There is no clear answer to the question of what mathematics is even today, despite the fact that this science originated quite a long time ago, almost at the dawn of civilization. Throughout time, it has been enriched, becoming more and more established and updated as the laws of the surrounding world.
Thanks to the expansion and change of the multifaceted connections between mathematics and practice, humanity is provided with unique opportunity discover and use certain laws of nature. IN present time it is truly a mighty and powerful engine of technology and science.
Many people are interested in this, but to answer this question not easy. Of course, everyone is able to give their own answer, which will depend on the level of their mathematical knowledge. For the student high school this is a generalized name for arithmetic, algebra, geometry and the principles of analysis. For student technical university this is a science consisting of several dozen separate sections.
It should be noted that the number of such sections is constantly increasing over time, since, as modern mathematics develops, it is constantly enriched with new information. Well, for small child this science lies in the ability to count. However, our whole life is inextricably linked with solving various mathematical problems.
Similar to the definition of what mathematics is, there is no generally accepted clear definition of the subject of this science. In the past, it was believed that the solution to such problems was to measure quantities or numbers. But after some time, a definition of mathematics arose as the study of infinite quantities.
The modern world views mathematics as the science of mathematical structures. This term was introduced by the group French mathematicians, known to the world under the pseudonym Bourbaki.
This science is not an arbitrary creation of thought. It displays the objective world in a somewhat abstract way. Its studies are based on concepts obtained by abstracting from phenomena directly real world and, in addition, from previous abstractions.
The emergence of such abstractions is closely related to reality. Moreover, after deciding one or another mathematical problem its result is recorded and then applied to various phenomena, physical nature which differ significantly from each other.
For example, studying mathematics often comes down to solving specific tasks: bacterial growth, how it changes Atmosphere pressure, or how to determine speed radioactive decay. In this case, the solution to all these problems comes down to the same thing. differential equation.
Such abstractness is quite difficult not only to understand, but also to feel for an adult, and even more so for a student. This is why it is so important to make the study of mathematics accessible to everyone. And this requires maintaining a balance of specificity and abstraction, intuitiveness and rigor, without losing the ease of explanation complex concepts.
Of course, today it is difficult to find someone who would not have an idea of \u200b\u200bwhat mathematics is. But, as a rule, many people mistakenly believe that this is just arithmetic, which involves the study of numbers and certain operations with their help, such as multiplication or division.
But if you go deeper into this science, you can understand that in fact this concept is much more comprehensive. After all, mathematics is a unique way of describing the world and combining some of its parts with others. IN mathematical symbols, describing the Universe, the relationships between numbers are expressed.
But this is a separate question. Such a process requires patience, desire and attention. However, everything is not so complicated. It is common for everyone to excel in mathematics as “number sense” has been proven to be an innate ability.
Unfortunately, memorizing axioms, theorems and formulas will not give any results. The main thing is to understand the essence of mathematical theory and its laws. AND special attention requires the ability to draw conclusions from the statements that have been made.
Mathematics arose a very long time ago. The man collected fruits, dug up fruits, caught fish and stored it all for the winter. To understand how much food was stored, man invented counting. This is how mathematics began to emerge.
Then man began to engage in farming. It was necessary to measure plots of land, build houses, and measure time.
That is, it became necessary for a person to use the quantitative relation of the real world. Determine how much harvest has been harvested, what is the size of the building plot, or how large is the area of the sky with a certain number of bright stars.
In addition, man began to determine the shapes: a round sun, a square box, an oval lake, and how these objects are located in space. That is, a person became interested in the spatial forms of the real world.
Thus, the concept mathematics can be defined as the science of quantitative relationships and spatial forms of the real world.
Currently, there is not a single profession where one could do without mathematics. The famous German mathematician Carl Friedrich Gauss, who was called the “King of Mathematics,” once said:
“Mathematics is the queen of sciences, arithmetic is the queen of mathematics.”
The word “arithmetic” comes from the Greek word “arithmos” - “number”.
Thus, arithmetic is a branch of mathematics that studies numbers and operations on them.
IN primary school First of all, they study arithmetic.
How did this science develop, let's explore this question.
The birth period of mathematics
The main period of accumulation of mathematical knowledge is considered to be the time before the 5th century BC.
