Mathematics 1. Where did the word mathematics come from 2. Who invented mathematics? 3. Main themes. 4. Definition 5. Etymology On the last slide.

Where did the word come from (go to the previous slide) Mathematics from Greek - study, science) is the science of structures, order and relationships, historically based on the operations of counting, measuring and describing the shape of objects. Mathematical objects are created by idealizing the properties of real or other mathematical objects and writing these properties in a formal language.

Who invented mathematics (go to the menu) The first mathematician is usually called Thales of Miletus, who lived in the VI century. BC e. , one of the so-called Seven Wise Men of Greece. Be that as it may, it was he who was the first to structure the entire knowledge base on this subject, which has long been formed within the world known to him. However, the author of the first treatise on mathematics that has come down to us was Euclid (III century BC). He, too, deservedly be considered the father of this science.

Main topics (go to the menu) The field of mathematics includes only those sciences in which either order or measure is considered, and it does not matter at all whether these are numbers, figures, stars, sounds, or anything else in which this measure is found . Thus, there must be some general science that explains everything pertaining to order and measure, without entering into the study of any particular subjects, and this science must be called not by the foreign, but by the old, already common name of General Mathematics.

Definition (go to menu) Based on classical mathematical analysis modern analysis, which is considered as one of the three main areas of mathematics (along with algebra and geometry). At the same time, the term "mathematical analysis" in the classical sense is used mainly in curricula and materials. In the Anglo-American tradition, classical mathematical analysis corresponds to the course programs with the name "calculus"

Etymology (go to the menu) The word "mathematics" comes from other Greek. , which means study, knowledge, science, etc. -Greek, originally meaning receptive, successful, later related to study, later related to mathematics. Specifically, in Latin, it means the art of mathematics. The term is other -Greek. in the modern meaning of this word, “mathematics” is already found in the works of Aristotle (4th century BC). in "The Book of Selected Briefly on the Nine Muses and on the Seven Free Arts" (1672)

Mathematics is the science of quantitative relations and spatial forms of the real world. In close connection with the demands of science and technology, the stock of quantitative relations and spatial forms studied by mathematics is constantly expanding, so that the above definition must be understood in the most general sense.

The purpose of studying mathematics is to increase the general outlook, the culture of thinking, the formation of a scientific worldview.

Understanding the independent position of mathematics as a special science became possible after the accumulation of a fairly large amount of factual material and arose for the first time in Ancient Greece in the 6th-5th centuries BC. This was the beginning of the period of elementary mathematics.

During this period, mathematical research dealt only with a rather limited stock of basic concepts that arose with the simplest demands of economic life. At the same time, a qualitative improvement of mathematics as a science is already taking place.

Modern mathematics is often compared to a big city. This is an excellent comparison, because in mathematics, as in a big city, there is a continuous process of growth and improvement. New areas are emerging in mathematics, elegant and deep new theories are being built, like the construction of new neighborhoods and buildings. But the progress of mathematics is not limited to changing the face of the city due to the construction of a new one. We have to change the old. Old theories are included in new, more general ones; there is a need to strengthen the foundations of old buildings. New streets have to be laid in order to establish connections between the distant quarters of the mathematical city. But this is not enough - architectural design requires considerable effort, since the difference in styles of various areas of mathematics not only spoils general impression from science, but also hinders the understanding of science as a whole, the establishment of links between its various parts.

Another comparison is often used: mathematics is likened to a large branched tree, which, systematically, gives new shoots. Each branch of the tree is one or another area of ​​mathematics. The number of branches does not remain unchanged, as new branches grow, grow together at first growing separately, some of the branches dry up, deprived of nourishing juices. Both comparisons are successful and very well convey the actual state of affairs.

Undoubtedly, the demand for beauty plays an important role in the construction of mathematical theories. It goes without saying that the perception of beauty is very subjective and there are often quite ugly ideas about this. And yet one has to be surprised at the unanimity that mathematicians put into the concept of "beauty": the result is considered beautiful if from a small number of conditions it is possible to obtain a general conclusion relating to a wide range of objects. A mathematical derivation is considered beautiful if it is possible to prove a significant mathematical fact in it by simple and short reasoning. The maturity of a mathematician, his talent is guessed by how developed his sense of beauty is. Aesthetically complete and mathematically perfect results are easier to understand, remember and use; it is easier to identify their relationship with other areas of knowledge.

Mathematics in our time has become a scientific discipline with many areas of research, a huge number of results and methods. Mathematics is now so great that it is not possible for one person to cover it in all its parts, there is no possibility of being a universal specialist in it. The loss of connections between its separate directions is certainly a negative consequence of the rapid development of this science. However, at the basis of the development of all branches of mathematics there is a common thing - the origins of development, the roots of the tree of mathematics.

Euclid's geometry as the first natural science theory

• In the 3rd century BC, a book of Euclid with the same name appeared in Alexandria, in the Russian translation of "Beginnings". From the Latin name "Beginnings" came the term "elementary geometry". Although the writings of Euclid's predecessors have not come down to us, we can form some opinion about these writings from Euclid's Elements. In the "Beginnings" there are sections that are logically very little connected with other sections. Their appearance is explained only by the fact that they were introduced according to tradition and copy the "Beginnings" of Euclid's predecessors.

  Euclid's Elements consists of 13 books. Books 1 - 6 are devoted to planimetry, books 7 - 10 are about arithmetic and incommensurable quantities that can be built using a compass and straightedge. Books 11 to 13 were devoted to stereometry.

  The "Beginnings" begin with a presentation of 23 definitions and 10 axioms. The first five axioms are "general concepts", the rest are called "postulates". The first two postulates determine actions with the help of an ideal ruler, the third - with the help of an ideal compass. The fourth, "all right angles are equal to each other," is redundant, since it can be deduced from the rest of the axioms. The last, fifth postulate read: "If a line falls on two lines and forms interior one-sided angles in the sum of less than two lines, then, with an unlimited continuation of these two lines, they will intersect on the side where the angles are less than two lines."

