In the language, everything is subject to strict rules, often similar to mathematical ones. For example, the relations between phonemes resemble mathematical proportions in Russian [b] is related to [p] as [e] is to [t] (see Articulatory classification of sounds) By three members of such a "proportion" one can "calculate" the fourth one. In the same way, from one form of a word one can usually "calculate" its remaining forms, if all forms of some other "similar" words are known, such "calculations" are constantly made by children when they learn to speak (see Analogy in grammar) It is thanks to its strict rules that language can serve as a means of communication; if there were none, it would be difficult for people to understand each other

The similarity of these rules with mathematical ones is explained by the fact that mathematics ultimately originated from a language and itself is a special kind of language for describing quantitative relations and the mutual arrangement of objects. Such languages ​​are specially designed to describe some separate "parts" or aspects of reality. , are called specialized as opposed to universal ones, in which you can talk about anything. People have created many specialized languages, for example, the system of road signs, the language of chemical formulas, the notation of music. But among all these languages, the mathematical language is closest to the universal ones, because the relations that are expressed with its help are found everywhere - in nature, and in human life, and, moreover, these are the simplest and most important relations (more, less, closer, farther, inside, outside, between, immediately follows, etc. ), on the model of which people did not learn to talk about other, more complex

Many mathematical expressions resemble sentences in ordinary, natural language in their structure. For example, in such expressions as 2< 3 или 2 + 3=5, знаки < и = играют такую же роль, как глагол (сказуемое) в предложениях естественною языка, а роль знаков 2, 3, 5 похожа на роль существительного (подлежащего) Но особен но похожи на предложения естественного язы ка формулы математической логики - наукн, в которой изучается строение точных рассуж дений, в первую очередь математических, н при этом используются математические же методы Наука эта сравнительно молода она возникла в XIX в и бурно развивалась в течение первой половины XX в Примерно в то же время воз никла и развилась абстрактная алгебра - ма тематическая наука, изучающая всевозможные отношения и всевозможные действия, которые можно производить над чем угодно (а не только над числами и многочленами, как в элементарной алгебре, которую изучают в школе)

With the development of these two sciences, as well as some other branches of mathematics closely related to them, it became possible to use mathematical tools to study the structure of natural languages, and since the middle of this century, mathematical tools have actually been used for this purpose. Ready-made methods suitable for linguistic applications , did not exist in mathematics, they had to be created anew, and the methods of mathematical logic and abstract algebra served as a model for them, first of all, so a new science arose - mathematical linguistics And although this is a mathematical discipline, the concepts and methods developed by it are used in linguistics play an ever greater role in it, gradually becoming one of its main tools

Why are mathematical tools used in linguistics? Language can be imagined as a kind of mechanism by which the speaker transforms the “meanings” in his brain (i.e., his thoughts, feelings, desires, etc.) into “texts” (i.e., chains of sounds or written characters), and then transforms "texts" back into "meanings" It is convenient to study these transformations mathematically. Formal grammars serve for their study - complex mathematical systems that are not at all like ordinary grammars, in order to truly understand how they are arranged and learn how to use them. For example, it is desirable to first get acquainted with mathematical logic. But among the mathematical methods used in linguistics, there are some rather simple ones, for example, various ways of accurately describing the syntactic structure of a sentence using graphs.

A graph in mathematics is a figure consisting of points - they are called nodes of a graph - connected by arrows a graph whose nodes are people. When using graphs to describe the structure of a sentence, it is easiest to take words as nodes and draw arrows from subordinate words to subordinate ones. For example, for the sentence Volga flows into the Caspian Sea, we get the following graph:

The Volga flows into the Caspian Sea.

In formal grammars, it is customary to assume that the predicate subordinates not only all additions and circumstances, if any, but also the subject, because the predicate is the “semantic center” of the sentence: the whole sentence as a whole describes some “situation”, and the predicate, as a rule, , is the name of this situation, and the subject and objects are the names of its "participants". For example, the sentence Ivan bought a cow from Peter for a hundred rubles describes a "purchase" situation with four participants - a buyer, a seller, a product and a price, and the Volga sentence flows into the Caspian Sea - a "flow" situation with two participants. Consider, moreover, that the noun is subject to the preposition, because the verb controls the noun through the preposition. Already such a simple mathematical representation, which seems to add a little to the usual, “school” analysis of a sentence, allows us to notice and formulate many important patterns accurately.

It turned out that for sentences without homogeneous members and not complex, the graphs constructed in this way are trees. A tree in graph theory is a graph in which: 1) there is a node, and moreover, only one - called the root - which does not include one arrow (in a sentence tree, as a rule, the predicate serves as the root); 2) each node except the root contains exactly one arrow; 3) it is impossible, moving from some node in the direction of the arrows, to return to this node. Trees built for sentences as done in the example are called syntactic subordination trees. Some stylistic features of the sentence depend on the type of tree of syntactic subordination. In sentences of the so-called neutral style (see Functional styles of the language), as a rule, the law of projectivity is observed, which consists in the fact that if in the syntactic subordination tree all arrows are drawn above the straight line on which the sentence is written, then no two of them intersect (more precisely, you can draw them so that no two intersect) and not a single arrow passes over the root. With the exception of a small number of special cases, when there are some special words and phrases in the sentence (for example, complex forms of verbs: Children will play here), non-observance of the law of projectivity in a neutral sentence is a sure sign of insufficient literacy:

"The assembly discussed the proposals put forward by Sidorov."

In the language of fiction, especially in poetry, violations of the law of projectivity are permissible; there, onn most often give the sentence some special stylistic coloring, for example, solemnity, elation:

One more last word

And my chronicle is over.