The first who began to prove mathematical propositions - ancient Greek thinker, who lived in the 7th century BC, presumably 625 - 545. This philosopher traveled to the countries of the East. Traditions say that he studied with the Egyptian priests and the Babylonian Chaldeans.
Thales of Miletus brought the first concepts of elementary geometry from Egypt to Greece: what is a diameter, what determines a triangle, and so on. He predicted solar eclipse, designed engineering structures.
During this period, arithmetic gradually developed, astronomy and geometry developed. Algebra and trigonometry are born.
Period of elementary mathematics
This period begins from VI BC. Now mathematics emerges as a science with theories and proofs. The theory of numbers, the doctrine of quantities and their measurement, appears.
Most famous mathematician this time is Euclid. He lived in the 3rd century BC. This man is the author of the first theoretical treatise on mathematics that has come down to us.
In the works of Euclid, the foundations of the so-called Euclidean geometry are given - these are axioms that rest on basic concepts, such as.
During elementary mathematics the theory of numbers was born, as well as the doctrine of quantities and their measurement. For the first time negative and irrational numbers.
At the end of this period, the creation of algebra as literal calculus is observed. The science of “algebra” itself appears among the Arabs as the science of solving equations. The word "algebra" translated from Arabic means "restoration", that is, transfer negative values to the other side of the equation.
Period of mathematics of variables
The founder of this period is considered to be Rene Descartes, who lived in the 17th century AD. In his writings, Descartes first introduced the concept of a variable quantity.
Thanks to this, scientists are moving from studying constant values to the study of dependencies between variables and to mathematical description movements.
This period was most vividly characterized by Friedrich Engels, in his writings he wrote:
“The turning point in mathematics was the Cartesian variable. Thanks to this, movement and thereby dialectics entered mathematics, and thanks to this, differential and integral calculus immediately became necessary, which immediately arises, and which was, by and large, completed and not invented by Newton and Leibniz.”
Period of modern mathematics
IN 20 years XIX century Nikolai Ivanovich Lobachevsky becomes the founder of the so-called non-Euclidean geometry.
From this moment the development of the most important sections begins modern mathematics. Such as probability theory, set theory, math statistics and so on.
All these discoveries and research find wide application in the most different areas Sciences.
And at present, the science of mathematics is rapidly developing, the subject of mathematics is expanding, including new forms and relationships, new theorems are being proven, and basic concepts are deepening.
MATHEMATICS – the science of quantitative relations and spatial forms of the real world; Greek word(mathematics) comes from the Greek word (mathema), meaning “knowledge”, “science”.
Mathematics arose in ancient times from the practical needs of people. Its content and character have changed throughout history and continue to change now. From the primary subject concepts of a positive whole number, as well as from the concept of a line segment as the shortest distance between two points, mathematics went through a long development path before it became an abstract science with specific methods research.
The modern understanding of spatial forms is very broad. It includes, along with geometric objects of three-dimensional space (straight line, circle, triangle, cone, cylinder, ball, etc.), also numerous generalizations - the concepts of multidimensional and infinite-dimensional space, as well as geometric objects in them, and much more. In the same way, quantitative relations are now expressed not only by positive integers or rational numbers, but also with the help complex numbers, vectors, functions etc. The development of science and technology forces mathematics to continuously expand its ideas about spatial forms and quantitative relationships.
The concepts of mathematics are abstracted from specific phenomena and objects; they are obtained as a result of abstraction from quality features, specific to of this circle phenomena and objects. This circumstance is extremely important for applications of mathematics. The number 2 is not inextricably linked with any specific subject content. It can refer to two apples, or two books, or two thoughts. It treats all of these and countless other objects equally well. Similar geometric properties the ball does not change because it is made of glass, steel or stearin. Of course, abstracting from the properties of an object impoverishes our knowledge about this object, about its characteristic material features. At the same time, it is precisely this distraction from special properties individual objects gives generality to concepts, makes possible use mathematics to the most diverse phenomena of material nature. Thus, the same laws of mathematics, the same mathematical apparatus can be quite satisfactorily applied to the description of natural phenomena, technical, as well as economic and social processes.
