). Aryabhata (p. 476) went further than Diophantus in the use of continued fractions in solving indefinite equations.

Our modern system Notation based on the positional principle of writing numbers and zero as a cardinal number and the use of the empty digit notation is called Indo-Arabic. On the wall of a temple built in India ca. 250 BC, several figures were discovered that resemble our modern figures in their outlines.

Around 800 Indian mathematics reached Baghdad. The term "algebra" comes from the beginning of the book's title Al-jabr wa-l-muqabala (Replenishment and opposition), written in 830 by the astronomer and mathematician al-Khwarizmi. In his essay he paid tribute to the merits of Indian mathematics. Al-Khwarizmi's algebra was based on the works of Brahmagupta, but Babylonian and Greek influences are clearly discernible. Another prominent Arab mathematician, Ibn al-Haytham (c. 965–1039), developed a method for obtaining algebraic solutions to quadratic and cubic equations. Arab mathematicians, including Omar Khayyam, were able to solve some cubic equations using geometric methods using conic sections. Arab astronomers introduced the concept of tangent and cotangent into trigonometry. Nasireddin Tusi (1201–1274) in Treatise on the Complete Quadrangle systematically outlined plane and spherical geometry and was the first to consider trigonometry separately from astronomy.

Yet the most important contribution of the Arabs to mathematics was their translations and commentaries on the great works of the Greeks. Europe became acquainted with these works after the Arab conquest of North Africa and Spain, and later the works of the Greeks were translated into Latin.

MIDDLE AGES AND RENAISSANCE

Medieval Europe.

Roman civilization did not leave a noticeable mark on mathematics because it was too concerned with solving practical problems. The civilization that developed in early Middle Ages Europe (c. 400–1100) was not productive for exactly the opposite reason: intellectual life focused almost exclusively on theology and the afterlife. The level of mathematical knowledge did not rise above arithmetic and simple sections from Began Euclid. Astrology was considered the most important branch of mathematics in the Middle Ages; astrologers were called mathematicians. And since medical practice was based primarily on astrological indications or contraindications, doctors had no choice but to become mathematicians.

Around 1100, Western European mathematics began an almost three-century period of mastering the heritage preserved by the Arabs and Byzantine Greeks Ancient world and East. Since the Arabs owned almost all the works of the ancient Greeks, Europe received an extensive mathematical literature. The translation of these works into Latin contributed to the rise of mathematical research. All the great scientists of the time admitted that they drew inspiration from the works of the Greeks.

The first European mathematician worth mentioning was Leonardo of Pisa (Fibonacci). In his essay Book of abacus(1202) he introduced the Europeans to Indo-Arabic numerals and methods of calculation, as well as Arabic algebra. Over the next few centuries, mathematical activity in Europe waned. The body of mathematical knowledge of the era, compiled by Luca Pacioli in 1494, did not contain any algebraic innovations that Leonardo did not have.

Revival.

Among the best geometers of the Renaissance were artists who developed the idea of ​​perspective, which required a geometry with converging parallel lines. The artist Leon Battista Alberti (1404–1472) introduced the concepts of projection and section. Straight rays of light from the observer's eye to various points in the depicted scene form a projection; the section is obtained by passing the plane through the projection. In order for the painted picture to look realistic, it had to be such a cross-section. The concepts of projection and section gave rise to purely mathematical questions. For example, what common geometric properties do the section and the original scene have, and what are the properties of two different sections of the same projection formed by two different planes intersecting the projection at different angles? From such questions projective geometry arose. Its founder, J. Desargues (1593–1662), using evidence based on projection and section, unified the approach to various types conic sections, which the great Greek geometer Apollonius considered separately.

THE BEGINNING OF MODERN MATHEMATICS

Advance of the 16th century. in Western Europe was marked by important achievements in algebra and arithmetic. Decimal fractions and rules for arithmetic operations with them were introduced. A real triumph was the invention of logarithms in 1614 by J. Napier. By the end of the 17th century. the understanding of logarithms as exponents with any positive number other than one as the base has finally emerged. From the beginning of the 16th century. Irrational numbers began to be used more widely. B. Pascal (1623–1662) and I. Barrow (1630–1677), I. Newton’s teacher at Cambridge University, argued that a number such as , can only be interpreted as a geometric quantity. However, in those same years, R. Descartes (1596–1650) and J. Wallis (1616–1703) believed that irrational numbers are acceptable on their own, without reference to geometry. In the 16th century Controversy continued over the legality of introducing negative numbers. Complex numbers that arose when solving quadratic equations, such as those called “imaginary” by Descartes, were considered even less acceptable. These numbers were under suspicion even in the 18th century, although L. Euler (1707–1783) used them with success. Complex numbers were finally recognized only at the beginning of the 19th century, when mathematicians became familiar with their geometric representation.

Advances in algebra.

In the 16th century Italian mathematicians N. Tartaglia (1499–1577), S. Dal Ferro (1465–1526), ​​L. Ferrari (1522–1565) and D. Cardano (1501–1576) found general solutions to equations of the third and fourth degrees. To make algebraic reasoning and notation more precise, many symbols were introduced, including +, –, ґ, =, > and<.>b 2 – 4 ac] quadratic equation, namely, that the equation ax 2 + bx + c= 0 has equal real, different real, or complex conjugate roots, depending on whether the discriminant b 2 – 4ac equal to zero, greater than or less than zero. In 1799, K. Friedrich Gauss (1777–1855) proved the so-called. fundamental theorem of algebra: every polynomial n-th degree has exactly n roots.

The main task of algebra—the search for a general solution to algebraic equations—continued to occupy mathematicians at the beginning of the 19th century. When talking about the general solution of a second degree equation ax 2 + bx + c= 0, mean that each of its two roots can be expressed using a finite number of addition, subtraction, multiplication, division and rooting operations performed on the coefficients a, b And With. The young Norwegian mathematician N. Abel (1802–1829) proved that it is impossible to obtain common decision equations of degree above 4 using a finite number of algebraic operations. However, there are many equations of a special form of degree higher than 4 that admit such a solution. On the eve of his death in a duel, the young French mathematician E. Galois (1811–1832) gave a decisive answer to the question of which equations are solvable in radicals, i.e. the roots of which equations can be expressed through their coefficients using a finite number of algebraic operations. Galois theory used substitutions or permutations of roots and introduced the concept of a group, which found wide application in many areas of mathematics.