  The five "general concepts" of Euclid are the principles of measuring lengths, angles, areas, volumes: "equal to the same are equal to each other", "if equals are added to equals, the sums are equal to each other", "if equals are subtracted from equals, the remainders are equal among themselves", "combining with each other are equal to each other", "the whole is greater than the part".

  Then came the criticism of Euclid's geometry. Euclid was criticized for three reasons: for the fact that he considered only such geometric quantities that can be built using a compass and straightedge; for breaking up geometry and arithmetic and proving for integers what he had already proved for geometric quantities, and, finally, for the axioms of Euclid. The fifth postulate, Euclid's most difficult postulate, has been most strongly criticized. Many considered it superfluous, and that it can and should be deduced from other axioms. Others believed that it should be replaced by a simpler and more illustrative one, equivalent to it: "Through a point outside a straight line, no more than one straight line can be drawn in their plane that does not intersect this straight line."

  Criticism of the gap between geometry and arithmetic led to the extension of the concept of number to a real number. Disputes about the fifth postulate led to the fact that at the beginning of the 19th century, N.I. Lobachevsky, J. Bolyai and K.F. Gauss built a new geometry in which all the axioms of Euclid's geometry were fulfilled, with the exception of the fifth postulate. It was replaced by the opposite statement: "In a plane through a point outside a line, more than one line can be drawn that does not intersect the given one." This geometry was as consistent as the geometry of Euclid.

  The Lobachevsky planimetry model on the Euclidean plane was built French mathematician Henri Poincaré in 1882.

  Draw a horizontal line on the Euclidean plane. This line is called the absolute (x). The points of the Euclidean plane lying above the absolute are the points of the Lobachevsky plane. The Lobachevsky plane is an open half-plane lying above the absolute. Non-Euclidean segments in the Poincaré model are arcs of circles centered on the absolute or line segments perpendicular to the absolute (AB, CD). The figure on the Lobachevsky plane is the figure of an open half-plane lying above the absolute (F). Non-Euclidean motion is a composition of a finite number of inversions centered on the absolute and axial symmetries whose axes are perpendicular to the absolute. Two non-Euclidean segments are equal if one of them can be translated into the other by a non-Euclidean movement. These are the basic concepts of the axiomatics of Lobachevsky's planimetry.

  All axioms of Lobachevsky's planimetry are consistent. "A non-Euclidean line is a semicircle with ends on the absolute, or a ray with origin on the absolute and perpendicular to the absolute." Thus, the assertion of Lobachevsky's axiom of parallelism holds not only for some line a and a point A not lying on this line, but also for any line a and any point A not lying on it.

  Behind Lobachevsky's geometry, other consistent geometries arose: projective geometry separated from Euclidean, multidimensional Euclidean geometry formed, Riemannian geometry arose ( general theory spaces with an arbitrary law of measurement of lengths), etc. From the science of figures in one three-dimensional Euclidean space, geometry in 40 - 50 years has turned into a set of various theories, only somewhat similar to its progenitor - the geometry of Euclid.

  The main stages of the formation of modern mathematics. Structure of modern mathematics

• Academician A.N. Kolmogorov identifies four periods in the development of mathematics Kolmogorov A.N. - Math, Math encyclopedic Dictionary, Moscow, Soviet Encyclopedia, 1988: the birth of mathematics, elementary mathematics, mathematics of variables, modern mathematics.

  During the development of elementary mathematics, the theory of numbers gradually grows out of arithmetic. Algebra is created as a literal calculus. And the system of presentation of elementary geometry, the geometry of Euclid, created by the ancient Greeks, became a model of deductive construction for two millennia ahead. mathematical theory.

  In the 17th century, the demands of natural science and technology led to the creation of methods that allow the mathematical study of movement, the processes of changing quantities, and the transformation of geometric figures. With the use of variables in analytic geometry and the creation of differential and integral calculus, the period of mathematics of variables begins. The great discoveries of the 17th century are the concept of an infinitesimal quantity introduced by Newton and Leibniz, the creation of the foundations for the analysis of infinitesimal quantities ( mathematical analysis).

  The concept of a function comes to the fore. Function becomes the main subject of study. The study of a function leads to the basic concepts of mathematical analysis: limit, derivative, differential, integral.

  The appearance of the brilliant idea of ​​R. Descartes on the method of coordinates also belongs to this time. Analytical geometry is created, which allows studying geometric objects by methods of algebra and analysis. On the other hand, the coordinate method opened up the possibility of a geometric interpretation of algebraic and analytic facts.

  Further development of mathematics led at the beginning of the 19th century to the formulation of the problem of studying possible types of quantitative relations and spatial forms from a fairly general point of view.

  The connection between mathematics and natural science is becoming more and more complex. New theories arise and they arise not only as a result of the demands of natural science and technology, but also as a result of the inner need of mathematics. A remarkable example of such a theory is the imaginary geometry of N.I. Lobachevsky. The development of mathematics in the 19th and 20th centuries allows us to attribute it to the period of modern mathematics. The development of mathematics itself, the mathematization of various fields of science, the penetration mathematical methods in many areas of practical activity, the progress of computer technology has led to the emergence of new mathematical disciplines, for example, operations research, game theory, mathematical economics, and others.

  The main methods in mathematical research are mathematical proofs - rigorous logical reasoning. Mathematical thinking is not limited to logical reasoning. Mathematical intuition is necessary for the correct formulation of the problem, for evaluating the choice of the method for solving it.

  In mathematics, mathematical models of objects are studied. The same mathematical model can describe the properties of real phenomena that are far from each other. Yes, same thing differential equation can describe the processes of population growth and the decay of radioactive material. For a mathematician, it is not the nature of the objects under consideration that is important, but the relations existing between them.

  There are two types of reasoning in mathematics: deduction and induction.

  Induction is a research method in which a general conclusion is built on the basis of particular premises.

  Deduction is a method of reasoning by means of which a conclusion of a particular nature follows from general premises.