The abstractness of concepts is not exceptional feature mathematics; any scientific and general concepts carry in themselves an element of abstraction from the properties of specific things. But in mathematics the process of abstraction goes further than in natural sciences; In mathematics, the process of constructing abstractions at different levels is widely used. Yes, the concept groups arose by abstracting from some properties of the collection of numbers and other abstract concepts. Mathematics is also characterized by the method of obtaining its results. If a natural scientist constantly resorts to experience to prove his positions, then a mathematician proves his results only through logical reasoning. In mathematics, not a single result can be considered proven until it needs a logical proof, and this is even if special experiments provide confirmation of this result. At the same time the truth mathematical theories also passes the test of practice, but this test is special character: the basic concepts of mathematics are formed as a result of their long-term crystallization from particular needs of practice; the rules of logic themselves were developed only after thousands of years of observing the flow of processes in nature; The formulation of theorems and the formulation of problems in mathematics also arise from the needs of practice. Mathematics arose from practical needs, and its connections with practice became more and more diverse and deep over time.
In principle, mathematics can be applied to the study of any type of movement, a wide variety of phenomena. In fact, its role in various areas scientific and practical activities not the same. The role of mathematics in the development of modern physics, chemistry, many fields of technology, in general when studying those phenomena where even a significant abstraction from their specifically qualitative features allows one to quite accurately grasp the quantitative and spatial patterns inherent in them. For example - mathematical study movement celestial bodies, based on significant abstractions from their real features (bodies, for example, is considered material points), led and leads to an excellent coincidence with their real movement. On this basis, it is possible not only to pre-calculate celestial phenomena(eclipses, positions of planets, etc.), but also by deviations of true movements from calculated ones to predict the existence of planets that had not been observed before (Pluto was discovered in this way in 1930, Neptune in 1846). A smaller, but still significant place is occupied by mathematics in such sciences as economics, biology, and medicine. The qualitative uniqueness of the phenomena studied in these sciences is so great and so strongly influences the nature of their flow that mathematical analysis can still play only a subordinate role. Of particular importance for social and biological sciences acquires math statistics. Mathematics itself also develops under the influence of the requirements of natural science, technology, and economics. Yes for last years A number of mathematical disciplines were formed that arose on the basis of practical needs: information theory, game theory and etc.
It is clear that the transition from one stage of knowledge of phenomena to the next, more accurate one, places new demands on mathematics and leads to the creation of new concepts and new research methods. Thus, the requirements of astronomy, moving from purely descriptive knowledge to precise knowledge, led to the development of basic concepts trigonometry: in the 2nd century BC the ancient Greek scientist Hipparchus compiled tables of chords corresponding to modern tables of sines; ancient Greek scientists in the 1st century Menelaus and in the 2nd century Claudius Ptolemy created the foundations spherical trigonometry. Increased interest in the study of movement, brought about by the development of manufacturing, navigation, artillery, etc., led in the 17th century to the creation of the concepts mathematical analysis, the development of new mathematics. Widespread implementation mathematical methods in the study of natural phenomena (primarily astronomical and physical) and the development of technology (especially mechanical engineering) led to rapid development in the 18th and 19th centuries theoretical mechanics and theories differential equations. Development of ideas molecular structure matter has caused rapid development probability theory. Currently, we can trace the emergence of new directions using many examples. mathematical research. The successes must be recognized as particularly significant computational mathematics and computer technology and the transformations they produce in many branches of mathematics.
Historical sketch. In the history of mathematics, four periods with significant qualitative differences can be identified. It is difficult to divide these periods precisely, since each subsequent one developed within the previous one and therefore there were quite significant transitional stages when new ideas were just emerging and had not yet become guiding either in mathematics itself or in its applications.
1) The period of the birth of mathematics as an independent scientific discipline; the beginning of this period is lost in the depths of history; it lasted until approximately 6-5 centuries BC. e.
2) The period of elementary mathematics, mathematics of constant quantities; it continued until approximately the end of the 17th century, when the development of new, “higher” mathematics had progressed quite far.
3) Mathematics period variables; characterized by the creation and development of mathematical analysis, the study of processes in their movement and development.