Analytic geometry.

Analytical, or coordinate, geometry was created independently by P. Fermat (1601–1665) and R. Descartes in order to expand the capabilities of Euclidean geometry in construction problems. However, Fermat considered his work only as a reformulation of the work of Apollonius. The real discovery - the realization of the full power of algebraic methods - belongs to Descartes. Euclidean geometric algebra required the invention of its own original method for each construction and could not offer the quantitative information necessary for science. Descartes solved this problem: he formulated geometric problems algebraically, solved the algebraic equation, and only then constructed the desired solution - a segment that had the appropriate length. Analytical geometry itself arose when Descartes began to consider indeterminate construction problems whose solutions were not one, but many possible lengths.

Analytic geometry uses algebraic equations to represent and study curves and surfaces. Descartes considered an acceptable curve that could be written using a single algebraic equation with respect to X And at. This approach was important step forward, because he not only included such curves as the conchoid and cissoid among the permissible ones, but also significantly expanded the range of curves. As a result, in the 17th–18th centuries. many new important curves, such as the cycloid and catenary, entered scientific use.

Apparently, the first mathematician who used equations to prove the properties of conic sections was J. Wallis. By 1865 he had obtained algebraically all the results presented in Book V Began Euclid.

Analytical geometry completely reversed the roles of geometry and algebra. As the great French mathematician Lagrange noted, “As long as algebra and geometry went their separate ways, their progress was slow and their applications limited. But when these sciences united their efforts, they borrowed new vital forces from each other and since then have moved quickly towards perfection.” see also ALGEBRAIC GEOMETRY; GEOMETRY ; GEOMETRY REVIEW.

Mathematical analysis.

Founders modern science– Copernicus, Kepler, Galileo and Newton – approached the study of nature as mathematics. By studying motion, mathematicians developed such a fundamental concept as function, or the relationship between variables, for example d = kt 2 where d is the distance traveled by a freely falling body, and t– the number of seconds that the body is in free fall. The concept of function immediately became central in determining the speed at a given moment in time and the acceleration of a moving body. The mathematical difficulty of this problem was that at any moment the body travels zero distance in zero time. Therefore, determining the value of the speed at an instant of time by dividing the path by the time, we arrive at the mathematically meaningless expression 0/0.

The problem of determining and calculating instantaneous rates of change of various quantities attracted the attention of almost all mathematicians of the 17th century, including Barrow, Fermat, Descartes and Wallis. The disparate ideas and methods they proposed were combined into a systematic, universally applicable formal method by Newton and G. Leibniz (1646–1716), the creators of differential calculus. There were heated debates between them on the issue of priority in the development of this calculus, with Newton accusing Leibniz of plagiarism. However, as research by historians of science has shown, Leibniz created mathematical analysis independently of Newton. As a result of the conflict, the exchange of ideas between mathematicians in continental Europe and England was interrupted for many years, to the detriment of the English side. English mathematicians continued to develop the ideas of analysis in a geometric direction, while mathematicians of continental Europe, including I. Bernoulli (1667–1748), Euler and Lagrange achieved incomparably greater success following the algebraic, or analytical, approach.

The basis of all mathematical analysis is the concept of limit. The speed at a moment in time is defined as the limit to which it tends average speed d/t when the value t getting closer to zero. Differential calculus provides a computationally convenient general method for finding the rate of change of a function f (x) for any value X. This speed is called derivative. From the generality of the record f (x) it is clear that the concept of derivative is applicable not only in problems related to the need to find speed or acceleration, but also in relation to any functional dependence, for example, to some relation from economic theory. One of the main applications of differential calculus is the so-called. maximum and minimum tasks; Another important range of problems is finding the tangent to a given curve.

It turned out that with the help of a derivative, specially invented for working with motion problems, it is also possible to find areas and volumes limited by curves and surfaces, respectively. The methods of Euclidean geometry did not have the necessary generality and did not allow obtaining the required quantitative results. Through the efforts of mathematicians of the 17th century. Numerous private methods were created that made it possible to find the areas of figures bounded by curves of one type or another, and in some cases the connection between these problems and problems of finding the rate of change of functions was noted. But, as in the case of differential calculus, it was Newton and Leibniz who realized the generality of the method and thereby laid the foundations of integral calculus.

MODERN MATHEMATICS

The creation of differential and integral calculus marked the beginning of “higher mathematics”. The methods of mathematical analysis, in contrast to the concept of limit that underlies it, seemed clear and understandable. For many years mathematicians, including Newton and Leibniz, tried in vain to give a precise definition of the concept of limit. And yet, despite numerous doubts about the validity of mathematical analysis, it found increasingly widespread use. Differential and integral calculus became the cornerstones of mathematical analysis, which eventually included such subjects as theory differential equations, ordinary and partial derivatives, infinite series, calculus of variations, differential geometry and much more. A strict definition of the limit was obtained only in the 19th century.

Non-Euclidean geometry.

By 1800, mathematics rested on two pillars - the number system and Euclidean geometry. Since many properties of the number system were proven geometrically, Euclidean geometry was the most reliable part of the edifice of mathematics. However, the axiom of parallels contained a statement about straight lines extending to infinity, which could not be confirmed by experience. Even Euclid's own version of this axiom does not at all state that some lines will not intersect. It rather formulates a condition under which they intersect at some end point. For centuries, mathematicians have tried to find a suitable replacement for the parallel axiom. But in each option there was certainly some gap. The honor of creating non-Euclidean geometry fell to N.I. Lobachevsky (1792–1856) and J. Bolyai (1802–1860), each of whom independently published his own original presentation of non-Euclidean geometry. In their geometries, an infinite number of parallel lines could be drawn through a given point. In the geometry of B. Riemann (1826–1866), no parallel can be drawn through a point outside a straight line.