(A.S. Pushkin)

or, conversely, ease, colloquialism:

Some Chef, literate, From the kitchen ran his To a tavern (he was pious rules)

(I.A. Krylov)

The stylistic coloring of the sentence is also associated with the presence in the tree of syntactic subordination of nests - sequences of arrows nested in each other and having no common ends (the number of arrows forming a nest is called its depth). A sentence in which the tree contains nests is felt as cumbersome, ponderous, and the depth of the nest can serve as a "measure of bulkiness". Compare, for example, the sentences:

A writer (whose tree contains slots of depth 3) has arrived and is collecting the information needed for a new book.

A writer has arrived, collecting information needed for a new book (whose tree has no nests, more precisely, there are no nests of depth greater than 1).

The study of the features of trees of syntactic subordination can give a lot of interesting things for studying the individual style of writers (for example, violations of projectivity are less common in A. S. Pushkin than in I. A. Krylov).

With the help of trees of syntactic subordination, syntactic homonymy is studied - a phenomenon consisting in the fact that a sentence or phrase has two different meanings - or more - but not due to the ambiguity of its constituent words, but due to differences in the syntactic structure. For example, the sentence Schoolchildren from Kostroma went to Yaroslavl can mean either “Kostroma schoolchildren went from somewhere (not necessarily from Kostroma) to Yaroslavl”, or “some (not necessarily Kostroma) schoolchildren went from Kostroma to Yaroslavl”. The first meaning corresponds to the tree Schoolchildren from Kostroma went to Yaroslavl, to the second - Schoolchildren from Kostroma went to Yaroslavl.

There are other ways to represent the syntactic structure of a sentence using graphs. If we represent its structure with the help of a tree, the constituent nodes will be phrases and words; arrows are drawn from larger phrases to the smaller ones contained in them and from phrases to the words contained in them.

The use of exact mathematical methods makes it possible, on the one hand, to penetrate deeper into the content of the "old" concepts of linguistics, on the other hand, to explore the language in new directions that would have been difficult to even outline before.

Mathematical methods of language research are important not only for theoretical linguistics, but also for applied linguistic problems, especially those related to the automation of individual language processes (see Automatic translation), automatic search for scientific and technical books and articles on a given topic, and etc. The technical basis for solving these problems are electronic computers. To decide! any task on such a machine, you must first write a program that clearly and unambiguously determines the order of operation of the machine, and in order to write a program, you must present the initial data in a clear and precise form. In particular, to compile programs that solve linguistic problems, you need an accurate description of the language (or at least those aspects of it that are important for this task) - and it is mathematical methods that make it possible to build such a description.

Not only natural, but also artificial languages ​​(see Artificial languages) can be explored with the help of tools developed by mathematical linguistics. Some artificial languages ​​can be completely described by these means, which is not possible and, presumably, will never be possible for natural languages, which are incomparably more complex. In particular, formal grammars are used in the construction, description and analysis of the input languages ​​of computers, on which the information entered into the machine is recorded, and in solving many other problems related to the so-called communication between a person and a machine (all ethnic problems are reduced to the development of some artificial languages)

Gone are the days when a linguist could do without knowledge of mathematics. Every year this ancient science, which combines the features of the natural sciences and the humanities, becomes more and more necessary for scientists involved in the theoretical study of language and the practical application of the results of this study. Therefore, in our time, every student who wants to thoroughly get acquainted with linguistics or is going to study it himself in the future should pay the most serious attention to the study of mathematics.

Mathematics is a language.

David Gilbert

Mathematics is a language. Language is needed for communication, in order to convey the meaning that arose from one person to another person. For this, sentences of this language, compiled according to certain rules, serve. Why do people learn different languages, what does this give them besides the opportunity to communicate in other countries? The answer is that each language has words that do not exist in other languages, therefore it allows you to describe (and see) such phenomena that a person would never see if he did not know this language. Knowing one more language allows you to get another, different from others, vision of the world. (The Eskimos have 20 different words for snow in their language, unlike Russian, where there is only one. Although, for example, in Russian there is such a word “nast” to refer to a crust that forms on the snow after a thaw, followed immediately by frost. There are probably other words describing special states of snow.)

Mathematics as the language of science

Representing a type of formal knowledge, mathematics occupies a special place in relation to the factual sciences. It turns out to be well suited for the quantitative processing of any scientific information, regardless of its content. Moreover, in many cases mathematical formalism turns out to be the only possible way to express the physical characteristics of phenomena and processes, since their natural properties and especially relationships are not directly observable. Let's say, how to describe in physical terms gravity, the effects of electromagnetism, etc.? They can only be represented mathematically as certain numerical ratios in laws fixed by quantitative indicators. Modern science in the face of quantum mechanics and a little earlier the theory of relativity only added to the abstractness of theoretical objects, completely depriving them of visibility. It only remains to appeal to mathematics. L. Landau once declared that it is not at all necessary for a modern physicist to know physics, it is enough for him to know mathematics.

The considered circumstance also puts forward mathematics to the role of the language of science. Perhaps for the first time this was clearly heard by G. Galileo, one of the decisive characters in the creation of mathematical natural science, which has dominated for more than three hundred years. Galileo wrote: “Philosophy is written in a majestic book (I mean the Universe), which is constantly open to our gaze, but only those who first learned to comprehend its language and interpret the signs with which it is written can understand it. It is written in the language of mathematics ".