4) Period of modern mathematics; characterized by conscious and systematic study possible types quantitative relationships and spatial forms. Geometry studies not only the real three-dimensional space, but also spatial forms similar to it. IN mathematical analysis variables are considered that depend not only on numeric argument, but also from some line (function), which leads to the concepts functionality And operator. Algebra turned into a theory of algebraic operations on elements of arbitrary nature. If only these operations could be performed on them. The beginning of this period can naturally be attributed to the 1st half of the 19th century.
IN Ancient world mathematical information were originally included as an integral part of the knowledge of priests and government officials. The supply of this information, as can be judged from already deciphered clay Babylonian tablets and Egyptian mathematical papyri, was relatively large. There is evidence that a thousand years before the ancient Greek scientist Pythagoras, in Mesopotamia not only was Pythagoras’ theory known, but the problem of finding all right triangles with integer sides was also solved. However, the overwhelming majority of documents of that time are collections of rules for the production of the simplest arithmetic operations, as well as for calculating the areas of figures and volumes of bodies. Tables have also been preserved various kinds to facilitate these calculations. In all manuals, the rules are not formulated, but are explained in frequent examples. Transformation of mathematics into a formalized science with an established deductive method construction occurred in Ancient Greece. There, mathematical creativity ceased to be nameless. Practical arithmetic and geometry in ancient Greece they had high level development. The beginning of Greek geometry is associated with the name of Thales of Miletus (late 7th century BC - early 6th century BC), who brought primary knowledge from Egypt. In the school of Pythagoras of Samos (6th century BC), the divisibility of numbers was studied, the simplest progressions were summed up, perfect numbers were studied, and introduced into consideration Various types averages (arithmetic mean, geometric mean, harmonic mean), found again Pythagorean numbers(triples of integers that can be sides right triangle). In the 5th-6th centuries BC. famous problems of antiquity arose - squaring a circle, trisection of an angle, doubling a cube, and the first irrational numbers were constructed. The first systematic textbook of geometry is attributed to Hippocrates of Chios (2nd half of the 5th century BC). The significant success of the Platonic school associated with attempts to rational explanation structure of matter in the Universe, - searching for all regular polyhedra. On the border of the 5th and 4th centuries BC. Democritus, based on atomic concepts, proposed a method for determining the volumes of bodies. This method can be considered a prototype of the infinitesimal method. In the 4th century BC. Eudoxus of Cnidus developed the theory of proportions. The 3rd century BC is characterized by the greatest intensity of mathematical creativity. (1st century of the so-called Alexandrian era). In the 3rd century BC. mathematicians such as Euclid, Archimedes, Apollonius of Perga, Eratosthenes worked; later – Heron (1st century AD) Diophantus (3rd century). In his Elements, Euclid collected and subjected to final logical revision the achievements in the field of geometry; at the same time, he laid the foundations of number theory. Archimedes' main achievement in geometry was the determination of various areas and volumes. Diophantus studied primarily the solution of equations in rational positive numbers. From the end of the 3rd century, the decline of Greek mathematics began.
Mathematics achieved significant development in ancient China and India. Chinese mathematicians are characterized by high computing techniques and an interest in the development of general algebraic methods. In the 2nd-1st centuries BC. "Mathematicians in Nine Books" was written. It contains the same extraction techniques square root, which are set out in modern school: methods for solving linear systems algebraic equations, an arithmetic formulation of the Pythagorean theorem.
Indian mathematics, the heyday of which dates back to the 5th-12th centuries, is credited with the use of modern decimal numbering, as well as zero to indicate the absence of units of a given rank, and the merit of a much wider development of algebra than that of Diophantus, operating not only with positive rational numbers, but also with negative and irrational numbers.
The Arab conquests led to Central Asia before Iberian Peninsula scientists during the 9th-15th centuries used Arabic. In the 9th century, the Central Asian scientist al-Khwarizmi first expounded algebra as independent science. During this period many geometric problems received an algebraic formulation. The Syrian al-Battani introduced trigonometric functions sine, tangent and cotangent. The Samarkand scientist al-Kashi (15th century) introduced into consideration decimals and gave a systematic presentation, formulated the Newton binomial formula.