Nobody seriously thought about physical applications of non-Euclidean geometry. The creation by A. Einstein (1879–1955) of the general theory of relativity in 1915 awakened scientific world to an awareness of the reality of non-Euclidean geometry.

Mathematical rigor.

Until about 1870, mathematicians believed that they were acting as the ancient Greeks had designed, applying deductive reasoning to mathematical axioms, thereby providing their conclusions with a reliability no less than that possessed by the axioms. Non-Euclidean geometry and quaternions (an algebra that does not obey the commutative property) forced mathematicians to realize that what they took to be abstract and logically consistent statements were in fact based on an empirical and pragmatic basis.

The creation of non-Euclidean geometry was also accompanied by the awareness of the existence of logical gaps in Euclidean geometry. One of the disadvantages of Euclidean Began was the use of assumptions that were not explicitly stated. Apparently, Euclid did not question the properties that his geometric figures possessed, but these properties were not included in his axioms. In addition, when proving the similarity of two triangles, Euclid used the superposition of one triangle on another, implicitly assuming that the properties of the figures do not change when moving. But besides such logical gaps, in Beginnings There was also some erroneous evidence.

The creation of new algebras, which began with quaternions, gave rise to similar doubts regarding the logical validity of arithmetic and the algebra of the ordinary number system. All numbers previously known to mathematicians had the property of commutativity, i.e. ab = ba. Quaternions, which revolutionized traditional ideas about numbers, were discovered in 1843 by W. Hamilton (1805–1865). They turned out to be useful for solving a number of physical and geometric problems, although the commutativity property did not hold for quaternions. Quaternions forced mathematicians to realize that, apart from the part dedicated to integers and far from perfect, the Euclidean Began, arithmetic and algebra do not have their own axiomatic basis. Mathematicians freely handled negative and complex numbers and performed algebraic operations, guided only by the fact that they worked successfully. Logical rigor gave way to demonstrating the practical benefits of introducing dubious concepts and procedures.

Almost from the very beginning of mathematical analysis, attempts have been made repeatedly to provide a rigorous foundation for it. Mathematical analysis introduced two new complex concepts - derivative and definite integral. Newton and Leibniz struggled with these concepts, as well as mathematicians of subsequent generations, who turned differential and integral calculus into mathematical analysis. However, despite all efforts, much uncertainty remained in the concepts of limit, continuity and differentiability. In addition, it turned out that the properties of algebraic functions cannot be transferred to all other functions. Almost all mathematicians of the 18th century. and the beginning of the 19th century. efforts have been made to find a rigorous basis for mathematical analysis, and all have failed. Finally, in 1821, O. Cauchy (1789–1857), using the concept of number, provided a strict basis for all mathematical analysis. However, later mathematicians discovered logical gaps in Cauchy. The desired rigor was finally achieved in 1859 by K. Weierstrass (1815–1897).

Weierstrass initially considered the properties of real and complex numbers self-evident. Later, like G. Cantor (1845–1918) and R. Dedekind (1831–1916), he realized the need to build a theory of irrational numbers. They gave a correct definition of irrational numbers and established their properties, but they still considered the properties of rational numbers to be self-evident. Finally, the logical structure of the theory of real and complex numbers acquired its complete form in the works of Dedekind and J. Peano (1858–1932). The creation of the foundations of the numerical system also made it possible to solve the problems of substantiating algebra.

The task of increasing the rigor of the formulations of Euclidean geometry was relatively simple and boiled down to listing the terms being defined, clarifying the definitions, introducing missing axioms, and filling gaps in the proofs. This task was completed in 1899 by D. Gilbert (1862–1943). Almost at the same time, the foundations of other geometries were laid. Hilbert formulated the concept of formal axiomatics. One of the features of the approach he proposed is the interpretation of undefined terms: they can be understood as any objects that satisfy the axioms. The consequence of this feature was the increasing abstractness of modern mathematics. Euclidean and non-Euclidean geometries describe physical space. But in topology, which is a generalization of geometry, the undefined term "point" can be free of geometric associations. For a topologist, a point can be a function or a sequence of numbers, as well as anything else. Abstract space is a set of such “points” ( see also TOPOLOGY).

Hilbert's axiomatic method was included in almost all branches of mathematics of the 20th century. However, it soon became clear that this method had certain limitations. In the 1880s, Cantor tried to systematically classify infinite sets (for example, the set of all rational numbers, the set of real numbers, etc.) by comparatively quantifying them, attributing to them the so-called. transfinite numbers. At the same time, he discovered contradictions in set theory. Thus, by the beginning of the 20th century. mathematicians had to deal with the problem of their resolution, as well as with other problems of the foundations of their science, such as the implicit use of the so-called. axioms of choice. And yet nothing could compare with the destructive impact of K. Gödel's (1906–1978) incompleteness theorem. This theorem states that any consistent formal system rich enough to contain number theory must necessarily contain an undecidable proposition, i.e. a statement that can neither be proven nor disproved within its framework. It is now generally accepted that there is no absolute proof in mathematics. Opinions differ as to what evidence is. However, most mathematicians tend to believe that the problems of the foundations of mathematics are philosophical. Indeed, not a single theorem has changed as a result of the newly discovered logically rigorous structures; this shows that mathematics is based not on logic, but on sound intuition.

If the mathematics known before 1600 can be characterized as elementary, then in comparison with what was created later, this elementary mathematics is infinitesimal. Old areas expanded and new ones emerged, both pure and applied branches of mathematical knowledge. About 500 mathematical journals are published. The huge number of published results does not allow even a specialist to familiarize himself with everything that is happening in the field in which he works, not to mention the fact that many results are understandable only to a specialist of a narrow profile. No mathematician today can hope to know more than what is going on in a very small corner of science. see also articles about scientists - mathematicians.