  Mathematics plays an important role in natural science, engineering and humanities research. The reason for the penetration of mathematics into various branches of knowledge is that it offers very clear models for studying the surrounding reality, in contrast to the less general and more vague models offered by other sciences. Without modern mathematics, with its developed logical and computational apparatus, progress in various areas of human activity would be impossible.

  Mathematics is not only a powerful tool for solving applied problems and a universal language of science, but also an element of a common culture.

  Basic features of mathematical thinking

• On this issue, of particular interest is the characteristic of mathematical thinking given by A.Ya. Khinchin, or rather, its specific historical form - the style of mathematical thinking. Revealing the essence of the style of mathematical thinking, he singles out four features common to all epochs that noticeably distinguish this style from the styles of thinking in other sciences.

  First, the mathematician is characterized by the dominance of the logical scheme of reasoning brought to the limit. A mathematician who loses sight of this scheme, at least temporarily, loses the ability to think scientifically altogether. This peculiar feature of the style of mathematical thinking has a lot of value in itself. Obviously, to the maximum extent it allows you to monitor the correctness of the flow of thought and guarantees against errors; on the other hand, it forces the thinker to have before his eyes the totality of available possibilities during analysis and obliges him to take into account each of them without missing a single one (such omissions are quite possible and, in fact, are often observed in other styles of thinking).

  Secondly, conciseness, i.e. the conscious desire to always find the shortest logical path leading to a given goal, the merciless rejection of everything that is absolutely necessary for the impeccable validity of the argument. A mathematical essay of good style, does not tolerate any “water”, no embellishing, weakening the logical tension of ranting, distraction to the side; extreme stinginess, severe strictness of thought and its presentation are an integral feature of mathematical thinking. This feature is of great value not only for mathematical, but also for any other serious reasoning. Laconism, the desire not to allow anything superfluous, helps both the thinker himself and his reader or listener to fully concentrate on this course thoughts without being distracted by side ideas and without losing direct contact with the main line of reasoning.

  The luminaries of science, as a rule, think and express themselves succinctly in all fields of knowledge, even when their thought creates and sets out fundamentally new ideas. What a majestic impression, for example, the noble stinginess of thought and speech of the greatest creators of physics: Newton, Einstein, Niels Bohr! Perhaps it is difficult to find a more striking example of what a profound effect the style of thinking of its creators can have on the development of science.

  For mathematics, the conciseness of thought is an indisputable law, canonized for centuries. Any attempt to burden the presentation with not necessarily necessary (even if pleasant and exciting for the listeners) pictures, distractions, oratory is placed under legitimate suspicion in advance and automatically causes critical alertness.

  Thirdly, a clear dissection of the course of reasoning. If, for example, when proving a proposition, we must consider four possible cases, each of which can be broken down into one or another number of subcases, then at each moment of reasoning, the mathematician must clearly remember in which case and subcase his thought is now being acquired and which cases and subcases he still has to consider. With any kind of ramified enumerations, the mathematician must at every moment be aware of for which generic concept he enumerates the species concepts that make up it. In ordinary, non-scientific thinking, we very often observe confusion and jumps in such cases, leading to confusion and errors in reasoning. It often happens that a person begins to enumerate the species of one genus, and then, imperceptibly to the listeners (and often to himself), using the insufficient logical distinctness of the reasoning, jumped into another genus and ends with the statement that both genera are now classified; and listeners or readers do not know where the boundary lies between species of the first and second kind.

  In order to make such confusions and jumps impossible, mathematicians have long made extensive use of simple external methods of numbering concepts and judgments, sometimes (but much less often) used in other sciences. Those possible cases or those generic concepts that should be considered in this reasoning are renumbered in advance; within each such case, those subcases to be considered that it contains are also renumbered (sometimes, for distinction, using some other numbering system). Before each paragraph, where the consideration of a new subcase begins, the designation accepted for this subcase is put (for example: II 3 - this means that the consideration of the third subcase of the second case begins here, or the description of the third type of the second kind, if we are talking about classification). And the reader knows that until he comes across a new numerical rubric, everything that is presented applies only to this case and subcase. It goes without saying that such numbering is only an external device, very useful, but by no means obligatory, and that the essence of the matter is not in it, but in that distinct division of argumentation or classification, which it both stimulates and marks by itself.

  Fourthly, scrupulous accuracy of symbols, formulas, equations. That is, “each mathematical symbol has a strictly defined meaning: replacing it with another symbol or rearranging it to another place, as a rule, entails a distortion, and sometimes complete destruction of the meaning of this statement.”

  Having singled out the main features of the mathematical style of thinking, A.Ya. Khinchin notes that mathematics (especially the mathematics of variables) by its nature has a dialectical character, and therefore contributes to the development of dialectical thinking. Indeed, in the process of mathematical thinking there is an interaction between visual (concrete) and conceptual (abstract). “We cannot think of lines,” Kant wrote, “without drawing it mentally, we cannot think of three dimensions for ourselves without drawing three lines perpendicular to each other from one point.”

  The interaction of concrete and abstract “led” mathematical thinking to the development of new and new concepts and philosophical categories. In ancient mathematics (mathematics of constants), these were “number” and “space”, which were originally reflected in arithmetic and Euclidean geometry, and later in algebra and various geometric systems. The mathematics of variables was “based” on concepts that reflected the movement of matter - “finite”, “infinite”, “continuity”, “discrete”, “infinitely small”, “derivative”, etc.

  If we talk about the current historical stage in the development of mathematical knowledge, then it goes in line with the further development of philosophical categories: the theory of probability “masters” the categories of the possible and the random; topology - categories of relation and continuity; catastrophe theory - jump category; group theory - categories of symmetry and harmony, etc.

