What is mathematics. History of mathematics

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 by comparing various parts of the body with them, mainly ... ... Collier Encyclopedia

History of science ... Wikipedia

This article is part of the review History of Mathematics. Contents 1 Antiquity and the Middle Ages 2 XVII century 3 ... Wikipedia

The doctrine of the essence of mathematical knowledge and the basic principles of mathematical proofs, a section of the philosophy of science; it can also be called "metamathematics". Contents 1 Possibility of foundations of mathematics 2 Literature ... Wikipedia

This article is part of the review History of Mathematics. Scientific achievements Indian mathematics are wide and varied. Already in ancient times, the scientists of India, on their own, in many respects, the original path of development reached high level mathematical knowledge.… … Wikipedia

Scientific Research Institute of Mathematics and Mechanics named after Academician V. I. Smirnov (NIIMM St. Petersburg State University), a structural subdivision of the St. state university. Performs an organizational role, is the material basis for ... ... Wikipedia

Euclid. Detail of the "School of Athens" by Raphael Mathematician (from other Greek ... Wikipedia

Discrete mathematics is a branch of mathematics concerned with the study of discrete structures that arise both within mathematics itself and in its applications. Finite groups, finite graphs, and ... ... Wikipedia

This term has other meanings, see Analysis. Mathematical analysis is a set of branches of mathematics devoted to the study of functions and their generalizations by methods of differential and integral calculus. With such a common ... ... Wikipedia

A method of constructing a theory, while it is based on some of its provisions - axioms or postulates - from which all other provisions of the theory (theorems) are derived by reasoning, called proofs m i. Rules, by the way ... ... Philosophical Encyclopedia

Books

• Special sections of mathematics. Workshop, V. A. Kramar, V. A. Karapetyan, V. V. Alchakov. Special sections of mathematics are considered, which are used in the study of a number of specialized disciplines in the direction of Control in technical systems. The main…
• Probabilistic sections of mathematics: A textbook for bachelors of technical fields (under the general editorship of Maksimov Yu. D.), Amosova N.N., Kuklin B.A., Makarova S.B. and etc.. …

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Mathematics in translation from ancient Greek means - study, science. This is the science of structures, order and relationships that have historically developed on the basis of the operations of counting, measuring and describing the shape of an object or object. Mathematical objects are founded by idealizing the properties of real or other mathematical objects and writing these properties in a formal language.

Mathematics is not a natural science, but it is widely used in them both for the exact formulation of their content and for obtaining new results.

Mathematics is a fundamental science providing (general) language tools other sciences; thus, it reveals their structural relationship and contributes to the definition of the most general laws of the universe. This science is associated with many calculations, formulas, equations and terms. Comprehending mathematics, it is very difficult not to get lost in all these endless numbers and calculations. The complexity of this science also lies in its versatility, because it includes many sections:

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Traditionally, mathematics is divided into:

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Mathematics is the queen of all sciences
Gauss Carl Friedrich

Mathematics is a science historically based on solving problems of quantitative and spatial relationships real world by idealizing the properties of objects necessary for this and formalizing these tasks. The science concerned with the study of numbers, structures, spaces, and transformations.

As a rule, people think that mathematics is just arithmetic, that is, the study of numbers and operations with them, such as multiplication and division. In fact, mathematics is much more than that. It is a way of describing the world and how one part of it fits into another. Relationships of numbers are expressed in mathematical symbols that describe the universe in which we live. Any normal child can do well in math because "number sense" is an inborn ability. True, for this you need to make some efforts and spend a little time.

The ability to count is not everything. The child needs to be able to express his thoughts well in order to understand tasks and make connections between the facts that are stored in memory. In order to learn the multiplication table, you need memory and speech. This is why some people with brain damage find it difficult to multiply, although other types of counting are not difficult for them.

In order to know geometry well and understand form and space, other kinds of thinking are also required. With the help of mathematics, we solve problems in life, for example, dividing a chocolate bar equally or finding the right shoe size. Thanks to the knowledge of mathematics, the child knows how to save pocket money and understands what can be bought and how much money he will have left. Mathematics is also the ability to count the right amount of seeds and sow them in a pot, measure the right amount of flour for a cake or fabric on a dress, understand the score football game and many other daily activities. Everywhere: in the bank, in the store, at home, at work - we need the ability to understand and handle numbers, shapes and measures. Numbers are only part of a special mathematical language, and The best way to learn any language is to apply it. And it is better to start from an early age.

