The science of the quantitative relations of the real world. Mathematics as a science of quantitative relations and spatial forms of the real world. Period of mathematics of variables
The idealized properties of the objects under study are either formulated as axioms or listed in the definition of the corresponding mathematical objects. Then, according to strict rules of logical inference, other true properties (theorems) are deduced from these properties. This theory together forms a mathematical model of the object under study. Thus, initially proceeding from spatial and quantitative relations, mathematics obtains more abstract relations, the study of which is also the subject of modern mathematics.
Traditionally, mathematics is divided into theoretical, which performs an in-depth analysis of intra-mathematical structures, and applied, which provides its models to other sciences and engineering disciplines, and some of them occupy a position bordering on mathematics. In particular, formal logic can be considered both as part of the philosophical sciences and as part of the mathematical sciences; mechanics - both physics and mathematics; computer science, computer technology and algorithmics refer to both engineering and mathematical sciences, etc. Many different definitions of mathematics have been proposed in the literature.
Etymology
The word "mathematics" comes from other Greek. μάθημα, which means the study, knowledge, the science, etc. - Greek. μαθηματικός, originally meaning receptive, prolific, later studyable, subsequently pertaining to mathematics. In particular, μαθηματικὴ τέχνη , in Latin ars mathematica, means art of mathematics. The term other Greek. μᾰθημᾰτικά in the modern sense of the word "mathematics" is already found in the writings of Aristotle (4th century BC). According to Fasmer, the word came to the Russian language either through Polish. matematyka, or through lat. mathematica.
Definitions
One of the first definitions of the subject of mathematics was given by Descartes:
The field of mathematics includes only those sciences in which either order or measure is considered, and it does not matter at all whether these are numbers, figures, stars, sounds, or anything else in which this measure is sought. Thus, there must be some general science that explains everything pertaining to order and measure, without entering into the study of any particular subjects, and this science must be called not by the foreign, but by the old, already common name of General Mathematics.
The essence of mathematics ... is now presented as a doctrine of relations between objects, about which nothing is known, except for some properties that describe them - precisely those that are put as axioms at the basis of the theory ... Mathematics is a set of abstract forms - mathematical structures.
Branches of mathematics
1. Mathematics as academic discipline
Notation
Since mathematics deals with extremely diverse and rather complex structures, its notation is also very complex. The modern system of writing formulas was formed on the basis of the European algebraic tradition, as well as the needs of later branches of mathematics - mathematical analysis, mathematical logic, set theory, etc. Geometry has used a visual (geometrical) representation from time immemorial. In modern mathematics, complex graphic notation systems (for example, commutative diagrams) are also common, and notation based on graphs is also often used.
Short story
Philosophy of mathematics
Goals and Methods
Space R n (\displaystyle \mathbb (R) ^(n)), at n > 3 (\displaystyle n>3) is a mathematical invention. However, a very ingenious invention that helps to mathematically understand complex phenomena».
Foundations
intuitionism
Constructive mathematics
clarify
Main topics
Quantity
The main section dealing with the abstraction of quantity is algebra. The concept of "number" originally originated from arithmetic representations and referred to natural numbers. Later, with the help of algebra, it was gradually extended to integer, rational, real, complex and other numbers.
Transformations
The phenomena of transformations and changes are considered in the most general form by analysis.
structures
Spatial Relations
Geometry considers the basics of spatial relations. Trigonometry considers the properties of trigonometric functions. The study of geometric objects through mathematical analysis deals with differential geometry. The properties of spaces that remain unchanged under continuous deformations and the very phenomenon of continuity are studied by topology.
Discrete Math
| ∀ x (P (x) ⇒ P (x ′)) (\displaystyle \forall x(P(x)\Rightarrow P(x"))) | |||
Mathematics has been around for a very long time. Man gathered fruits, dug up fruits, fished and stored them all for the winter. To understand how much food is stored, a person invented the account. This is how mathematics began.
Then the man began to engage in agriculture. It was necessary to measure plots of land, build dwellings, measure time.
That is, it became necessary for a person to use the quantitative ratio of the real world. Determine how much crops have been harvested, what is the size of the building plot, or how large is the area of the sky with a certain number of bright stars.
In addition, a person began to determine the forms: the sun is round, the box is square, the lake is oval, and how these objects are located in space. That is, a person became interested in the spatial forms of the real world.
Thus the concept maths can be defined as the science of quantitative relations and spatial forms of the real world.
At present, there is not a single profession where one could do without mathematics. The famous German mathematician Carl Friedrich Gauss, who was called the "King of Mathematics", once said:
"Mathematics is the queen of sciences, arithmetic is the queen of mathematics."
The word "arithmetic" comes from the Greek word "arithmos" - "number".
In this way, arithmetic is a branch of mathematics that studies numbers and operations on them.
In primary school, first of all, they study arithmetic.
How did this science develop, let's explore this issue.
The period of the birth of mathematics
The main period of accumulation of mathematical knowledge is considered to be the time before the 5th century BC.
The first who began to prove mathematical positions was an ancient Greek thinker who lived in the 7th century BC, presumably 625-545. This philosopher traveled through the countries of the East. Tradition says that he studied with the Egyptian priests and the Babylonian Chaldeans.
Thales of Miletus brought from Egypt to Greece the first concepts of elementary geometry: what is a diameter, what determines a triangle, and so on. He predicted a solar eclipse, designed engineering structures.
During this period, arithmetic gradually develops, astronomy and geometry develop. Algebra and trigonometry are born.
Period of elementary mathematics
This period begins with VI BC. Now mathematics is emerging as a science with theories and proofs. The theory of numbers appears, the doctrine of quantities, of their measurement.
The most famous mathematician of this time is Euclid. He lived in the III century BC. This man is the author of the first theoretical treatise on mathematics that has come down to us.
In the works of Euclid, the foundations of the so-called Euclidean geometry are given - these are axioms that rest on basic concepts, such as.
During the period of elementary mathematics, the theory of numbers was born, as well as the doctrine of quantities and their measurement. For the first time, negative and irrational numbers appear.
At the end of this period, the creation of algebra, as a literal calculus, is observed. The very science of "algebra" appears among the Arabs as the science of solving equations. The word "algebra" in Arabic means "recovery", that is, the transfer of negative values to another part of the equation.
Period of mathematics of variables
The founder of this period is Rene Descartes, who lived in the 17th century AD. In his writings, Descartes for the first time introduces the concept of a variable.
Thanks to this, scientists move from the study of constant quantities to the study of relationships between variables and to the mathematical description of motion.
Friedrich Engels characterized this period most clearly, in his writings he wrote:
“The turning point in mathematics was the Cartesian variable. Thanks to this, movement and thus dialectics entered mathematics, and thanks to this, differential and integral calculus immediately became necessary, which immediately arises, and which was by and large completed, and not invented by Newton and Leibniz.
Period of modern mathematics
In the 20s of the 19th century, Nikolai Ivanovich Lobachevsky became the founder of the so-called non-Euclidean geometry.
From this moment begins the development of the most important sections of modern mathematics. Such as probability theory, set theory, mathematical statistics and so on.
All these discoveries and studies are widely used in various fields of science.
And at present, the science of mathematics is rapidly developing, the subject of mathematics is expanding, including new forms and relationships, new theorems are being proved, and the basic concepts are deepening.
The idealized properties of the objects under study are either formulated as axioms or listed in the definition of the corresponding mathematical objects. Then, according to strict rules of logical inference, other true properties (theorems) are deduced from these properties. This theory together forms a mathematical model of the object under study. Thus, initially, proceeding from spatial and quantitative relations, mathematics obtains more abstract relations, the study of which is also the subject of modern mathematics.
Traditionally, mathematics is divided into theoretical, which performs an in-depth analysis of intra-mathematical structures, and applied, which provides its models to other sciences and engineering disciplines, and some of them occupy a position bordering on mathematics. In particular, formal logic can be considered both as part of the philosophical sciences and as part of the mathematical sciences; mechanics - both physics and mathematics; computer science, computer technology, and algorithmics refer to both engineering and mathematical sciences, etc. Many different definitions of mathematics have been proposed in the literature (see).
Etymology
The word "mathematics" comes from other Greek. μάθημα ( mathema), which means the study, knowledge, the science, etc. - Greek. μαθηματικός ( mathematicos), originally meaning receptive, prolific, later studyable, subsequently pertaining to mathematics. In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), in Latin ars mathematica, means art of mathematics.
Definitions
The field of mathematics includes only those sciences in which either order or measure is considered, and it does not matter at all whether these are numbers, figures, stars, sounds, or anything else in which this measure is sought. Thus, there must be some general science that explains everything pertaining to order and measure, without entering into the study of any particular subjects, and this science must be called not by the foreign, but by the old, already common name of General Mathematics.
In Soviet times, the definition from the TSB given by A. N. Kolmogorov was considered classic:
Mathematics ... the science of quantitative relations and spatial forms of the real world.
The essence of mathematics ... is now presented as a doctrine of relations between objects, about which nothing is known, except for some properties that describe them - precisely those that are put as axioms at the basis of the theory ... Mathematics is a set of abstract forms - mathematical structures.
Here are some more modern definitions.
Modern theoretical ("pure") mathematics is the science of mathematical structures, mathematical invariants of various systems and processes.
Mathematics is a science that provides the ability to calculate models that can be reduced to a standard (canonical) form. The science of finding solutions to analytical models (analysis) by means of formal transformations.
Branches of mathematics
1. Mathematics as academic discipline is subdivided in the Russian Federation into elementary mathematics studied in secondary school and formed by the following disciplines:
- elementary geometry: planimetry and stereometry
- theory of elementary functions and elements of analysis
4. The American Mathematical Society (AMS) has developed its own standard for classifying branches of mathematics. It's called Mathematics Subject Classification. This standard is updated periodically. The current version is MSC 2010. The previous version is MSC 2000.
Notation
Due to the fact that mathematics deals with extremely diverse and rather complex structures, the notation is also very complex. The modern system of writing formulas was formed on the basis of the European algebraic tradition, as well as mathematical analysis (the concept of a function, derivative, etc.). From time immemorial, geometry has used a visual (geometrical) representation. In modern mathematics, complex graphic notation systems (for example, commutative diagrams) are also common, and notation based on graphs is also often used.
Short story
The development of mathematics relies on writing and the ability to write down numbers. Probably, ancient people first expressed quantity by drawing lines on the ground or scratching them on wood. The ancient Incas, having no other writing system, represented and stored numerical data using a complex system of rope knots, the so-called quipu. There were many different number systems. The first known records of numbers were found in the Ahmes Papyrus, created by the Egyptians of the Middle Kingdom. The Indian civilization developed the modern decimal number system incorporating the concept of zero.
Historically, the major mathematical disciplines emerged under the influence of the need to make calculations in the commercial field, in measuring the land and for predicting astronomical phenomena and, later, for solving new physical problems. Each of these areas plays a large role in the broad development of mathematics, which consists in the study of structures, spaces and changes.
Philosophy of mathematics
Goals and Methods
Mathematics studies imaginary, ideal objects and the relationships between them using a formal language. In general, mathematical concepts and theorems do not necessarily correspond to anything in the physical world. The main task of the applied branch of mathematics is to create a mathematical model that is adequate enough for the real object under study. The task of the theoretical mathematician is to provide a sufficient set of convenient means to achieve this goal.
The content of mathematics can be defined as a system of mathematical models and tools for their creation. The object model does not take into account all its features, but only the most necessary for the purposes of study (idealized). For example, when studying the physical properties of an orange, we can abstract from its color and taste and represent it (albeit not perfectly accurately) as a ball. If we need to understand how many oranges we get if we add two and three together, then we can abstract away from the form, leaving the model with only one characteristic - quantity. Abstraction and the establishment of relationships between objects in the most general form is one of the main areas of mathematical creativity.
Another direction, along with abstraction, is generalization. For example, generalizing the concept of "space" to the space of n-dimensions. " The space at is a mathematical fiction. However, a very ingenious invention that helps to mathematically understand complex phenomena».
The study of intramathematical objects, as a rule, takes place using the axiomatic method: first, a list of basic concepts and axioms is formulated for the objects under study, and then meaningful theorems are obtained from the axioms using inference rules, which together form a mathematical model.
Foundations
The question of the essence and foundations of mathematics has been discussed since the time of Plato. Since the 20th century, there has been comparative agreement on what should be considered a rigorous mathematical proof, but there has been no agreement on what is considered true in mathematics. This gives rise to disagreements both in questions of axiomatics and the interconnection of branches of mathematics, and in the choice of logical systems that should be used in proofs.
In addition to the skeptical, the following approaches to this issue are known.
Set-theoretic approach
It is proposed to consider all mathematical objects within the framework of set theory, most often with the Zermelo-Fraenkel axiomatics (although there are many others that are equivalent to it). This approach has been considered prevailing since the middle of the 20th century, however, in reality, most mathematical works do not set themselves the task of translating their statements strictly into the language of set theory, but operate with concepts and facts established in some areas of mathematics. Thus, if a contradiction is found in set theory, this will not entail the invalidation of most of the results.
logicism
This approach assumes strict typing of mathematical objects. Many paradoxes avoided in set theory only by special tricks turn out to be impossible in principle.
Formalism
This approach involves the study of formal systems based on classical logic.
intuitionism
Intuitionism presupposes at the foundation of mathematics an intuitionistic logic that is more limited in the means of proof (but, it is believed, also more reliable). Intuitionism rejects proof by contradiction, many non-constructive proofs become impossible, and many problems of set theory become meaningless (non-formalizable).
Constructive mathematics
Constructive mathematics is a trend in mathematics close to intuitionism that studies constructive constructions [ clarify] . According to the criterion of constructibility - " to exist means to be built". The constructivity criterion is a stronger requirement than the consistency criterion.
Main topics
Numbers
The concept of "number" originally referred to natural numbers. Later it was gradually extended to integer, rational, real, complex and other numbers.
| Numerical systems | |
|---|---|
| Counting sets |
Natural numbers () Integers () Rationals () Algebraic () Periods Computable Arithmetic |
| Real numbers and their extensions |
Real () Complex () Quaternions () Cayley numbers (octaves, octonions) () Sedenions () Alternions Cayley-Dixon procedure Dual Hypercomplex Superreal Hyperreal Surreal number ( English) |
| Other number systems |
Cardinal numbers Ordinal numbers (transfinite, ordinal) p-adic Supernatural numbers |
| see also | Double numbers Irrational numbers Transcendent Number ray Biquaternion |
Transformations
Discrete Math
Codes in knowledge classification systems
Online Services
There are a large number of sites that provide services for mathematical calculations. Most of them are in English. Of the Russian-speaking ones, the service of mathematical queries of the search engine Nigma can be noted.
see also
Popularizers of ScienceNotes
- Encyclopedia Britannica
- Webster's Online Dictionary
- Chapter 2. Mathematics as the language of science. Siberian Open University. Archived from the original on February 2, 2012. Retrieved October 5, 2010.
- Large Ancient Greek Dictionary (αω)
- Dictionary of the Russian language of the XI-XVII centuries. Issue 9 / Ch. ed. F. P. Filin. - M.: Nauka, 1982. - S. 41.
- Descartes R. Rules to guide the mind. M.-L.: Sotsekgiz, 1936.
- See: TSB Mathematics
- Marx K., Engels F. Works. 2nd ed. T. 20. S. 37.
- Bourbaki N. The architecture of mathematics. Essays on the history of mathematics / Translated by I. G. Bashmakova, ed. K. A. Rybnikova. M.: IL, 1963. S. 32, 258.
- Kaziev V. M. Introduction to Mathematics
- Mukhin O. I. Modeling Systems Tutorial. Perm: RCI PSTU.
- Herman Weil // Kline M.. - M.: Mir, 1984. - S. 16.
- State educational standard of higher professional education. Specialty 01.01.00. "Maths". Qualification - Mathematician. Moscow, 2000 (Compiled under the guidance of O. B. Lupanov)
- The nomenclature of specialties of scientific workers, approved by the order of the Ministry of Education and Science of Russia dated February 25, 2009 No. 59
- UDC 51 Mathematics
- Ya. S. Bugrov, S. M. Nikolsky. Elements of linear algebra and analytic geometry. M.: Nauka, 1988. S. 44.
- N. I. Kondakov. Logical dictionary-reference book. M.: Nauka, 1975. S. 259.
- G. I. Ruzavin. On the nature of mathematical knowledge. M.: 1968.
- http://www.gsnti-norms.ru/norms/common/doc.asp?0&/norms/grnti/gr27.htm
- For example: http://mathworld.wolfram.com
Literature
encyclopedias- // Encyclopedic Dictionary of Brockhaus and Efron: In 86 volumes (82 volumes and 4 additional). - St. Petersburg. , 1890-1907.
- Mathematical Encyclopedia (in 5 volumes), 1980s. // General and special math references on EqWorld
- Kondakov N.I. Logical dictionary-reference book. Moscow: Nauka, 1975.
- Encyclopedia of the Mathematical Sciences and their Applications (German) 1899-1934 (the largest review of 19th century literature)
- G. Korn, T. Korn. Handbook of mathematics for scientists and engineers M., 1973
- Kline M. Maths. Loss of certainty. - M.: Mir, 1984.
- Kline M. Maths. The search for truth. M.: Mir, 1988.
- Klein F. Elementary mathematics from a higher point of view.
- Volume I. Arithmetic. Algebra. Analysis M.: Nauka, 1987. 432 p.
- Volume II. Geometry M.: Nauka, 1987. 416 p.
- R. Courant, G. Robbins. What is mathematics? 3rd ed., rev. and additional - M.: 2001. 568 p.
- Pisarevsky B. M., Kharin V. T. About mathematics, mathematicians and not only. - M.: Binom. Knowledge Laboratory, 2012. - 302 p.
- Poincare A. Science and method (rus.) (fr.)
Mathematics is one of the oldest sciences. It is not at all easy to give a brief definition of mathematics, its content will vary greatly depending on the level of mathematical education of a person. A primary school student who has just begun to study arithmetic will say that mathematics is studying the rules for counting objects. And he will be right, because it is with this that he gets acquainted at first. Older students will add to what has been said that the concept of mathematics includes algebra and the study of geometric objects: lines, their intersections, plane figures, geometric bodies, various kinds of transformations. High school graduates, however, will include in the definition of mathematics the study of functions and the action of passing to the limit, as well as the related concepts of derivative and integral. Graduates of higher technical educational institutions or natural science departments of universities and pedagogical institutes will no longer be satisfied with school definitions, since they know that other disciplines are also part of mathematics: probability theory, mathematical statistics, differential calculus, programming, computational methods, as well as applications of these disciplines for modeling production processes, processing of experimental data, transmission and processing of information. However, what is listed does not exhaust the content of mathematics. Set theory, mathematical logic, optimal control, the theory of random processes and much more are also included in its composition.
Attempts to define mathematics by listing its constituent branches lead us astray, because they do not give an idea of what exactly mathematics studies and what its relation to the world around us is. If such a question were put to a physicist, biologist or astronomer, each of them would give a very brief answer, not containing a listing of the parts that make up the science they study. Such an answer would contain an indication of the phenomena of nature that she investigates. For example, a biologist would say that biology is the study of the various manifestations of life. Although this answer is not completely complete, since it does not say what life and life phenomena are, nevertheless, such a definition would give a fairly complete idea of the content of the science of biology itself and of the different levels of this science. And this definition would not change with the expansion of our knowledge of biology.
There are no such phenomena of nature, technical or social processes that would be the subject of study of mathematics, but would not be related to physical, biological, chemical, engineering or social phenomena. