Mathematical sciences and their metamorphoses. Mathematical disciplines

The oldest mathematical activity was counting. An account was needed to keep track of livestock and trade. Some primitive tribes counted the number of objects by comparing different parts of the body to them, mainly ... ... Collier's Encyclopedia

History of Science ... Wikipedia

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

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

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

Discrete mathematics is a field of mathematics that deals with the study of discrete structures that arise both within mathematics itself and in its applications. Such structures can include finite groups, finite graphs, and ... ... Wikipedia

This term has other meanings, see Analysis. Mathematical analysis is a set of sections 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, with some of its provisions - axioms or postulates - from which all other provisions of the theory (theorems) are deduced by reasoning, called a proof m and. Rules, by the eye ... ... Philosophical Encyclopedia

Books

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

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. Math doesn't apply to natural sciences, but is widely used in them both for the precise formulation of their content and for obtaining new results. Mathematics is a fundamental science providing (general) linguistic means to other sciences; thereby, it reveals their structural relationship and contributes to finding the most general laws of nature.

History of mathematics.

Academician A.N. Kolmogorov proposed the following structure for 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, beginning in the VI-V centuries BC. e. ending at the end of the 16th century (“The stock of concepts that mathematics dealt with until the beginning of the 17th century still constitutes the basis of“ elementary mathematics ”taught in primary and secondary schools”);

3. The period of mathematics of variable quantities, covering the XVII-XVIII centuries, "which can be conventionally called the period of" higher mathematics "";

4. The period of modern mathematics - mathematics of the 19th-20th centuries, during which mathematicians had to "deliberately treat the process of expanding the subject of mathematical research, setting themselves the task of systematic study from a fairly general point of view of possible types of quantitative relations and spatial forms."

The development of mathematics began at the same time as man began to use abstractions of any higher level. A simple abstraction is numbers; comprehension of the fact 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. Besides the fact that the ancient people learned how to count concrete objects, they also understood how to calculate abstract quantities, such as time: days, seasons, years. From elementary counting, arithmetic naturally began to develop: addition, subtraction, multiplication and division of numbers.

The development of mathematics relies on writing and the ability to write numbers. Probably, the ancient people first expressed the 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 kipu. There were many different number systems. The first known records of numbers were found in the Ahmes papyrus created by the Egyptians in the Middle Kingdom. The Inca civilization developed the modern decimal number system, incorporating the concept of zero.

Historically, the basic mathematical disciplines have emerged under the influence of the need to carry out calculations in the commercial sphere, in the measurement of lands and to predict astronomical phenomena and, later, to solve new physical problems. Each of these areas plays a large role in the wider development of mathematics, which consists in the study of structures, spaces and changes.

Mathematics studies imaginary, ideal objects and relationships between them using 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 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 creating them. The model of an object does not take into account all of 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 imagine it (albeit not perfectly accurately) with a ball. If we need to understand how many oranges will turn out if we add two and three together, then we can abstract from the form, leaving the model with only one characteristic - quantity. Abstraction and the establishment of connections between objects in the most general form is one of the main directions of mathematical creativity.

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

The role of mathematics in chemistry

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

Chemists usually define mathematics in a simplistic way - as the science of numbers. Many properties of substances and characteristics of chemical reactions are expressed in numbers. 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 is physics and where is mathematics. Hence it follows that chemistry is unthinkable without mathematics.

For chemists, mathematics 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 are the main tool in chemical kinetics, and topology and differential geometry methods are used in chemical thermodynamics.

The expression "mathematical chemistry" has become part of the lexicon of chemists. Many articles in serious chemical journals do not contain a single chemical formula, but are replete with 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 have symmetry of some kind themselves, or contain symmetric fragments. So it’s probably harder to find an asymmetric molecule in chemistry than a symmetric 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 less rigorous sciences like “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 this or that subject of analysis.

Mathematics in medicine is most often used in modeling as a method of scientific analysis. However, this method began to be used in ancient times in such industries as: architecture, astronomy, physics, biology, and since recent years - medicine. Currently, a very rich stock of knowledge has been accumulated about infectious diseases, not only the symptomatology, but also the course of the disease, the results of fundamental analyzes concerning the mechanism of interaction of 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 and obstetrics.

And how many counting methods exist in the course of antibiotic use. In pharmaceuticals, mathematics is especially important. After all, it is necessary to accurately calculate how much the drug needs to be administered to a certain person, depending on his personal characteristics, and even the composition of the medicinal substance itself must be calculated so as not to make a mistake anywhere. Pharmacists are racking their brains to find the one or the most beneficial component for the formula chain of any drug.

The role of mathematics in medicine is invaluable, without this science (in general) nothing is possible, it is not for nothing that it is considered a "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 problems in a chess game is nothing more than a mathematical exercise, and the game itself is a whistling of mathematical tunes. 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 of exact sciences, there are many strong chess players, for example, mathematician Academician A.A. Markov, mechanic Academician A. Yu. Ishlinsky, physicist Academician, Nobel Prize laureate P. L. Kapitsa.

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

Chess mathematics is one of the most popular genres of 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 mathematical problems or a book of puzzles and mathematical leisure, you can find beautiful and witty problems with the participation of 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. Chess occupies an important place in "computer science".

Without knowledge of mathematics, it is impossible to solve many problems on the 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 sciences differ from each other in the subject of research, first of all, in that each of them studies one of the sides of the real world, one or several closely related and passing into each other forms of movement of objective reality.

