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Mathematical analysis is a set of branches of mathematics devoted to the study of functions and their generalizations using the methods of differential and integral calculus.
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exhaustion method
An ancient method for studying the area or volume of curvilinear figures.
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The method was as follows: to find the area (or volume) of a certain figure, a monotonous sequence of other figures was inscribed in this figure and it was proved that their areas (volumes) indefinitely approach the area (volume) of the desired figure.
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In 1696, L'Hopital wrote the first textbook, setting forth the new method as applied to the theory of plane curves. He called it analysis of infinitesimals, thus giving one of the names to the new branch of mathematics. In the introduction, Lopital outlines the history of the emergence of a new analysis, dwelling on the works of Descartes, Huygens, Leibniz, and also expresses his gratitude to the latter and the Bernoulli brothers.
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The term "function" first appears only in 1692 by Leibniz, but it was Euler who put it forward to the first roles. The original interpretation of the concept of a function was that a function is an expression for counting or an analytic expression.
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"The 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.
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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 founders of analytic geometry, mathematical analysis, probability theory and number theory. Fermat, practically according to modern rules, found tangents to algebraic curves.
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Rene Descartes (March 31, 1596 - February 11, 1650) - French mathematician, philosopher, physicist and physiologist, creator of analytic geometry and modern algebraic symbolism. In 1637, the main mathematical work of Descartes, "Discourse on the method" was published. This book outlined analytical geometry, and in applications - numerous results in algebra, geometry, optics and much more. Of particular note is Vieta's revised mathematical symbolism: he introduced now generally accepted signs for variables and sought values (x, y, z, ...) and for literal coefficients. (a, b, c, ...)
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François Viet (1540-1603) - French mathematician, founder of symbolic algebra. By education and main profession - a lawyer. In 1591, he introduced letter designations not only for unknown quantities, but also for the coefficients of equations. He established a uniform method for solving equations of the 2nd, 3rd, and 4th degrees. Among the discoveries, Viet himself especially appreciated the establishment of a relationship between the roots and coefficients of equations.
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Galileo Galilei (February 15, 1564, Pisa - January 8, 1642) - Italian physicist, mechanic, astronomer, philosopher and mathematician, who had a significant impact on the science of his time, formulated the "Galileo paradox": there are as many natural numbers as 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.
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"New stereometry of wine barrels"
When Kepler was buying 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 in a given barrel. So the scientist was the first to pay attention to the class of problems, the study of which led to the creation of integral calculus.
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So, for example, to find the formula for the volume of a torus, Kepler divided it by meridional sections into an infinite number of circles, the thickness of which on the outside was somewhat 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 the cylinder, in which the base area is equal to the area of the section of the torus, and the height is equal to the length of the circle, which is described by the point F - the center of the section of the torus.
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Method of indivisibles
The theoretical substantiation of a new method for finding areas and volumes was proposed in 1635 by Cavalieri. He put forward the following thesis: Figures are related to each other, like all their lines, taken along any regular [base of parallels], and bodies - like all their planes, taken along any regular.
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For example, let's calculate the area of a circle. The formula for the circumference of a circle is assumed to be known. Let's break the circle (on the left in Fig. 1) into infinitely small rings. Consider also a triangle (on the right in Fig. 1) with a base length L and a height R, which we also divide by 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. Finding the area of a triangle is easy:
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Zharkov Alexander Kiseleva Marina Ryasov Mikhail Cherednichenko Alina
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Philosophy is considered the center of all sciences, since it was it that included the first shoots of literature, astronomy, literature, natural science, mathematics and other areas. Over time, each area has developed independently, mathematics is no exception. The first "hint" to analysis is considered to be the theory of decomposition into infinitesimal quantities, which many minds tried to approach, but it was vague and had no basis. This is due to the attachment to the old school of science, which was strict in its formulations. Isaac Newton came very close to forming the foundations, but he was too late. As a result, mathematical analysis owes its appearance as a separate system to the philosopher Gottfried Leibniz. It was he who presented to the scientific world in his works such concepts as minimum and maximum, inflection points and convexity of the graph of a function, formulated the foundations of differential calculus. Since then, mathematics has been officially divided into elementary and higher.
Mathematical analysis. Our days
Any specialty, be it technical or humanitarian, includes analysis in the course of study. The depth of study varies, but the essence remains the same. Despite all the "abstractness", it is one of the pillars on which natural science in its modern sense is based. With his help, physics and economics were developed, he is able to describe and predict the activities of the stock exchange, help in building an optimal stock portfolio. Introduction to mathematical analysis is based on elementary concepts:
- sets;
- basic operations on sets;
- properties of operations on sets;
- functions (otherwise - mappings);
- function types;
- sequences;
- number lines;
- sequence limit;
- limit properties;
- function continuity.
