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

The abstractness of mathematics allows it to apply it in a wide variety of areas, it is a powerful tool for knowledge of nature.

Forms of knowledge are divided into two groups.

First group The forms of sensual knowledge carried out with the help of various senses: view, hearing, smell, touch, taste.

Ko second group The forms of abstract thinking, primarily the concepts, statements and conclusion.

Forms of sensual knowledge are feel, perception and representation.

Each item has not one, but many properties, and we will learn them with the help of sensations.

Feeling - This is a reflection of the individual properties of objects or phenomena of the material world, which are directly (i.e., at the moment) affect our senses. It is a feeling of red, warm, round, green, sweet, smooth and other individual properties of objects [Hetmanova, p. 7].

From individual sensations there is a perception of a whole subject. For example, the perception of the apple is composed of such sensations: spherical, red, sour-sweet, fragrant, etc.

Perception There is a holistic reflection of the external material subject, directly affected by our senses [Hetmanov, p. eight]. For example, an image of a plate, cups, spoons, other dishes; The image of the river, if we are now sailing on it or are on its shore; Forest image, if we now came to the forest, etc.

Perceptions, although they are a sensual reflection of reality in our consciousness, largely depend on the human experience. For example, a biologist will perceive meadow in one way (he will see different types of plants), and the tourist or the artist is very different.

Representation - This is a sensual image of the subject, at the moment we are not perceived, but who previously perceived in one form or another [Hetmanova, p. 10]. For example, we can visually imagine the faces of acquaintances, our room in the house, birch or mushroom. These are examples reproducing Presentations, since we saw these items.

Representation can be creative, including fantastic. We present the beautiful princess Swan, or the king of Saltan, or the Golden Cockerel, and many other characters from the fairy tales A.S. Pushkin, who have never seen and see. These examples of creative presentation on verbal description. We also imagine the Snow Maiden, Santa Claus, mermaid, etc.

So, the forms of sensual knowledge are sensations, perception and presentation. With their help, we learn the external sides of the subject (its signs, including properties).

The forms of abstract thinking are concepts, statements and conclusion.

Concepts. The volume and content of concepts

The term "concept" is usually used to designate a whole class of objects of arbitrary nature, which have a certain characteristic (distinctive, essential) property or a set of such properties, i.e. Properties inherent in only elements of this class.

In terms of logic, the concept is a special form of thinking characteristic of which the following is: 1) The concept is a product of highly organized matter; 2) the concept reflects the material world; 3) the concept appears in consciousness as a means of generalization; 4) the concept means specifically human activity; 5) the formation of the concept in the consciousness of a person is inseparable from its expression by speech, record or symbol.

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

The process of forming some concept is a gradual process in which several consecutive stages can be obtained. Consider this process on the simplest example - the formation of children's concepts about number 3.

1. At the first stage of knowledge, children get acquainted with various specific sets, while subject images are used and various sets of three elements are demonstrated (three apples, three books, three pencils, etc.). Children not only see each of these sets, but can also be born (touch) those objects from which these sets consist. This process of "vision" creates in the mind of the child a special form of reflection of real reality, which is called perception (sensation).

2. We will remove objects (objects), constituting each set, and offer children to determine whether something has been common characterizing each set. In the consciousness of children, the number of items in each set should be captured, the fact that everywhere was "three". If so, then in the minds of the children a new form was created - the idea of \u200b\u200bthe number "Three".

3. In the next stage, on the basis of a mental experiment, children should see that the property expressed in the word "three" characterizes any set of different elements of the form (a; b; c). Thus, there will be significant general feature such sets - "Have three elements." Now we can say that in the minds of children formed the 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 language form of concept is a word or a group of words. For example, a "triangle", "number three", "point", "straight", "anoscele triangle", "plant", "coniferous tree", "River Yenisei", "table", etc.

Mathematical concepts have a number of features. The main thing is that mathematical objects that need to be a concept do not exist in reality. Mathematical objects are created by the mind of a person. These are ideal objects reflecting real objects or phenomena. For example, in geometry study the shape and size of objects, without taking into account the other properties: color, mass, hardness, etc. From all this is distracted, abstracts. Therefore, in geometry instead of the word "subject" they say "Geometric Figure". The result of abstraction is both mathematical concepts as "number" and "value".

Basic characteristics anyone concepts are Next: 1) volume; 2) content; 3) relationship between concepts.

When they talk about a mathematical concept, they usually mean the entire set (set) of objects denoted by one term (word or group of words). So, speaking of the 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 There are many objects or items to which this concept is applicable.

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

Any mathematical object has certain properties. For example, the square has four sides, four straight corners equal to the diagonal, the diagonal of the intersection point is divided by half. You can also specify other properties, but among the properties of the object distinguish significant (distinctive) and insignificant.

Property is called essential (distinctive) for the object, if it is inherent in this object and without it it cannot exist; Property is called irrelevant For an object, if it can exist without it.

For example, for a square, all properties listed above are essential. The "AD Horizontal Side" will be insignificant for the AVD Square (Fig. 1). If you turn this square, then the AD side will be vertical.

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

Describe a figure.

Little black triangle. Fig. 2.

Large white triangle.

What are the figures like?

What are the figures different?

Color, magnitude.

What is the triangle?

3 sides, 3 corners.

Thus, children find out the essential and insignificant properties of the concept of "triangle". Essential properties - "have three sides and three angle", insignificant properties - color and sizes.

The combination of all essential (distinctive) properties of an object or subject reflected in this concept call concept content .

For example, for the concept of "parallelograms", a set of properties are: has four sides, has four angle, opposite sides are parallel, the opposite sides are equal, opposite angles are equal to the diagonal in the intersection points are divided by half.

There is a connection between the scope of the concept and its content: if the scope of the concept increases, its content is reduced, and vice versa. Thus, for example, the scope of the concept of "an elevated triangle" is part of the concept of the concept of "triangle", and in the content of the concept "an equallyced triangle" includes more properties than the concept of the concept of "triangle", because An equally chaired triangle has not only all the properties of the triangle, but also by other, inherent in equally feasible triangles ("two sides are equal", "two corners are equal," two medians are equal, etc.).

By volume, the concepts are divided into single, commonand categories.

The concept whose amount is 1, called single concept .

For example, the concepts: "River Yenisei", "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 in children are formed, mainly general concepts. For example, in primary classes, students get acquainted with such concepts as "figure", "number", "unambiguous numbers", "double-digit numbers", "multi-valued numbers", "fraction", "share", "addition", "Society" , "Amount", "subtraction", "subtracted", "diminished", "difference", "multiplication", "multiplier", "work", "division", "divisible", "divider", "private", " Ball »," Cylinder "," cone "," cube "," parallelepiped "," Pyramid "," Corner "," Triangle "," Quadrangle "," Square "," Rectangle "," Polygon "," Circle " , "Circle", "Curve", "Loaven", "Cut", "Cut Length", "Light", "Direct", "Point", "Length", "Width", "Height", "Perimeter", "Figure Square", "Volume", "Time", "Speed", "Mass", "Price", "Cost" and many others. All these concepts are common concepts.

Science, learning magnitude, quantitative relations and spatial forms

The first letter "M"

The second letter "A"

Third letter "T"

Last beech letter "A"

Answer to the question "Science, studying values, quantitative relations and spatial forms", 10 letters:
mathematics

Alternative Questions in Crosswords for Mathematics

The representative of this science was bought by Nobel a bride, and therefore for success in her Nobel Prize do not give

"Tower" in the Polytechnic Program

Exact science, learning magnitude, quantitative relations and spatial forms

Science of values, quantitative relations, spatial forms

It was this subject that I taught at the School "Dear Elena Sergeevna" performed by Marina Nelaova

Determination of the word mathematics in dictionaries

Explanatory dictionary of the living Great Russian, Dal Vladimir The meaning of the word in the dictionary Explanatory dictionary of the living Great Russian, Dal Vladimir
g. Science of values \u200b\u200band quantities; All that can be expressed digitally belongs to mathematics. - Clean, engaged in the values \u200b\u200bof abstract; - Applied, makes the first to business, to subjects. Mathematics is divided into arithmetic and geometry, the first one has ...

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

Great Soviet Encyclopedia The meaning of the word in the Dictionary Big Soviet Encyclopedia
I. Determination of the subject of mathematics, connection with other sciences and technology. Mathematics (Greek. Mathematike, from Máthema ≈ Knowledge, science), science of quantitative relations and spatial forms of valid world. "Clean mathematics has its own object ...

A new intelligent-word-formational dictionary of the Russian language, T. F. Efremova. The meaning of the word in the dictionary is a new intelligent-word-formational dictionary of the Russian language, T. F. Efremova.
g. Scientific discipline on spatial forms and quantitative relations of the actual world. Tutorial theoretical basis This scientific discipline. . A textbook that sets the content of this educational subject. . . Accurate,...

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

First, Trediakovsky sheltered Vasily Adadurov - mathematician, a student of the Great Jacob Bernoulli, and for this prudence the poet of a scientist in French instructed.

Embodied mathematician Adadurov, Mechanic Ladyzhensky, Architect Ivan Blank, clasped on the light of the Assators in various boards, doctors and gardeners, officers Army and Fleet.

Behind the long polished table of walnuts were sitting in chairs two: Axel Brigs and mathematician Brodsky, whom I learned on a powerful Socratian Lysin.

Pontryagin, the efforts of which the new section was created mathematics - topological algebra, - studying various algebraic structures endowed with topology.