  In mathematical thinking, the main patterns of constructing logical connections similar in form are expressed. With its help, the transition from the singular (say, from certain mathematical methods - axiomatic, algorithmic, constructive, set-theoretic and others) to the special and general, to generalized deductive constructions is carried out. The unity of the methods and the subject of mathematics determines the specifics of mathematical thinking, allows us to speak of a special mathematical language that not only reflects reality, but also synthesizes, generalizes, and predicts scientific knowledge. The power and beauty of mathematical thought lies in the utmost clarity of its logic, the elegance of constructions, and the skilful construction of abstractions.

  Fundamentally new possibilities of mental activity opened up with the invention of the computer, with the creation of machine mathematics. Significant changes have taken place in the language of mathematics. If the language of classical computational mathematics consisted of formulas of algebra, geometry and analysis, focused on the description of the continuous processes of nature, studied primarily in mechanics, astronomy, physics, then its modern language is the language of algorithms and programs, including the old language of formulas as a particular case.

  The language of modern computational mathematics is becoming more and more universal, capable of describing complex (multi-parameter) systems. At the same time, I would like to emphasize that no matter how perfect the mathematical language, enhanced by electronic computing technology, it does not break ties with the diverse “living”, natural language. Little of, colloquial is the basis of an artificial language. In this regard, the recent discovery of scientists is of interest. The point is that the ancient language of the Aymara Indians, which is spoken by about 2.5 million people in Bolivia and Peru, turned out to be extremely convenient for computer technology. As early as 1610, the Italian Jesuit missionary Ludovico Bertoni, who compiled the first Aymara dictionary, noted the genius of its creators, who achieved high logical purity. In Aymara, for example, there are no irregular verbs and no exceptions to the few clear grammatical rules. These features of the Aymara language allowed the Bolivian mathematician Ivan Guzman de Rojas to create a system of simultaneous computer translation from any of the five European languages ​​included in the program, the “bridge” between which is the Aymara language. The computer "Aymara", created by a Bolivian scientist, was highly appreciated by specialists. Summarizing this part of the question about the essence of the mathematical style of thinking, it should be noted that its main content is the understanding of nature.

  Axiomatic Method

• Axiomatics is the main way of constructing a theory, from antiquity to the present day, confirming its universality and all applicability.

  The construction of a mathematical theory is based on the axiomatic method. The scientific theory is based on some initial provisions, called axioms, and all other provisions of the theory are obtained as logical consequences of the axioms.

  The axiomatic method appeared in ancient Greece, and is currently used in almost all theoretical sciences, and, above all, in mathematics.

  Comparing three, in a certain respect, complementary geometries: Euclidean (parabolic), Lobachevsky (hyperbolic), and Riemannian (elliptic), it should be noted that, along with some similarities, there is a big difference between spherical geometry, on the one hand, and the geometries of Euclid and Lobachevsky - on the other.

  The fundamental difference between modern geometry is that it now embraces the "geometries" of an infinite number of different imaginary spaces. However, it should be noted that all these geometries are interpretations of Euclidean geometry and are based on the axiomatic method first used by Euclid.

  On the basis of research, the axiomatic method has been developed and widely used. As a special case of applying this method is the method of traces in stereometry, which allows solving problems on the construction of sections in polyhedra and some other positional problems.

  The axiomatic method, first developed in geometry, has now become an important tool of study in other branches of mathematics, physics, and mechanics. Currently, work is underway to improve and study the axiomatic method of constructing a theory in more depth.

  The axiomatic method of constructing a scientific theory consists in highlighting the basic concepts, formulating the axioms of theories, and all other statements are derived in a logical way, based on them. It is known that one concept must be explained with the help of others, which, in turn, are also defined with the help of some well-known concepts. Thus, we arrive at elementary concepts that cannot be defined in terms of others. These concepts are called basic.

  When we prove a statement, a theorem, we rely on premises that are considered already proven. But these premises were also proved, they had to be substantiated. In the end, we come to unprovable statements and accept them without proof. These statements are called axioms. The set of axioms must be such that, relying on it, one can prove further statements.

  Having singled out the main concepts and formulated the axioms, then we derive theorems and other concepts in a logical way. This is the logical structure of geometry. Axioms and basic concepts form the foundations of planimetry.

  Since it is impossible to give a single definition of the basic concepts for all geometries, the basic concepts of geometry should be defined as objects of any nature that satisfy the axioms of this geometry. Thus, in the axiomatic construction of a geometric system, we start from a certain system of axioms, or axiomatics. These axioms describe the properties of the basic concepts of a geometric system, and we can represent the basic concepts in the form of objects of any nature that have the properties specified in the axioms.

  After formulating and proving the first geometric statements, it becomes possible to prove some statements (theorems) with the help of others. The proofs of many theorems are attributed to Pythagoras and Democritus.

  Hippocrates of Chios is credited with compiling the first systematic course geometry based on definitions and axioms. This course and its subsequent processings were called "Elements".

  Axiomatic method of constructing a scientific theory

• The creation of a deductive or axiomatic method of constructing science is one of the greatest achievements of mathematical thought. It required the work of many generations of scientists.

  A remarkable feature of the deductive system of presentation is the simplicity of this construction, which makes it possible to describe it in a few words.

  The deductive system of presentation is reduced to:

  1) to the list of basic concepts,

  2) to the presentation of definitions,

  3) to the presentation of the axioms,

  4) to the presentation of theorems,

  5) to the proof of these theorems.

  An axiom is a statement accepted without proof.

  A theorem is a statement that follows from axioms.

  Proof - component deductive system, it is reasoning which shows that the truth of a statement follows logically from the truth of previous theorems or axioms.

  Within a deductive system, two questions cannot be resolved: 1) about the meaning of the basic concepts, 2) about the truth of the axioms. But this does not mean that these questions are generally unsolvable.

  The history of natural science shows that the possibility of an axiomatic construction of a particular science appears only at a fairly high level of development of this science, on the basis of a large amount of factual material, which makes it possible to clearly identify the main connections and relationships that exist between the objects studied by this science.

  An example of the axiomatic construction of mathematical science is elementary geometry. The system of axioms of geometry was expounded by Euclid (about 300 BC) in the work "Beginnings" unsurpassed in its significance. This system has largely survived to this day.