It is worth highlighting separately such concepts as set, point, line, plane. All of them have no definitions, as they are the basic concepts on which all mathematics is built. All that can be done in the course of work is to explain what exactly they mean in individual cases.
Limit as a continuation
The limit is one of the fundamentals of mathematical analysis. In practice, it represents the value to which the sequence or function tends, comes as close as you like, but does not reach it. It is denoted as lim, consider special case function limit: lim (x-1)= 0 for x→1. From this simplest example, it can be seen that when x→1, the entire function tends to 0, since if we substitute the limit into the function itself, we get (1-1)=0. In more detail, from elementary to complicated special cases, the information is presented in a kind of "Bible" of analysis - the works of Fikhtengol'ts. There, mathematical analysis, limits, their derivation and further application are considered in the context. For example, the derivation of the number e (Euler's constant) would be impossible without the theory of limits. Despite the dynamic abstractness of the theory, the limits are actively used in practice in the same economics and sociology. For example, they cannot be dispensed with when calculating interest on a bank deposit.
In the history of mathematics, two main periods can be conventionally distinguished: elementary and modern mathematics. The milestone, from which it is customary to count the era of new (sometimes they say - higher) mathematics, was the 17th century - the century of the emergence of mathematical analysis. By the end of the XVII 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 vast area of mathematics with a characteristic object of study (a variable), a peculiar research method (analysis by means of infinitesimals or by passing to the limit), a certain system of basic concepts (function, limit, derivative, differential, integral, series) and constantly improving and developing apparatus, which is based on differential and integral calculus.
Let's try to give an idea of what kind of mathematical revolution took place in the 17th century, what characterizes the transition from elementary mathematics associated with the birth of mathematical analysis to the one that 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 running ashore: a powerful stooped back, a steep but slightly sunken chest, already tilted forward and ready to fall head with a gray mane torn by the wind. You have stopped the moment, you have managed to catch the wave, and now you can carefully study it in all its details without haste. A wave can be measured, and using the means of elementary mathematics, you will draw many important conclusions about this wave, and therefore all its oceanic sisters. But by stopping the wave, you have deprived it of movement and life. Its origin, development, run, the force with which it falls on the shore - all this turned out to be out of 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 interrelations.
"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 are all around us. Various types of motion and their regularities constitute the main object of study of specific sciences: physics, geology, biology, sociology, etc. Therefore, the exact language and appropriate mathematical methods for describing and studying variables turned out to be necessary in all areas of knowledge approximately to the same extent as numbers and arithmetic are necessary in describing quantitative relationships. So, mathematical analysis is the basis of the language and mathematical methods for describing variables and their relationships. Today, without mathematical analysis, it is impossible not only to calculate space trajectories, the operation of nuclear reactors, the running of an ocean wave and the patterns of cyclone development, but also to economically manage production, resource distribution, organization of technological processes, predict the course of chemical reactions or changes in the number of various species interconnected in nature. animals and plants, because all these are dynamic processes.
Elementary mathematics was basically the mathematics of constants, it studied mainly the relations between the elements of geometric figures, the arithmetic properties of numbers, and algebraic equations. To some extent, her attitude to reality can 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 on a separate frame and which can be observed only by looking tape as a whole. But just as cinema is unthinkable without photography, so modern mathematics is impossible without that part of it, which we conditionally call elementary, without the ideas and achievements of many outstanding scientists, sometimes separated by tens of centuries.
Mathematics is one, and its “higher” part is connected with the “elementary” one 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 up to us in the world around us depends on which floor of this building we managed to reach. rise. Born in the 17th century mathematical analysis opened up possibilities for scientific description, quantitative and qualitative study of variables and motion in the broadest sense of the word.
What are the prerequisites for the emergence of mathematical analysis?
By the end of the XVII century. the following situation has arisen. First, within the framework of mathematics itself, long 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 have appeared for solving them in various special cases. Secondly, it turned out that these problems are closely related to the problems of describing an arbitrary (not necessarily uniform) mechanical motion, and in particular with the calculation of its instantaneous characteristics (velocity, acceleration at any time), as well as with finding the distance traveled for movement at a given variable speed. The solution of these problems was necessary for the development of physics, astronomy, and technology.
Finally, thirdly, by the middle of the XVII 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 dependences, or, as we now say, numerical functions.