As the abstraction of natural science grew, this idea found an ever wider implementation, and on the slope of the 19th century. century has already entered the practice of scientific research as a kind of methodological maxim. That is how the words of the famous American theoretical physicist D. Gibbs sounded when once, when discussing the issue of teaching English at school, he, as usual silent at such meetings, unexpectedly said: "Mathematics is also a language." They say that you are here all about English and about English, mathematics is also a language. The expression has become catchy. And now, after that, the English physical chemist, Nobel Prize winner (received, by the way, together with our N. Semenov) Hanschelwood announces that scientists should know mathematics like their native language.

Characteristic is the reasoning of the remarkable domestic researcher V. Nalimov, who worked in the field of scientometrics, the theory of mathematical experiment, who proposed probabilistic models of the language. Good science, he writes, speaks the language of mathematics. For some reason, we humans are arranged in such a way that we perceive the Universe through space, time and number. This means that we are prepared to turn to mathematics, prepared by the evolution of the living, that is, a priori. Trying to reveal the secret underlying reason of the mathematical power over the scientist, Nalimov remarks further: “I am often accused of using mathematics in the study of consciousness, linguistics, biological evolution. But is there mathematics as such? Hardly. I use mathematics as an Observer. it is more convenient to think, otherwise I cannot. Space, time, number and logic are the prerogative of the Observer."

The situation sometimes develops in science in such a way that without the use of an appropriate mathematical language, it is impossible to understand the nature of physical, chemical, etc. process is not possible. It is no coincidence that P. Dirac recognized that each new step in the development of physics requires ever higher mathematics. Such a fact. Creating a planetary model of the atom, the famous English physicist of the XX century. E. Rutherford experienced mathematical difficulties. At first, his theory was not accepted: it did not sound convincing, and the reason for this was Rutherford's ignorance of the theory of probability, on the basis of the mechanism of which it was only possible to understand the model representation of atomic interactions. Realizing this, already by that time an outstanding scientist, the owner of the Nobel Prize, enrolled in the seminar of the mathematician Professor Lamb and for two years, together with the students, attended a course and worked out a workshop on the theory of probability. Based on it, Rutherford was able to describe the behavior of the electron, giving his structural model convincing accuracy and gaining recognition.

This begs the question, what is so mathematical in objective phenomena, thanks to which they can be described in the language of mathematics, in the language of quantitative characteristics? These are homogeneous units of matter distributed in space and time. Those sciences that have gone farther than others towards the isolation of homogeneity, and turn out to be better suited for the use of mathematics in them. In particular, most of all - physics. V. Lenin, noting the serious successes of natural science and, above all, physical knowledge at the turn of the 19th-20th centuries, saw one of the reasons precisely in the fact that nature was brought closer "to such homogeneous elements of matter, the laws of motion of which allowed mathematical processing."

Physics is followed by chemical disciplines, where they also operate with atoms and molecules, and where many homogeneous units of matter and field flow from physics by the method of "paradigm grafting" along with the corresponding methods of research. Mathematical chemistry is becoming more and more established. Mathematical language has so far entered biology much weaker, since the units of the substrate have not yet been singled out here, except for genetics. The humanitarian sections of scientific knowledge are even less prepared for this. A breakthrough is observed only in linguistics with the creation and successful development of mathematical linguistics, as well as in logic (mathematical logic). The sciences of society, of course, are difficult to quantify due to the specific nature of the phenomena and processes taking place here, since they are marked by originality and uniqueness. L. Tolstoy made an interesting attempt to identify homogeneous elements in historical processes. In the novel "War and Peace", the writer introduces the concept of "differential of historical action" and explains that only by assuming an infinitely small unit - the differential of history, that is, "homogeneous inclinations of people", and then learning to integrate them (taking the sums of these infinitesimal ones), one can hope to understand history.

However, such homogeneity turns out to be very conditional, since "people's attractions" are always colored by individual uniqueness, psychologically variable, which will impose perturbations that are difficult to take into account on the postulated homogeneity. In general, each event in the history of society is rather peculiar and cannot be leveled into homogeneous units. A good illustration of this is one reasoning of A. Poincaré. Once he read from a famous English historian of the XIX century. T. Carlyle's statement: "John the Landless passed here, and this fact is dearer to me than all historical theories." Poincaré remarked on this: “This is the language of a historian. A physicist would not say so. A physicist would say: “John Landless passed here, and it makes no difference to me, because he will not pass here again.” The mathematician Poincaré’s objection is understandable: a physicist needs repeatability only then will he be able to deduce laws.On the contrary, the uniqueness of the event is the material that feeds the historical description.

Note that the understanding of homogeneity as a condition for the applicability of the mathematical description of phenomena came to science rather late. Until a certain time, it was considered impossible to digress from objective meanings in order to move on to numerical characteristics. So, even G. Galileo, one of the founders of mathematical natural science, did not want to accept the speed of uniform rectilinear motion in the form. He believed that the action of dividing the path by time is physically incorrect, since it was necessary to divide kilometers, meters, etc. for hours, minutes, etc. That is, he considered it unacceptable to carry out the division operation with qualitatively inhomogeneous quantities. For Galileo, the velocity equation had a purely meaningful meaning, but by no means a mathematical relation of quantities. And only centuries later, Academician of the St. Petersburg Academy of Sciences L. Euler, introducing the formula into scientific use, explained that we do not divide the path into time and, therefore, not kilometers or meters into hours or minutes, but one quantitative dimension into another, one abstract numerical value to another. As M. Rozov remarks, by this act Euler performed a sign-subject inversion, translating a meaningful description into an algebraically abstract one 63 . That is, Euler accepts qualitatively given kilometers, meters, hours, minutes, etc. as an abstract measure for units of measurement, and then we already have, say, not 10 meters, but 10 abstract units, which we divide, let's say, not by 2 seconds, but into two equally abstract units. With this technique, we manage to invert qualitatively heterogeneous objects that have spatial and temporal certainty into homogeneity, which allows us to apply the mathematical quantitative language of description.

Shapovalova Anna

The paper tells about the development and universality of the language of mathematics.

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Section Mathematics

"The Language of Mathematics"

Report.

Made by Anna Shapovalova

scientific adviser

Romanchuk Galina Anatolyevna

mathematics teacher of the highest qualification category.

Introduction.

Seeing in the office G. Galileo's statement “The Book of Nature is written in the language of mathematics”, I became interested: what kind of language is this?

It turns out that Galileo was of the opinion that nature was created according to a mathematical plan. He wrote: “The philosophy of nature is written in the greatest book ... but only those who first learn the language and comprehend the writings with which it is inscribed can understand it. And this book is written in the language of mathematics.”

And so, in order to find the answer to the question about the mathematical language, I studied a lot of literature, materials from the Internet.

In particular, I found on the Internet “History of Mathematics” by Stroyka D.Ya., where I learned the stages of development of mathematics and mathematical language.

I tried to answer the questions:

1. how did mathematical language originate;
2. what is a mathematical language;
3. where it is distributed;
4. Is it really universal?

I think it will be interesting not only for me, because We all use the language of mathematics.

Therefore, the purpose of my work was to study such a phenomenon as "mathematical language" and its distribution.

Naturally, the object of study will be mathematical language.

I will make an analysis of the application of mathematical language in various fields of science (natural science, literature, music); in everyday life. I will prove that this language is indeed universal.

Brief history of the development of mathematical language.

Mathematics is convenient for describing the most diverse phenomena of the real world and thus can perform the function of a language.

The historical components of mathematics - arithmetic and geometry - grew, as you know, from the needs of practice, from the need to inductively solve various practical problems of agriculture, navigation, astronomy, tax collection, debt collection, sky observation, crop distribution, etc. When creating the theoretical foundations of mathematics, the foundations of mathematics as a scientific language, the formal language of sciences, and various theoretical constructions, various generalizations and abstractions coming from these practical problems and their tools have become important elements.

The language of modern mathematics is the result of its long development. In the period of its birth (before the 6th century BC), mathematics did not have its own language. In the process of the formation of writing, mathematical signs appeared to denote some natural numbers and fractions. The mathematical language of ancient Rome, including the system of notation for integers that has survived to this day, was poor:

I, II, III, IV, V, VI, VII, VIII, IX, X, XI,..., L,..., C,..., D,..., M.

The unit I symbolizes the notch on the staff (not the Latin letter I - this is a later rethinking). The effort that goes into each notch, and the space it occupies on, say, a shepherd's stick, makes it necessary to move from a simple numbering system

I, II, III, IIII, IIIII, IIIIII, . . .

to a more complex, economical system of "names" rather than symbols:

I=1, V=5, X=10, L=50, C=100, D=500, M=1000.

In Russian, numbers were written in letters with a special sign "titlo"

The first nine letters of the alphabet were units, the next 9 were tens, and the last 9 were hundreds.

To designate large numbers, the Slavs came up with their own original way: ten thousand - darkness, ten topics - legion, ten legions - leodr, ten leods - raven, ten - raven - deck. And there is nothing more for the human mind to understand, i.e. there are no names for large numbers.

In the next period of development of elementary mathematics (VI century BC - XVII century AD), the main language of science was the language of geometry. With the help of segments, figures, areas and volumes, objects were depicted that were accessible to the mathematics of that time. That is why the famous "Principles" of Euclid (III century BC) were subsequently perceived as a geometric work, although most of them are a presentation in the geometric language of the principles of algebra, number theory and analysis. However, the possibilities of the geometric language turned out to be insufficient to ensure the further development of mathematics, which led to the emergence of the symbolic language of algebra.

The penetration of the set-theoretical concept into science (the end of the 19th century) begins the period of modern mathematics. The construction of mathematics on a set-theoretic basis caused a crisis of its foundations (beginning of the 20th century), since contradictions were discovered in set theory. Attempts to overcome the crisis stimulated research into the problems of proof theory, which, in turn, required the development of new, more precise means of expressing the logical component of a language. Under the influence of these needs, the language of mathematical logic, which appeared in the middle of the 19th century, was further developed. At present, it penetrates into various branches of mathematics and becomes an integral part of its language.

The basis for the development of mathematics in the 20th century was the formed formal language of numbers, symbols, operations, geometric images, structures, relationships for the formal-logical description of reality - that is, the formal, scientific language of all branches of knowledge, primarily natural sciences, was formed. This language is successfully used at the present time in other, "non-natural science" areas.

The language of mathematics is an artificial, formal language, with all its shortcomings (for example, low figurativeness) and advantages (for example, brevity of description).

The development of an artificial language of symbols and formulas was the greatest achievement of science, which largely determined the further development of mathematics. At present, it becomes obvious that mathematics is not only a set of facts and methods, but also a language for describing the facts and methods of various fields of science and practice.

Spread of mathematical language

Thus, a mathematical language is the totality of all means by which mathematical content can be expressed. Such means include logical-mathematical symbols, graphic diagrams, geometric drawings, a system of scientific terms along with elements of a natural (ordinary) language.

Mathematical language, unlike natural language, is symbolic, although natural language also uses certain symbols - letters and punctuation marks. There are significant differences in the use of symbols in mathematical and natural languages. In mathematical language, one sign denotes what in natural language is denoted by a word. This achieves a significant reduction in the "length" of linguistic expressions.