Literature:

Van der Waerden B.L. Awakening Science. Mathematics of Ancient Egypt, Babylon and Greece. M., 1959
Yushkevich A.P. History of mathematics in the Middle Ages. M., 1961
Daan-Dalmedico A., Peiffer J. Paths and labyrinths. Essays on the history of mathematics. M., 1986
Klein F. Lectures on the development of mathematics in the 19th century. M., 1989



In the next 10 years natural Sciences will get closer to the humanities to answer the complex questions of humanity. And the language of mathematics will play a huge role in this. It will become possible to discover new trends in history, explain them, and in the future even predict what will happen. So says history researcher Jean-Baptiste Michel, who in February of this year, speaking at TED, outlined his point of view on how mathematics can be useful to historians.

In his short (6 min.) talk, Jean-Baptiste Michel talks about how digitized history is on the way to revealing deep underlying trends, such as changes in language or the lethality of wars.

Text of the speech

It turns out that the language of mathematics is a powerful tool. He contributed to significant progress in physics, biology and economics, but not in humanities and history. Perhaps people think it is impossible - it is impossible to count the deeds of mankind or measure history. However, I think differently. Here are some examples.

My colleague Erez and I were thinking about this: two kings living in different centuries speak completely different languages. This is a powerful historical force. For example, the vocabulary and grammar rules used by King Alfred the Great of England were very different from the speech of hip-hop king Jay-Z. (Laughter) Nothing can be done. Over time, language changes, and this is an influential factor.

Erez and I wanted to know more about this. Therefore, we turned to the class of past tense conjugation, where the ending “-ed” on the verb denotes action in the past tense. "Today I walk." [I'm walking today] "Yesterday I walked." [I walked yesterday]. But not all verbs are regular. For example, "Yesterday I thought." [Yesterday I was thinking]. It's interesting that today, in Jay-Z's time, we have more regular verbs than we had in Alfred's time. For example, the verb "to wed" became correct.

Erez and I traced the fates of more than 100 irregular verbs over 12 centuries of history in English and noticed that this complex historical change can be summarized by a fairly simple mathematical formula: if a verb is used 100 times more often than others, it becomes correct 10 times slower. Here is a historical fact in a mathematical wrapper.

In some cases, mathematics helps explain or suggest versions for historical events. Together with Steve Pinker, we reflected on the scale of the wars of the past two centuries. There is a well-known pattern: wars that claimed 100 times more lives happened 10 times less often. For example, there are 30 wars as deadly as the Six Day War, and only 4 wars that have claimed 100 times as many lives as World War I did. So what historical mechanism leads to this? What is the root cause?

Using mathematical analysis, Steve and I believe that it is based on a very simple property of our brains. This is a well-known property of understanding relative quantities such as intensity luminous flux or loudness. For example, if we need to mobilize 10,000 soldiers for a battle, the figure will seem huge to us, especially if last time only 1,000 soldiers were mobilized. But this is not a lot at all, relatively few, no one will notice if to at this moment 100,000 soldiers were mobilized. Because of the way we represent values, as the war continues, the number of mobilized and wounded will increase not linearly - 10,000, 11,000, 12,000, but exponentially: 10,000, 20,000, 40,000. This explains the model about which we talked earlier.

Mathematics can link the known properties of the human brain to a long-term historical pattern that extends across centuries and continents.

I think these couple of examples will become commonplace in the next 10 years. This will be possible thanks to the high speed of digitization of historical documents. Since the beginning of time, about 130 million books have been written. Many books have been digitized by companies like Google - more than 20 million books. When historical facts available in digital form, one can quickly and easily view trends in our history and culture using mathematical analysis.

Therefore, I think that in the next 10 years the natural sciences will converge with the humanities to answer the complex questions of humanity. And the language of mathematics will play a huge role in this. It will become possible to discover new trends in history, explain them, and in the future even predict what will happen.

Thank you very much.

(Applause)

Translation: Olga Dmitrochenkova

Slide 2

Mathematical analysis is a set of branches of mathematics devoted to the study of functions and their generalizations by methods of differential and integral calculus.

Slide 3

Exhaustion method

An ancient method for studying the area or volume of curved figures.

Slide 4

The method was as follows: to find the area (or volume) of a certain figure, a monotonic sequence of other figures was fit into this figure and it was proved that their areas (volumes) indefinitely approach the area (volume) of the desired figure.

Slide 5

In 1696, L'Hopital wrote the first textbook, setting out a new method as applied to the theory of plane curves. He called it Analysis of Infinitesimals, thereby giving one of the names to the new branch of mathematics. In the introduction, L'Hopital outlines the history of the emergence of the new analysis, dwelling on the works of Descartes, Huygens, Leibniz, and also expresses his gratitude to the latter and the Bernoulli brothers.

Slide 6

The term “function” first appears only in 1692 in Leibniz, but it was Euler who brought it to the fore. The original interpretation of the concept of a function was that a function is an expression for counting or an analytical expression.

Slide 7

“Theory of analytic functions” (“Th.orie des fonctions analytiques”, 1797). In The Theory of Analytic Functions, Lagrange sets out his famous interpolation formula, which inspired Cauchy to develop a rigorous foundation for analysis.

Slide 8

Fermat's important lemma can be found in calculus textbooks. He also formulated the general law of differentiation of fractional powers.

Pierre de Fermat (August 17, 1601 - January 12, 1665) was a French mathematician, one of the creators of analytical geometry, mathematical analysis, probability theory and number theory. Fermat, using almost modern rules, found tangents to algebraic curves.

Slide 9

Rene Descartes (March 31, 1596 - February 11, 1650) - French mathematician, philosopher, physicist and physiologist, creator of analytical geometry and modern algebraic symbolism. In 1637, Descartes's main mathematical work, Discourse on Method, was published. This book presented analytical geometry, and in its appendices numerous results in algebra, geometry, optics, and much more. Particularly noteworthy is the mathematical symbolism of Vieta that he reworked: he introduced the now generally accepted signs for variables and required quantities (x, y, z, ...) and for letter coefficients. (a, b, c, ...)

