The sign of the function f. Mathematical signs
As you know, mathematics loves accuracy and brevity - it is not without reason that a single formula can occupy a paragraph in verbal form, and sometimes an entire page of text. Thus, the graphic elements used throughout the world in science are designed to increase the speed of writing and the compactness of data presentation. In addition, standardized graphic images can recognize a native speaker of any language who has basic knowledge in the relevant field.
The history of mathematical signs and symbols goes back many centuries - some of them were invented randomly and were intended to denote other phenomena; others have become the product of the activities of scientists purposefully shaping artificial language and guided by purely practical considerations.
Plus and minus
The history of the origin of symbols denoting the simplest arithmetic operations is not known for certain. However, there is a fairly probable hypothesis of the origin of the plus sign, which looks like crossed horizontal and vertical lines. In accordance with it, the addition symbol originates in the Latin union et, which is translated into Russian as "and". Gradually, in order to speed up the writing process, the word was reduced to a vertically oriented cross, resembling the letter t. The earliest reliable example of such a reduction dates from the 14th century.
The generally accepted minus sign appeared, apparently, later. In the 14th and even 15th century scientific literature used a number of symbols denoting the operation of subtraction, and only to XVI century"plus" and "minus" in their modern form began to meet in mathematical works together.
Multiplication and division
Ironically, the mathematical signs and symbols for these two arithmetic operations are not fully standardized today. A popular notation for multiplication is the diagonal cross proposed by the mathematician Oughtred in the 17th century, which can be seen, for example, on calculators. In mathematics lessons at school, the same operation is usually represented as a point - this method proposed in the same century by Leibniz. Another way of representation is the asterisk, which is most often used in computer representation of various calculations. It was proposed to use it all in the same 17th century, Johann Rahn.
For the division operation, a slash sign (proposed by Ougtred) and a horizontal line with dots above and below (the symbol was introduced by Johann Rahn) are provided. The first version of the designation is more popular, but the second is also quite common.
Mathematical signs and symbols and their meanings sometimes change over time. However, all three methods of graphical representation of multiplication, as well as both methods for division, are to some extent consistent and relevant today.
Equality, identity, equivalence
As with many other mathematical signs and symbols, the notation for equality was originally verbal. For quite a long time, the generally accepted designation was the abbreviation ae from the Latin aequalis (“equal”). However, in the 16th century, a Welsh mathematician named Robert Record proposed two horizontal lines, one below the other, as a symbol. According to the scientist, it is impossible to come up with anything more equal to each other than two parallel segments.
Despite the fact that a similar sign was used to indicate the parallelism of lines, the new equality symbol gradually gained popularity. By the way, such signs as "more" and "less", depicting ticks turned in different directions, appeared only in the 17th-18th centuries. Today, they seem intuitive to any student.
A few more complex signs equivalences (two wavy lines) and identities (three horizontal parallel lines) came into use only in the second half of the 19th century.
Sign of the unknown - "X"
The history of the emergence of mathematical signs and symbols knows and is very interesting cases rethinking graphics as science advances. The symbol for the unknown, today called "x", originates in the Middle East at the dawn of the last millennium.
Back in the 10th century in the Arab world, famous for that historical period by their scientists, the concept of the unknown was denoted by a word that literally translates as “something” and begins with the sound “Sh”. In order to save materials and time, the word in the treatises began to be reduced to the first letter.
After many decades, the written works of Arab scientists ended up in the cities of the Iberian Peninsula, on the territory modern Spain. Scientific treatises began to be translated into the national language, but a difficulty arose - there is no "Sh" phoneme in Spanish. Borrowed Arabic words beginning with it were written according to a special rule and were preceded by the letter X. scientific language At that time there was Latin, in which the corresponding sign is called "X".
Thus, the sign, which at first glance is only a randomly selected symbol, has a deep history and is originally an abbreviation Arabic word"something".
Notation of other unknowns
Unlike "X", Y and Z, familiar to us from school, as well as a, b, c, have a much more prosaic history of origin.
In the 17th century, a book by Descartes called "Geometry" was published. In this book, the author proposed to standardize the symbols in equations: in accordance with his idea, the last three letters of the Latin alphabet (starting from "X") began to denote unknown, and the first three - known values.
Trigonometric terms
The history of such a word as "sine" is truly unusual.
Initially relevant trigonometric functions named in India. The word corresponding to the concept of sine literally meant "string". In the heyday of Arabic science, Indian treatises were translated, and the concept, which had no analogue in Arabic, transcribed. By coincidence, what happened in the letter resembled the real-life word "hollow", the semantics of which had nothing to do with the original term. As a result, when Arabic texts were translated into Latin in the 12th century, the word "sine" arose, meaning "depression" and fixed as a new mathematical concept.
But the mathematical signs and symbols for tangent and cotangent are still not standardized - in some countries they are usually written as tg, and in others - as tan.
Some other signs
As can be seen from the examples described above, the emergence of mathematical signs and symbols largely took place in the 16th-17th centuries. The same period saw the emergence of today's usual forms of recording such concepts as percentage, square root, degree.
Percentage, i.e. hundredth, for a long time was designated as cto (short for Latin cento). It is believed that the sign generally accepted today appeared as a result of a misprint about four hundred years ago. The resulting image was perceived as a good way to reduce and took root.
The root sign was originally a stylized letter R (short for the Latin word radix, "root"). The upper line, under which the expression is written today, served as brackets and was a separate character, separate from the root. Parentheses were invented later - they entered widespread circulation thanks to the activities of Leibniz (1646-1716). Thanks to his own work, the integral symbol was also introduced into science, looking like an elongated letter S - an abbreviation for the word "sum".
Finally, the exponentiation sign was invented by Descartes and refined by Newton in the second half of the 17th century.
Later designations
Considering that the familiar graphic images of “plus” and “minus” were put into circulation only a few centuries ago, it does not seem surprising that mathematical signs and symbols denoting complex phenomena began to be used only in the century before last.
So, the factorial, which has the form of an exclamation mark after a number or a variable, appeared only in early XIX century. Approximately at the same time, the capital “P” appeared to denote the work and the symbol of the limit.
It is somewhat strange that the signs for the number pi and algebraic sum appeared only in the 18th century - later than, for example, the integral symbol, although intuitively it seems that they are more common. The graphic representation of the ratio of circumference to diameter comes from the first letter Greek words, meaning "circumference" and "perimeter". And the sign "sigma" for the algebraic sum was proposed by Euler in the last quarter of the 18th century.
Symbol names in different languages
As you know, the language of science in Europe for many centuries was Latin. Physical, medical and many other terms were often borrowed in the form of transcriptions, much less often in the form of tracing paper. Thus, many mathematical signs and symbols in English are called almost the same as in Russian, French or German. How harder essence phenomena, the more likely it is that different languages it will have the same name.
Computer notation of mathematical symbols
The simplest mathematical signs and symbols in the Word are indicated by the usual key combination Shift + a number from 0 to 9 in the Russian or English layout. Separate keys are reserved for some widely used signs: plus, minus, equality, slash.
If you want to use graphic representations of the integral, algebraic sum or product, Pi number, etc., you need to open the "Insert" tab in Word and find one of the two buttons: "Formula" or "Symbol". In the first case, a constructor will open that allows you to build an entire formula within one field, and in the second, a symbol table where you can find any mathematical symbols.
How to remember math symbols
Unlike chemistry and physics, where the number of symbols to remember can exceed a hundred units, mathematics operates with a relatively small number of symbols. We learn the simplest of them in early childhood, learning to add and subtract, and only at the university in certain specialties do we get acquainted with a few complex mathematical signs and symbols. Pictures for children help in a matter of weeks to achieve instant recognition of the graphic image of the required operation, much more time may be needed to master the skill of the very implementation of these operations and understand their essence.
Thus, the process of memorizing characters occurs automatically and does not require much effort.
Finally
The value of mathematical signs and symbols lies in the fact that they are easily understood by people who speak different languages and are carriers of different cultures. For this reason, it is extremely useful to understand and be able to reproduce graphic representations of various phenomena and operations.
The high level of standardization of these signs leads to their use in the most various fields: in finance, information technologies, engineering, etc. For anyone who wants to do business related to numbers and calculations, knowledge of mathematical signs and symbols and their meanings becomes a vital necessity.
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Mathematical notation("language of mathematics") - a complex graphical notation that serves to present abstract mathematical ideas and judgments in a human-readable form. It makes up (in its complexity and diversity) a significant proportion of non-speech sign systems used by mankind. This article describes the generally accepted international system designations, although different cultures of the past had their own, and some of them even have limited use until now.
Note that mathematical notation, as a rule, is used in conjunction with the written form of some of the natural languages.
In addition to fundamental and applied mathematics, mathematical notation has wide application in physics, as well as (in its incomplete scope) in engineering, computer science, economics, and indeed in all areas of human activity where mathematical models are used. Differences between the proper mathematical and applied style of notation will be discussed in the course of the text.
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Hello! This video is not about mathematics, but rather about etymology and semiotics. But I'm sure you'll like it. Go! You are aware that the search for a solution to cubic equations in general view took mathematicians several centuries? This is partly why? Because there were no clear symbols for clear thoughts, whether it's our time. There are so many characters that you can get confused. But you can't fool us, let's figure it out. This is an inverted capital letter A. This is actually an English letter, listed first in the words "all" and "any". In Russian, this symbol, depending on the context, can be read like this: for anyone, everyone, everyone, everyone, and so on. Such a hieroglyph will be called a universal quantifier. And here is another quantifier, but already existence. The English letter e was reflected in Paint from left to right, thus hinting at the overseas verb "exist", in our opinion we will read: exists, there is, there is another similar way. An exclamation mark would add uniqueness to such an existential quantifier. If this is clear, we move on. You probably came across indefinite integrals in the eleventh class, so I would like to remind you that this is not just some kind of antiderivative, but the collection of all antiderivatives of the integrand. So don't forget about C - the constant of integration. By the way, the integral icon itself is just an elongated letter s, an echo of the Latin word sum. This is precisely the geometric meaning of a definite integral: the search for the area of \u200b\u200bthe figure under the graph by summing infinitesimal values. To me, this is the most romantic activity in calculus. But school geometry is most useful because it teaches logical rigor. By the first course, you should have a clear understanding of what a consequence is, what an equivalence is. Well, you can't get confused between necessity and sufficiency, you understand? Let's even try to dig a little deeper. If you decide to take up higher mathematics, then I imagine how bad things are with your personal life, but that is why you will surely agree to overcome a small exercise. There are three points here, each has a left and right side, which you need to connect with one of the three drawn symbols. Please pause, try it out for yourself, and then listen to what I have to say. If x=-2, then |x|=2, but from left to right, so the phrase is already built. In the second paragraph, absolutely the same thing is written on the left and right sides. And the third point can be commented as follows: every rectangle is a parallelogram, but not every parallelogram is a rectangle. Yes, I know that you are no longer small, but still my applause to those who have coped with this exercise. Well, okay, enough, let's remember the number sets. Natural numbers are used in counting: 1, 2, 3, 4 and so on. In nature, -1 apple does not exist, but, by the way, integers allow you to talk about such things. The letter ℤ screams to us about the important role of zero, the set of rational numbers is denoted by the letter ℚ, and this is no coincidence. AT English word"quotient" means "attitude". By the way, if somewhere in Brooklyn an African American approaches you and says: "Keep it real!", you can be sure that you are a mathematician, an admirer of real numbers. Well, you should read something about complex numbers, it will be more useful. We will now roll back, return to the first grade of the most ordinary Greek school. In short, let's remember the ancient alphabet. The first letter is alpha, then betta, this hook is gamma, then delta, followed by epsilon, and so on, up to the last letter omega. You can be sure that the Greeks also have capital letters, but we will not talk about sad things now. We are better about cheerful - about limits. But here there are just no riddles, it is immediately clear from which word the mathematical symbol appeared. Well, therefore, we can move on to the final part of the video. Please try to sound out the definition of the limit of the number sequence, which is now written in front of you. Click rather pause and think, and may you have the happiness of a one-year-old child who has learned the word "mother." If for any epsilon greater than zero there is a natural number N, such that for all numbers of the numerical sequence greater than N, the inequality |xₙ-a|<Ɛ (эпсилон), то тогда предел числовой последовательности xₙ , при n, стремящемся к бесконечности, равен числу a. Такие вот дела, ребята. Не беда, если вам не удалось прочесть это определение, главное в свое время его понять. Напоследок отмечу: множество тех, кто посмотрел этот ролик, но до сих пор не подписан на канал, не является пустым. Это меня очень печалит, так что во время финальной музыки покажу, как это исправить. Ну а остальным желаю мыслить критически, заниматься математикой! Счастливо! [Музыка / аплодиминнты]
General information
The system evolved like natural languages, historically (see the history of mathematical notation), and is organized like the writing of natural languages, borrowing many symbols from there as well (primarily from the Latin and Greek alphabets). Symbols, as well as in ordinary writing, are depicted with contrasting lines on a uniform background (black on white paper, light on a dark board, contrasting on a monitor, etc.), and their meaning is determined primarily by the shape and relative position. Color is not taken into account and is usually not used, but when using letters, their characteristics such as style and even typeface, which do not affect the meaning in ordinary writing, can play a semantic role in mathematical notation.
Structure
Ordinary mathematical notation (in particular, the so-called mathematical formulas) are written in general in a string from left to right, but do not necessarily constitute a consecutive string of characters. Separate blocks of characters can be located in the upper or lower half of the line, even in the case when the characters do not overlap vertically. Also, some parts are located entirely above or below the line. On the grammatical side, almost any "formula" can be considered a hierarchically organized tree-type structure.
Standardization
Mathematical notation represents a system in terms of the relationship of its components, but, in general, not constitute a formal system (in the understanding of mathematics itself). They, in any complicated case, cannot even be disassembled programmatically. Like any natural language, the “language of mathematics” is full of inconsistent designations, homographs, different (among its speakers) interpretations of what is considered correct, etc. There is not even any foreseeable alphabet of mathematical symbols, and in particular because the question is not always unambiguously resolved whether to consider two designations as different characters or as different spellings of one character.
Some of the mathematical notation (mainly related to measurements) is standardized in ISO 31 -11, but in general, there is rather no standardization of notation.
Elements of mathematical notation
Numbers
If necessary, apply a number system with a base less than ten, the base is written in a subscript: 20003 8 . Number systems with bases greater than ten are not used in the generally accepted mathematical notation (although, of course, they are studied by science itself), since there are not enough numbers for them. In connection with the development of computer science, the hexadecimal number system has become relevant, in which the numbers from 10 to 15 are indicated by the first six Latin letters from A to F. Several different approaches are used to designate such numbers in computer science, but they are not transferred to mathematics.
Superscript and subscript characters
Parentheses, similar symbols, and delimiters
Parentheses "()" are used:
Square brackets "" are often used in grouping meanings when you have to use many pairs of brackets. In this case, they are placed on the outside and (with neat typography) have a greater height than the brackets that are inside.
Square "" and round "()" brackets are used to denote closed and open spaces, respectively.
Curly braces "()" are usually used for , although the same caveat applies to them as for square brackets. Left "(" and right ")" brackets can be used separately; their purpose is described.
Angle bracket symbols " ⟨ ⟩ (\displaystyle \langle \;\rangle )» with neat typography should have obtuse angles and thus differ from similar ones that have a right or acute angle. In practice, one should not hope for this (especially when manually writing formulas) and one has to distinguish between them with the help of intuition.
Pairs of symmetric (with respect to the vertical axis) symbols, including those other than those listed, are often used to highlight a piece of a formula. The purpose of paired brackets is described.
Indices
Depending on the location, superscripts and subscripts are distinguished. The superscript can mean (but does not necessarily mean) exponentiation to , about other uses of .
Variables
In the sciences, there are sets of quantities, and any of them can take either a set of values and be called variable value (variant), or only one value and be called a constant. In mathematics, quantities are often diverted from the physical meaning, and then the variable turns into abstract(or numeric) variable, denoted by some symbol not occupied by the special notation mentioned above.
Variable X is considered given if the set of values it takes is specified (x). It is convenient to consider a constant value as a variable for which the corresponding set (x) consists of one element.
Functions and Operators
Mathematically, there is no significant difference between operator(unary), mapping and function.
However, it is understood that if to record the value of the mapping from the given arguments, it is necessary to specify , then the symbol of this mapping denotes a function, in other cases it is more likely to speak of an operator. Symbols of some functions of one argument are used with and without brackets. Many elementary functions, for example sin x (\displaystyle \sin x) or sin (x) (\displaystyle \sin(x)), but elementary functions are always called functions.
Operators and Relations (Unary and Binary)
Functions
A function can be referred to in two senses: as an expression of its value with given arguments (written f (x) , f (x , y) (\displaystyle f(x),\ f(x,y)) etc.) or actually as a function. In the latter case, only the function symbol is put, without brackets (although they often write it randomly).
There are many notations for common functions used in mathematical work without further explanation. Otherwise, the function must be described somehow, and in fundamental mathematics it does not fundamentally differ from and is also denoted by an arbitrary letter in the same way. The letter f is the most popular for variable functions, g and most Greek are also often used.
Predefined (reserved) designations
However, single-letter designations can, if desired, be given a different meaning. For example, the letter i is often used as an index in a context where complex numbers are not used, and the letter can be used as a variable in some combinatorics. Also, set theory symbols (such as " ⊂ (\displaystyle \subset )" and " ⊃ (\displaystyle \supset )”) and propositional calculus (such as “ ∧ (\displaystyle \wedge )" and " ∨ (\displaystyle\vee )”) can be used in another sense, usually as an order relation and a binary operation, respectively.
Indexing
Indexing is plotted (usually bottom, sometimes top) and is, in a sense, a way to expand the content of a variable. However, it is used in three slightly different (though overlapping) senses.
Actually numbers
You can have multiple different variables by denoting them with the same letter, similar to using . For example: x 1 , x 2 , x 3 … (\displaystyle x_(1),\ x_(2),\ x_(3)\ldots ). Usually they are connected by some commonality, but in general this is not necessary.
Moreover, as "indexes" you can use not only numbers, but also any characters. However, when another variable and expression is written as an index, this entry is interpreted as "a variable with a number determined by the value of the index expression."
In tensor analysis
In linear algebra, tensor analysis, differential geometry with indices (in the form of variables) are written
“Symbols are not only a record of thoughts,
means of its image and fixation, -
no, they affect the very thought,
they... guide her, and that's enough
move them on paper... in order to
unmistakably reach new truths.
L. Carnot
Mathematical signs serve primarily for accurate (uniquely defined) recording of mathematical concepts and sentences. Their totality in the real conditions of their application by mathematicians constitutes what is called the mathematical language.
Mathematical signs allow you to write in a compact form sentences that are cumbersomely expressed in ordinary language. This makes them easier to remember.
Before using certain signs in reasoning, the mathematician tries to say what each of them means. Otherwise, they may not understand it.
But mathematicians cannot always say right away what this or that symbol that they have introduced for any mathematical theory reflects. For example, for hundreds of years, mathematicians operated with negative and complex numbers, but the objective meaning of these numbers and the operation with them were discovered only at the end of the 18th and at the beginning of the 19th century.
1. Symbolism of mathematical quantifiers
Like ordinary language, the language of mathematical signs allows the exchange of established mathematical truths, but being only an auxiliary tool attached to ordinary language and cannot exist without it.
Mathematical definition:
In regular language:
function limit F (x) at some point X0 is called a constant number A, such that for an arbitrary number E>0 there is a positive d(E) such that from the condition |X - X 0 | Notation in quantifiers (in mathematical language) 2. Symbolism of mathematical signs and geometric figures. 1) Infinity is a concept used in mathematics, philosophy and the natural sciences. The infinity of some concept or attribute of some object means the impossibility of specifying boundaries or a quantitative measure for it. The term infinity corresponds to several different concepts, depending on the field of application, whether it be mathematics, physics, philosophy, theology, or everyday life. In mathematics, there is no single concept of infinity; it is endowed with special properties in each section. Moreover, these various "infinities" are not interchangeable. For example, set theory implies different infinities, and one can be greater than the other. Say, the number of integers is infinitely large (it is called countable). To generalize the concept of the number of elements for infinite sets, the concept of cardinality of a set is introduced in mathematics. In this case, there is no one "infinite" power. For example, the cardinality of the set of real numbers is greater than the cardinality of integers, because a one-to-one correspondence cannot be built between these sets, and integers are included in the real numbers. Thus, in this case, one cardinal number (equal to the cardinality of the set) is "infinite" than the other. The founder of these concepts was the German mathematician Georg Cantor. In mathematical analysis, two symbols, plus and minus infinity, are added to the set of real numbers, which are used to determine boundary values and convergence. It should be noted that in this case we are not talking about "tangible" infinity, since any statement containing this symbol can be written using only finite numbers and quantifiers. These symbols (as well as many others) were introduced to shorten the notation of longer expressions. Infinity is also inextricably linked with the designation of the infinitely small, for example, even Aristotle said: Infinity in most cultures appeared as an abstract quantitative designation for something incomprehensibly large, applied to entities without spatial or temporal boundaries. 2) Circle - the locus of points in the plane, the distance from which to a given point, called the center of the circle, does not exceed a given non-negative number, called the radius of this circle. If the radius is zero, then the circle degenerates into a point. A circle is a locus of points in a plane that are equidistant from a given point, called the center, at a given non-zero distance, called its radius. 3) Square (rhombus) - is a symbol of the combination and ordering of four different elements, for example, the four main elements or the four seasons. Symbol of the number 4, equality, simplicity, directness, truth, justice, wisdom, honor. Symmetry is the idea through which a person tries to comprehend harmony and has long been considered a symbol of beauty. Symmetry is possessed by the so-called “curly” verses, the text of which has the shape of a rhombus. We - (E. Martov, 1894) 4) Rectangle. Of all geometric forms, this is the most rational, most reliable and regular figure; empirically this is explained by the fact that always and everywhere the rectangle was the favorite shape. With the help of it, a person adapted a space or any object for direct use in his life, for example: a house, a room, a table, a bed, etc. 5) The Pentagon is a regular pentagon in the form of a star, a symbol of eternity, perfection, the universe. Pentagon - an amulet of health, a sign on the door to drive away witches, the emblem of Thoth, Mercury, Celtic Gawain, etc., a symbol of the five wounds of Jesus Christ, prosperity, good luck among the Jews, the legendary key of Solomon; a sign of high position in society among the Japanese. 6) Regular hexagon, hexagon - a symbol of abundance, beauty, harmony, freedom, marriage, a symbol of the number 6, the image of a person (two arms, two legs, head and torso). 7) The cross is a symbol of the highest sacred values. The cross models the spiritual aspect, the ascent of the spirit, the aspiration to God, to eternity. The cross is a universal symbol of the unity of life and death. 8) A triangle is a geometric figure that consists of three points that do not lie on the same straight line, and three segments connecting these three points. 9) Six-pointed Star (Star of David) - consists of two equilateral triangles superimposed on one another. One of the versions of the origin of the sign associates its shape with the shape of the White Lily flower, which has six petals. The flower was traditionally placed under the temple lamp, in such a way that the priest lit the fire, as it were, in the center of Magen David. In Kabbalah, the two triangles symbolize the duality inherent in man: good versus evil, spiritual versus physical, and so on. The upward pointing triangle symbolizes our good deeds, which ascend to heaven and cause a stream of grace to descend back into this world (which symbolizes the downward pointing triangle). Sometimes the Star of David is called the Star of the Creator and each of its six ends is associated with one of the days of the week, and the center with Saturday. 