About mathematics "smartly"

Usually, the idealized properties of the objects and processes under study are formulated in the form of axioms, then strict rules logical inference, other true properties (theorems) are deduced from them. This theory together forms a mathematical model of the object under study. That. Initially, based on spatial and quantitative relationships, mathematics receives more abstract relationships, the study of which is also the subject of modern mathematics.

Traditionally, mathematics is divided into theoretical, which performs an in-depth analysis of intra-mathematical structures, and applied, which provides its models to other sciences and engineering disciplines, and some of them occupy a position bordering on mathematics. In particular, formal logic can be considered both as part of the philosophical sciences and as part of the mathematical sciences; mechanics - both physics and mathematics; computer science, computer technology and algorithmics are both engineering and mathematical sciences, etc. There are many different definitions of mathematics in the literature.

Branches of mathematics

• Mathematical analysis.
• Algebra.
• Analytic geometry.
• Linear algebra and geometry.
• Discrete Math.
• Mathematical logic.
• Differential equations.
• Differential geometry.
• Topology.
• Functional analysis and integral equations.
• Theory of functions of a complex variable.
• Equations with partial derivatives.
• Probability Theory.
• Math statistics.
• Theory of random processes.
• Calculus of variations and optimization methods.
• Calculation methods, that is, numerical methods.
• Number theory.

Goals and Methods

Mathematics studies imaginary, ideal objects and the relationships between them using a formal language. In general mathematical concepts and theorems do not necessarily correspond to anything in physical world. the main task applied mathematician - to create a mathematical model that is sufficiently adequate to the real object under study. The task of the theoretical mathematician is to provide a sufficient set of convenient means to achieve this goal.

The content of mathematics can be defined as a system of mathematical models and tools for their creation. The object model does not take into account all its features, but only the most necessary for the purposes of study (idealized). For example, studying physical properties orange, we can abstract from its color and taste and represent it (albeit not perfectly accurately) as a ball. If we need to understand how many oranges we get if we add two and three together, then we can abstract away from the form, leaving the model with only one characteristic - quantity. Abstraction and establishing links between objects in the general view- one of the main directions of mathematical creativity.

Another direction, along with abstraction, is generalization. For example, generalizing the concept of "space" to the space of n-dimensions. The space R n , for n>3 is a mathematical invention. However, a very ingenious invention, which helps to mathematically understand complex phenomena.

The study of intramathematical objects, as a rule, takes place using the axiomatic method: first, a list of basic concepts and axioms is formulated for the objects under study, and then meaningful theorems are obtained from the axioms using inference rules, which together form a mathematical model.

Video lecture by Smirnov S.K. and Yashchenko I.V. "What is Mathematics":

Maths- the science of structures, order and relationships, which historically developed on the basis of 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. Mathematics does not apply natural sciences, but is widely used in them both for the exact formulation of their content and for obtaining new results. Mathematics is a fundamental science that provides (general) linguistic means to other sciences; thus, it reveals their structural interrelation and contributes to the discovery of the most general laws of nature.

History of mathematics.

Academician A. N. Kolmogorov proposed the following structure of the history of mathematics:

1. The period of the birth of mathematics, during which a fairly large amount of factual material was accumulated;

2. The period of elementary mathematics, starting in the VI-V centuries BC. e. ending at the end of the 16th century (“The stock of concepts with which mathematics dealt before the beginning of the 17th century constitutes to this day the basis of “elementary mathematics” taught in elementary and high school»);

3. The period of mathematics variables, covering the XVII-XVIII centuries, "which can also be conditionally called the period of" higher mathematics "";

4. The period of modern mathematics - mathematics of the XIX-XX centuries, during which mathematicians had to "refer to the process of expanding the subject mathematical research consciously, setting himself the task of systematically studying, from a fairly general point of view, the possible types of quantitative relations and spatial forms.