Slide 10

François Viête (1540 -1603) - French mathematician, founder of symbolic algebra. By education and main profession - lawyer. In 1591 he introduced letter notation not only for unknown quantities, but also for the coefficients of equations. He was responsible for establishing a uniform method for solving equations of the 2nd, 3rd and 4th degrees. Among the discoveries, Viète himself especially highly valued the establishment of the relationship between the roots and coefficients of equations.

Slide 11

GalileoGalilei (February 15, 1564, Pisa - January 8, 1642) - Italian physicist, mechanic, astronomer, philosopher and mathematician, who had a significant influence on the science of his time Formulated the “Galileo's paradox”: there are as many natural numbers as there are their squares, although most of the numbers are not squares . This prompted further research into the nature of infinite sets and their classification; The process ended with the creation of set theory.

Slide 12

"New stereometry of wine barrels"

When Kepler bought wine, he was amazed at how the merchant determined the capacity of the barrel. The seller took the stickus in divisions, and with its help determined the distance from the filling hole to the farthest point of the barrel. Having done this, he immediately said how many liters of wine were in a given barrel. Thus, the scientist was the first to draw attention to a class of problems, the study of which led to the creation of integral calculus.

Slide 13

So, for example, to find the formula for the volume of a torus, Kepler divided it with meridional sections into an infinite number of circles, the thickness of which on the outside was slightly greater than on the inside. The volume of such a circle is equal to the volume of a cylinder with a base equal to the cross-section of the torus and a height equal to the thickness of the circle in its middle part. From here it immediately turned out that the volume of the torus is equal to the volume of a cylinder, the base area of ​​which is equal to the cross-sectional area of ​​the torus, and the height is equal to the length of the circle, which is described by point F - the center of the torus cross-section.

Slide 14

Indivisible method

The theoretical justification for the new method of finding areas and volumes was proposed in 1635 by Cavalieri. He put forward the following thesis: Figures are related to each other as all their lines, taken according to any regular [base of parallels], and bodies - as all their planes, taken according to any regular.

Slide 15

For example, let's calculate the area of ​​a circle. Formula for circumference: considered known. Let's divide the circle (on the left in Fig. 1) into infinitesimal rings. Let us also consider a triangle (on the right in Fig. 1) with base length L and height R, which is also divided into sections parallel to the base. Each ring of radius R and length can be associated with one of the sections of a triangle of the same length. Then, according to Cavalieri's principle, their areas are equal. And the area of ​​a triangle is easy to find: .

Slide 16

Worked on the presentation:

Zharkov Alexander Kiseleva Marina Ryasov Mikhail Cherednichenko Alina

View all slides

The general goal of the course is to reveal to students completing general mathematical education some historical aspects of mathematics and to show, to some extent, the nature of mathematical creativity. The general panorama of the development of mathematical ideas and theories, from the Babylonian and Egyptian periods to the beginning of the 20th century, is examined in a concise form. The course includes a section “Mathematics and Computer Science”, which provides an overview of the milestones in the history of computer technology, fragments of the history of the development of computers in Russia, and fragments of the history of computer science. A fairly large list of references and some reference material for independent work and for preparing abstracts are offered as teaching materials.

• The period of accumulation of mathematical knowledge.
  Formation of primary concepts: numbers and geometric shapes. Mathematics in the countries of ancient civilizations - in Ancient Egypt, Babylon, China, India. Basic types of number systems. The first achievements of arithmetic, geometry, algebra.
• Mathematics of constant quantities.
  Formation of mathematical science (VI century BC – VI century AD). The creation of mathematics as an abstract deductive science in Ancient Greece. Conditions for the development of mathematics in Ancient Greece. School of Pythagoras. Discovery of incommensurability and creation of geometric algebra. Famous problems of antiquity. Exhaustion method, infinitesimal methods of Eudoxus and Archimedes. Axiomatic construction of mathematics in Euclid's Elements. "Conic Sections" by Apollonius. Science of the first centuries of our era: “Mechanics” of Heron, “Almagest” of Ptolemy, his “Geography”, the emergence of a new letter algebra in the works of Diophantus and the beginning of the study of indefinite equations. The decline of ancient science.
  Mathematics of the peoples of Central Asia and the Arab East in the 7th-16th centuries. Separation of algebra into an independent field of mathematics. Formation of trigonometry in applications of mathematics to astronomy. State of mathematical knowledge in countries Western Europe and in Russia in the Middle Ages. "The Book of Abacus" by Leonardo of Pisa. Opening of the first universities. Advances in mathematics of the Renaissance.
• Panorama of the development of mathematics in the XVII-XIX centuries.
  Scientific revolution of the 17th century. and the creation of the mathematics of variables. The first academies of sciences. Mathematical analysis and its connection with mechanics in the 17th-18th centuries. Works of Euler, Lagrange, Laplace. The heyday of mathematics in France during the Revolution and the opening of the Polytechnic School.
• Algebra XVI-XIX centuries.
  Advances in algebra in the 16th century: solving algebraic equations of the third and fourth degree and the introduction of complex numbers. The creation of literal calculus by F. Viète and the beginning of the general theory of equations (Viète, Descartes). Euler's fundamental theorem of algebra and its proof. The problem of solving equations in radicals. Abel's theorem on the unsolvability of equations of degree n > 4 in radicals. Abel's results. Galois theory; introduction of group and field. The triumphant march of group theory: its role in algebra, geometry, analysis and mathematical science. The concept of n-dimensional vector space. Dedekind's axiomatic approach and the creation of abstract algebra.
• Development of mathematical analysis.
  The formation of mathematics of variable quantities in the 17th century, connection with astronomy: Kepler’s laws and the works of Galileo, developing the ideas of Copernicus. Invention of logarithms. Differential forms and integration methods in the works of Kepler, Cavalieri, Fermat, Descartes, Pascal, Wallis, N. Mercator. Creation of mathematical analysis by Newton and Leibniz. Mathematical analysis in the 18th century. and its connection with natural science. Euler's work. The doctrine of functions. Creation and development of the calculus of variations, the theory of differential equations and the theory of integral equations. Power series and trigonometric series. General theory of functions of a complex variable by Riemann and Weierstrass. Formation of functional analysis. Problems of substantiation of mathematical analysis. Its construction is based on the doctrine of limits. Works by Cauchy, Bolzano and Weierstrass. Theories of the real number (from Eudoxus to Dedekind). Creation of the theory of infinite sets by Cantor and Dedekind. The first paradoxes and problems of the foundations of mathematics.
• Mathematics in Russia (review).
  Math knowledge until the 17th century Reforms of Peter I. Founding of the St. Petersburg Academy of Sciences and Moscow University. St. Petersburg Mathematical School (M.V. Ostrogradsky, P.L. Chebyshev, A.A. Markov, A.M. Lyapunov). The main directions of Chebyshev's creativity. Life and work of S.V. Kovalevskaya. Organization of a mathematical society. Mathematical collection. The first scientific schools in the USSR. Moscow school of function theory (N.N. Luzin, D.F. Egorov and their students). Mathematics at Moscow University. Mathematics at the Ural University, Ural mathematical schools (P.G. Kontorovich, G.I. Malkin, E.A. Barbashin, V.K. Ivanov, S.B. Stechkin, A.F. Sidorov).
• Mathematics and Computer Science (overview)
  Milestones of computer technology from Leonardo da Vinci's sketch machine to the first computers.
  Fragments of the history of computers. The problem of automating complex calculations (aircraft design, atomic physics, etc.). Connecting electronics and logic: Leibniz's binary system, J. Boole's algebra of logic. "Computer Science" and "Informatics". Theoretical and applied computer science. New information technologies: scientific direction - artificial intelligence and its applications (using logical methods to prove the correctness of programs, providing an interface in professional natural language with application software packages, etc.).
  Fragments of the history of the development of computers in Russia. Developments by S.A. Lebedev and his students, their application (calculating the orbits of small planets, drawing up maps from geodetic surveys, creating dictionaries and translation programs, etc.). The creation of domestic machines (A.A. Lyapunov, A.P. Ershov, B.I. Rameev, M.R. Shura-Bura, G.P. Lopato, M.A. Kartsev and many others), the emergence of personal computers. Multifaceted use of machines: control of space flights, observation of outer space, in scientific work, for control of technological processes, processing of experimental data, electronic dictionaries and translators, economic tasks, teacher and student machines, household computers, etc.).