We also note in passing that the epoch, described by us, witnessed the development of algebra, relatively abstract department mathematicsThrough the connection of less abstract departments of it, geometry and arithmetic, is the fact proven by the most ancient from the manifestations of algebra, half algebraic, half geometric.

The idealized properties of the objects under study are either formulated in the form of axioms, or listed in the definition of relevant mathematical objects. Then, by strict rules of logical output, other true properties (theorems) are displayed from these properties. This theory in the aggregate forms the mathematical model of the object under study. Thus, initially based on spatial and quantitative relations, mathematics receives more abstract ratios, the study of which is also the subject of modern mathematics.

Traditionally, mathematics is divided into a theoretical, performing an in-depth analysis of intramathematical structures, and applied, providing its models to other sciences and engineering disciplines, and some of them occupy a border with mathematics. In particular, formal logic can be considered as part philosophical sciences, and as part of the Mathematical Sciences; mechanics - both physics, and mathematics; Computer science, computer technologies and algorithm are both engineering and mathematical sciences, etc. In the literature, many different definitions of mathematics were proposed.

Etymology

The word "mathematics" occurred from Dr. Greek. άάθημα, which means study, knowledge, the science, and other Greek. μαθηματικός, originally meaning susceptible, successful Later targetedSubsequently mathematical. In particular, μαθηματικὴ τέχνη , Latin ars Mathematica.means art of mathematics. The term dr.-Greek. ᾰᾰθημᾰτικά B. modern meaning This word "mathematics" is found already in the writings of Aristotle (IV century BC. er). According to the Fasmere in the Russian language, the Word came either through Polish. Matematyka, either through a lat. Mathematica.

Definitions

One of the first definitions of the subject of mathematics gave Descartes:

The field of mathematics includes only those sciences in which either order or measure are considered, and will not be completely significant, whether these numbers, figures, stars, sounds or something else, will find this measure. Thus, there must be a certain overall science, explaining all related to the procedure and least, without entering the study of any private subjects, and this science should be called not foreign, but the old one who has already included in the use of universal mathematics.

The essence of mathematics ... It seems now as the doctrine of relationships between objects, which are not known about, except for the describing them of some properties, is precisely those who are as an axiom in the base of the theory ... Mathematics is a set of abstract forms - mathematical structures.

Sections of mathematics

1. Mathematics as academic discipline

Designations

Since mathematics works with extremely diverse and fairly complex structures, the system of designations in it is also very complex. The modern system of recording formulas was formed on the basis of the European algebraic tradition, as well as the needs of the later sections of mathematics - mathematical analysis, mathematical logic, the theory of sets, etc. Geometry of the century, used visual (geometric) representation. In modern mathematics, complex graphical records of recording systems are also common (for example, switching charts), indications based on graphs are also used.

Short story

Philosophy Mathematics

Goals and methods

Space R n (\\ displaystyle \\ mathbb (r) ^ (n)), P. N\u003e 3 (\\ DisplayStyle N\u003e 3) It is mathematical fiction. However, very brilliant fiction, which helps to mathematically understand complex phenomena».

Basis

Intuitionism

Constructive mathematics

clarify

Main topics

number

The main section considering the abstraction of the number of algebra. The concept of "number" originally originated from the arithmetic representations and was related to natural numbers. In the future, it, with the help of algebra, was gradually distributed to integer, rational, valid, complex and other numbers.

1, - 1, 1 2, 2, 3, 0, 12, ... (\\ displaystyle 1, \\; - 1, \\; (\\ FRAC (1) (2)), \\; (\\ FRAC (2) (3) ), \\; 0 (,) 12, \\; \\ ldots) Rational numbers 1, - 1, 1 2, 0, 12, π, 2, ... (\\ displaystyle 1, \\; - 1, \\; (\\ FRAC (1) (2)), \\; 0 (,) 12, \\; \\ pi, \\; (\\ sqrt (2)), \\; \\ ldots) Real numbers - 1, 1 2, 0, 12, π, 3 i + 2, Ei π / 3, ... (\\ displaystyle -1, \\; (\\ FRAC (1) (2)), \\; 0 (,) 12, \\; \\ pi, \\; 3i + 2, \\; e ^ (i \\ pi / 3), \\; \\ ldots) 1, i, j, k, π j - 1 2 k, ... (\\ displaystyle 1, \\; i, \\; j, \\; k, \\; \\ pi j - (\\ FRAC (1) (2)) k , \\; \\ dots) Complex numbers Quaternions

Conversion

The phenomena of transformations and changes in the general form considers the analysis.

Structures

Spatial relations

The basics of spatial relationships considers geometry. Trigonometry considers the properties of trigonometric functions. The study of geometric objects through mathematical analysis is engaged in differential geometry. The properties of spaces remaining unchanged with continuous deformations and the very phenomenon of continuity is studied topology.

Discrete Math

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Mathematics arose for a long time. The man collected fruit, digging the fruits, caught fish and reached all this for the winter. To understand how much food is a man invented an account. So began to emerge mathematics.

Then the man began to engage in agriculture. It was necessary to measure the plots of land, build housing, measuring time.

That is, a person has become necessary to use the quantitative relationship of the real world. Determine how many harvest assembled, what are the sizes of the construction site or as a large section of the sky, on which a certain number of bright stars.

In addition, a person began to define forms: the sun round, the box is square, the lake oval, and how these items are located in space. That is, the person has become interested in the spatial forms of the real world.

Thus, the concept mathematics You can define as a science on quantitative relations and spatial forms of the real world.

Currently, there is not a single profession, where it would be possible to do without mathematics. The famous German mathematician Karl Friedrich Gauss, who was called "King of Mathematics" somehow said:

"Mathematics - Queen of science, arithmetic - Queen of Mathematics."

The word "arithmetic" comes from the Greek word "arithmos" - "Number".

In this way, arithmetic This is a section of mathematics learning numbers and actions on them.

In elementary school, first of all, learn arithmetic.

How to develop this science, let's explore this question.

The emergence period of mathematics

The main period of accumulation of mathematical knowledge is the time to the V century to our era.

The first who began to prove mathematical provisions - an ancient Greek thinker who lived in the VII century BC is presumably 625 - 545. This philosopher traveled through the countries of the East. Traditions say he studied from Egyptian priests and Babylonian chaldeys.

Falez Miletsky brought from Egypt to Greece the first concepts of elementary geometry: what a diameter is what the triangle is determined and so on. He predicted a solar eclipse, engineering structures designed.

During this period, arithmetic is gradually folded, astronomy develops, geometry. Algebra and trigonometry emerge.

Period of elementary mathematics

This period begins with the VI to our era. Now mathematics arises like science with theories and evidence. There appears the theory of numbers, the doctrine of magnitude, about their dimension.

The most famous mathematician of this time is Euclide. He lived in the III century BC. This man is the author of the first of the theoretical treatise in mathematics that came to us.

In the works of Euclidea, the foundations are given, the so-called Euclidean geometry are axioms, resting on basic concepts, such as.

During the elementary mathematics, the theory of numbers is born, as well as the doctrine of values \u200b\u200band measurement. Negative and irrational numbers appear for the first time.

At the end of this period, the creation of algebra is observed, as an alphabetic calculus. The science of "Algebra" appears in Arabs, as a science on solving equations. The word "algebra" translated from Arabic means "recovery", that is, the transfer of negative values \u200b\u200bto another part of the equation.

The period of mathematics variables

The founder of this period is considered to be Rene Descartes, who lived in the XVII century of our era. In his writings, Decartes first introduces the concept of variable value.

Due to this, scientists transfers from the study of constant values \u200b\u200bto the study of dependencies between variables and to mathematical description Movement.

This period was characterized by Frederick Engels, he wrote in his writings:

"A swivel point in mathematics was a decartian variable. Thanks to this, the mathematics entered the Mathematics and thus the dialectic, and due to the same it became necessary to the differential and integral calculus, which immediately arises, and, which was generally completed, and not invented by Newton and Leibnian. "

The period of modern mathematics

In 20 years of the XIX century, Nikolai Ivanovich Lobachevsky becomes the founder, the so-called non-child geometry.

From that moment on, the development of the most important sections of modern mathematics begins. Such as theory of probability, the theory of sets, mathematical statistics, and so on.

All these discoveries and research find extensive use in various fields of science.

And at present, the science mathematics is growing rapidly, the subject of mathematics, including new forms and relations, are proved by new theorems, the main concepts are deepened.

The idealized properties of the objects under study are either formulated in the form of axioms, or listed in the definition of relevant mathematical objects. Then, by strict rules of logical output, other true properties (theorems) are displayed from these properties. This theory in the aggregate forms the mathematical model of the object under study. Thus, initially, based on spatial and quantitative relations, mathematics receives more abstract ratios, the study of which is also the subject of modern mathematics.

Traditionally, mathematics is divided into a theoretical, performing an in-depth analysis of intramathematical structures, and applied, providing its models to other sciences and engineering disciplines, and some of them occupy a border with mathematics. In particular, formal logic can also be considered as part of philosophical sciences, and as part of the Mathematical Sciences; mechanics - both physics, and mathematics; Informatics, computer technologies and algorithm are related to both engineering and mathematical sciences, etc. In the literature, many different definitions of mathematics were proposed (see).