Application of mathematical language in natural science.

“... All laws are derived from experience. But to express them, a special language is needed. Everyday language is too poor, besides, it is too indefinite to express such precise and subtle relationships rich in content. This is the first reason why the physicist cannot dispense with mathematics; it gives him the only language in which he is able to express himself." "The mechanism of mathematical creativity, for example, does not differ significantly from the mechanism of any other kind of creativity." (A. Poincaré).

Mathematics is the science of the quantitative relations of reality. "Genuinely realistic mathematics is a fragment of the theoretical construction of the same real world." (G. Weyl) It is an interdisciplinary science. Its results are used in natural science and social sciences. The role of mathematics and the language it speaks in modern natural science is manifested in the fact that a new theoretical interpretation of a phenomenon is considered complete if it is possible to create a mathematical apparatus that reflects the basic laws of this phenomenon. In many cases, mathematics plays the role of a universal language of natural science, specially designed for concise and precise recording of various statements.

In natural science, it is increasingly using mathematical language to explain natural phenomena, these are:

1. quantitative analysis and quantitative formulation of qualitatively established facts, generalizations and laws of specific sciences;
2. building mathematical models and even creating such areas as mathematical physics, mathematical biology, etc.;

Considering a mathematical language that differs from a natural language, where, as a rule, they use concepts that characterize certain qualities of things and phenomena (therefore they are often called qualitative). This is where the knowledge of new objects and phenomena begins. The next step in the study of the properties of objects and phenomena is the formation of comparative concepts, when the intensity of any property is displayed using numbers. Finally, when the intensity of a property or quantity can be measured, i.e. represented as the ratio of a given quantity to a homogeneous quantity taken as a unit of measurement, then quantitative, or metric, concepts arise.

Let's remember the cartoon "38 parrots". Fragment of the cartoon

The boa constrictor was measured by monkeys, elephants and parrots. Since the values ​​\u200b\u200bare heterogeneous, the boa constrictor concludes: “And in parrots, then I am longer ...”

But if its length is translated into mathematical language; to translate the measurements into the same-named values, then the conclusion is completely different: that in monkeys, that in elephants, that in parrots, the length of the boa constrictor will be the same.

The advantages of the quantitative language of mathematics over natural language are as follows:

Such language is very short and precise. For example, to express the intensity of any property using ordinary language, you need several dozen adjectives. When numbers are used for comparison or measurement, the procedure is simplified. By constructing a scale for comparison or choosing a unit of measure, all relationships between quantities can be translated into the exact language of numbers. With the help of the mathematical language (formulas, equations, functions, and other concepts), it is possible to express much more accurately and briefly the quantitative relationships between the most diverse properties and relationships that characterize the processes that are studied in natural science.

Here the mathematical language performs two functions:

1. with the help of mathematical language, quantitative patterns are precisely formulated that characterize the phenomena under study; the exact formulation of laws and scientific theories in the language of mathematics makes it possible, when deriving consequences from them, to apply a rich mathematical and logical apparatus.

All this shows that in any process of scientific knowledge there is a close relationship between the language of qualitative descriptions and the quantitative mathematical language. This relationship is concretely manifested in the combination and interaction of natural science and mathematical research methods. The better we know the qualitative features of phenomena, the more successfully we can use quantitative mathematical methods of research for their analysis, and the more advanced quantitative methods are used to study phenomena, the more fully their qualitative features are known.

Ex. A cartoon about characters already familiar to us: a boa constrictor, a monkey, a parrot and an elephant calf.

A bunch of nuts is a lot. And "a lot" is how much?

Mathematical language plays the role of a universal language, specially designed for concise and precise writing of various statements. Of course, everything that can be described in the language of mathematics can be expressed in ordinary language, but then the explanation may be too long and confusing.

2. serves as a source of models, algorithmic schemes for displaying connections, relationships and processes that make up the subject of natural science. On the one hand, any mathematical scheme or model is a simplifying idealization of the object or phenomenon under study, and on the other hand, simplification allows you to clearly and unambiguously reveal the essence of the object or phenomenon.

Since certain general properties of the real world are reflected in mathematical formulas and equations, they are repeated in its different areas.

Here are the tasks about completely different things.

1. There were 48 cars in two garages. One garage has twice as many cars as the other. How many cars are in the first garage?
2. In the poultry yard there were half as many geese as ducks. How many geese were there if there were 48 birds in the poultry yard.

You can come up with a lot of such problems, but they are all described using a mathematical one model:

2x+x=48., understandable to all mathematicians of the world.

Mathematical language in literature.

Since the language of mathematics is universal, it is not in vain that the expression “believed harmony by algebra” exists.

Here are some examples.

Meters and dimensions of the verse.

Verse size	Stressed syllables	Mathematical dependency	Mat. model
Dactyl	1,4,7,10…		Arith progression
Anapaest	3,6,9,12…		Arith progression
Amphibrachius	2,5,8,11…		Arith progression
Yamb	2,4,6,8,10…		Arith progression
Chorey	1,3,5,7…		Arith progression

In literature, there is a technique called "euphonics", where the sonority of a poem is described with the help of mathematical language.

Listen to two excerpts from the poems.

Dactyl - 1,4,7,10,13…

How good are you, O night sea, -

It's radiant here, it's gray-dark there...

In the moonlight, as if alive,

It walks and breathes and shines.

Anapaest - 3,6,9,12 ...

Sounded over a clear river,

Rang out in the faded meadow,

It swept over the mute grove,

It lit up on the other side.

If we take the entire sound composition as a whole, then the picture will be as follows (in%):

Here is their description using mathematical language.

Mathematical language in music.

The musical system was based on two laws, which bear the names of two great scientists - Pythagoras and Archytas.

1. Two sounding strings determine consonance if their lengths are related as integers forming a triangular number 10=1+2+3+4, i.e. like 1:2, 2:3, 3:4. Moreover, the smaller the number n in relation to n/(n+1) (n=1,2,3), the more consonant the resulting interval.

2. Oscillation frequency w sounding string is inversely proportional to its length l

w = a/l , (a is the coefficient characterizing the physical properties of the string).

Interval coefficients and their corresponding intervals in the Middle Ages were called perfect consonances and received the following names: octave ( w 2 / w 1 \u003d 2/1, l 2 / l 1 \u003d 1/2); fifth (w 2 / w 1 \u003d 3/2, l 2 / l 1 \u003d 2/3); quart (w 2 / w 1 \u003d 4/3, l 2 / l 1 \u003d 3/4).

(3/2) 1 \u003d 3/2 - salt, (3/2) 2: 2 \u003d 9/8 - re, (3/2) 3: 2 \u003d 27/16 - la, (3/2) 4: 2 2 \u003d 81/64 - mi, (3/2) 5: 2 2 \u003d 243/128 - si, (3/2) -1: 2 \u003d 4/3 - fa.

To construct a gamma, it turns out that it is much more convenient to use the logarithms of the corresponding frequencies:

log 2 w 0 , log 2 w 1 ... log 2 w m

So, music written in mathematical language is understandable to all musicians regardless of their spoken language.

In everyday life

Without noticing it ourselves, we constantly operate with mathematical terms: numbers, concepts (area, volume), ratio.

We constantly read in mathematical language and say: determining the mileage of the car, reporting the price of the goods, time; describing the dimensions of the room, etc.

In the youth environment, the expression “parallel to me” has now appeared - which means “I don’t care, it doesn’t concern me”

And this is associated with parallel lines, probably because they do not intersect, so this problem “does not intersect” with me. That is, it does not concern me.

In contrast, the answer follows: “So I will make it perpendicular to you.”

And again: the perpendicular intersects with the line, i.e. it means that this problem will concern you - will intersect with you.

So the language of mathematics penetrated into youth slang.

Versatility.

If you see this phrase written in different languages, you will not understand what it is about, but if you write it in the language of mathematics, it will immediately become clear to everyone.

Deux fois trios font six (French)

Two multiply three equals six (English)

Zwei mal drei ist secks (German)

Tlur shche pshteme mekhu hy (Adyghe)

2∙3=6

Conclusion.

“If you can measure and express in numbers what you are talking about, then you know something about it. If you cannot do this, then your knowledge is poor. They represent the first steps of research, but they are not real knowledge." Lord Kelvin

The Book of Nature is written in the language of mathematics. Everything essential in nature can be measured, turned into numbers and described mathematically. Mathematics is a language that allows you to create a concise model of reality; it is an organized statement that makes it possible to quantitatively predict the behavior of objects of any nature. The greatest discovery of all time is that information can be written down using a mathematical code. After all, formulas are designations of words with signs, which leads to huge savings in time, space, and symbols. The formula is compact, clear, simple, rhythmic.

Mathematical language is potentially the same for all worlds. The orbit of the Moon and the trajectory of a stone falling on the Earth are special cases of the same mathematical object - an ellipse. The universality of differential equations makes it possible to apply them to objects of different nature: string vibrations, the process of propagation of an electromagnetic wave, etc.

Mathematical language today describes not only the properties of space and time, particles and their interaction, physical and chemical phenomena, but also more and more processes and phenomena in the fields of biology, medicine, economics, computer science; mathematics is widely used in applied fields and engineering.

Mathematical knowledge and skills are necessary in almost all professions, first of all, of course, in those related to the natural sciences, technology and economics. Mathematics is the language of natural science and technology, and therefore the profession of a natural scientist and engineer requires a serious mastery of many professional information based on mathematics. Galileo said this very well: ``Philosophy (we are talking about natural philosophy, in our modern language, about physics) is written in a majestic book that is constantly open to your gaze, but only those who first learn to understand its language and interpret it can understand it. signs with which it is written. It was written in the language of mathematics. "" But now the need for the application of mathematical knowledge and mathematical thinking to a doctor, linguist, historian is undeniable, and it is difficult to cut off this list, the knowledge of mathematical language is so important.

Understanding and knowledge of the mathematical language is necessary for the intellectual development of the individual. In 1267, the famous English philosopher Roger Bacon said: ``He who does not know the language of mathematics cannot know any other science and cannot even show his ignorance.'

With the development of knowledge over the past hundreds of years, the effectiveness of mathematical methods for describing the surrounding world and its properties, including the structure, transformation and interaction of matter, has become more and more obvious. Many systems for describing the phenomena of gravitation, electromagnetism, as well as the forces of interaction between elementary particles were built - all the fundamental forces of nature known to science; particles, materials, chemical processes. At present, the mathematical language is in fact the only effective language in which this description is made, which raises a natural question whether this circumstance is not a consequence of the initially mathematical nature of the world around us, which would thus be reduced to the action of purely mathematical laws (“substance disappears, only equations remain.