SUBJECTS OF ABSTRACTS

1. Biographical series.
2. The history of the formation and development of a specific branch of mathematics in a specific period. The history of the formation and development of mathematics in a specific historical period in a specific state.
3. History of origin scientific centers and their role in the development of specific branches of mathematics.
4. History of the formation and development of computer science for specific time periods.
5. The founders of some areas of computer science.
6. Specific outstanding scientists and world culture in different periods.
7. From the history Russian mathematics(specific historical era and specific individuals).

1. Ancient mechanics (" Combat vehicles antiquities").
2. Mathematics during the Arab Caliphate.
3. Foundations of geometry: From Euclid to Hilbert.
4. The remarkable mathematician Niels Henrik Abel.
5. 15th century encyclopedist Gerolamo Cardano.
6. The great Bernoulli family.
7. Prominent figures in the development of probability theory (from Laplace to Kolmogorov).
8. The period of the forerunner of the creation of differential and integral calculus.
9. Newton and Leibniz are the creators of differential and integral calculus.
10. Alexey Andreevich Lyapunov is the creator of the first computer in Russia.
11. "Passion for Science" (S.V. Kovalevskaya).
12. Blaise Pascal.
13. From the abacus to the computer.
14. “To be able to give direction is a sign of genius.” Sergei Alekseevich Lebedev. Developer and designer of the first computer in the Soviet Union.
15. The pride of Russian science is Pafnutiy Lvovich Chebyshev.
16. François Viète is the father of modern algebra and a brilliant cryptographer.
17. Andrei Nikolaevich Kolmogorov and Pavel Sergeevich Alexandrov are unique phenomena of Russian culture, its national treasure.
18. Cybernetics: neurons – automata – perceptrons.
19. Leonhard Euler and Russia.
20. Mathematics in Russia from Peter I to Lobachevsky.
21. Pierre Fermat and René Descartes.
22. How the personal computer was invented.
23. From the history of cryptography.
24. Generalization of the concept of geometric space. History of the creation and development of topology.
25. The golden ratio in music, astronomy, combinatorics and painting.
26. Golden ratio in the solar system.
27. Programming languages, their classification and development.
28. Probability theory. Aspect of history.
29. History of the development of non-Euclidean geometry (Lobachevsky, Gauss, Bolyai, Riemann).
30. The king of number theory is Carl Friedrich Gauss.
31. Three famous problems of antiquity as a stimulus for the emergence and development of various branches of mathematics.
32. Aryabhata, "Copernicus of the East".
33. David Gilbert. 23 Hilbert problems.
34. Development of the concept of number from Eudoxus to Dedekind.
35. Integral methods in Eudoxus and Archimedes.
36. Questions of mathematics methodology. Hypotheses, laws and facts.
37. Questions of mathematics methodology. Methods of mathematics.
38. Questions of mathematics methodology. Structure, driving forces, principles and patterns.
39. Pythagoras is a philosopher and mathematician.
40. Galileo Galilei. Formation of classical mechanics.
41. Life path and scientific activity M.V. Ostrogradsky.
42. Contribution of Russian scientists to the theory of probability.
43. Development of mathematics in Russia in the 18th and 19th centuries.
44. The history of the discovery of logarithms and their connection with areas.
45. From the history of the development of computer technology.
46. Computers before the electronic era. The first computers.
47. Milestones in the history of Russian computing technology and computer mathematics.
48. History of the development of operating systems. Chronology of the appearance of WINDOWS 98.
49. B. Pascal, G. Leibniz, P. Chebyshev.
50. Norbert Wiener, Claude Shannon and the theory of computer science.
51. From the history of mathematics in Russia.
52. Life and work of Gauss.
53. Formation and development of topology.
54. Évariste Galois – mathematician and revolutionary.
55. The golden ratio from Leonardo Fibonacci and Leonardo da Vinci to the 21st century.
56. Mathematics in Russia in the 18th-19th centuries.
57. Computer Science, history issues.
58. From the history of Russian mathematics: N.I. Lobachevsky, M.V. Ostrogradsky, S.V. Kovalevskaya.
59. Ancient mathematics VI-IV centuries. BC.
60. Programming languages: historical issues.
61. Pierre Fermat and René Descartes.
62. Leonard Euler.
63. The history of the creation of integral and differential calculus by I. Newton and G. Leibniz.
64. Mathematics of the 17th century as a forerunner of the creation of mathematical analysis.
65. Mathematical analysis after Newton and Leibniz: criticism and justification.
66. Mathematics of the 17th, 18th centuries: the formation of analytical, projective and differential geometries.

In the history of mathematics, we can roughly distinguish two main periods: elementary and modern mathematics. The milestone from which it is customary to count the era of new (sometimes called higher) mathematics was the 17th century - the century of the appearance of mathematical analysis. By the end of the 17th century. I. Newton, G. Leibniz and their predecessors created the apparatus of a new differential calculus and integral calculus, which forms the basis of mathematical analysis and even, perhaps, the mathematical basis of all modern natural science.

Mathematical analysis is a broad field of mathematics with a characteristic object of study (variable quantity), a unique research method (analysis by means of infinitesimals or by means of passages to limits), a certain system of basic concepts (function, limit, derivative, differential, integral, series) and constantly improving and a developing apparatus, the basis of which is differential and integral calculus.

Let's try to give an idea of ​​what kind of mathematical revolution occurred in the 17th century, what characterizes the transition associated with the birth of mathematical analysis from elementary mathematics to what is now the subject of research in mathematical analysis, and what explains its fundamental role in the entire modern system of theoretical and applied knowledge .

Imagine that in front of you is a beautifully executed color photograph of a stormy ocean wave rushing onto the shore: a powerful stooped back, a steep but slightly sunken chest, a head already tilted forward and ready to fall with a gray mane tormented by the wind. You stopped the moment, you managed to catch the wave, and you can now carefully study it in every detail without haste. A wave can be measured, and using the tools of elementary mathematics, you can draw many important conclusions about this wave, and therefore all its ocean sisters. But by stopping the wave, you deprived it of movement and life. Its origin, development, running, the force with which it hits the shore - all this turned out to be outside your field of vision, because you do not yet have either a language or a mathematical apparatus suitable for describing and studying not static, but developing, dynamic processes, variables and their relationships.

“Mathematical analysis is no less comprehensive than nature itself: it determines all tangible relationships, measures times, spaces, forces, temperatures.” J. Fourier

Movement, variables and their relationships surround us everywhere. Various types of motion and their patterns constitute the main object of study of specific sciences: physics, geology, biology, sociology, etc. Therefore, precise language and corresponding mathematical methods for describing and studying variable quantities turned out to be necessary in all areas of knowledge to approximately the same extent as numbers and arithmetic are necessary when describing quantitative relationships. So, mathematical analysis forms the basis of the language and mathematical methods for describing variables and their relationships. Nowadays, without mathematical analysis it is impossible not only to calculate space trajectories, the operation of nuclear reactors, the movement of ocean waves and the patterns of cyclone development, but also to economically manage production, distribution of resources, organization technological processes, predict the course of chemical reactions or changes in the numbers of various interconnected species of animals and plants in nature, because all of these are dynamic processes.

Elementary mathematics was mainly the mathematics of constant quantities, it studied mainly the relationships between elements geometric shapes, arithmetic properties of numbers and algebraic equations. Its attitude to reality can to some extent be compared with an attentive, even thorough and complete study of each fixed frame of a film that captures the changing, developing living world in its movement, which, however, is not visible in a separate frame and which can only be observed by looking the tape as a whole. But just as cinema is unthinkable without photography, so modern mathematics is impossible without that part of it that we conventionally call elementary, without the ideas and achievements of many outstanding scientists, sometimes separated by tens of centuries.

Mathematics is united, and the “higher” part of it is connected with the “elementary” part in much the same way as the next floor of a house under construction is connected with the previous one, and the width of the horizons that mathematics opens to us in the world, depends on which floor of this building we managed to climb to. Born in the 17th century. mathematical analysis has opened up opportunities for us to scientifically describe, quantitatively and qualitatively study variables and motion in the broad sense of the word.

What are the prerequisites for the emergence of mathematical analysis?

By the end of the 17th century. The following situation has arisen. Firstly, within the framework of mathematics itself over many years, some important classes of problems of the same type have accumulated (for example, problems of measuring areas and volumes of non-standard figures, problems of drawing tangents to curves) and methods for solving them in various special cases have appeared. Secondly, it turned out that these problems are closely related to the problems of describing arbitrary (not necessarily uniform) mechanical motion, and in particular with the calculation of its instantaneous characteristics (speed, acceleration at any time), as well as with finding the distance traveled for movement occurring at a given variable speed. The solution to these problems was necessary for the development of physics, astronomy, and technology.

Finally, thirdly, by the middle of the 17th century. the works of R. Descartes and P. Fermat laid the foundations of the analytical method of coordinates (the so-called analytical geometry), which made it possible to formulate geometric and physical problems of heterogeneous origin in the general (analytical) language of numbers and numerical dependencies, or, as we now say, numerical functions.