Etymology

The word "mathematics" occurred from Dr. Greek. άάθημα ( máthēma.), which means study, knowledge, the science, and other Greek. μαθηματικός ( mathēmatikós.), originally meaning susceptible, successful Later targetedSubsequently mathematical. In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē.), in Latin ars Mathematica.means art of mathematics.

Definitions

The field of mathematics includes only those sciences in which either order, or measure and absolutely not essentially, will these numbers, figures, stars, sounds, or something else, what this measure is found. Thus, there must be a certain overall science, explaining all related to the procedure and least, without entering the study of any private subjects, and this science should be called not foreign, but the old one who has already included in the use of universal mathematics.

In Soviet times, the definition of BSE was considered to be classically, A. N. Kolmogorov:

Mathematics ... Science of quantitative relations and spatial forms of valid world.

The essence of mathematics ... It seems now as the doctrine of relationships between objects, which are not known about, except for the describing them of some properties, is precisely those who are as an axiom in the base of the theory ... Mathematics is a set of abstract forms - mathematical structures.

We give a few more modern definitions.

Modern theoretical ("net") mathematics is a science of mathematical structures, mathematical invariants of various systems and processes.

Mathematics - science that provides the ability to calculate the models given to the standard (canonical) mind. Science about finding solutions of analytical models (analysis) by means of formal transformations.

Sections of mathematics

1. Mathematics as academic discipline It is divided into the Russian Federation to elementary mathematics studied in high school and disciplines formed:

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

4. American Mathematical Society (AMS) has developed its standard for classifying mathematics sections. It is called Mathematics SUBJECT CLASSIFICATION. This standard is periodically updated. The current version is MSC 2010. Previous version - MSC 2000.

Designations

Due to the fact that mathematics works with extremely diverse and fairly complex structures, the system of designation is also very complex. The modern formula recording system was formed on the basis of a European algebraic tradition, as well as mathematical analysis (concept of function, derivative, etc.). Geometry Impact of the century enjoyed visual (geometric) representation. In modern mathematics, complex graphical records of recording systems are also common (for example, switching charts), indications based on graphs are also used.

Short story

The development of mathematics is based on writing and the ability to record numbers. Probably ancient people first expressed the amount by drawing cereals on Earth or scratched them on wood. Ancient Incas, having a different writing system, represented and maintained numerical data using a complex system of rope nodes, the so-called kip. There were many different number systems. The first known records of the numbers were found in the Akhmes papyrus created by the Egyptians of the Middle Kingdom. India civilization has developed modern decimal system Number, including zero concept.

Historically, basic mathematical disciplines have appeared under the influence of the need to conduct calculations in the commercial sphere, when measuring land and for predicting astronomical phenomena and, later, to solve new physical problems. Each of these areas plays a large role in the wide development of mathematics, which consists in the study of structures, spaces and changes.

Philosophy Mathematics

Goals and methods

Mathematics studies imaginary, ideal objects and ratios between them using a formal language. In general, mathematical concepts and theorems do not necessarily comply with anything in the physical world. The main task of the applied section of mathematics is to create a mathematical model, a fairly adequate to the real object under study. Task Mathematics-Theority - to provide a sufficient set of comfortable means to achieve this goal.

The content of mathematics can be defined as a system of mathematical models and tools for their creation. The model of the object takes into account not all its features, but only the most necessary for the purposes of study (idealized). For example, studying the physical properties of the orange, we can abstract from its color and taste and present it (even if not perfectly for sure) ball. If we need to understand how many oranges it turns out if we fold together two and three, then you can abstract and from the form, leaving the model only one characteristic - the amount. The abstraction and establishment of links between objects in the most general form is one of the main directions of mathematical creativity.

Another direction, along with abstraction - generalization. For example, summarizing the concept of "space" to the N-measurement space. " Space, with a mathematical fiction. However, very brilliant fiction, which helps to mathematically understand complex phenomena».

The study of intramathematics objects, as a rule, occurs using an axiomatic method: First, the list of basic concepts and axioms are formulated for the objects under study, and then content theorems are obtained from the axiom of the output rules, in the aggregate forming mathematical model.

Basis

The issue of the essence and grounds of mathematics was discussed from the time of Plato. Since the 20th century, there is a comparative agreement on the matter, which should be considered strict mathematical proof, but there is no consent in understanding that in mathematics it is originally true. From here, disagreements arise both in the questions of axiomatics and the relationship of the industries of mathematics and in the choice of logical systems that should be used in evidence.

In addition to skeptical, the following approaches to this issue are known.

Multiple approach

It is proposed to consider all mathematical objects within the framework of the theory of sets, most often with the axiomatics of Cermelo - Frankel (although there are many other equivalent to it). This approach is considered from the middle of the 20th century by the predominant, but in reality most mathematical works do not set the tasks to translate their statements strictly into the language of the theory of sets, but operate with concepts and facts established in some areas of mathematics. Thus, if a contradiction is detected in the theory of sets, this will not affect the depreciation of most results.

Logicism

This approach implies the strict typing of mathematical objects. Many paradoxes avoiding in the theory of sets only by special tricks are impossible in principle.

Formalism

This approach involves the study of formal systems based on classical logic.

Intuitionism

Intuitionism suggests intuitionist logic at the base of mathematics, more limited in evidence (but, as it is considered more reliable). Intuitionism rejects proof of the opposite, many non-constructive evidence becomes impossible, and many problems of the theory of sets are meaningless (informalizable).

Constructive mathematics

Constructive mathematics - close to intuitionism in mathematics, studying structural constructions [ clarify]. According to the criterion of constructiveness - " exist - it means to be built" Constructive criteria - a stronger demand than the criterion of consistency.

Main topics

Numbers

The concept of "number" was originally related to natural numbers. In the future, it was gradually distributed to integer, rational, real, complex and other numbers.

Whole numbers Rational numbers Real numbers Complex numbers Quaternions

Numeric systems
Accounts Set	Natural numbers () whole () rational () algebraic () periods computable arithmetic
Real numbers And their expansion	Real () complex () quaternions () Cali numbers (octaves, octonions) () Sedenions () Alternions Cali procedure - Dixon Dual hypercomplex superreal hypereral surreal Number ( english)
Others Numeric systems	Cardinal numbers ordinal numbers (transfinite, ordinal) P-adic supernatural numbers
see also	Double numbers irrational numbers transcendental numeric beam Bikwaternion

Conversion

Discrete Math

Codes in knowledge classification systems

Online services

There are a large number of sites providing service for mathematical calculations. Most of them are English-speaking. From Russian-speaking, you can note the service of mathematical queries of the Nigma search engine.

see also

Popularizers of science

Notes

1. Encyclopedia Britannica.
2. Webster's Online Dictionary
3. Chapter 2. Mathematics as a language of science. Siberian open university. Archived from the original source February 2, 2012. Checked October 5, 2010.
4. Large ancient Greek dictionary (αω)
5. Dictionary of the Russian language XI-XVII centuries. Issue 9 / ch. ed. F. P. Filin. - M.: Science, 1982. - P. 41.
6. Descartes R. Rules for the leadership of the mind. M.-L.: SOCHEKGISIS, 1936.
7. See: Mathematics BSE
8. Marx K., Engels F. Works. 2nd ed. T. 20. P. 37.
9. Burbaki N. Mathematics architecture. Essays on the history of mathematics / translation of I. G. Bashmakova ed. K. A. Rybnikova. M.: Il, 1963. P. 32, 258.
10. Kaziev V. M. Introduction to mathematics
11. Mukhin O. I. System modeling training manual. Perm: Rzi PSTU.
12. Herman Veil // Klein M. . - M.: Mir, 1984. - P. 16.
13. State educational standard Higher professional education. Specialty 01.01.00. "Mathematics". Qualification - mathematician. Moscow, 2000 (compiled under the leadership of O. B. Lupanova)
14. Nomenclature of specialties of scientists, approved by the Order of the Ministry of Education and Science of Russia of February 25, 2009 No. 59
15. UDC 51 mathematics
16. Ya. S. Bugrov, S. M. Nikolsky. Elements of linear algebra and analytical geometry. M.: Nauka, 1988. P. 44.
17. N. I. Kondakov. Logic Dictionary-Directory. M.: Science, 1975. P. 259.
18. G. I. Ruzavin. About nature mathematical knowledge. M.: 1968.
19. http://www.gsnti-norms.ru/norms/common/doc.asp?0&/norms/grnti/gr27.htm
20. For example: http://mathworld.wolfram.com

Literature

Encyclopedia

• // Encyclopedic Dictionary of Brockhaus and Efron: in 86 volumes (82 tons and 4 extra). - St. Petersburg. , 1890-1907.
• Mathematical encyclopedia (in 5 volumes), 1980s. // General and special reference books in mathematics on Eqworld
• Kondakov N. I. Logic Dictionary-Directory. M.: Science, 1975.
• Encyclopedia of Mathematical Sciences and their applications (it.) 1899-1934. (the largest literature review of the XIX century)

Directories

• Korn, T. Corn. Mathematics reference for scientists and engineers M., 1973

Books

• Klein M. Mathematics. Loss of certainty. - M.: Mir, 1984.
• Klein M. Mathematics. Search for truth. M.: Mir, 1988.
• Klein F. Elementary mathematics from the point of view of the highest.

• Tom I. Arithmetic. Algebra. Analysis M.: Science, 1987. 432 p.
• Volume II. Geometry M.: Science, 1987. 416 p.