The development of mathematics began with the fact that man began to use abstractions of any high level. Simple abstraction - numbers; understanding that two apples and two oranges, despite all their differences, have something in common, namely, they occupy both hands of one person, is a qualitative achievement of human thinking. In addition to learning how to count concrete objects, ancient people also understood how to calculate abstract quantities such as time: days, seasons, years. From the elementary account, arithmetic naturally began to develop: addition, subtraction, multiplication and division of numbers.

The development of mathematics relies on writing and the ability to write down numbers. Probably, ancient people first expressed quantity by drawing lines on the ground or scratching them on wood. The ancient Incas, having no other writing system, represented and stored numerical data using a complex system rope knots, the so-called quipu. There were many different number systems. The first known records of numbers were found in the Ahmes Papyrus, created by the Egyptians of the Middle Kingdom. The Inca civilization developed the modern decimal number system, incorporating the concept of zero.

Historically, the major mathematical disciplines have been influenced by the need to make calculations in the commercial sphere, in land measurement and for prediction. astronomical phenomena and, later, to solve new physical problems. Each of these areas plays big role in the broad development of mathematics, which consists in the study of structures, spaces and changes.

Mathematics studies imaginary, ideal objects and the relationships between them using a formal language. In general, mathematical concepts and theorems do not necessarily correspond to anything in the physical world. The main task of the applied branch of mathematics is to create a mathematical model that is adequate enough for the real object under study. The task of the theoretical mathematician is to provide a sufficient set of convenient means to achieve this goal.

The content of mathematics can be defined as a system of mathematical models and tools for their creation. The object model does not take into account all its features, but only the most necessary for the purposes of study (idealized). For example, when studying the physical properties of an orange, we can abstract from its color and taste and represent it (albeit not perfectly accurately) as a ball. If we need to understand how many oranges we get if we add two and three together, then we can abstract away from the form, leaving the model with only one characteristic - quantity. Abstraction and the establishment of relationships between objects in the most general form is one of the main areas of mathematical creativity.

Consider the role of mathematics in chemistry, medicine and chess.

The role of mathematics in chemistry

Chemistry widely uses the achievements of other sciences for its own purposes, primarily physics and mathematics.

Chemists usually define mathematics in a simplistic way, as the science of numbers. Numbers express many properties of substances and characteristics chemical reactions. To describe substances and reactions, physical theories are used, in which the role of mathematics is so great that it is sometimes difficult to understand where physics is and where mathematics is. From this it follows that chemistry is unthinkable without mathematics.

Mathematics for chemists is, first of all, a useful tool for solving many chemical problems. It is very difficult to find any branch of mathematics that is not used at all in chemistry. Functional analysis and group theory are widely used in quantum chemistry, probability theory is the basis of statistical thermodynamics, graph theory is used in organic chemistry to predict the properties of complex organic molecules, differential equations- the main tool of chemical kinetics, methods of topology and differential geometry are used in chemical thermodynamics.

The expression "mathematical chemistry" has firmly entered the lexicon of chemists. Many articles in serious chemical journals do not contain any chemical formula, but abound in mathematical equations.

Symmetry is one of the basic concepts in modern science. It underlies the fundamental laws of nature, such as the law of conservation of energy. Symmetry is a very common phenomenon in chemistry: almost all known molecules either themselves have symmetry of some kind, or contain symmetrical fragments. So, perhaps, in chemistry, it is more difficult to detect an asymmetric molecule than a symmetrical one.

The interaction of chemists and mathematicians is not limited to solving only chemical problems. Sometimes abstract problems arise in chemistry, which even lead to the emergence of new areas of mathematics.

The role of mathematics in medicine

No wonder many people called mathematics the queen of sciences, since the applications of this science can be found in any field of human activity. However, the value of mathematics in such less rigorous sciences as "medicine and biology" is often questioned. Since the chance to achieve the most accurate results of analyzes or experiments is zero. This factor can be explained by the fact that our world as a whole is very changeable, and it is difficult to predict what will happen to one or another subject of analysis.

Mathematics in medicine is most often used in modeling as a method of scientific analysis. However, this method has been used since ancient times in such areas as architecture, astronomy, physics, biology, and, more recently, medicine. At present, a very rich stock of knowledge about infectious diseases has been accumulated, not only the symptoms, but also the course of the disease, the results of fundamental analyzes regarding the mechanism of interaction between antigens and antibodies at various levels of detail: macroscopic, microscopic, up to the genetic level. These research methods made it possible to approach the construction of mathematical models of immune processes.