Bibliography:

1. Languages ​​of mathematics or mathematics of languages. Report at the conference within the framework of the "Days of Science" (organizer - the Dynasty Foundation, St. Petersburg, May 21–23, 2009)
2. Perlovsky L. Consciousness, language and mathematics. "Russian Journal"[email protected]
3. Green F. Mathematical harmony of nature. Magazine New Faces #2 2005
4. Bourbaki N. Essays on the history of mathematics, M.: IL, 1963.
5. Stroyk D.Ya "History of Mathematics" - M .: Nauka, 1984.
6. Euphonics of "The Stranger" by A.M.Finkel Publication, preparation of the text and comments by Sergei GINDIN
7. Euphonics of the "Winter Road" by A.S. Pushkin. Supervisor Khudayeva L.G. - teacher of the Russian language

Section Mathematics

"The Language of Mathematics"

Made by Anna Shapovalova

scientific adviser

mathematics teacher of the highest qualification category.

Introduction.

When I saw G. Galileo's statement “The Book of Nature is written in the language of mathematics” in the office, I became interested: what kind of language is this?

It turns out that Galileo was of the opinion that nature was created according to a mathematical plan. He wrote: “The philosophy of nature is written in the greatest book ... but only those who first learn the language and comprehend the writings with which it is inscribed can understand it. And this book is written in the language of mathematics.”

And so, in order to find the answer to the question about the mathematical language, I studied a lot of literature, materials from the Internet.

In particular, I found the History of Mathematics on the Internet, where I learned the stages in the development of mathematics and the mathematical language.

I tried to answer the questions:

How did mathematical language originate?

What is mathematical language?

Where is it distributed?

Is it really universal?

I think it will be interesting not only for me, because we all use the language of mathematics.

Therefore, the purpose of my work was to study such a phenomenon as "mathematical language" and its distribution.

Naturally, the object of study will be mathematical language.

I will make an analysis of the application of mathematical language in various fields of science (natural science, literature, music); in everyday life. I will prove that this language is indeed universal.

Brief history of the development of mathematical language.

Mathematics is convenient for describing the most diverse phenomena of the real world and thus can perform the function of a language.

The historical components of mathematics - arithmetic and geometry - grew, as is known, from the needs of practice, from the need to inductively solve various practical problems of agriculture, navigation, astronomy, tax collection, debt collection, sky observation, crop distribution, etc. When creating the theoretical foundations of mathematics, the foundations of mathematics as a scientific language, the formal language of sciences, various theoretical constructions have become important elements of various generalizations and abstractions emanating from these practical problems, and their tools.

The language of modern mathematics is the result of its long development. During its inception (before the 6th century BC), mathematics did not have its own language. In the process of the formation of writing, mathematical signs appeared to denote some natural numbers and fractions. The mathematical language of ancient Rome, including the system of notation for integers that has survived to this day, was poor:

I, II, III, IV, V, VI, VII, VIII, IX, X, XI,..., L,..., C,..., D,..., M.

The unit I symbolizes the notch on the staff (not the Latin letter I - this is a later rethinking). The effort that goes into each notch, and the space it occupies on, say, a shepherd's stick, makes it necessary to move from a simple numbering system

I, II, III, IIII, IIIII, IIIIII, . . .

to a more complex, economical system of "names" rather than symbols:

I=1, V=5, X=10, L=50, C=100, D=500, M=1000.

2. Perlovsky L. Consciousness, language and mathematics. "Russian Journal" *****@***ru

3. Green F. Mathematical harmony of nature. Magazine New Faces #2 2005

4. Bourbaki N. Essays on the history of mathematics, Moscow: IL, 1963.

5. Stroyk D. I "History of Mathematics" - M .: Nauka, 1984.

6. Euphonics of "The Stranger" by A. M. FINKEL Publication, preparation of the text and comments by Sergei GINDIN

7. Euphonics of the "Winter Road". Scientific adviser - teacher of the Russian language

Mathematics 7th grade.

Theme of the lesson: "What is a mathematical language."

Fedorovtseva Natalya Leonidovna

Cognitive UUD: develop the ability to translatemathematical word expressions into literal expressions and explain the meaning of literal expressions

Communicative UUD: cultivate a love for mathematics, participate in a collective discussion of problems, respect for each other, the ability to listen, discipline, independence of thought.Regulatory UUD: the ability to process information and translate the problem from the native language into mathematical.Personal UUD: to form learning motivation, adequate self-esteem, the need to acquire new knowledge, to cultivate responsibility and accuracy.
Work with text. In mathematical language, many statements look clearer and more transparent than in ordinary language. For example, in ordinary language they say: "The sum does not change from a change in the places of the terms." Hearing this, the mathematician writes (or speaks)a + b \u003d b + a.He translates the stated statement into a mathematical one, which uses different numbers, letters (variables), signs of arithmetic operations, and other symbols. The notation a + b = b + a is economical and convenient to use.Let's take another example. In ordinary language they say: "To add two ordinary fractions with the same denominators, you need to add their numerators, and leave the denominator unchanged."

The mathematician performs "simultaneous translation" into his own language:

And here is an example of a reverse translation. The distributive law is written in mathematical language:

Translating into ordinary language, we get a long sentence: "To multiply the number a by the sum of the numbers b and c, you need to multiply the number a in turn by each term and add the resulting products."

Every language has written and spoken language. Above we talked about written speech in mathematical language. And oral speech is the use of special terms, for example: “term”, “equation”, “inequality”, “graph”, “coordinate”, as well as various mathematical statements expressed in words.

To master a new language, it is necessary to study its letters, syllables, words, sentences, rules, grammar. This is not the most fun activity, it is more interesting to read and speak right away. But this does not happen, you have to be patient and learn the basics first. And, of course, as a result of such study, your understanding of the mathematical language will gradually expand.

Tasks. 1. Acquaintance. Read the text on your own and write down the types of mathematical language.2. Understanding. Give an example (not from the text) of oral and written speech in mathematical language.3.Application. Conduct an experiment confirming that mathematical language, like any other language, is a means of communication, thanks toto which we can transfer information, describe this or that phenomenon, law or property.

4. Analysis. Expand the features of mathematical speech.

5. Synthesis. Come up with a game for the 6th grade "Rules for actions with positive and negative numbers." Formulate them in ordinary language and try to translate these rules into mathematical language.

“How often are mathematical terms used in everyday life?”

In Chubais' speeches, we often hear the words
"Unification of subjects, and the energy industry is intact", And some strict leader constantly says: "It's time to divide Russia, that's when we will live" President Vladimir Putin always assures us: "There will never be a turn to the past!" Here are our leaders, made sure They often speak mathematical language.

"In medicine, mathematical language is indispensable."

In medicine, degrees, parameters, pressure.

Everyone who works there knows these terms.

math language at school

History and chemistry and physics teachers
They cannot but use the language of mathematics. It is needed in biology, where the flower has a root, It is needed in zoology, there are many vertebrae, And our writers, reading the biography Famous writer, all dates are indicated. And your classmates, asking for time, They can't live two minutes before the change.

newspapers use mathematical language:

Yes, if you open our newspapers,
They are all full of numbers. From there you will know, the budget is decreasing, And the prices are rising as they want.

Mathematical language on the street, in football training:

Mathematical language is always used
Passers-by on the street “How do you feel? Affairs?" “I work all the time, I took five acres of the garden, What kind of health is there, to live for two years. And the football coach yells at the boys: “You pick up speed, the ball is already flying to the center.

Let's conclude this from today's lesson
We all need the language of mathematics, it is very convincing. He is clear and specific, strict, unambiguous, Helps everyone in life to solve their problems. This makes him very attractive. And I think that in our life it is simply mandatory

Operations with negative and positive numbers

Absolute value (or absolute value) is the positive number obtained by changing its sign(-) to the reverse(+) . Absolute value-5 there is+5 , i.e.5 . The absolute value of a positive number (as well as the number0 ) is called the number itself. The sign of the absolute value is two straight lines that enclose the number whose absolute value is taken. For example,
|-5| = 5,
|+5| = 5,
| 0 | = 0. Adding numbers with the same sign. a) When Two numbers with the same sign are added together with their absolute values ​​and the sum is preceded by their common sign.Examples. (+8) + (+11) = 19; (-7) + (-3) = -10.
6) When adding two numbers with different signs, the absolute value of one of them is subtracted from the absolute value of the other (the smaller one from the larger one), and the sign of the number whose absolute value is greater is put.Examples. (-3) + (+12) = 9;
(-3) + (+1) = -2. Subtraction of numbers with different signs. one number from another can be replaced by addition; in this case, the minuend is taken with its sign, and the subtrahend with the reverse.Examples. (+7) - (+4) = (+7) + (-4) = 3;
(+7) - (-4) = (+7) + (+4) = 11;
(-7) - (-4) = (-7) + (+4) = -3;
(-4) - (-4) = (-4) + (+4) = 0;
Comment. When doing addition and subtraction, especially when dealing with multiple numbers, the best thing to do is: 1) release all numbers from brackets, while putting the sign “” before the number + ", if the previous character before the parenthesis was the same as the character in the parenthesis, and " - "" if it was the opposite of the sign in the parenthesis; 2) add up the absolute values ​​of all numbers that now have a sign on the left + ; 3) add up the absolute values ​​of all numbers that now have a sign on the left - ; 4) subtract the smaller amount from the larger amount and put the sign corresponding to the larger amount.
Example. (-30) - (-17) + (-6) - (+12) + (+2);
(-30) - (-17) + (-6) - (+12) + (+2) = -30 + 17 - 6 - 12 + 2;
17 + 2 = 19;
30 + 6 + 12 = 48;
48 - 19 = 29.

The result is a negative number

-29 , since a large amount(48) was obtained by adding the absolute values ​​​​of those numbers that were preceded by minuses in the expression-30 + 17 – 6 -12 + 2. This last expression can also be viewed as the sum of numbers -30, +17, -6, -12, +2, and as a result of successive addition to the number-30 numbers17 , then subtracting the number6 , then subtraction12 and finally additions2 . In general, the expressiona - b + c - d etc., you can also look at the sum of numbers(+a), (-b), (+c), (-d), and as a result of such sequential actions: subtractions from(+a) numbers(+b) , additions(+c) , subtraction(+d) etc.Multiplication of numbers with different signs At two numbers are multiplied by their absolute values ​​and the product is preceded by a plus sign if the signs of the factors are the same, and a minus sign if they are different.
Scheme (sign rule for multiplication):

+

Examples. (+ 2,4) * (-5) = -12; (-2,4) * (-5) = 12; (-8,2) * (+2) = -16,4.

When multiplying several factors, the sign of the product is positive if the number of negative factors is even, and negative if the number of negative factors is odd.

Examples. (+1/3) * (+2) * (-6) * (-7) * (-1/2) = 7 (three negative factors);
(-1/3) * (+2) * (-3) * (+7) * (+1/2) = 7 (two negative factors).

Division of numbers with different signs

At one number by another, the absolute value of the first is divided by the absolute value of the second, and a plus sign is placed in front of the quotient if the signs of the dividend and divisor are the same, and minus if they are different (the scheme is the same as for multiplication).

Examples. (-6) : (+3) = -2;
(+8) : (-2) = -4; (-12) : (-12) = + 1.