• Kuralt R., G. Robbins. What is mathematics? 3-E ed., Act. and add. - M.: 2001. 568 p.
• Pisarevsky B. M., Kharin V. T. About mathematics, mathematicians and not only. - M.: Binom. Laboratory of Knowledge, 2012. - 302 p.
• Poincare A. Science and method (Rus) (Fr.)

Mathematics is one of the oldest sciences. To give quick definition Mathematics is not at all, its content will vary very much depending on the level of human mathematical education. Schoolboy primary classesOn just started studying arithmetic, says that mathematics studies the rules of the counting items. And he will be right, because it is with this that he meets at first. Older schoolchildren add to what was said that the concept of mathematics includes an algebra and study of geometric objects: lines, their intersections, flat figures, geometric bodies, various transforms. Graduates of the same secondary school will be included in the definition of mathematics to still study the functions and the action of the transition to the limit, as well as the concepts of the derivative and integral associated with it. Graduates of higher technical educational institutions or Natural Scientific Facilities of Universities and Pedagogical Institutions will no longer satisfy school definitions, as they know that the composition of mathematics includes other disciplines: probability theory, mathematical statistics, differential calculus, programming, computational methods, as well as applications of these disciplines for modeling production processes, processing experienced data, transmission and processing information. However, the fact that it is listed, the content of mathematics is not exhausted. The theory of sets, mathematical logic, optimal control, the theory of random processes and much more also included in its composition.

Attempts to identify mathematics by transferring the components of its branches will lead to us aside, because they do not give ideas that mathematics are studying and what is her attitude towards the world around us. If such a question was set to physics, a biologist or astronomer, then each of them would give a very short response that does not contain the listing of parts, of which the science studied by them. Such an answer would contain an indication of the phenomena of nature, which it explores. For example, a biologist would say that biology studies various manifestations of life. Let this answer are not fully completed because it does not say that such life and life phenomena are, but nevertheless, such a definition would have given a fairly complete picture of the content of the science of biology and about different levels of this science. And this definition would not change with the expansion of our biology knowledge.

There are no such phenomena of nature, technical or social processes, which would be the subject of studying mathematics, but did not relate to the phenomena of physical, biological, chemical, engineering or social. Each natural science discipline: biology and physics, chemistry and psychology - is determined by the material feature of its subject, specific features of the region of the real world, which it studies. The object itself or phenomenon can be studied by different methods, including mathematical, but, changing methods, we still remain within the limits of this discipline, since the content of this science is a real object, and not a research method. For mathematics, the material subject of research does not have a decisive value, the method used is important. For example, trigonometric functions You can also use for the study of the oscillatory movement, and to determine the height of the inaccessible item. And what phenomena of the real world can be explored using a mathematical method? These phenomena are not determined by their material nature, but exclusively formal structural properties, and above all those quantitative relations and spatial forms in which they exist.

So, mathematics studies not material objects, but research methods and structural properties of the object of study that allow you to apply some operations to it (summation, differentiation, etc.). However, a significant part of mathematical problems, concepts and theories has real phenomena and processes with its primary source. For example, arithmetic and the theory of numbers were mediated from the primary practical task - counting objects. The elementary geometry had its source problems associated with comparing distances, calculating the areas of flat figures or spatial bodies. All this was required to find, since it was necessary to redistribute land Between users, calculate the size of the granaries or the volume of earthworks during the construction of defense structures.

The mathematical result has the property that it can not only be used when studying a single particular phenomenon or process, but also use to study other phenomena, the physical nature of which is fundamentally different from previously considered. So, the rules of arithmetic applicable in the tasks of the economy, and in technical issues, and when solving problems agriculture, and in scientific research. The arithmetic rules have been developed by the Millennium Back, but they have retained applied value for eternal times. The arithmetic is an integral part of mathematics, its traditional part is no longer subject to creative development within the framework of mathematics, but it finds and will continue to find numerous new applications. These applications may be of great importance for humanity, but the contribution itself in mathematics will not be made.

Mathematics, as a creative force, is intended to develop general ruleswhich should be used in numerous special cases. The one who creates these rules creates a new one, creates. The one who applies ready-made rules is no longer creating in mathematics itself, but it is quite possible, it creates new values \u200b\u200bwith the help of mathematical rules in other areas of knowledge. For example, today, these campaign decryption data, as well as information on the composition and age of rock, geochemical and geophysical anomalies is processed using computers. There is no doubt that the use of a computer in geological studies leaves these studies with geological. The principles of the work of computers and their mathematical support were designed without taking into account the possibility of their use in the interests of geological science. This feature itself is determined by the fact that the structural properties of geological data are in accordance with the logic of certain programs of the computer.

Two definitions of mathematics were widely distributed. The first of them was given by F. Engels in the work "Anti-Dühring", another - a group of French mathematicians known as Nicola Burbaki, in the article "Architecture of Mathematics" (1948).

"Clean mathematics has its own object spatial forms and quantitative relations of the real world." This definition does not only describe the object of studying mathematics, but also indicates its origin - the actual world. However, this definition of F. Engels significantly reflects the state of mathematics in the second half of the XIX century. And it does not take into account those of its new areas that are not directly related to any quantitative relations or geometric forms. This is, first of all, mathematical logic and disciplines associated with programming. Therefore, this definition needs some clarification. Perhaps it should be said that mathematics has its own object of studying spatial forms, quantitative relations and logical structures.

Bombaki argue that "the only mathematical objects become, in fact, mathematical structures." In other words, mathematics should be defined as science on mathematical structures. This definition is essentially tautology, since it approves only one thing: mathematics is engaged in those objects that it studies. Another defect of this definition is that it does not find out the relationship of mathematics to the world around us. Moreover, Bombaki emphasizes that mathematical structures are created regardless of the real world and its phenomena. That is why Bombaki was forced to state that "the main problem is in the relationship between the world of experimental and the world mathematical. The fact that there is a close connection between experimental phenomena and mathematical structures - it seems to be quite unexpectedly confirmed by discoveries. modern physics, but we are completely unknown deep reasons for this ... and, perhaps, we will never know them. "

From the definition of F. Engels, such a disappointing output cannot occur, since it already provides a statement that mathematical concepts are abstractions from some relations and forms of the real world. These concepts are taken from the real world and are connected with it. In essence, it is precisely that the striking applicability of mathematics results to the phenomena of the world around us, and at the same time the success of the knowledge mathematization process.

Mathematics is not an exception to all areas of knowledge - concepts arising from practical situations and subsequent abstragments are also formed in it; It allows you to study reality also approximately. But it should be borne in mind that mathematics studies not the real world things, but abstract concepts and that the logical conclusions are absolutely strict and accurate. Its approach is not internal character, but is associated with the preparation of the mathematical model of the phenomenon. We note also that the rules of mathematics do not have absolute applicability, for them there is also a limited area of \u200b\u200bapplication, where they dominate it is undivided. Let us clarify the thought expressed example: it turns out that two and two are not always equal to four. It is known that when mixing 2 liters of alcohol and 2 l of water, less than 4 liters of mixtures are obtained. In this mixture, the molecules are arranged compact, and the volume of the mixture is less than the sum of the volume components. The rule of arithmetic is broken. You can still give examples in which other truths of arithmetic are disturbed, for example, when adding some objects, it turns out that the amount depends on the order of summation.

Many mathematicians consider mathematical concepts not as the creation of a pure mind, but as abstraction from actually existing things, phenomena, processes, or abstractions from the already established abstractions (up abstraction of higher orders). In the "Dialectics of Nature" F. Engels wrote that "All the so-called pure mathematics is engaged in abstractions ... All its values, strictly speaking, imaginary values \u200b\u200b..." These words clearly reflect the opinion of one of the founders of Marxist philosophy on the role of abstractions in mathematics. We only need to add that all these "imaginary values" are taken from real reality, and not designed arbitrarily, free flight of thought. That is how the concept of the number has been included in the universal use. At first, these were numbers within units, and moreover only entire positive numbers. Then the experience made expanding the arsenal of numbers up to dozen and hundreds. The idea of \u200b\u200bthe unlimited range of integers was born already in historically close to us: Archimedes in the book "Psammith" ("Calculating the grains") showed how to design numbers even more than those specified. At the same time, the concept of fractional numbers was born from practical needs. The calculations associated with the simplest geometric figures led humanity to new numbers - irrational. So gradually the idea of \u200b\u200bthe set of all valid numbers was formed.

The same path can be traced for any other concepts of mathematics. All of them arose of practical needs and gradually formed in abstract concepts. You can again remember the words of F. Engels: "... Clean mathematics is important, independent of the special experience of each individual personality ... But it is completely incorrect that in a clean mathematics Mind dealt only with products of its own creativity and imagination. The concepts of numbers and figures are not taken from somewhere, but only from the actual world. Ten fingers, in which people learned to count, that is, to produce the first arithmetic operation, represent anything, just not a product of free creativity of the mind. To consider, it is necessary to have not only items to be invalid, but have the ability to be distracted when considering these items from all other properties, other than the number, and this ability is the result of a long historical development based on experience. As the concept of the number and the concept of the figure is borrowed exclusively from the outside world, and did not appear in the head of pure thinking. There should have been things that have a certain shape, and these forms were supposed to be compared before it was possible to come to the concept of the figure. "

Consider whether there are concepts in science that are created without communication with the past progress of science and current progress of practice. We know perfectly well that scientific mathematical creativity is preceded by the study of many subjects at school, university, reading books, articles, conversations with specialists in their own field and in other areas of knowledge. Mathematics lives in society, and from books, on the radio, from other sources, he learns about the problems arising in science, engineering, public life. In addition, the thinking of the researcher is under the influence of the entire preceding evolution of scientific thought. Therefore, it turns out to be prepared by. Solving certain problems necessary for the progress of science. That is why the scientist cannot put forward the problems of arbitrariness, by whim, and should create mathematical concepts and theories that would be valuable for science, for other researchers, for humanity. But mathematical theories retain their importance in conditions of various public formations and historical epochs. In addition, often the same ideas arise from scientists who are in no way interconnected. This is an additional argument against those who adhere to the concept of free creativity of mathematical concepts.