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NIKOLAI NIKOLAEVICH LUZIN
N. N. Luzin - Soviet mathematician, founder of the Soviet school of function theory, academician (1929). Luzin was born in Tomsk, studied at the Tomsk gymnasium. The formalism of the gymnasium course in mathematics alienated the talented young man, and only a capable tutor could reveal to him the beauty and grandeur of mathematical science. In 1901, Luzin entered the mathematical department of the Faculty of Physics and Mathematics of Moscow University. From the first years of study, questions related to infinity fell into the circle of his interests. AT late XIX in. the German scientist G. Kantor created the general theory of infinite sets, which has 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 his return to Russia, Luzin graduated from the university and was left to prepare for a professorship. Soon he again went to Paris, and then to Göttingen, where he became close to many scientists and wrote his first scientific papers. The main problem that interested the scientist was the question of whether there can be sets containing more elements than the set of natural numbers, but less than the set of points of the 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 true, and in order to solve the problem, it was necessary to find out what other ways of constructing sets were. At the same time, Luzin studied the question of whether it is possible to represent any periodic function, even if it has infinitely many discontinuity points, as the sum of a trigonometric series, i.e. sums of an infinite set of harmonic oscillations. Luzin obtained a number of significant results on these issues and in 1915 he defended his dissertation "The Integral and the Trigonometric Series", for which he was immediately awarded the 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 heyday in the first post-revolutionary years. Luzin's students formed a creative team, which was jokingly called "Luzitania". Many of them received first-class scientific results during their student days. For example, P. S. Aleksandrov and M. Ya. Suslin (1894-1919) discovered a new method for constructing sets, which initiated the development of a new direction - descriptive set theory. Research in this area, conducted by Luzin and his students, showed that the usual methods of set theory are not enough to solve many of the problems that arose in it. Luzin's scientific predictions were fully confirmed in the 1960s. 20th century Many students of N. N. Luzin later became academicians and corresponding members of the Academy of Sciences of the USSR. Among them P. S. Aleksandrov. A. N. Kolmogorov. M. A. Lavrentiev, 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 combination of these circumstances led to the late XVII in. two scientists - I. Newton and G. Leibniz - independently 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. interrelations of 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 a general mathematical, but also a general scientific meaning.
Initial information about the basic concepts and the 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 that is common to all mathematics and characteristic of analysis, and in this connection to 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 some explanatory examples and analogies.
We sometimes no longer realize that, for example, a mathematical ratio, 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 experience has shown to be applicable to various concrete objects. So, studying in mathematics general properties abstract, abstract numbers, we thereby study the quantitative relationships of the real world.
For example, it is known from a school mathematics course that, therefore, in a specific situation, you could say: “If two six-ton dump trucks are not allocated to me for transporting 12 tons of soil, then you can request three four-ton dump trucks and the work will be done, and if they give only one four-ton dump truck, then she will have to make three flights. Thus, the abstract numbers and numerical regularities that are now familiar to us are connected with their concrete manifestations and applications.
Approximately in the same way, the laws of change of concrete variable quantities and developing processes of nature are connected with the abstract, abstract form-function in which they appear and are studied in mathematical analysis.
For example, an abstract ratio may be a reflection of the dependence of the box office at the cinema on the number of tickets sold, if 20 is 20 kopecks - the price of one ticket. But if we are cycling on a highway at 20 km per hour, then the same ratio can be interpreted as the relationship of the time (hours) of our bike ride and the distance covered during this time (kilometers), you can always argue that, for example, a change by several times leads to a proportional (i.e., by the same number of times) change in the value of , and if , then the opposite conclusion is also true. So, in particular, to double the box office revenue of a cinema, you have to attract twice as many viewers, and to ride a bike at the same speed twice as far, you have to ride twice as long.
Mathematics studies both the simplest dependence and other, much more complex dependences in an abstract, general, abstract form abstracted from private interpretation. The properties of a function identified in such a study or methods for studying these properties will be in the nature of general mathematical techniques, conclusions, laws and conclusions applicable to each specific phenomenon in which the function studied in an abstract form occurs, regardless of which field 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 variables.
With the advent of mathematical analysis, it became possible for mathematics to study and reflect the developing processes of the real world; variables and motion entered mathematics.
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MATHEMATICS HISTORY. The most ancient mathematical activity was counting. The account was necessary to keep track of livestock and trade. Some primitive tribes counted the number of objects, correlating them with different parts of the body, mainly fingers and toes. The rock drawing, preserved to our times from the Stone Age, depicts the number 35 in the form of 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 a straight line and a circle. 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 is well-preserved clay tablets covered with so-called. cuneiform texts that date from 2000 BC. and up to 300 AD Mathematics on cuneiform tablets was mainly related to housekeeping. Arithmetic and simple algebra were used in the exchange of money and settlements for goods, the calculation of simple and compound interest, taxes, and the share of the crop 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 timing of agricultural work and religious holidays. The division of the circle into 360, and the degrees and minutes into 60 parts originate 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. For 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 more, the Babylonians introduced a positional number system with base 60. A significant advance was the positional principle, according to which the same numerical sign (symbol) has different meanings depending on the place where it is located. An example is the values of the six in the (modern) notation of the number 606. However, there was no zero in the number system of the ancient Babylonians, because of which the same set of characters could mean both the number 65 (60 + 5) and the number 3605 (60 2 + 0 + 5). There were also ambiguities in the interpretation of fractions. For example, the same symbols could mean both the number 21 and the fraction 21/60 and (20/60 + 1/60 2). The ambiguity was resolved depending on the specific context.
The Babylonians compiled tables of reciprocals (which were used when doing 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 devoted to the solution of algebraic and geometric problems indicate that they used the quadratic formula to solve quadratic equations and could solve some special types problems that included up to ten equations with ten unknowns, as well as individual varieties of cubic equations and equations of the fourth degree. 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 in 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 the angle inscribed in a semicircle is a right one. They also had rules for calculating the areas of simple flat 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 under crops and the volume of granaries, the amount of taxes and the number of stones required to build certain structures. In the papyri, one can also find problems related to determining the amount of grain needed to prepare a given number of mugs of beer, as well as more complex problems related to the difference in grain grades; for these cases, conversion factors were calculated.
But the main area of application of mathematics was astronomy, more precisely, calculations related to the calendar. The calendar was used to determine the dates of religious holidays and predict the annual floods of the Nile. However, the level of development of astronomy in Ancient Egypt was 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. The Egyptians used a non-positional decimal system, in which the numbers from 1 to 9 were denoted by the corresponding number of vertical lines, and individual symbols were introduced for successive powers of 10. By sequentially combining these symbols, it was possible to write down any number. 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 from 1 to 9 and for each of the first nine multiples of 10, 100, etc. used a special identification symbol. Fractions were written as a sum of fractions with a numerator equal to one. With such fractions, the Egyptians produced all four arithmetic operations, but the procedure for such calculations remained very cumbersome.
The geometry of the Egyptians was reduced to calculating the areas of rectangles, triangles, trapezoids, circles, as well as formulas for calculating the volumes of certain bodies. I must say that the mathematics that the Egyptians used in the construction of the pyramids was simple and primitive.
The tasks and solutions given in the papyri are formulated purely by prescription, without any explanation whatsoever. The Egyptians dealt only with the simplest types of quadratic equations and arithmetic and geometric progressions, and therefore those general rules that they were able to deduce were also of the simplest kind. Neither Babylonian nor Egyptian mathematicians had general methods; the entire body of mathematical knowledge was a collection of empirical formulas and rules.
Although the Maya who lived in Central America did not influence the development of mathematics, their achievements, dating back to about the 4th century, deserve attention. The Maya seem to have been the first to use a special character to represent zero in their base 20 system. They had two number systems: in one, hieroglyphs were used, and in the other, more common, the dot denoted one, the horizontal line denoted the number 5, and the symbol denoted zero. Positional designations began with the number 20, and the 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, which existed in the earlier period, was a set of empirical conclusions. On the contrary, in deductive reasoning, a new statement is deduced from the 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 come up with the idea of drawing conclusions solely on the basis of deductive reasoning from explicitly stated axioms. One of the explanations for the adherence of the Greeks to the methods of deduction we find in the structure of the Greek society of the classical period. Mathematicians and philosophers (often they were the same persons) belonged to the upper strata of society, where any practical activity was considered as an unworthy occupation. Mathematicians preferred abstract reasoning about numbers and spatial relationships to the solution of 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 developed by the time of Plato and Aristotle. The invention of deductive mathematics is usually 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 become acquainted with Babylonian and Egyptian mathematics during his long wanderings. Pythagoras founded a movement that flourished in the period ca. 550–300 BC The Pythagoreans created pure mathematics in the form of number theory and geometry. They represented integers in the form of configurations of dots or pebbles, classifying these numbers in accordance with the shape of the emerging 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, because corresponding number 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.
Some properties of integers arose from simple geometric configurations. 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 for this number itself, was called perfect by the Pythagoreans. Examples of perfect numbers are integers such as 6, 28 and 496. 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 therefore was identified with opinion. Four represented justice, since it is the first number equal to the product of two identical 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. Triplets 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 triple are equated to the lengths of the legs right triangle, then the third number will be equal to the length of its hypotenuse. This interpretation seems to have 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 found that the length of its hypotenuse is equal to , and this plunged them into confusion, because they tried in vain to represent the number as a ratio of two whole numbers, which was extremely important for their philosophy. Values that cannot be represented as a ratio of integers were called incommensurable by the Pythagoreans; 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 and are considered the lengths 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 speak of the number 25 as the square of 5, and the number 27 as the cube of 3.
The ancient Greeks solved equations with unknowns through geometric constructions. Special constructions were developed for performing 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 a 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 with the help of geometric methods. Geometry became the basis of almost all rigorous mathematics until at least 1600. And even in the 18th century, when algebra and mathematical analysis were already sufficiently developed, rigorous mathematics was treated as geometry, and the word "geometr" was equivalent to the word "mathematician".
It is to the Pythagoreans that we owe much of the mathematics that was then systematically presented and proved in Beginnings Euclid. There is reason to believe that it was they who discovered what is now known as the 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 is comprehensible only through mathematics. It is believed that he is credited with the invention of the analytical method of proof. ( Analytical method begins with the statement to be proved, and then the consequences are successively deduced from it until some known fact is reached; the proof is obtained by an inverse procedure.) It is generally accepted that the followers of Plato invented a method of proof, called "proof by contradiction." A prominent place in the history of mathematics is occupied by Aristotle, a student of Plato. 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 in importance only to Archimedes, 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 justified the Pythagorean method of dealing with irrational numbers.
The works of Eudoxus made it possible to establish the deductive structure of mathematics on the basis of explicitly formulated axioms. He also owns the first step in the creation of mathematical analysis, since it was he who invented the method of calculating areas and volumes, which was called the "method of exhaustion". This method consists in 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 study. Eudoxus also owns the first astronomical theory explaining the observed movement of the planets. The theory proposed by Eudoxus was purely mathematical; she showed how combinations of rotating spheres with different radii and axes of rotation could explain the seemingly irregular motions of the sun, moon, and planets.
Around 300 BC the results of many Greek mathematicians were brought together by Euclid, who wrote a mathematical masterpiece Beginnings. From a few astutely 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 line, angle, and circle. He then formulated 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. For mathematicians text Started Euclid served as a model of rigor for a long time, until in the 19th century. it was not found to have serious flaws, 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 keeping with the classical traditions. His analysis of conic sections - the 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, the mathematicians of the Alexandrian period were more inclined towards solving purely technical problems than towards philosophy. The great Alexandrian mathematicians - Eratosthenes, Archimedes, Hipparchus, Ptolemy, Diophantus, and Pappus - demonstrated the power of the Greek genius in theoretical abstraction, but were equally willing to apply their talent to solving practical problems and purely quantitative problems.
Eratosthenes (c. 275-194 BC) found a simple method for accurately calculating the circumference of the Earth, he also owns a calendar in which every fourth year has one day more than others. The astronomer Aristarchus (c. 310–230 BC) wrote an essay About the sizes and distances of the Sun and the Moon, containing one of the first attempts to determine these dimensions and distances; in character, Aristarchus's work was geometric.
The greatest mathematician of antiquity was Archimedes (c. 287–212 BC). He owns the formulations of many theorems on the areas and volumes of complex figures and bodies, quite rigorously proved by him by the method of exhaustion. Archimedes always sought to obtain exact solutions and found upper and lower bounds for ir rational numbers. For example, working with a regular 96-gon, he irreproachably proved that the exact value of the number p is between 3 1/7 and 3 10/71. Archimedes also proved several theorems containing new results in geometric algebra. He owns 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 isosceles hyperbola.
Archimedes was the greatest mathematical physicist of antiquity. To prove the theorems of mechanics, he used geometric considerations. His essay About floating bodies laid the foundations of hydrostatics. According to legend, Archimedes discovered the law bearing his name, according to which a buoyant force equal to the weight of the liquid displaced by him acts on a body immersed in water. into the street shouting "Eureka!" ("Opened!")
At the time of Archimedes, they were no longer limited geometric constructions, feasible only with a compass and straightedge. Archimedes used a spiral in his constructions, and Diocles (end of the 2nd century BC) solved the problem of doubling the cube with the help of a curve he introduced, called the cissoid.
During the Alexandrian period, arithmetic and algebra were considered independently of geometry. The Greeks of the classical period had a logically based 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 Hero 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 Nicomachus (c. 100 AD). In the history of arithmetic, its role is comparable to that of Started Euclid in the history of geometry. For over 1,000 years, it has served as the standard textbook because it clearly, concisely, and comprehensively expounds the doctrine of whole numbers (prime, composite, coprime, and also proportions). 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 connected with the introduction of the principles of symbolism into algebra. In his works, Diophantus did not offer general methods, he dealt with specific positive rational numbers, and not with their letter designations. He laid the foundations for the so-called. diophantine analysis - studies of indefinite equations.
The highest achievement of the Alexandrian mathematicians was the creation of quantitative astronomy. Hipparchus (c. 161-126 BC) we owe the invention of trigonometry. 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 the two corresponding sides of the other. In particular, the ratio of the length of the leg opposite the acute angle BUT in a right triangle, the length of the hypotenuse must be the same for all right triangles having the same acute angle BUT. This ratio is known as the sine of the angle BUT. The ratio of the lengths of the other sides of a right triangle is called the cosine and tangent of the angle BUT. 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 circumference 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 the latitudes and longitudes.
Greek trigonometry and its applications in astronomy reached their peak in Almagest Egyptian Claudius Ptolemy (died 168 AD). AT Almagest introduced the theory of motion celestial bodies, which dominated 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 is just a convenient mathematical description of astronomical phenomena, consistent with observations. The Copernican theory prevailed precisely because, as a model, it turned out to be simpler.
The decline of Greece.
After the conquest of Egypt by the Romans in 31 BC. the great Greek Alexandrian civilization fell into decay. Cicero proudly asserted that, unlike the Greeks, the Romans were not dreamers, and therefore put their mathematical knowledge into practice, deriving real benefits from them. 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 in the form of IX, came into wide use only after the invention of typesetting letters in the 15th century. Roman designations for numbers were 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 deal with proofs, but they introduced original concepts and a series 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 the 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 (born in 1114), he also owns the rules for operations on irrational numbers. The Indians introduced the concept of negative numbers (to denote debts). We find the earliest use of them in Brahmagupta (c. 630). Aryabhata (p. 476) went further than Diophantus in using continued fractions to solve indefinite equations.
Our modern number system, based on the positional principle of writing numbers and zero as a cardinal number and using the designation of an empty bit, is called Hindu-Arabic. On the wall of a temple built in India c. 250 BC, several figures were found, reminiscent of our modern numbers in their outlines.
Around 800 Indian mathematics reached Baghdad. The term "algebra" comes from the beginning of the title of the book Al-jabr wal-muqabala (Completion 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 writings of Brahmagupta, but Babylonian and Greek influences are clearly distinguishable in it. 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. Nasir al-Din Tusi (1201–1274) c Treatise on complete quadrilateral systematically expounded plane and spherical geometries 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 conquest by the Arabs North Africa and Spain, and later the works of the Greeks were translated into Latin.
MIDDLE AGES AND RENAISSANCE
Medieval Europe.
Roman civilization left little mark on mathematics because it was too preoccupied with solving practical problems. The civilization that developed in Europe in the early Middle Ages (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 Started Euclid. Astrology was considered the most important branch of mathematics in the Middle Ages; astrologers were called mathematicians. And since the practice of medicine was based predominantly on astrological indications or contraindications, the physicians had no choice but to become mathematicians.
Around 1100, Western European mathematics began an almost three-century period of development of 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 that time admitted that they drew inspiration from the writings of the Greeks.
The first European mathematician worth mentioning was Leonardo of Pisa (Fibonacci). In his essay abacus book(1202) he introduced Europeans to Indo-Arabic numerals and methods of calculation, as well as to Arabic algebra. Over the next few centuries, mathematical activity in Europe waned. The body of mathematical knowledge of that era, compiled by Luca Pacioli in 1494, did not contain any algebraic innovations that Leonardo did not have.
Renaissance.
Among the best geometers of the Renaissance were the artists who developed the idea of perspective, which required geometry with converging parallel lines. The artist Leon Battista Alberti (1404–1472) introduced the concepts of projection and section. Rectilinear rays of light from the eye of the observer to various points of the depicted scene form a projection; the section is obtained when the plane passes through the projection. In order for the painted picture to look realistic, it had to be such a section. The concepts of projection and section gave rise to purely mathematical questions. For example, what are the general geometric properties of the section and the original scene, what are the properties of two different sections of the same projection formed by two different planes intersecting the projection at different angles? Projective geometry arose from such questions. Its founder, J. Desargues (1593–1662), using proofs based on projection and section, unified the approach to various types of conic sections, which the great Greek geometer Apollonius considered separately.
THE BEGINNING OF MODERN MATHEMATICS
The offensive 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 put into circulation. A real triumph was the invention in 1614 of logarithms by J. Napier. By the end of the 17th century. the understanding of logarithms as exponents with any positive number other than one as a base has finally developed. 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 such a number 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 admissible on their own, without reference to geometry. In the 16th century controversy continued over the legality of introducing negative numbers. Even less acceptable were the complex numbers that appeared when solving quadratic equations, such as those called "imaginary" by Descartes. These numbers were under suspicion even in the 18th century, although L. Euler (1707-1783) successfully used them. 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 writing more precise, a variety of symbols have been 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 is 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 of algebraic equations - continued to occupy mathematicians in the early 19th century. When talking about the general solution of an equation of the second degree ax 2 + bx + c= 0, mean that each of its two roots can be expressed using a finite number of operations of addition, subtraction, multiplication, division and extraction of roots 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 higher than 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 in terms of their coefficients using a finite number of algebraic operations. In Galois theory, substitutions or permutations of roots were used and the concept of a group was introduced, which has found wide application in many areas of mathematics.
Analytic geometry.
Analytic, or coordinate, geometry was created independently by P. Fermat (1601–1665) and R. Descartes in order to expand the possibilities of Euclidean geometry in construction problems. However, Fermat considered his work only as a reformulation of the work of Apollonius. True discovery - the realization of all power algebraic methods belongs to Descartes. Euclidean geometric algebra for each construction required the invention of its original method and could not offer the quantitative information needed by science. Descartes solved this problem: he formulated geometric problems algebraically, solved an algebraic equation, and only then built the desired solution - a segment that had the appropriate length. Properly analytic geometry arose when Descartes began to consider indefinite construction problems, the solutions of which are not one, but a set of possible lengths.
Analytic geometry uses algebraic equations to represent and study curves and surfaces. Descartes considered acceptable a curve that could be written with a single algebraic equation for X and at. This approach was an important step forward, because it not only included such curves as the conchoid and cissoid, but also significantly expanded the range of curves. As a result, in the 17-18 centuries. many important new curves, such as the cycloid and the 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 algebraically obtained all the results presented in Book V Started Euclid.
Analytic geometry completely reversed the roles of geometry and algebra. As the great French mathematician Lagrange remarked, “While algebra and geometry have gone their separate ways, their progress has been slow and their applications limited. But when these sciences united their efforts, they borrowed new vitality from each other, and since then they have been heading towards perfection with rapid steps. see also ALGEBRAIC GEOMETRY; GEOMETRY; GEOMETRY REVIEW.
Mathematical analysis.
The founders of modern science - Copernicus, Kepler, Galileo and Newton - approached the study of nature as mathematics. While studying motion, mathematicians developed such a fundamental concept as a function, or a relationship between variables, for example d = kt 2 , where d is the distance traveled by a freely falling body, and t is the number of seconds the body is in free fall. The concept of function immediately became central to the definition of speed in this moment time and 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 a moment of time by dividing the path by the time, we will come to the mathematically meaningless expression 0/0.
The task of definition and calculation instant speeds changes in various quantities attracted the attention of almost all mathematicians of the 17th century, including Barrow, Fermat, Descartes, and Wallis. The disparate ideas and methods proposed by them were combined into a systematic, universally applicable formal method by Newton and G. Leibniz (1646–1716), the creators of the differential calculus. There was a heated debate between them over the priority in developing this calculus, with Newton accusing Leibniz of plagiarism. However, as studies of historians of science have shown, Leibniz created mathematical analysis independently of Newton. As a result of the conflict, the exchange of ideas between the mathematicians of continental Europe and England was interrupted for many years, to the detriment of the British side. English mathematicians continued to develop the ideas of analysis in a geometric direction, while the 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 a limit. Speed at a point in time is defined as the limit towards which the average speed tends d/t when the value t getting closer to zero. The differential calculus provides a convenient general method for finding the rate of change of a function f (x) for any value X. This speed is called the derivative. From the generality of the record f (x) it is clear that the concept of a derivative is applicable not only in problems related to the need to find the speed or acceleration, but also in relation to any functional dependence, for example, to some ratio from economic theory. One of the main applications of differential calculus is the so-called. tasks for maximum and minimum; Another important range of problems is finding the tangent to a given curve.
It turned out that with the help of the derivative, specially invented for working with problems of motion, one can also find areas and volumes bounded by curves and surfaces, respectively. The methods of Euclidean geometry did not have the proper 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 kind or another, and in some cases a connection was noted between these problems and problems of finding the rate of change of functions. But, as in the case of differential calculus, it was Newton and Leibniz who realized the generality of the method and thus 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 the limit underlying it, looked 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 has been increasingly widely used. Differential and integral calculus became the cornerstones of mathematical analysis, which eventually included such subjects as the theory of 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" - on the number system and Euclidean geometry. Since many properties of the number system were proved geometrically, Euclidean geometry was the most reliable part of the building of mathematics. Nevertheless, the axiom of parallels contained a statement about 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. Rather, it formulates a condition under which they intersect at some end point. For centuries, mathematicians have tried to find an appropriate replacement for the parallel axiom. But in every variant there was bound to be some kind of 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 exposition of non-Euclidean geometry. In their geometries through given point infinitely many parallel lines could be drawn. In the geometry of B. Riemann (1826–1866), not a single parallel line can be drawn through a point outside a straight line.
No one seriously thought about the physical applications of non-Euclidean geometry. The creation by A. Einstein (1879–1955) of the general theory of relativity in 1915 awakened the scientific world to the realization of the reality of non-Euclidean geometry.
mathematical rigor.
Until about 1870, mathematicians were convinced that they were acting according to the plans of the ancient Greeks, applying deductive reasoning to mathematical axioms, thereby providing their conclusions with no less reliability than that possessed by axioms. Non-Euclidean geometry and quaternions (an algebra in which the property of commutativity does not hold) made mathematicians realize that what they took for abstract and logically consistent statements, in reality rests on an empirical and pragmatic basis.
The creation of non-Euclidean geometry was also accompanied by the realization of the existence of logical gaps in Euclidean geometry. One of the shortcomings of the Euclidean Started was the use of assumptions not explicitly stated. Apparently, Euclid did not question the properties that his geometric figures, but these properties were not included in his axioms. In addition, proving the similarity of two triangles, Euclid used the imposition of one triangle on another, implicitly assuming that the properties of the figures do not change during movement. But apart from such logical gaps, in Beginnings There were also some erroneous proofs.
The creation of new algebras, which began with quaternions, gave rise to similar doubts about the logical validity of arithmetic and the algebra of the usual number system. All numbers previously known to mathematicians had the property of commutativity, i.e. ab = ba. Quaternions, which made a revolution in 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. The quaternions forced mathematicians to realize that, apart from the integer and far from perfect part of the Euclidean Started, 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 work successfully. Logical rigor has given way to demonstrating the practical utility of introducing dubious concepts and procedures.
Almost from the very beginning of mathematical analysis, there have been repeated attempts to give it a rigorous foundation. Mathematical analysis introduced two new complex concepts - derivative and definite integral. Newton and Leibniz fought over these concepts, as well as mathematicians of subsequent generations, who turned differential and integral calculus into mathematical analysis. However, despite all efforts, there was much that was not clear 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 calculus, and they have all failed. Finally, in 1821, O. Cauchy (1789-1857), using the concept of number, brought a strict basis for all mathematical analysis. However, later mathematicians discovered logical gaps in Cauchy. The desired severity was finally achieved in 1859 by K. Weierstrass (1815-1897).
Weierstrass initially considered the properties of real and complex numbers self-evident. Later, he, like G. Kantor (1845–1918) and R. Dedekind (1831–1916), realized the need to build a theory irrational numbers. They gave a correct definition of irrational numbers and established their properties, but the properties of rational numbers were still considered self-evident. Finally, the logical structure of the theory of real and complex numbers acquired its final form in the works of Dedekind and J. Peano (1858–1932). The creation of the foundations of a numerical system also made it possible to solve the problems of substantiating algebra.
The task of strengthening the rigor of the formulations of Euclidean geometry was relatively simple and amounted to listing the defined terms, clarifying the definitions, introducing the missing axioms, and filling in the 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 his approach is the interpretation of undefined terms: they can mean 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 from geometric associations. For a topologist, a point can be a function or a sequence of numbers, or anything else. Abstract space is a set of such "points" ( see also TOPOLOGY).
Hilbert's axiomatic method entered almost all branches of mathematics in the 20th century. However, it soon became clear that this method had certain limitations. In the 1880s, Cantor attempted to systematically classify infinite sets (e.g., the set of all rational numbers, the set real numbers etc.) by their comparative quantitative assessment, 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 solution, 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 the incompleteness theorem of K. Gödel (1906-1978). This theorem states that any consistent formal system rich enough to contain number theory necessarily contains an undecidable sentence, i.e. a statement that can neither be proved nor disproved within its framework. It is now generally accepted that there is no absolute proof in mathematics. As to what evidence is, opinions differ. However, most mathematicians are inclined 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 found logically rigorous structures; this shows that mathematics is not based on logic, but on sound intuition.
If mathematics known before 1600 can be characterized as elementary, then compared to what was created later, this elementary mathematics is infinitesimal. Old areas have expanded and new, both pure and applied branches of mathematical knowledge have appeared. Approximately 500 mathematical journals are published. A huge number of published results does not allow even a specialist to get acquainted with everything that happens in the field in which he works, not to mention the fact that many results are understandable only to a specialist with a narrow profile. No mathematician today can hope to know more than what goes on in a very small corner of science. see also articles about mathematicians.
Literature:
Van der Waerden B.L. Awakening Science. Mathematics 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
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हिन्दी: विकिपीडिया साइट को और अधिक सुरक्षित बना रहा है। आप एक पुराने वेब ब्राउज़र का उपयोग कर रहे हैं जो भविष्य में विकिपीडिया से कनेक्ट नहीं हो पाएगा। कृपया अपना डिवाइस अपडेट करें या अपने आईटी व्यवस्थापक से संपर्क करें। नीचे अंग्रेजी में एक लंबा और अधिक तकनीकी अद्यतन है।
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