Consider one of the possible options for the classification of sciences:

Natural Sciences, studying subjects, phenomena and laws of nature. Among them are distinguished: mechanics, astronomy, physics, chemistry, paleontology, biology and other sciences.

Social Sciences, studying the phenomena of social life. Such sciences are historical science, political economy, etc.

Technical sciencestudying the functioning of technical devices and systems. For example, the theory of machines and mechanisms, resistance of materials, etc. etc.

Cognitive Sciences: philosophy, logic, psychology, etc.

Previously, scientists and philosophers often considered mathematics to be a natural science discipline. Now it is usually said that mathematics is an independent science, by the degree of generality located between philosophy and natural science.

Mathematics, like other sciences, studies the real, material world, objects of this world and the relationship between them. However, unlike the sciences of nature, which study various forms of motion of matter (mechanics, physics, chemistry, biology, etc.) or forms of information transfer (computer science, the theory of automata, and other branches of cybernetics), mathematics studies the forms and relationships of the material world. taken in abstraction from their content. Therefore, mathematics does not study any special form of motion of matter and, therefore, cannot be considered as one of the natural sciences.

In the second half of the XIX century. F. Engels gave the following definition of the subject of mathematics: "Pure mathematics has as its object the spatial forms and quantitative relations of the real world, therefore it is a very real material." At the same time, he pointed out: “But in order to be able to investigate these forms and relations in their pure form, it is necessary to completely separate them from their content, to leave this last aside, as something indifferent; in this way we get points devoid of measurements, lines devoid of thickness and width, different a and b , x and y , constant and variable quantities "

From these words of Engels it follows that the original concepts of mathematics, which were the subject of study from the very inception of mathematical science - natural number, magnitude and geometric figure - are borrowed from the real world, are the results of abstraction of individual features of material objects, and did not arise through "pure thinking" divorced from reality. At the same time, in order to become the subject of mathematical research, the properties and relationships of material objects must be abstracted from their material content.

Thus, the specificity of mathematics lies in the fact that it singles out quantitative relations and spatial forms inherent in all objects and phenomena, regardless of their material content, abstracts these relations and forms and makes them the object of its research.

However, F. Engels's definition largely reflects the state of mathematics in the second half of the 19th century. and does not take into account those of its new areas that are not directly related either to quantitative relations or to geometric forms. These are, first of all, mathematical logic and disciplines related to computer programming. Therefore, the definition of F. Engels needs some clarification. Perhaps it should be said that mathematics has as its object of study spatial forms, quantitative relations and logical constructions.

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

Traditionally, mathematics is divided into theoretical, which performs 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 borderline position with 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; Informatics, computer technology and algorithms 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 ancient Greek. μάθημα which means study of, knowledge, the science, and other Greek. μαθηματικός , originally meaning receptive, successful later studying, subsequently mathematics... In particular, μαθηματικὴ τέχνη , in Latin ars mathematicameans art of mathematics... The term ancient Greek. μᾰθημᾰτικά in the modern sense of the word "mathematics" is found already in the works of Aristotle (IV century BC). According to Vasmer, 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 is completely irrelevant whether it will be numbers, figures, stars, sounds or something else, in which this measure is sought. Thus, there should be a certain general science that explains everything related to order and measure, without entering into the study of any particular subjects, and this science should not be called foreign, but the old, already used name of General Mathematics.

In Soviet times, the definition from the TSB, given by A.N.Kolmogorov, was considered classical:

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

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

Sections of mathematics

1. Mathematics as academic discipline is subdivided in the Russian Federation into elementary mathematics studied in secondary school and educated in the following disciplines:

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

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

Designations

Since mathematics deals with extremely diverse and rather complex structures, the notation system in it is also very complex. The modern system of writing formulas was formed on the basis of the European algebraic tradition, as well as the needs of the later branches of mathematics - mathematical analysis, mathematical logic, set theory, etc. Geometry from time immemorial has used a visual (geometric) representation. 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 numbers. Probably, the ancient people first expressed the 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 kipu. There were many different number systems. The first known records of numbers were found in the Ahmes papyrus created by the Egyptians in the Middle Kingdom. The Indian civilization developed the modern decimal number system, incorporating the concept of zero.

Historically, the basic mathematical disciplines have emerged under the influence of the need to carry out calculations in the commercial sphere, in the measurement of lands and to predict astronomical phenomena and, later, to solve new physical problems. Each of these areas plays a large role in the wider development of mathematics, which consists in the study of structures, spaces and changes.

Philosophy of Mathematics

Objectives and methods

Mathematics studies imaginary, ideal objects and relationships between them using 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 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 creating them. The model of an object does not take into account all of its features, but only the most necessary for the purposes of study (idealized). For example, by studying the physical properties of an orange, we can abstract from its color and taste and imagine it (albeit not perfectly accurately) as a ball. If we need to understand how many oranges will turn out if we add two and three together, then we can abstract from the form, leaving the model with only one characteristic - quantity. Abstraction and the establishment of connections between objects in the most general form is 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. " Space R n (\\ displaystyle \\ mathbb (R) ^ (n)), at n\u003e 3 (\\ displaystyle n\u003e 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, takes place using the axiomatic method: first, for the objects under study, a list of basic concepts and axioms is formulated, and then from the axioms, using inference rules, meaningful theorems are obtained, which together form a mathematical model.

Foundations

Intuitionism

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

Constructive mathematics

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

Main topics

amount

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

Transformations

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

Structures

Spatial relationships

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

Geometry	Trigonometry	Differential geometry	Topology	Fractals	Measure theory

Discrete Math

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