Mathematics in medicine does not stop there, it is also used in such narrow specialties as pediatrics, obstetrics.

And how many counting methods exist during the use of antibiotics. Mathematics is especially important in pharmaceuticals. After all, you need to accurately calculate how much you need to administer the drug to a certain person, depending on his personal characteristics, and even the composition of the medicinal substance must be calculated so as not to be mistaken anywhere. Doctors and pharmacists rack their brains to find one or the most beneficial component for the formula chain of any drug.

The role of mathematics in medicine is invaluable, without this science (as a whole) nothing is possible, it is not for nothing that it is considered the “queen”. Now even many authors write books about mathematics, about what an invaluable contribution it made.

The role of mathematics in chess

Chess and mathematics have a lot in common. The eminent mathematician Godfrey Harald Hardy once remarked that solving the problems of a chess game is nothing more than a mathematical exercise, and the game itself is whistling mathematical melodies. The forms of thinking of a mathematician and a chess player are very close, and it is no coincidence that mathematicians are often capable chess players.

Among prominent scientists, experts in the field exact sciences, many strong chess players are known, for example, mathematician academician A. A. Markov, mechanic academician A. Yu. Ishlinsky, physicist academician, laureate Nobel Prize P. L. Kapitsa.

Chess is constantly used to illustrate various mathematical concepts and ideas. Chess examples and terms can be found in the literature, game theory, etc. Important.

Chess mathematics is one of the most popular genres entertaining mathematics, logic games and entertainment. However, some chess-mathematical puzzles are so complex that prominent mathematicians developed a special mathematical apparatus for them.

In almost every collection of Olympiad math problems or a book of puzzles and mathematical leisure, you can find beautiful and witty problems involving a chessboard and pieces. Many of them have interesting story, attracted the attention of famous scientists.

Chess is constantly used to illustrate various mathematical concepts and ideas. Chess examples and terms can be found in literature, game theory, etc. important place occupy chess in "computer science".

Without knowledge of mathematics, it is impossible to solve many problems on a chessboard. Without mastering mathematical knowledge, it is difficult to understand what is happening in the field of mathematics now, in the field of other sciences. So the role of mathematics in the life of society is increasing every day.

The idealized properties of the objects under study are either formulated as axioms or listed in the definition of the corresponding mathematical objects. Then, according to strict rules of logical inference, other true properties (theorems) are deduced from these properties. This theory together forms a mathematical model of the object under study. Thus, initially starting from spatial and quantitative relations, mathematics receives more abstract relations, the study of which is also the subject of modern mathematics.

Traditionally, mathematics is divided into theoretical, which performs an in-depth analysis of intra-mathematical structures, and applied, which provides its models to other sciences and engineering disciplines, and some of them occupy a position bordering on mathematics. In particular, formal logic can be considered both as part of the philosophical sciences and as part of the mathematical sciences; mechanics - both physics and mathematics; computer science, computer technology and algorithmics refer to both engineering and mathematical sciences, etc. Many different definitions of mathematics have been proposed in the literature.

Etymology

The word "mathematics" comes from other Greek. μάθημα , which means the study, knowledge, the science, etc. - Greek. μαθηματικός , originally meaning receptive, prolific, later studyable, subsequently pertaining to mathematics. In particular, μαθηματικὴ τέχνη , in Latin ars mathematica, means art of mathematics. The term other Greek. μᾰθημᾰτικά in modern meaning this word "mathematics" is already found in the writings of Aristotle (4th century BC). According to Fasmer, the word came to the Russian language either through Polish. matematyka, or through lat. mathematica.

Definitions

One of the first definitions of the subject of mathematics was given by Descartes:

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 sought. 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.

AT Soviet time the definition from TSB given by A. N. Kolmogorov was considered classic:

Mathematics ... the science of quantitative relations and spatial forms of the real world.

The essence of mathematics ... is now presented as a doctrine of relations between objects, about which nothing is known, except for some properties that describe them - precisely those that are put as axioms at the basis of the theory ... Mathematics is a set of abstract forms - mathematical structures.

Branches of mathematics

1. Mathematics as academic discipline subdivided in the Russian Federation into elementary mathematics, studied in high school and educated by disciplines:

• elementary geometry: planimetry and solid geometry
• theory of elementary functions and elements of analysis

4. The American Mathematical Society (AMS) has developed its own standard for classifying branches of mathematics. It's called Mathematics Subject Classification. This standard is updated periodically. The current version is MSC 2010. previous version- MSC 2000 .

Notation

Since mathematics deals with extremely diverse and rather complex structures, its notation is also very complex. Modern system formula writing was formed on the basis of the European algebraic tradition, as well as the needs of later branches of mathematics - mathematical analysis, mathematical logic, set theory, etc. Geometry has used a visual (geometrical) representation from time immemorial. In modern mathematics, complex graphical notation systems (for example, commutative diagrams) are also common, and graph-based notation is also often used.

Short story

The development of mathematics relies on writing and the ability to write down numbers. Probably, ancient people first expressed quantity by drawing lines on the ground or scratching them on wood. The ancient Incas, having no other writing system, represented and stored numerical data using a complex system of rope knots, the so-called quipu. There were many different number systems. The first known records of numbers were found in the Ahmes Papyrus, produced by the Egyptians of the Middle Kingdom. The Indian civilization developed the modern decimal number system, incorporating the concept of zero.

Historically, the major mathematical disciplines emerged under the influence of the need to make calculations in the commercial field, in measuring the land and for predicting astronomical phenomena and, later, for solving new physical problems. Each of these areas plays a large role in the broad development of mathematics, which consists in the study of structures, spaces and changes.

Philosophy of mathematics

Goals and Methods

Mathematics studies imaginary, ideal objects and the relationships between them using a formal language. In general, mathematical concepts and theorems do not necessarily correspond to anything in the physical world. The main task of the applied section of mathematics is to create a mathematical model that is adequate enough for the real object under study. The task of the theoretical mathematician is to provide a sufficient set of convenient means to achieve this goal.

The content of mathematics can be defined as a system of mathematical models and tools for their creation. The object model does not take into account all its features, but only the most necessary for the purposes of study (idealized). For example, when studying the physical properties of an orange, we can abstract from its color and taste and represent it (albeit not perfectly accurately) as a ball. If we need to understand how many oranges we get if we add two and three together, then we can abstract away from the form, leaving the model with only one characteristic - quantity. Abstraction and the establishment of relationships between objects in the most general form is one of the main areas of mathematical creativity.

Another direction, along with abstraction, is generalization. For example, generalizing the concept of "space" to the space of n-dimensions. " Space R n (\displaystyle \mathbb (R) ^(n)), at n > 3 (\displaystyle n>3) is a mathematical invention. However, a very ingenious invention that helps to mathematically understand complex phenomena».

The study of intra-mathematical objects, as a rule, occurs using the axiomatic method: first, for the objects under study, a list of basic concepts and axioms is formulated, and then meaningful theorems are obtained from the axioms using inference rules, which together form a mathematical model.

Foundations

intuitionism

Intuitionism presupposes at the foundation of mathematics intuitionistic logic, which is more limited in the means of proof (but, as it is believed, also more reliable). Intuitionism rejects proof by contradiction, many non-constructive proofs become impossible, and many problems in set theory become meaningless (non-formalizable).

Constructive mathematics

Constructive mathematics is a trend in mathematics close to intuitionism that studies constructive constructions [ clarify] . According to the criterion of constructibility - " to exist means to be built". The constructivity criterion is a stronger requirement than the consistency criterion.

Main topics

Quantity

The main section dealing with the abstraction of quantity is algebra. The concept of "number" originally originated from arithmetic representations and referred to natural numbers. Later, with the help of algebra, it was gradually extended to integer, rational, real, complex and other numbers.

Transformations

The phenomena of transformations and changes are considered in the most general form by analysis.

structures

Spatial Relations

The basics of spatial relations are considered by geometry. Trigonometry considers the properties of trigonometric functions. The study of geometric objects through mathematical analysis deals with differential geometry. The properties of spaces that remain unchanged under continuous deformations and the very phenomenon of continuity are studied by topology.

Geometry	Trigonometry	Differential geometry	Topology	fractals	measure theory

Discrete Math

∀ x (P (x) ⇒ P (x ′)) (\displaystyle \forall x(P(x)\Rightarrow P(x")))