  Basic concepts: point, line, plane basic images; lie between, belong, move.

  Elementary geometry has 13 axioms, which are divided into five groups. In the fifth group, there is one axiom about parallels (V postulate of Euclid): only one straight line can be drawn through a point on a plane that does not intersect this straight line. This is the only axiom that caused the need for proof. Attempts to prove the fifth postulate occupied mathematicians for more than 2 millennia, up to the first half of the 19th century, i.e. until the moment when Nikolai Ivanovich Lobachevsky proved in his writings the complete hopelessness of these attempts. At present, the unprovability of the fifth postulate is a strictly proven mathematical fact.

  Axiom about parallel N.I. Lobachevsky replaced the axiom: Let a straight line and a point lying outside the straight line be given in a given plane. Through this point, at least two parallel lines can be drawn to the given line.

  From the new system of axioms N.I. Lobachevsky, with impeccable logical rigor, deduced a coherent system of theorems that constitute the content of non-Euclidean geometry. Both geometries of Euclid and Lobachevsky are equal as logical systems.

  Three great mathematicians in the 19th century almost simultaneously, independently of each other, came to the same results of the unprovability of the fifth postulate and to the creation of non-Euclidean geometry.

  Nikolai Ivanovich Lobachevsky (1792-1856)

  Carl Friedrich Gauss (1777-1855)

  Janos Bolyai (1802-1860)

  Mathematical proof

• The main method in mathematical research is mathematical proof - rigorous logical reasoning. By virtue of objective necessity, points out Corresponding Member of the Russian Academy of Sciences L.D. Kudryavtsev Kudryavtsev L.D. - Modern mathematics and its teaching, Moscow, Nauka, 1985, logical reasoning (which by its nature, if correct, is also rigorous) is a method of mathematics, mathematics is unthinkable without them. It should be noted that mathematical thinking is not limited to logical reasoning. For the correct formulation of the problem, for the evaluation of its data, for the selection of significant ones from them and for the choice of a method for solving it, mathematical intuition is also necessary, which makes it possible to foresee the desired result before it is obtained, to outline the path of research with the help of plausible reasoning. But the validity of the fact under consideration is proved not by checking it on a number of examples, not by conducting a number of experiments (which in itself plays a big role in mathematical research), but in a purely logical way, according to the laws of formal logic.

  It is believed that mathematical proof is the ultimate truth. A decision that is based on pure logic simply cannot be wrong. But with the development of science and the tasks before mathematicians are put more and more complex.

  “We have entered an era when the mathematical apparatus has become so complex and cumbersome that at first glance it is no longer possible to say whether the problem encountered is true or not,” believes Keith Devlin from Stanford University, California, USA. He cites as an example the “classification of simple finite groups”, which was formulated back in 1980, but a complete exact proof has not yet been imparted. Most likely, the theorem is true, but it is impossible to say for sure about this.

  A computer solution cannot be called exact either, because such calculations always have an error. In 1998, Hales proposed a computer-assisted solution to Kepler's theorem, formulated back in 1611. This theorem describes the densest packing of balls in space. The proof was presented on 300 pages and contained 40,000 lines of machine code. 12 reviewers checked the solution for a year, but they never achieved 100% confidence in the correctness of the proof, and the study was sent for revision. As a result, it was published only after four years and without full certification of the reviewers.

  All the latest calculations for applied problems are made on a computer, but scientists believe that for greater reliability, mathematical calculations should be presented without errors.

  The theory of proof is developed in logic and includes three structural components: thesis (what is supposed to be proved), arguments (a set of facts, generally accepted concepts, laws, etc. of the relevant science) and demonstration (the procedure for deploying evidence itself; a consistent chain of inferences when The nth inference becomes one of the premises of the n+1th inference). The rules of proof are distinguished, possible logical errors are indicated.

  Mathematical proof has much in common with the principles established by formal logic. Moreover, the mathematical rules of reasoning and operations obviously served as one of the foundations in the development of the proof procedure in logic. In particular, researchers of the history of the formation of formal logic believe that at one time, when Aristotle took the first steps to create laws and rules of logic, he turned to mathematics and to the practice of legal activity. In these sources, he found material for the logical constructions of the conceived theory.

  In the 20th century, the concept of proof lost its strict meaning, which happened in connection with the discovery of logical paradoxes hidden in set theory and especially in connection with the results that K. Gödel's theorems on the incompleteness of formalization brought.

  First of all, this affected mathematics itself, in connection with which it was believed that the term "proof" has no exact definition. But if such an opinion (which still holds today) affects mathematics itself, then they come to the conclusion that the proof should be accepted not in the logico-mathematical, but in the psychological sense. Moreover, a similar view is found in Aristotle himself, who believed that to prove means to conduct a reasoning that would convince us to such an extent that, using it, we convince others of the correctness of something. We find a certain shade of the psychological approach in A.E. Yesenin-Volpin. He sharply opposes the acceptance of truth without proof, linking it with an act of faith, and further writes: "I call the proof of a judgment an honest method that makes this judgment undeniable." Yesenin-Volpin reports that his definition still needs to be clarified. At the same time, does not the very characterization of evidence as an "honest method" betray an appeal to a moral-psychological assessment?

  At the same time, the discovery of set-theoretic paradoxes and the appearance of Godel's theorems just contributed to the development of the theory of mathematical proof undertaken by intuitionists, especially the constructivist direction, and D. Hilbert.

  Sometimes it is believed that mathematical proof is universal and represents an ideal version of scientific proof. However, it is not the only method; there are other methods of evidence-based procedures and operations. It is only true that the mathematical proof has a lot in common with the formal logical one implemented in natural science, and that the mathematical proof has certain specifics, as well as the set of techniques-operations. This is where we will stop, omitting the general thing that makes it related to other forms of evidence, that is, without expanding the algorithm, rules, errors, etc. in all steps (even the main ones). proof process.