So, we told what enters the concept of "mathematics". But there is also such a concept as applied mathematics. Under it understand the combination of all mathematical methods and disciplines that are applications outside of mathematics. In antiquity, geometry and arithmetic imagined all mathematics and, since the other found numerous applications during trade exchanges, measuring areas and volumes, in navigation issues, all mathematics was not only theoretical, but also applied. Later, B. Ancient Greece, the separation of mathematics and mathematics applied. However, all outstanding mathematicians were engaged in applications, and not just purely theoretical studies.

The further development of mathematics was continuously connected with the progress of natural science, technology, with the emergence of new social needs. By the end of the XVIII century. There was a need (first of all in connection with the problems of navigation and artillery) the creation of mathematical theory of movement. This was done in their works G. V. Leibnitz and I. Newton. Applied mathematics replenished with a new very powerful study method - mathematical analysis. Almost at the same time, the needs of demography, insurance led to the formation began the theory of probabilities (see probabilities theory). XVIII and XIX centuries. The content of applied mathematics was expanded by adding the theory of differential equations of ordinary and with private derivatives, equations of mathematical physics, elements of mathematical statistics, differential geometry. XX century Brought new methods of mathematical research practical tasks: the theory of random processes, the theory of graphs, functional analysis, optimal control, linear and nonlinear programming. Moreover, it turned out that the theory of numbers and abstract algebra found unexpected applications to the tasks of physics. As a result, the conviction began to make sure that applied mathematics as a separate discipline does not exist and all mathematics can be considered applied. Perhaps it is not necessary to say that mathematics is applied and theoretical, but that mathematics are divided into approvers and theorists. For some mathematics is the method of knowledge of the surrounding world and occurring in it phenomena, it is for this purpose that a scientist develops and expands mathematical knowledge. For others, mathematics in itself is a whole world, worthy of study and development. For the progress of science, scientists are needed and the other plan.

Mathematics, before studying with its methods, some phenomenon creates its mathematical model, i.e. it lists all the features of the phenomenon that will be taken into account. The model forces the researcher to choose those mathematics that will allow quite adequately to transfer the peculiarities of the studied phenomenon and its evolution. As an example, take the model of the planetary system: the sun and planets are treated as material points with the corresponding masses. The interaction of every two points is determined by the strength of attraction between them.

where M 1 and m 2 are the mass of interacting points, R is the distance between them, and F is constant. Despite all the simplicity of this model, it is already three hundred years of age with great accuracy.

Of course, each model coats reality, and the task of the researcher consists primarily in proposing a model transmitting, on the one hand, the most fully actual side of the case (as it is customary to speak, its physical features), and on the other - gives a significant approximation to reality. Of course, for the same phenomenon you can offer several mathematical models. All of them have the right to exist until the significant discrepancy between the model and reality begins to affect.

Mathematics is a science of quantitative relations and spatial forms of valid world. In an inextricable connection with the requests of science and technology, the margin of quantitative relations and spatial forms studied by mathematics are continuously expanding, so the above definition must be understood in the general sense.

The purpose of the study of mathematics is to increase the overall outlook, culture of thinking, the formation of scientific worldview.

Understanding the independent position of mathematics as special science has become possible after the accumulation of a sufficiently large actual material and arose for the first time in ancient Greece in the VI-V centuries to our era. It was the beginning of a period of elementary mathematics.

During this period, mathematical studies deal only with a fairly limited reserve of basic concepts that have arisen with the most simple demands of economic life. At the same time, the qualitative improvement of mathematics as science already occurs.

Modern mathematics are often compared with a large 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 arise in mathematics, graceful and deep new theories are built, similar to the construction of new quarters and buildings. But the progress of mathematics is not reduced only to the change in the face of the city due to the construction of a new one. You have to change the old one. Old theories are included in new, more general; There is a need to strengthen the foundations of old buildings. It is necessary to lay new streets to establish links between the distant quarters of the mathematical city. But this is not enough - architectural design requires considerable effort, since the difference in various mathematics regions not only spoils the overall impression of science, but also interferes with the understanding of science in general, the establishment of connections between its various parts.

Another comparison is often used: mathematics are like a large branching tree, which systematically gives new shoots. Each branch of a tree is the one or another region of mathematics. The number of branches does not remain unchanged, because new branches grow, they grow together first grew apart, some of the branches dry out, devoid of nutritional juices. Both comparisons are successful and very well transmit the actual situation.

There is no doubt that the requirement of beauty plays a large role in building mathematical theories. It goes without saying that the feeling of beauty is very subjective and often there are enough ugly ideas on this. And yet it is necessary to be surprised by the Unanimity, which is invested by mathematicians to 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. The mathematical conclusion is considered beautiful if there is simple and short reasoning in it to prove a significant mathematical fact. Mature mathematics, his talent is guessed by how developed he has a sense of beauty. Aesthetically completed 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 turned into a scientific discipline with a variety of research directions, a huge number of results and methods. Mathematics is now so great that there is no possibility for one person to cover it in all its parts, there is no possibility to be in it a universal specialist. The loss of links between its individual directions is certainly a negative effect of the rapid development of this science. However, the development of all industries of mathematics is the common - the origins of the development, the roots of the Tree of mathematics.

Euclidean geometry as the first natural science theory

• In the III century BC, the Book of Euclideus appeared in Alexandria with the same name, in Russian translation "began". From the Latin name "began" the term "elementary geometry" occurred. Despite the fact that the compositions of Euclide's predecessors have not reached us, we can make some opinion about these essays on the "beginning of" Euclidea. In the "beginning" there are sections, logically very little associated with other sections. Their appearance is explained only by the fact that they are made by tradition and copy the "beginning" of the predecessors of Euclide.

  "Beginning" Euclid consists of 13 books. 1 - 6 books are devoted to planimetry, 7 - 10 books - about arithmetic and incommensurable values \u200b\u200bthat can be built using a circulation and ruler. Books from 11 to 13 were devoted to stereometry.

  "Beginning" begin with statement of 23 definitions and 10 axioms. The first five axioms are "common concepts", the rest are called "postulates". The first two postulates determine the actions with the help of an ideal line, the third - with the help of an ideal circulation. Fourth, "all straight corners are equal to each other," is unnecessary, as it can be removed from the other axioms. The latter, the fifth postulate read: "If direct drops into two straight lines and forms internal unilateral angles in the amount less than two direct, then, with an unlimited continuation of these two straight lines, they will cross from the other side where the corners are less than two direct."

  Five " common concepts"Euclidea is the principles of measurement of lengths, corners, areas, volumes:" equal to the same are equal to each other "," if it is equal to equal equal, the sums are among themselves "," if equal to equal, the remains are equal to each other ", "Comcommended with each other are equal to each other", "whole more part".

  Next began the criticism of the Euclidean geometry. Euclides were criticized for three reasons: for considering only such geometric values \u200b\u200bthat can be constructed using a circulation and a ruler; For the fact that he ruptured geometry and arithmetic and argued for integers, what has already proven for geometric values, and finally for Euclidea axioms. The fifth postulate is most criticized, the most difficult Euclid post. Many considered him superfluous, and that it can and should be removed from other axioms. Others believed that it should be replaced with a simpler and visual, equivalent to him: "After the point outside the straight, you can spend in their plane no more than one direct, which does not cross this straight."

  The criticism of the gap between geometry and arithmetic led to the expansion of the concept of the number to the actual number. Disputes about the fifth postulate led to the fact that early XIX. A century N.I.Lobachevsky, I. Bayyai and K.F.Gauss built a new geometry, in which all the axioms of the Euclidean geometry were carried out, with the exception of the fifth postulate. It was replaced by the opposite statement: "In the plane through a point outside the straight, you can spend more than one direct, not intersecting this." This geometry was as consistent as the geometry of Euclid.

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

  In the Euclidean plane, we draw a horizontal straight line. This direct is called the Absolute (x). The points of the Euclidean plane underlying the above absolute are the points of the Lobachevsky plane. The Lobachevsky plane is the open half-plane, which is above the absolute. Nevklidovy Segments in the Poincaré model are arcs of circles with a center on the absolute or segments of direct, perpendicular absolute (AB, CD). Figure on the Lobachevsky plane - the figure of the open half-plane underlying the above absolute (F). Neevklidovo Movement is a composition of a finite number of inversions with a center on the absolute and axial symmetries whose axes are perpendicular to the absolute. Two non-child segments are equal if one of them is non-child movement can be translated into another. These are the basic concepts of axiomatics of the planimetry of Lobachevsky.

  All axioms of planimetry Lobachevsky are consisting. "Nevklidova is direct - this is a semi-rapidness with the ends on the absolute or beam with the beginning of the absolute and perpendicular absolute." Thus, the assertion of the parallelism of Lobachevsky is performed not only for some direct A and point A, which is not lying on this straight, but also for any direct A and anyone who does not lie on it. A.