NIKOLAY NIKOLAEVICH LUZIN (1883-1950) N. N. Luzin - Soviet mathematician, founder of the Soviet school of function theory, academician (1929). Luzin was born in Tomsk and studied at the Tomsk gymnasium. The formalism of the gymnasium mathematics course alienated the talented young man, and only a capable tutor was able to reveal to him the beauty and greatness of mathematical science. In 1901, Luzin entered the mathematics department of the Faculty of Physics and Mathematics of Moscow University. From the first years of his studies, issues related to infinity fell into his circle of interests. At the end of the 19th century. The German scientist G. Cantor created the general theory of infinite sets, which received numerous applications in the study of discontinuous functions. Luzin began to study this theory, but his studies were interrupted in 1905. The student, who took part in revolutionary activities, had to leave for France for a while. There he listened to lectures by the most prominent French mathematicians of that time. Upon returning to Russia, Luzin graduated from the university and was left to prepare for a professorship. Soon he again left for Paris, and then to Göttingen, where he became close to many scientists and wrote his first scientific works. The main problem that interested the scientist was the question of whether there could be sets containing more elements than the set of natural numbers, but less than the set of points on a segment (the continuum problem). For any infinite set that could be obtained from segments using the operations of union and intersection of countable collections of sets, this hypothesis was satisfied, and in order to solve the problem, it was necessary to find out what other ways there were to construct sets. At the same time, Luzin studied the question of whether it is possible to represent any periodic function, even one with infinitely many discontinuity points, as a sum of a trigonometric series, i.e. the sum of an infinite number of harmonic vibrations. On these issues, Luzin obtained a number of significant results and in 1915 he defended his dissertation “Integral and trigonometric series,” for which he was immediately awarded the academic degree of Doctor of Pure Mathematics, bypassing the intermediate master’s degree that existed at that time. In 1917 Luzin became a professor at Moscow University. A talented teacher, he attracted the most capable students and young mathematicians. Luzin's school reached its peak in the first post-revolutionary years. Luzin’s students formed a creative team, which they jokingly called “Lusitania.” Many of them received first-class scientific results while still a student. For example, P. S. Aleksandrov and M. Ya. Suslin (1894-1919) discovered a new method for constructing sets, which served as the beginning of the development of a new direction - descriptive set theory. Research in this area carried out by Luzin and his students showed that the usual methods of set theory are not enough to solve many of the problems that arise in it. Luzin's scientific predictions were fully confirmed in the 60s. XX century Many of N. N. Luzin’s students later became academicians and corresponding members of the USSR Academy of Sciences. Among them is P. S. Alexandrov. A. N. Kolmogorov. M. A. Lavrentyev, L. A. Lyusternik, D. E. Menshov, P. S. Novikov. L. G. Shnirelman and others. Modern Soviet and foreign mathematicians in their works develop the ideas of N. N. Luzin.

The confluence of these circumstances led to the fact that at the end of the 17th century. two scientists - I. Newton and G. Leibniz - independently of each other managed to create a mathematical apparatus for solving these problems, summing up and generalizing individual results of their predecessors, including the ancient scientist Archimedes and contemporaries of Newton and Leibniz - B. Cavalieri, B. Pascal , D. Gregory, I. Barrow. This apparatus formed the basis of mathematical analysis - a new branch of mathematics that studies various developing processes, i.e. relationships between variables, which in mathematics are called functional dependencies or, in other words, functions. By the way, the term “function” itself was required and naturally arose precisely in the 17th century, and by now it has acquired not only general mathematical, but also general scientific significance.

Initial information about the basic concepts and mathematical apparatus of analysis is given in the articles “Differential calculus” and “Integral calculus”.

In conclusion, I would like to dwell on only one principle of mathematical abstraction, common to all mathematics and characteristic of analysis, and in this regard explain in what form mathematical analysis studies variables and what is the secret of such universality of its methods for studying all kinds of specific developing processes and their interrelations .

Let's look at a few illustrative examples and analogies.

Sometimes we no longer realize that, for example, a mathematical relation written not for apples, chairs or elephants, but in an abstract form abstracted from specific objects, is an outstanding scientific achievement. This is a mathematical law that, as experience shows, is applicable to various specific objects. So, studying in mathematics general properties abstract, abstract numbers, we thereby study quantitative relationships real world.

For example, from a school mathematics course it is known that, therefore, in a specific situation you could say: “If they don’t give me two six-ton ​​dump trucks to transport 12 tons of soil, then I can ask for three four-ton dump trucks and the work will be done, and if they give me only one four-ton dump truck, then she will have to make three flights.” Thus, the abstract numbers and numerical patterns that are now familiar to us are associated with their specific manifestations and applications.

The laws of change in specific variables and developing processes of nature are related in approximately the same way to the abstract, abstract form-function in which they appear and are studied in mathematical analysis.

For example, an abstract ratio may reflect the dependence of a cinema's box office on the number of tickets sold, if 20 is 20 kopecks - the price of one ticket. But if we are riding a bicycle on a highway, traveling 20 km per hour, then this same ratio can be interpreted as the relationship between the time (hours) of our cycling trip and the distance covered during this time (kilometers). You can always say that, for example, a change of several times leads to a proportional (i.e., the same number of times) change in the value of , and if , then the opposite conclusion is also true. This means, in particular, to double the box office of a movie theater, you will have to attract twice as many spectators, and in order to travel twice as far on a bicycle at the same speed, you will have to ride twice as long.

Mathematics studies both the simplest dependence and other, much more complex dependences in a general, abstract form, abstracted from a particular interpretation. The properties of a function or methods for studying these properties identified in such a study will be of the nature of general mathematical techniques, conclusions, laws and conclusions applicable to each specific phenomenon in which the function studied in abstract form occurs, regardless of what area of ​​knowledge this phenomenon belongs to .

So, mathematical analysis as a branch of mathematics took shape at the end of the 17th century. The subject of study in mathematical analysis (as it appears from modern positions) are functions, or, in other words, dependencies between variable quantities.

With the advent of mathematical analysis, mathematics became accessible to the study and reflection of developing processes in the real world; mathematics included variables and motion.