  Mathematical proof is a reasoning that has the task of substantiating the truth (of course, in the mathematical, that is, as deducibility, sense) of a statement.

  The set of rules used in the proof was formed along with the advent of axiomatic constructions of mathematical theory. This was realized most clearly and completely in the geometry of Euclid. His "Principles" became a kind of model standard for the axiomatic organization of mathematical knowledge, and for a long time remained such for mathematicians.

  Statements presented in the form of a certain sequence must guarantee a conclusion, which, subject to the rules of logical operation, is considered proven. It must be emphasized that a certain reasoning is a proof only with respect to some axiomatic system.

  When characterizing a mathematical proof, two main features are distinguished. First of all, the fact that mathematical proof excludes any reference to empirical evidence. The entire procedure for substantiating the truth of the conclusion is carried out within the framework of the accepted axiomatics. Academician A.D. Aleksandrov emphasizes in this regard. You can measure the angles of a triangle thousands of times and make sure that they are equal to 2d. But math doesn't prove anything. You will prove it to him if you deduce the above statement from the axioms. Let's repeat. Here mathematics is close to the methods of scholasticism, which also fundamentally rejects argumentation by experimentally given facts.

  For example, when the incommensurability of segments was discovered, when proving this theorem, an appeal to a physical experiment was excluded, since, firstly, the very concept of "incommensurability" is devoid of physical meaning, and, secondly, mathematicians could not, when dealing with abstraction, to bring to the aid material-concrete extensions, measurable by a sensory-visual device. The incommensurability, in particular, of the side and diagonal of a square, is proved based on the property of integers using the Pythagorean theorem on the equality of the square of the hypotenuse (respectively, the diagonal) to the sum of the squares of the legs (two sides of a right triangle). Or when Lobachevsky was looking for confirmation for his geometry, referring to the results of astronomical observations, then this confirmation was carried out by him by means of a purely speculative nature. Cayley-Klein and Beltrami's interpretations of non-Euclidean geometry also featured typically mathematical rather than physical objects.

  The second feature of mathematical proof is its highest abstractness, in which it differs from proof procedures in other sciences. And again, as in the case of the concept of a mathematical object, it is not just about the degree of abstraction, but about its nature. The fact is that high level The proof reaches abstraction in a number of other sciences, for example, in physics, cosmology and, of course, in philosophy, since the ultimate problems of being and thinking become the subject of the latter. Mathematics, on the other hand, is distinguished by the fact that variables function here, the meaning of which is in abstraction from any specific properties. Recall that, by definition, variables are signs that in themselves have no meanings and acquire the latter only when the names of certain objects are substituted for them (individual variables) or when specific properties and relations are indicated (predicate variables), or, finally, in cases of replacing a variable with a meaningful statement (propositional variable).

  The noted feature determines the nature of the extreme abstractness of the signs used in the mathematical proof, as well as statements, which, due to the inclusion of variables in their structure, turn into statements.

  The very procedure of proof, defined in logic as a demonstration, proceeds on the basis of the rules of inference, based on which the transition from one proven statement to another is carried out, forming a consistent chain of inferences. The most common are the two rules (substitution and derivation of conclusions) and the deduction theorem.

  substitution rule. In mathematics, substitution is defined as the replacement of each of the elements a of a given set by some other element F(a) from the same set. In mathematical logic, the substitution rule is formulated as follows. If a true formula M in the propositional calculus contains a letter, say A, then by replacing it wherever it occurs with an arbitrary letter D, we get a formula that is also true as the original one. This is possible, and admissible, precisely because in the calculus of propositions one abstracts from the meaning of propositions (formulas)... Only the values ​​"true" or "false" are taken into account. For example, in the formula M: A--> (BUA) we substitute the expression (AUB) in place of A, as a result we get a new formula (AUB) -->[(BU(AUB) ].

  The rule for inferring conclusions corresponds to the structure of the conditionally categorical syllogism modus ponens (affirmative mode) in formal logic. It looks like this:

  a .

  Given a proposition (a-> b) and also given a. It follows b.

  For example: If it is raining, then the pavement is wet, it is raining (a), therefore, the pavement is wet (b). In mathematical logic, this syllogism is written as follows (a-> b) a-> b.

  An inference is usually defined by separating for implication. If an implication (a-> b) and its antecedent (a) are given, then we have the right to add to the reasoning (proof) also the consequent of this implication (b). Syllogism is coercive, constituting an arsenal of deductive means of proof, that is, absolutely meeting the requirements of mathematical reasoning.

  An important role in mathematical proof is played by the deduction theorem - the general name for a number of theorems, the procedure of which makes it possible to establish the provability of the implication: A-> B, when there is a logical derivation of the formula B from the formula A. In the most common version of the propositional calculus (in the classical, intuitionistic and other types of mathematics), the deduction theorem states the following. If a system of premises G and a premise A are given, from which, according to the rules, B Г, A B (- sign of derivability) can be deduced, then it follows that only from the premises of G can one obtain the sentence A --> B.

  We have considered the type, which is a direct proof. At the same time, the so-called indirect evidence is also used in logic; there are non-direct evidence that unfolds according to the following scheme. Not having, due to a number of reasons (inaccessibility of the object of study, the loss of the reality of its existence, etc.) direct evidence the truth of any statement, thesis, build an antithesis. They are convinced that the antithesis leads to contradictions, and, therefore, is false. Then from the fact of the falsity of the antithesis one draws - on the basis of the law of the excluded middle (a v) - the conclusion about the truth of the thesis.

  In mathematics, one of the forms of indirect proof is widely used - proof by contradiction. It is especially valuable and, in fact, indispensable in the acceptance of fundamental concepts and provisions of mathematics, for example, the concept of actual infinity, which cannot be introduced in any other way.

  The operation of proof by contradiction is represented in mathematical logic as follows. Given a sequence of formulas G and the negation of A (G , A). If this implies B and its negation (G , A B, non-B), then we can conclude that the truth of A follows from the sequence of formulas G. In other words, the truth of the thesis follows from the falsity of the antithesis.