  Other consistent geometries arose for Lobachevsky geometry: projective geometry separated from Euclidean, the multidimensional Euclidean geometry emerged, the Riemannian geometry emerged (the overall theory of spaces with an arbitrary law measurement law) and others. From the science of figures in one three-dimensional Euclidean Space Geometry for 40 - 50 years has become a collection of various theories, only in something similar with his ancestor - Euclidean geometry.

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

• Academician A.N. Kolmogorov allocates four periods of development of mathematics Kolmogorov A.N. - mathematics, mathematical encyclopedic Dictionary, Moscow, Soviet Encyclopedia, 1988: the origin of mathematics, elementary mathematics, mathematics of variable values, modern mathematics.

  During the development of elementary mathematics from arithmetic, the theory of numbers gradually grows. A algebra is created as letter calculus. A created by the ancient Greeks, the system of presentation of elementary geometry - Euclidean geometry - for two millennium ahead was made a sample of the deductive construction of mathematical theory.

  In the XVII century, the requests of natural science and technology led to the creation of methods that allow mathematically to study the movement, changes in value changes, transformation geometric figures. With the use of variables in analytical geometry and the creation of differential and integral calculation, the period of mathematics of variables begins. The great discoveries of the XVII century is the concept of infinitely small magnitude introduced by Newton and Leibniz, the creation of the foundations of the analysis of infinitely small values \u200b\u200b(mathematical analysis).

  The concept of function is put forward on the fore. The function becomes the main subject of study. The study of the function leads to the basic concepts of mathematical analysis: the limit, derivative, differential, integral.

  By this time, the appearance of the brilliant ideas of R. Dekart about the coordinate method. Analytical geometry is created, which allows you to study geometric objects by algebra and analysis methods. On the other hand, the coordinate method has discovered the possibility of geometric interpretation of algebraic and analytical facts.

  Further development of mathematics led at the beginning of the XIX century to the formulation of the problem of studying the possible types of quantitative relations and spatial forms with a sufficiently general point of view.

  The connection of mathematics and natural science is becoming more and more complex forms. New theories arise and they arise not only as a result of requests of natural science and technology, but also as a result of the internal need of mathematics. A wonderful example of such the theory is the imaginary geometry of N.I.Lobachevsky. The development of mathematics in the XIX and XX centuries allows it to be attributed to the period of modern mathematics. The development of the mathematics itself, the mathematization of various science areas, the penetration of mathematical methods in many areas of practical activity, the progress of computational technology has led to the emergence of new mathematical disciplines, for example, a study of operations, game theory, mathematical economics and others.

  The main methods in mathematical studies are mathematical evidence - strict logical reasoning. Mathematical thinking is not reduced only to logical reasoning. For the proper formulation of the problem, mathematical intuition is necessary to assess the choice of the method of its solution.

  Mathematical models of objects are studied in mathematics. The same mathematical model can describe the properties of real phenomena from each other. So, the same differential equation It may describe the processes of population growth and the disintegration of a radioactive substance. For mathematics, the nature of the objects under consideration is important, but the relationship between them.

  In mathematics use two types of conclusions: deduction and induction.

  Induction - a study method in which general conclusion Constructed on the basis of private parcels.

  Deduction is a way of reasoning, through which a private conclusion is followed from common parcels.

  Mathematics plays an important role in natural scientific, engineering and humanitarian studies. The reason for the penetration of mathematics in 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 computing devices, progress would be impossible in various fields of human activity.

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

  The main features of mathematical thinking

• According to this issue, the characteristic of mathematical thinking is of particular interest, given A.Y. Khinchin, or rather, its concrete historical form - the style of mathematical thinking. Revealing the essence of the style of mathematical thinking, it highlights four common features for all eras, noticeably distinguishing this style from thinking styles in other sciences.

  First, for mathematics is characterized by the dominance of the logical scheme of reasoning. Mathematician, who lost, at least temporarily, out of sight, this scheme is generally deprived of the opportunity to scientifically think. This peculiar style of mathematical thinking has a lot of valuable. Obviously, it allows you to follow the correctness of the flow of thought and guarantees from errors; On the other hand, it forces the thinking when analyzing to have before his eyes the whole set of available opportunities and obliges it to take into account each of them, not missing anyone (this kind of passes is quite possible and are actually often observed with other thinking styles).

  Secondly, laconicism, i.e. A conscious desire to always find the shortest leading to this goal of a logical path, merciless discarding of everything that is absolutely necessary for the perfect fullness of the argument. The mathematical essay of a good style, does not tolerate any "water", no decorating, weakening the logical tension of ranting, distracts to the side; Maximum stiffness, the harsh severity of the thought and its presentation constitute an integral traction of mathematical thinking. This feature has greater value not only for mathematical, but also for any other serious reasoning. LAKONIS, the desire to prevent anything unnecessary, helps and the very thinking, and his reader or listener fully focus on this course of thoughts, without being distracted by side ideas and without losing direct contact with the main line of reasoning.

  Coriferations of science, as a rule, think and are concreted concisely in all areas of knowledge, even when the idea of \u200b\u200bthem creates and sets out fundamentally new ideas. What a majestic impression produces, for example, the noble misfortune of the thought and speech of the greatest creators of physics: Newton, Einstein, Nielsa Bor! It may be difficult to find a brighter example of how deep impact may have the style of thinking of her creators on the development of science.

  For mathematics, laconide of thoughts is a continued, canonized centuries by law. Any attempt to burden the presentation is not necessarily necessary (albeit even pleasant and fascinating for listeners) with paintings, distractions, ranting in advance to legal suspicion and automatically causes critical alertness.

  Third, clear dismemberment of the progress. If, for example, in case of proof of any sentence, we must consider four possible cases, of which everyone can be divided into a number of subhearders, then at every moment of reasoning the mathematician should clearly remember, in which case the subllity of his thought is now acquired and what Cases and subheard him still remains to consider. With any kind of branched transfers, the mathematician must pay a report at any time in what kind of concept he lists the components of its species concepts. In ordinary, not scientific thinking, we quite often observe in such cases of mixing and jumps, leading to confusion and errors in reasoning. It often happens that a person began to list the types of one kind of kind, and then imperceptibly for students (and often for himself), using the insufficient logical discrimination of reasoning, rearranged into another genus and finishes the statement that both kinds are now classified; And listeners or readers do not know where the boundary runs between the species of the first and second kind.

  In order to make such mixing and jumps impossible, mathematics have long been widely used by simple external taking numbers of the numbering of concepts and judgments, sometimes (but much less) applicable in other sciences. Those possible cases or those generic concepts that should be considered in this reasoning are renumbers in advance; Inside each such case, those subject to subheard, which it contains is also renumbered (sometimes, to distinguish with any other numbering system). Before each paragraph, where the consideration of a new subllitance begins, it is put to this expanding designation (for example: II 3 - this means that the third case of the third case is considered here, or a description of the third type of second kind, if it comes to classification). And the reader knows that until then, as long as he will not surpass to a new numeric heading, all the outlined applies only to this occasion and suberral. Self itself, of course, that such a numbering serves only by external reception, very useful, but it is not obligatory, and that the essence of the case is not in it, but in the distinct dismemberment of the argument or classification, which it stimulates, and marks it.

  Fourth, scrupulous accuracy of symbols, formulas, equations. That is, "Each mathematical symbol has a strictly defined value: replacing it with another symbol or permutation to another place, as a rule, entails distortion, and sometimes the complete destruction of the meaning of this statement."

  Having highlight the main features of the mathematical style of thinking, A.Ya.Hinchin notes that mathematics (especially mathematics of variable values) by its nature has a dialectical nature, and therefore contributes to the development of dialectical thinking. Indeed, in the process of mathematical thinking, the interaction of a visual (concrete) and conceptual (abstract). "We can't think of the lines," wrote Cant, "without spending her mentally, we cannot think of the three dimensions, without spending, from one point of three perpendicular to each other lines."

  The interaction of concrete and abstract "led" mathematical thinking to the development of new and new concepts and philosophical categories. In antique mathematics (mathematics of constant values) were "the number" and "space", which were originally reflected in arithmetic and Euclidean geometry, and later in algebra and various geometric systems. Mathematics of variables "based" on the concepts in which the movement of matter was reflected - "final", "infinite", "continuity", "discrete", "infinitely small", "derivative", etc.

  If we talk about the modern historical stage of the development of mathematical knowledge, it goes in line with further development of philosophical categories: the theory of probabilities "masters" categories of possible and random; topology - categories of relations and continuity; The theory of disasters - the category of jump; The theory of groups - categories of symmetry and harmony, etc.

  In mathematical thinking, the main patterns of constructing similar in the form of logical connections are expressed. With its help, the transition from one (say, from certain mathematical methods - axiomatic, algorithmic, constructive, theoretical and other) to a special and general, to generalized deductive buildings. The unity of methods and objects of mathematics determines the specifics of mathematical thinking, it allows you to talk about a special mathematical language, in which not only reality is reflected, but also synthesized, summarized, scientific knowledge is predicted. The power and beauty of mathematical thought - in the limiting clarity of its logic, the grace of structures, skilled building abstractions.

  Fundamentally new possibilities of mental activity opened with the invention of the computer, with the creation of machine mathematics. In the language of mathematics there were significant changes. If the language of classical computing mathematics consisted of the formulas of algebra, geometry and analysis, focused on the description of the continuous processes of nature studied, primarily in mechanics, astronomy, physics, its modern language is the language of algorithms and programs, including the old language formulas as private Case.