Essentially new period in the development of mathematics began in the 17th century, when the idea of movement and change clearly entered mathematics. Consideration of variables and connections between them led to the concepts of functions, derivative and integral Differential calculus, Integral calculus, to the emergence of a new mathematical discipline - mathematical analysis.
From the end of the 18th century to the beginning of the 19th century, a number of significantly new features were observed in the development of mathematics. The most characteristic of them was an interest in a critical revision of a number of issues in the substantiation of mathematics. Vague ideas about infinitesimals have been replaced by precise formulations associated with the concept of limit.
In algebra in the 19th century, the question of the possibility of solving algebraic equations in radicals was clarified (Norwegian scientist N. Abel, French scientist E. Galois).
In the 19th and 20th centuries numerical methods mathematicians are growing into an independent branch - computational mathematics. Important applications for the new computer technology found a branch of mathematics that developed in the 19th and 20th centuries - mathematical logic.
The material was prepared by O. V. Leshchenko, a mathematics teacher.
Nikolay Evgenievich, I still don’t agree with you.
Let's look at your points.
First– presence of a cognizable object
Mathematics has many objects. The difference between mathematics and other sciences is that it constructs its own objects of study. Moreover, initially mathematics took objects from reality. For example, trade, exchange, accounting require accounting operations. But even then, people discovered that no matter what they count, the counting rules are the same, which made it possible to create arithmetic in which they count not apples and pears, but abstract units.
By the way, chemistry is somewhat reminiscent of mathematics in this regard. After all, chemists not only search for and study ready-made substances in nature, but also actively synthesize new compounds that do not exist in nature. How active can be judged from Beilstein's directory: in the first edition of 1881 there were only 1500 compounds, but now there are more than 10 million, and this is only organic chemistry. Chemists and mathematicians themselves construct the objects of their research. Only chemists have to use the set of elements given by nature and given by nature the same “rules of the game” that determine the direction chemical reactions and the possibility of the existence of certain connections, and mathematicians themselves set the “rules of the game” by defining systems of axioms.
Second– the truth of judgments about it, verified by experience.
If experience is understood narrowly, only how scientific experiment, then the scientific nature of many sciences will be in doubt. And mathematics finds experimental confirmation daily. Arithmetic, for sure. Or here's more complex example. An artist painting a landscape or still life from life consciously or unconsciously uses the strict laws of mathematics, namely, the laws of projection of three-dimensional objects onto a plane. If he does not violate them, then the result is a realistic image, and if it is incorrect, then something also turns out, but no longer similar to the original.
If by experience we mean not only a scientific experiment in a laboratory, but the total experience of mankind, then mathematics has experimentally proven the truth of its judgments many times.
Third– generality (universality) and obligatory nature of established patterns.
Mathematics is universal. I would say that it is ideally universal, and its laws are binding on everyone. So you took 88 rubles worth of goods from the store, they gave you a hundred at the checkout, and the cashier gives you 10 in change. You can’t say that this is how it should be. You say: “Where are the other two rubles?” If a design engineer violates the laws of mathematics when creating drawings, then his drawings will not be used to assemble what he intended. If a chemist makes a mistake with the coefficients in the reaction equation and uses them to calculate how many reagents he should take and how much product he will get, then he will not receive the expected amount of product, which the scales will tell him completely impartially.
Fourth– systematicity, sequence of concepts arising from each other.
Nothing compares to mathematics here. Whatever geometry we take: Euclidean, Riemann or Lobachevsky, then all the judgments in them follow from a system of axioms that are defined absolutely strictly.
Mathematicians, when evaluating their work, rely on their “taste”, they say: “ Beautiful solution! And this is already art.
I'm smiling at you. Still, the correctness of solving the problem is put in first place, and if the solution is also beautiful, i.e. compact, then this is only a plus, but it is not beautiful crucial. Aircraft designer Tupolev liked to say: “Ugly planes don’t fly,” but it just so happens that planes that perfectly comply with the laws of aerodynamics have beautiful shape. Ivan Efremov in the novel “The Razor’s Edge,” through the mouth of one of the characters, gives the following definition of beauty: “Beauty is highest degree expediency, the degree of harmonious correspondence and combination of contradictory elements in every device, in every thing, in every organism." I like it. So the criterion of beauty is not so subjective.