10) Five-pointed Star - The main distinguishing emblem of the Bolsheviks is the red five-pointed star, officially installed in the spring of 1918. Initially, Bolshevik propaganda called it the “Mars Star” (allegedly belonging to the ancient god of war - Mars), and then began to declare that “The five rays of the star means the union of the workers of all five continents in the struggle against capitalism.” In reality, the five-pointed star has nothing to do with either the militant deity Mars or the international proletariat, it is an ancient occult sign (obviously of Middle Eastern origin) called the “pentagram” or “Star of Solomon”. It should be noted that the pentagram was often placed by the Bolsheviks on Red Army uniforms, in military equipment, various signs and all sorts of attributes of visual propaganda in a purely satanic way: with two “horns” up. 3. Masonic signs Masons Motto:"Freedom. Equality. Brotherhood". The social movement of free people who, on the basis of free choice, allow them to become better, to become closer to God, therefore, they are recognized to improve the world. Signs The radiant eye (delta) is an ancient, religious sign. He says that God oversees his creations. With the image of this sign, the Masons asked God for blessings for any grandiose actions, for their labors. The Radiant Eye is located on the pediment of the Kazan Cathedral in St. Petersburg. The combination of compass and square in the Masonic sign. For the uninitiated, this is a tool of labor (a bricklayer), and for the initiated, these are ways of knowing the world and the relationship between divine wisdom and human reason. For divine wisdom, there is nothing impossible, it can take on both the human form (-) and the divine form (0), it can accommodate everything. Thus, the human mind comprehends the divine wisdom, embraces it. In philosophy, this statement is a postulate about absolute and relative truth. Hexagonal Star (Bethlehem) The letter G is the designation of God (German - Got), the great geometer of the Universe. Conclusion Mathematical signs serve primarily to accurately record mathematical concepts and sentences. Their totality constitutes what is called the mathematical language.
“... it is always possible to come up with a larger number, because the number of parts into which a segment can be divided has no limit; therefore, infinity is potential, never real, and no matter how many divisions are given, it is always potentially possible to divide this segment into an even greater number. Note that Aristotle made a great contribution to the understanding of infinity, dividing it into potential and actual, and came close from this side to the foundations of mathematical analysis, also pointing out five sources of ideas about it:
Further, infinity was developed in philosophy and theology along with the exact sciences. For example, in theology, the infinity of God does not so much give a quantitative definition as it means unlimitedness and incomprehensibility. In philosophy, it is an attribute of space and time.
Modern physics comes close to the actuality of infinity denied by Aristotle - that is, accessibility in the real world, and not just in the abstract. For example, there is the concept of a singularity, closely related to black holes and the big bang theory: it is a point in space-time at which mass in an infinitely small volume is concentrated with infinite density. There is already solid circumstantial evidence for the existence of black holes, although the big bang theory is still under development.
The circle is a symbol of the Sun, the Moon. One of the most common characters. It is also a symbol of infinity, eternity, perfection.
The poem is a rhombus.
In the midst of darkness.
The eye is resting.
The darkness of the night is alive.
The heart sighs eagerly
The whisper of the stars flies at times.
And azure feelings are crowded by the crowd.
Everything was forgotten in the dewy brilliance.
Fragrant kiss!
Shine fast!
Whisper again
As then:
"Yes!"
Of course, one can disagree with these statements.
However, no one will deny that any image evokes associations in a person. But the problem is that some objects, plots or graphic elements evoke the same associations in all people (or rather, in many), while others are completely different.
Properties of a triangle as a figure: strength, immutability.
Axiom A1 of stereometry says: “Through 3 points of space that do not lie on one straight line, a plane passes, and moreover, only one!”
To check the depth of understanding of this statement, they usually set the backfill problem: “Three flies are sitting on the table, at three ends of the table. At a certain moment, they scatter in three mutually perpendicular directions with the same speed. When will they be on the same plane again? The answer is the fact that three points always, at any moment, define a single plane. And it is 3 points that define a triangle, so this figure in geometry is considered the most stable and durable.
The triangle is usually referred to as a sharp, "offensive" figure associated with the masculine principle. The equilateral triangle is a masculine and solar sign representing deity, fire, life, heart, mountain and ascent, prosperity, harmony and royalty. The inverted triangle is a female and lunar symbol, personifies water, fertility, rain, divine mercy.
US state symbols also contain the Six-pointed Star in various forms, in particular, it is on the Great Seal of the United States and on banknotes. The Star of David is depicted on the coats of arms of the German cities of Cher and Gerbstedt, as well as the Ukrainian Ternopil and Konotop. Three six-pointed stars are depicted on the flag of Burundi and represent the national motto: “Unity. Job. Progress".
In Christianity, the six-pointed star is a symbol of Christ, namely the union in Christ of divine and human nature. That is why this sign is inscribed in the Orthodox Cross.
Government”, which is under the complete control of Freemasonry.
Quite often, Satanists draw a pentagram with two ends up, so that it is easy to enter the devil's head "Pentagram of Baphomet" there. The portrait of the “Fiery Revolutionary” is placed inside the “Pentagram of Baphomet”, which is the central part of the composition of the special Chekist order “Felix Dzerzhinsky” designed in 1932 (the project was later rejected by Stalin, who deeply hates the “Iron Felix”).
The Marxist plans for a "world proletarian revolution" were clearly of Masonic origin, and a number of the most prominent Marxists were members of Freemasonry. L. Trotsky belonged to them, it was he who proposed to make the Masonic pentagram the identification emblem of Bolshevism.
International Masonic lodges secretly provided the Bolsheviks with comprehensive support, especially financial.
Freemasons are associates of the Creator, associates of social progress, against inertia, inertia and ignorance. Outstanding representatives of freemasonry - Karamzin Nikolai Mikhailovich, Suvorov Alexander Vasilyevich, Kutuzov Mikhail Illarionovich, Pushkin Alexander Sergeevich, Goebbels Joseph.
The square, as a rule, from below is a human knowledge of the world. From the point of view of Freemasonry, a person comes into the world to know the divine plan. And knowledge requires tools. The most effective science in the knowledge of the world is mathematics.
The square is the oldest mathematical tool known from time immemorial. The graduation of a square is already a big step forward in the mathematical tools of knowledge. Man cognizes the world with the help of the sciences of mathematics, the first of them, but not the only one.
However, the square is wooden, and it holds what it can hold. It cannot be moved. If you try to push it apart to fit more, you will break it.
So people who try to know the whole infinity of the divine plan either die or go crazy. "Know your limits!" - that's what this sign tells the World. Even if you are Einstein, Newton, Sakharov - the greatest minds of mankind! - understand that you are limited by the time in which you were born; in the knowledge of the world, language, brain size, a variety of human limitations, the life of your body. Therefore - yes, learn, but understand that you will never fully know!
And the circle? The compass is divine wisdom. A compass can describe a circle, and if you push its legs apart, it will be a straight line. And in symbolic systems, a circle and a straight line are two opposites. A straight line denotes a person, his beginning and end (like a dash between two dates - birth and death). The circle is a symbol of the deity, since it is a perfect figure. They oppose each other - the divine and human figures. Man is not perfect. God is perfect in everything.
People always know the truth, but always relative truth. And the absolute truth is known only to God.
Learn more and more, realizing that you will not be able to know the truth to the end - what depths we find in an ordinary compass with a square! Who would have thought!
This is the beauty and charm of Masonic symbolism, in its great intellectual depth.
Since the Middle Ages, the compass, as a tool for drawing perfect circles, has become a symbol of geometry, cosmic order and planned actions. At this time, the God of hosts was often painted in the image of the creator and architect of the universe with a compass in his hands (William Blake ‘‘The Great Architect’’, 1794).
The Hexagonal Star meant the Unity and Struggle of Opposites, the fight of Man and Woman, Good and Evil, Light and Darkness. One cannot exist without the other. The tension that arises between these opposites creates the world as we know it.
The triangle up means - "A person strives for God." Triangle down - "The Deity descends to Man." In their combination, our world exists, which is the combination of the Human and the Divine. The letter G here means that God lives in our world. He is really present in everything he created.
The decisive force in the development of mathematical symbolism is not the "free will" of mathematicians, but the requirements of practice, mathematical research. It is real mathematical research that helps to find out which sign system best reflects the structure of quantitative and qualitative relations, which can be an effective tool for their further use in symbols and emblems.
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