  The language of modern computing mathematics is becoming increasingly versatile, capable of describing complex (multi-parameter) systems. At the same time, I want to emphasize that anything perfect is a mathematical language, enhanced by electronic computing equipment, it does not impose connections with a diverse "alive", natural language. Moreover, the conversational language is an artificial language base. In this regard, it is of interest to the recent discovery of scientists. This is the fact that the ancient language of the Aimara Indians, which speaks about 2.5 million people in Bolivia and Peru, was extremely convenient for computer equipment. As early as 1610, the Italian missionary-Jesuit Louis Burtoni, who was the first dictionary of Aimar, noted the genius of his creators who achieved high logical purity. In Aimar, for example, there are no wrong verbs and no exceptions from a few clear grammatical rules. These features of the Aimar language allowed the Bolivian mathematics to create a system of synchronous computer translation from any of the five European languages \u200b\u200blaid in the program, the "bridge" between which is aimar. Emm "Aimara", created by the Bolivian scientist, received a high assessment of specialists. Summarizing this part of the question of 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 to build theory, with antiquity and until today confirming its versatility and all applicability.

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

  The axiomatic method appeared in ancient Greece, and at this time applies in almost all theoretical sciences, and, above all in mathematics.

  Comparing three, in a certain respect, complementing each other geometry: Euclidean (parabolic), Lobachevsky (hyperbolic) and Riemannov (elliptical), it should be noted that along with some similarities there are a great difference between spherical geometry, on the one hand, and Euclidean geometries and Lobachevsky - on the other.

  The indigenous difference of modern geometry is that now it covers "geometry" of an infinite multitude of different imaginary spaces. However, it should be noted that all these geometries are interpretations of Euclidean geometry and they are based on an axiomatic method for the first time used by Euclide.

  Based on research, the axiomatic method was developed and widespread. As a special case of applying this method, the method of traces in stereometry is used to solve problems on the construction of sections in polyhedra and some other positional tasks.

  The axiomatic method developed at the beginning in geometry has now become an important instrument of study and in other sections of mathematics, physics and mechanics. Currently, work is underway to improve and a deeper study of an axiomatic method for building the theory.

  The axiomatic method of constructing scientific theory is to allocate the main concepts, the formulation of axioms of theories, and all other statements are derived by the logical way, based on them. It is known that one concept should be explained with the help of others, which, in turn, are also determined using some known concepts. Thus, we come to elementary concepts that cannot be determined through others. These concepts are called basic.

  When we prove the approval, theorem, then rely on the prerequisites that are considered already proven. But these prerequisites were also proved, they needed to justify. In the end, we come to non-proven statements and accept them without proof. These statements are called axioms. The axiom set should be such that, relying on it, one could prove further allegations.

  Having highlight the basic concepts and formulating axioms, then we derive theorems and other concepts with a logical way. This is the logical structure of geometry. Axioms and basic concepts constitute the basis of the planimetry.

  Since it is impossible to give a unified definition of 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 the geometric system, we proceed from some axiom system, or axiomatics. These axioms describe the properties of the basic concepts of the geometric system, and we can present the basic concepts in the form of objects of any nature that have the properties specified in the axioms.

  After the wording and evidence of the first geometric statements, it becomes possible to prove some allegations (theorems) with the help of others. Proof of many theorems are attributed to Pythagora and Democritus.

  Hippocrata Chiosky is attributed to the preparation of the first systematic course of geometry based on definitions and axioms. This course and its subsequent processing were called "Elements".

  Axiomatic method for constructing scientific theory

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

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

  The deductive system of the presentation is reduced:

  1) to the listing of basic concepts,

  2) to the statement of definitions

  3) to the action of axioms,

  4) to present the theorems

  5) to the proof of these theorems.

  Axioma - approval taken without evidence.

  Theorem is a statement arising from the axiom.

  The proof is an integral part of the deductive system, it is a reasoning, which shows that the truth of the statement implies logically from the truth of the previous theorems or axioms.

  Inside the deductive system, two questions may not be solved: 1) on the meaning of basic concepts, 2) on the truth of the axiom. But this does not mean that these questions are generally insoluble.

  The history of natural science shows that the possibility of axiomatic construction of one or another science appears only at a fairly high level of development of this science, based on a large actual material, allows you to clearly identify the main connections and relations that exist between the objects studied by this science.

  A sample of axiomatic construction of mathematical science is elementary geometry. The system of axiom of geometry was set forth by Euclide (about 300 g. BC) In an unsurpassed labor "began". This system in the main features has been preserved to this day.

  Basic concepts: point, straight, plane Basic images; Lower between, belonging, movement.

  Elementary geometry has 13 axioms that are divided into five groups. In the fifth group, one axiom on parallel (V is the Euclid post): through the point on the plane, you can only spend one direct, which does not cross this direct. This is the only axiom that caused the need for evidence. Attempts to prove the fifth postulates occupied mathematicians of more than 2 thousand years, up to the first half of the 19th century, i.e. Until that Nikolai Ivanovich Lobachevsky proved in his writings a complete hopelessness of these attempts. Currently, the non-refusability of the fifth postulate is strictly proven mathematical fact.

  Axioma about parallel N.I. Lobachevsky replaced the axiom: let in this plane there is a straight and lying outside the straight point. After this point, you can spend on a given direct, at least two parallel straight.

  Of new system Aksiom N.I. Lobachevsky with impeccable logical rigor brought a slender system the theorems that make up the maintenance of non-child geometry. Both geometries of Euclidean and Lobachevsky, as logical systems are equal.

  Three great mathematics in the 19th century almost at the same time, independently of each other came to one results of the unprofitability of the fifth postulate and to the creation of non-child geometry.

  Nikolai Ivanovich Lobachevsky (1792-1856)

  Karl Friedrich Gauss (1777-1855)

  Janos Boyai (1802-1860)

  Mathematical evidence

• The main method in mathematical studies are mathematical evidence - strict logical reasoning. Due to objective necessity, the corresponding member of the Rounds of Ran L.D. Kudryavtsev Kudryavtsev L.D. - Modern mathematics and her teaching, Moscow, science, 1985., Logic arguments (which, by nature, if they are right, are both strict) represent the method of mathematics, without them, mathematics are unthinkable. It should be noted that mathematical thinking is not reduced only to logical reasoning. To correctly state the task, to evaluate its data, to allocate essential intuition, it is necessary to prevent its solution to its solution, which allows you to anticipate the desired result before it is obtained, outline the study path 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 a number of experiments (which in itself plays a large role in mathematical studies), but a purely logical path, according to the laws of formal logic.

  It is believed that mathematical proof is the truth in the last instance. The solution that is based on clean logic simply cannot be incorrect. But with the development of science and tasks in front of mathematics are increasingly complex.

  "We entered the era when the mathematical apparatus became so complicated and cumbersome that at first glance could not be said - truthful or not met the task," Kate Devlin believes from Stenford University of California, USA. It leads to an example of a "classification of simple finite groups", which was formulated back in 1980, and fully accurately attracted until now. Most likely, the theorem is faithful, but it is impossible to talk about it.

  Computer solution is also impossible to be called accurate, for such calculations always have an error. In 1998, Hales proposed a solution to the Kepler Theorem using a computer formulated back in 1611. This theorem describes the most dense packaging of balls in space. The proof was presented on 300 pages and contained 40000 lines of machine code. 12 reviewers tested the decision during the year, but they have never achieved one hundred percent confidence in the correctness of evidence, and the study was sent to refinement. As a result, it was published only four years and without complete certification of reviewers.

  All the last calculations for applied tasks are made on a computer, but scientists believe that for greater reliability, mathematical calculations must be represented without errors.

  The theory of proof was developed in logic and includes three structural components: the thesis (what is supposed to prove), arguments (a set of facts, generally accepted concepts, laws, etc. relevant science) and demonstration (the procedure for deploying evidence; consistent chain of conclusions when N-noise conclusion becomes one of the parcels N + 1st conclusion). The rules of evidence are allocated, possible logical errors are indicated.

  Mathematical proof has a lot in common with those principles that are established by formal logic. Moreover, mathematical rules of reasoning and operations obviously served as one of the basics in the development of the proof procedure in logic. In particular, researchers of the development of formal logic to believe that at one time, when Aristotle undertook the first steps to create laws and regulations of logic, he turned to mathematical and to the practice of legal activity. In these sources, he found material for the logical constructions of the intended theory.

  In the 20th centuries, the concept of evidence has lost a strict meaning, which happened due to the detection of logical paradoxes, taking place in the theory of sets and especially in connection with the results, which were brought by K. Gedel's theorems about the incomplete formalization.

  First of all, it touched on the mathematics itself, in connection with which the belief was expressed with that the term "proof" does not have an accurate definition. But if such an opinion (taking place and today) affects the mathematics itself, then they come to the conclusion, according to which evidence should be taken not in the logical and mathematical, but in a psychological sense. With that, this look is found at the very Aristotle, which considered that it would be to prove to conduct a reasoning that would convince us to such an extent that, using it, we convince others to be right anything. A certain shade of a psychological approach found A.E.I.Senin-Volpin. He sharply opposes the adoption of truth without proof, connecting it with the act of faith, and then writes: "I call proof of judgment, I call an honest reception that makes it an indisputable judgment." Yesenin-Volpin gives a report that his definition needs even in clarifications. At the same time, the characteristic of the proof as "honest reception" does the appeal of the moral and psychological assessment?