Mathematics is the queen of all sciences
Gauss Karl Friedrich
Mathematics is a science historically based on solving problems about quantitative and spatial relationships of the real world by idealizing the properties of objects necessary for this and formalizing these problems. The science concerned with the study of numbers, structures, spaces, and transformations.
Typically, people think that mathematics is just arithmetic, that is, the study of numbers and operations with them, such as multiplication and division. In fact, mathematics is much more than that. It is a way of describing the world and how one part of it fits together with another. The relationships of numbers are expressed in mathematical symbols that describe the Universe in which we live. Any normal child can excel in mathematics because "number sense" is an innate ability. True, this requires some effort and a little time.
The ability to count is not everything. The child needs to be able to express himself well in order to understand problems and make connections between facts that are stored in memory. In order to learn the multiplication table, you need memory and speech. This is why some people with brain damage find it difficult to multiply, although other types of calculation are not difficult for them.
Knowing geometry well and understanding shape and space requires other types of thinking. With the help of mathematics we solve problems in life, for example, dividing a chocolate equally or finding right size shoe Thanks to knowledge of mathematics, the child knows how to save pocket money and understands what can be bought and how much money he will then have. Mathematics is also the ability to count the right number of seeds and sow them in a pot, measure the right amount of flour for a cake or fabric for a dress, understand the score of a football game and many other everyday tasks. Everywhere: in a bank, in a store, at home, at work - we need the ability to understand and handle numbers, shapes and measures. Numbers are only part of the special mathematical language, A The best way to learn any language is to apply it. And it’s better to start from an early age.
About mathematics "smartly"
Typically, the idealized properties of the objects and processes under study are formulated in the form of axioms, then, according to strict rules of logical inference, other true properties (theorems) are derived from them. This theory together forms a mathematical model of the object under study. That. Initially based on spatial and quantitative relationships, mathematics receives more abstract relationships, the study of which is also the subject of modern mathematics.
Traditionally, mathematics is divided into theoretical, which performs an in-depth analysis of intra-mathematical structures, and applied, which provides its models to other sciences and engineering disciplines, some of which occupy a position bordering mathematics. In particular, formal logic can also be considered as part philosophical sciences, and as part of the mathematical sciences; mechanics - both physics and mathematics; Informatics, Computer techologies and algorithmics relate to both engineering and mathematical sciences etc. There are many in literature different definitions mathematics.
Sections of mathematics
- Mathematical analysis.
- Algebra.
- Analytic geometry.
- Linear algebra and geometry.
- Discrete Math.
- Mathematical logic.
- Differential equations.
- Differential geometry.
- Topology.
- Functional analysis and integral equations.
- Theory of functions of a complex variable.
- Partial differential equations.
- Probability theory.
- Math statistics.
- Theory of random processes.
- Calculus of variations and optimization methods.
- Computational methods, that is, numerical methods.
- Number theory.
Goals and methods
Mathematics studies imaginary, ideal objects and relationships between them using formal language. IN general case mathematical concepts and theorems do not necessarily correspond to anything in physical world. the main task applied mathematics - create a mathematical model that is sufficiently adequate to the subject under study real object. The task of a theoretical mathematician is to provide a sufficient set of convenient means to achieve this goal.
The content of mathematics can be defined as a system mathematical models and tools for creating them. The model of an object does not take into account all its features, but only those most necessary for the purposes of study (idealized). For example, studying physical properties orange, we can abstract from its color and taste and imagine it (even if not perfectly accurately) as a ball. If we need to understand how many oranges we will get if we add two and three together, then we can abstract from the shape, leaving the model with only one characteristic - quantity. Abstraction and establishment of connections between objects in the general view- one of the main directions of mathematical creativity.
Another direction, along with abstraction, is generalization. For example, generalizing the concept of “space” to a space of n-dimensions. The space R n, for n>3, is a mathematical invention. However, it is a very ingenious invention that helps to understand complex phenomena mathematically.
The study of intra-mathematical objects, as a rule, occurs with the help of axiomatic method: first, a list of basic concepts and axioms is formulated for the objects under study, and then meaningful theorems are obtained from the axioms using inference rules, which together form a mathematical model.
Video lecture by Smirnov S.K. and Yashchenko I.V. "What is mathematics":