  References:

• 1. N. Sh. Kremer, B. A. Putko, I. M. Trishin, M. N. Fridman, Higher Mathematics for Economists, textbook, Moscow, 2002;

  2. L.D. Kudryavtsev, Modern mathematics and its teaching, Moscow, Nauka, 1985;

  3. O. I. Larichev, Objective models and subjective decisions, Moscow, Nauka, 1987;

  4. A.Ya.Halamizer, “Mathematics? - It's funny! ”, Author's edition, 1989;

  5. P.K. Rashevsky, Riemannian geometry and tensor analysis, Moscow, 3rd edition, 1967;

  6. V.E. Gmurman, Probability Theory and Mathematical Statistics, Moscow, graduate School, 1977;

  7. World wide network Enternet.

Mathematics as a science of quantitative relations and spatial forms of reality studies the world around us, natural and social phenomena. But unlike other sciences, mathematics studies their special properties, abstracting from others. So, geometry studies the shape and size of objects, without taking into account their other properties: color, mass, hardness, etc. In general, mathematical objects (geometric figure, number, value) are created by the human mind and exist only in human thinking, in signs and symbols that form the mathematical language.

The abstractness of mathematics allows it to be applied in a variety of areas, it is a powerful tool for understanding nature.

Forms of knowledge are divided into two groups.

first group constitute forms of sensory cognition, carried out with the help of various sense organs: sight, hearing, smell, touch, taste.

Co. second group include forms of abstract thinking, primarily concepts, statements and inferences.

The forms of sensory cognition are Feel, perception and representation.

Each object has not one, but many properties, and we know them with the help of sensations.

Feeling- this is a reflection of individual properties of objects or phenomena of the material world, which are directly (i.e. now, in this moment) affect our senses. These are sensations of red, warm, round, green, sweet, smooth and other individual properties of objects [Getmanova, p. 7].

From individual sensations, the perception of the whole object is formed. For example, the perception of an apple is made up of such sensations: spherical, red, sweet and sour, fragrant, etc.

Perception is a holistic reflection of an external material object that directly affects our senses [Getmanova, p. eight]. For example, the image of a plate, cup, spoon, other utensils; the image of the river, if we are now sailing along it or are on its banks; the image of the forest, if we have now come to the forest, etc.

Perceptions, although they are a sensory reflection of reality in our minds, are largely dependent on human experience. For example, a biologist will perceive a meadow in one way (he will see different kinds plants), but a tourist or an artist is completely different.

Performance- this is a sensual image of an object that is not currently perceived by us, but which was previously perceived by us in one form or another [Getmanova, p. ten]. For example, we can visually imagine the faces of acquaintances, our room in the house, a birch tree or a mushroom. These are examples reproducing representations, as we have seen these objects.

The presentation can be creative, including fantastic. We present the beautiful Princess Swan, or Tsar Saltan, or the Golden Cockerel, and many other characters from the fairy tales of A.S. Pushkin, whom we have never seen and never will see. These are examples of creative presentation over verbal description. We also imagine the Snow Maiden, Santa Claus, a mermaid, etc.

So, the forms of sensory knowledge are sensations, perceptions and representations. With their help, we know outer sides object (its features, including properties).

Forms of abstract thinking are concepts, statements and conclusions.

Concepts. Scope and content of concepts

The term "concept" is usually used to refer to a whole class of objects of an arbitrary nature that have a certain characteristic (distinctive, essential) property or a whole set of such properties, i.e. properties that are unique to members of that class.

From the point of view of logic, the concept is a special form of thinking, which is characterized by the following: 1) the concept is a product of highly organized matter; 2) the concept reflects material world; 3) the concept appears in consciousness as a means of generalization; 4) the concept means specifically human activity; 5) the formation of a concept in the mind of a person is inseparable from its expression through speech, writing or symbol.

How does the concept of any object of reality arise in our minds?

The process of forming a certain concept is a gradual process in which several successive stages can be seen. Consider this process using the simplest example - the formation of the concept of the number 3 in children.

1. At the first stage of cognition, children get acquainted with various specific sets, using subject pictures and showing various sets of three elements (three apples, three books, three pencils, etc.). Children not only see each of these sets, but they can also touch (touch) the objects that make up these sets. This process of "seeing" creates in the mind of the child a special form of reflection of reality, which is called perception (feeling).

2. Let's remove the objects (objects) that make up each set, and invite the children to determine whether there was something in common that characterizes each set. The number of objects in each set was to be imprinted in the minds of the children, that there were “three” everywhere. If this is so, then a new form has been created in the minds of children - idea of ​​the number three.

3. At the next stage, on the basis of a thought experiment, children should see that the property expressed in the word "three" characterizes any set various elements of the form (a; b; c). This will highlight a significant common feature such sets "to have three elements". Now we can say that in the minds of children formed concept of number 3.

concept- this is a special form of thinking, which reflects the essential (distinctive) properties of objects or objects of study.

The linguistic form of a concept is a word or a group of words. For example, “triangle”, “number three”, “point”, “straight line”, “isosceles triangle”, “plant”, “coniferous tree”, “Yenisei River”, “table”, etc.

Mathematical concepts have a number of features. The main one is that the mathematical objects about which it is necessary to form a concept do not exist in reality. Mathematical objects are created by the human mind. These are ideal objects that reflect real objects or phenomena. For example, in geometry, the shape and size of objects are studied, without taking into account their other properties: color, mass, hardness, etc. From all this they are distracted, abstracted. Therefore, in geometry, instead of the word "object" they say "geometric figure". The result of abstraction are also such mathematical concepts as "number" and "value".

Main Features any concepts are the following: 1) volume; 2) content; 3) relationships between concepts.

When they talk about a mathematical concept, they usually mean the whole set (set) of objects denoted by one term (word or group of words). So, speaking of a square, they mean all geometric shapes that are squares. It is believed that the set of all squares is the scope of the concept of "square".