  At the same time, the detection of multiple paradox theoretical and the appearance of Gedel's theorems just contributed to the development of the theory of mathematical evidence undertaken by intuitionists, especially the constructivist direction, and D.Gilbert.

  Sometimes it is believed that mathematical proof is universal and represents the ideal version of scientific evidence. However, it is not the only method, there are other ways of evidentiary procedures and operations. It is true that the mathematical evidence has a lot of similar to formal-logical, realizable in natural science, and that mathematical evidence has a certain specificity, as well as a set of receiving operations. On this we will stop, omitting that general that it relates him to other forms of evidence, that is, without deploying in all steps (even the main) algorithm, rules, errors, etc. Process proof.

  Mathematical proof represents reasoning, having a task to substantiate the truth (of course, in mathematical, that is, as an derivability, sense) of any approval.

  The set of rules applied in the proof has been formed together with the advent of axiomatic constructions of mathematical theory. The most clearly and fully was implemented in the geometry of Euclide. His "beginning" became a kind of model standard of an axiomatic organization of mathematical knowledge, and for a long time remained as such for mathematicians.

  The statements submitted in the form of a specific sequence should guarantee the conclusion that, subject to the rules of logical operating and is considered proven. It must be emphasized that a certain reasoning is evidence only with respect to some axiomatic system.

  When characterizing mathematical evidence, two main features are allocated. First of all, the fact that mathematical evidence excludes any references to empirius. The whole procedure for substantiating the truth of the output is carried out within the framework of the accelerated axiomatics. Academician A.D. Alksandrov, in connection with this, emphasizes. You can measure the corners of the triangle thousand times and make sure that they are equal to 2D. But the mathematics will not prove anything. He will prove if you bring the aligned assertion from the axiom. Repeat. Here mathematics and close methods of scholasticism, which also fundamentally rejects the argument is experienced by these facts.

  For example, when the incomingness of segments was discovered, with the proof of this theorem, an appeal to the physical experiment was excluded, since, firstly, the very concept of "non-elementaryness" is deprived of physical meaning, and, secondly, mathematics and could not, dealing with abstraction, Attract for the aid of real-specific lengths, measured by a sensually visual reception. The incompleteness, in particular, the parties and diagonals of the square, is proved, based on the property of integers with the involvement of the Pythagorean theorem on the equality of the square of hypotenuse (respectively - diagonally) the sum of the squares of the cathets (two sides of the rectangular triangle). Or when Lobachevsky was looking for confirmation for his geometry, referring to the results of astronomical observations, this confirmation was carried out by them by means of a purely speculative nature. In the interpretations of nehvklide geometry carried out by Cali - Klein and Beltra, also appeared typical mathematical, and not physical objects.

  The second feature of mathematical evidence is its highest abstraction, which it differs from the proof procedures in the rest of the sciences. And again, as in the case of the concept of a mathematical object, we are not just about the degree of abstraction, but about its nature. The fact is that the high level of abstraction proof reaches both in a number of other sciences, for example, in physics, cosmology and, of course, in philosophy, since the subject of the latter becomes the limit problems of being and thinking. Mathematics is distinguished by the fact that there are variables, the meaning of which is in distraction from any specific properties. Recall that, by definition, variables - signs that themselves do not have the values \u200b\u200band acquire the last only when substituting the names of certain objects (individual variables), or when specifying specific properties and relationships (predicate variables), or finally Cases of replacement by a variable meaningful statement (propositional variable).

  Announced feature and is due to the nature of the extreme abbreviations used in mathematical proof of signs, as well as statements that, thanks to the inclusion of variables in their structure, are transformed into a statement function.

  The proof procedure itself, determined in logic as a demonstration, proceeds on the basis of the rules of the output, relying on which the transition from some proven statements to another, forming a serial chain of conclusions. The most common rules (substitutions and conclusions) and deduction theorem are most common.

  Rule of substitution. In mathematics, the substitution is defined as a replacement of each of the elements A of a given set by any other element F (A) from the same set. In mathematical logic, the substitution rule is formulated as follows. If the true formula M in the statement calculus contains the letter, say a, then, replacing it everywhere, where it occurs, an arbitrary letter D, we get a formula, also true as the original. This is possible, and permissible because it is that in the calculation of the statements are distracted by the meaning of statements (formulas) ... only the values \u200b\u200bof "truth" or "lie" are taken into account. For example, in the formula M: A -\u003e (BUA) in place a we substitute the expression (AUB), as a result we obtain a new formula (AUB) -\u003e [(BU (AUB)].

  The conclusion output rule corresponds to the structure of the conditionally categorical Slitogism of Modus Ponens (Modeus Approving) in the formal logic. It has the following form:

  a. .

  The statement is given (A-\u003e B) and is still given a. From this follows.

  For example: if it rains, then bridge wet, the rain is (a), therefore, the bridge wet (B). In mathematical logic, this syllogism is written in this way (A-\u003e B) A-\u003e b.

  The conclusion is defined, as a rule, offices for implication. If implication (A-\u003e B) and its antecedent (a) are given, then we are entitled to join the reasoning (evidence) also a consequent implication (b). Sillogism is compulsory, making the arsenal of deductive means of evidence, that is, absolutely answering the requirements of mathematical reasoning.

  A large role in mathematical proof plays the deduction theorem - the general name for a number of theorems, the procedure of which ensures the ability to establish the proof of the implux: A-\u003e B, when the logical output of the formula B is obvious in formula A. In the most common option for statements (in classical, intuitionist and other types of mathematics) The Deduction Theorem approves the following. If the parcel system is given and the parcel A, from which, according to the rules, derived b g, A B (- the sign of the output), it follows that only from the parcels G, it is possible to obtain an offer A -\u003e B.

  We looked at the type that is direct evidence. At the same time, the logic uses the so-called indirect, there are not direct evidence that are deployed according to the following scheme. Without having, due to a number of reasons (the inaccessibility of the object of the study, the loss of the reality of its existence, etc.) is the possibility of direct evidence of the truth of any approval, the thesis, build antithesis. They are convinced that the antithesis leads to contradictions, and, it became false. Then, from the fact of the flaunt of the antithesis, they do on the basis of the law of the excluded third (A V) - the conclusion about the truth of the thesis.

  In mathematics, one of the forms of indirect evidence is widely used - proof of nasty. It is especially valuable and, in fact, is indispensable in the adoption of the fundamental concepts and provisions of mathematics, for example, the concept of relevant infinity, which is impossible in any other way.

  Operation of evidence from the contrary is presented in mathematical logic as follows. The sequence of formulas G and denial A (G, A) is given. If it follows from this b and its denial (G, A B, non-b), then we can conclude that the truth is derived from the sequence of formula G. In other words, the truth of the thesis follows from the falsity of the antitesis.

  References:

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

  2. L.D. Cudryavtsev, modern mathematics and her teaching, Moscow, science, 1985;

  3. O.I. Larichev, objective models and subjective solutions, Moscow, Science, 1987;

  4. A.Ya. Khalamizer, "Mathematics? - Funny! ", The publication of the author, 1989;

  5. P.K.Rashevsky, Riemanova Geometry and Tensor Analysis, Moscow, 3 Edition, 1967;

  6. V.E.gmurman, theory of probability and mathematical statistics, Moscow, high school, 1977;

  7. World Enternet network.

Mathematics 1. Where did the word mathematics come from? Who came up with mathematics? 3. Basic topics. 4. Definition 5. Etymology on the last slide.

Where did the word come from (go to the previous slide) Matemaa Tika from Greek - study, science) - science of structures, order and relationship that has historically developed on the basis of calculation, measurement and descriptions of the form of objects. Mathematical objects are created by the idealization of the properties of real or other mathematical objects and record these properties in the formal language.

Who came up with mathematics (go to the menu) the first mathematician is made to call Falez Miletsky, who lived in the VI century. BC e. , one of the so-called seven wise men of Greece. Be that as it may, but it was he who was the first to structure the entire knowledge base for this expense, which has long been formed within the world known to him. However, the author of the first treatise in mathematics reached us was Euclide (III century. BC). It also can be deservedly considered by the father of this science.

The main topics (go to the menu) to the region of mathematics include only those sciences in which either order, or measure, and absolutely not essentially, will these numbers, figures, stars, sounds, or something else, what this measure will find . Thus, there must be a certain overall science, explaining all related to the procedure and least, without entering the study of any private subjects, and this science should be called not foreign, but the old one who has already included in the use of universal mathematics.

The definition (go to the menu) on a classic mathematical analysis is based on a modern analysis, which is considered as one of the three main directions of mathematics (along with algebra and geometry). At the same time, the term "mathematical analysis" in the classical understanding is mainly used in curriculum and materials. In the Anglo-American tradition, the classic mathematical analysis corresponds to the courses program with the name "Calculus"

Etymology (Go to the menu) The word "mathematics" occurred from other. What does learning, knowledge, science, and others. -Grech, initially meaning susceptible, successive, later related to the study, subsequently related to mathematics. In particular, in Latin, means the art of mathematics. The term dr. -Grech. In the modern meaning of this word "mathematics" is already found in the writings of Aristotle (IV century BC) in the texts in Russian, the word "mathematics" or "mailematics" is found, at least from the XVII century, for example, Nicholas spa In the "Book of Chosen in Brief about nine Musakh and the Sedmi High Free Arts" (1672)