The scope of the concept the set of objects or objects to which this concept is applicable is called.

For example, 1) the scope of the concept of "parallelogram" is the set of quadrilaterals such as parallelograms proper, rhombuses, rectangles and squares; 2) the scope of the concept of "unambiguous natural number» there will be a set - (1, 2, 3, 4, 5, 6, 7, 8, 9).

Any mathematical object has certain properties. For example, a square has four sides, four right angles equal to the diagonals, the diagonals are bisected by the intersection point. You can specify its other properties, but among the properties of an object there are essential (distinctive) and non-essential.

The property is called essential (distinctive) for an object if it is inherent in this object and without it it cannot exist; property is called insignificant for an object if it can exist without it.

For example, for a square, all the properties listed above are essential. The property “side AD is horizontal” will be irrelevant for the square ABCD (Fig. 1). If this square is rotated, then side AD will be vertical.

Consider an example for preschoolers using visual material (Fig. 2):

Describe the figure.

Small black triangle. Rice. 2

Big white triangle.

How are the figures similar?

How are the figures different?

Color, size.

What does a triangle have?

3 sides, 3 corners.

Thus, children find out the essential and non-essential properties of the concept of "triangle". Essential properties - "have three sides and three angles", non-essential properties - color and size.

The totality of all essential (distinctive) properties of an object or object reflected in this concept is called the content of the concept .

For example, for the concept of "parallelogram" the content is a set of properties: it has four sides, has four corners, opposite sides are pairwise parallel, opposite sides are equal, opposite angles are equal, the diagonals at the intersection points are divided in half.

There is a connection between the volume of a concept and its content: if the volume of a concept increases, then its content decreases, and vice versa. So, for example, the scope of the concept "isosceles triangle" is part of the scope of the concept "triangle", and the content of the concept "isosceles triangle" includes more properties than the content of the concept "triangle", because an isosceles triangle has not only all the properties of a triangle, but also others inherent only in isosceles triangles (“two sides are equal”, “two angles are equal”, “two medians are equal”, etc.).

Concepts are divided into single, common and categories.

A concept whose volume is equal to 1 is called single concept .

For example, the concepts: "Yenisei River", "Republic of Tuva", "city of Moscow".

Concepts whose volume is greater than 1 are called common .

For example, the concepts: "city", "river", "quadrilateral", "number", "polygon", "equation".

In the process of studying the foundations of any science, children generally form general concepts. For example, in primary school students get acquainted with such concepts as “number”, “number”, “single-digit numbers”, “two-digit numbers”, “multi-digit numbers”, “fraction”, “share”, “addition”, “term”, “sum”, "subtraction", "subtracted", "reduced", "difference", "multiplication", "multiplier", "product", "division", "divisible", "divisor", "quotient", "ball", "cylinder" ”, “cone”, “cube”, “parallelepiped”, “pyramid”, “angle”, “triangle”, “quadrilateral”, “square”, “rectangle”, “polygon”, “circle”, “circle”, “curve”, “polyline”, “segment”, “length of segment”, “ray”, “straight line”, “point”, “length”, “width”, “height”, “perimeter”, “shape area”, "volume", "time", "speed", "mass", "price", "cost" and many others. All these concepts are general concepts.

The science that studies quantities, quantitative relations and spatial forms

First letter "m"

Second letter "a"

Third letter "t"

The last beech is the letter "a"

Answer for the clue "Science that studies quantities, quantitative relations and spatial forms", 10 letters:
maths

Alternative questions in crossword puzzles for the word mathematics

The representative of this science beat off the bride from Nobel, and therefore the Nobel Prize is not given for success in it.

"Tower" in the program of the Polytechnic University

An exact science that studies quantities, quantitative relationships and spatial forms

The science of quantities, quantitative relations, spatial forms

It was this subject that was taught at school by "dear Elena Sergeevna" performed by Marina Neelova

Word definitions for mathematics in dictionaries

Dictionary living Great Russian language, Vladimir Dal The meaning of the word in the dictionary Explanatory Dictionary of the Living Great Russian Language, Vladimir Dal
and. the science of magnitudes and quantities; everything that can be expressed in numbers belongs to mathematics. - pure, deals with magnitudes abstractly; - applied, attaches the first to the case, to objects. Mathematics is divided into arithmetic and geometry, the first has ...

Wikipedia The meaning of the word in the Wikipedia dictionary
Maths (

Great Soviet Encyclopedia The meaning of the word in the dictionary Great Soviet Encyclopedia
I. Definition of the subject of mathematics, connection with other sciences and technology. Mathematics (Greek mathematike, from máthema ≈ knowledge, science), the science of quantitative relations and spatial forms of the real world. "Pure mathematics has as its object...

New explanatory and derivational dictionary of the Russian language, T. F. Efremova. The meaning of the word in the dictionary New explanatory and derivational dictionary of the Russian language, T. F. Efremova.
and. Scientific discipline about spatial forms and quantitative relations of the real world. An academic subject containing the theoretical foundations of a given scientific discipline. unfold A textbook that sets out the content of a given academic subject. trans. unfold Accurate,...

Examples of the use of the word mathematics in the literature.

At first, Trediakovsky was sheltered by Vasily Adadurov - mathematician, a student of the great Jacob Bernoulli, and for this shelter the poet instructed the scientist in French.

Went in mathematician Adadurov, mechanic Ladyzhensky, architect Ivan Blank, assessors from various colleges, doctors and gardeners, army and navy officers dropped in on the fire.

Two people sat in armchairs at a long, polished walnut table: Axel Brigov and mathematician Brodsky, whom I recognized by his powerful Socratic bald head.

Pontryagin, whose efforts created a new section mathematics- topological algebra, - studying various algebraic structures endowed with topology.

Let us also note in passing that the era we are describing witnessed the development of algebra, a comparatively abstract branch of mathematics, by combining its less abstract departments, geometry and arithmetic, a fact proved by the oldest manifestations of algebra that have come down to us, half algebraic, half geometric.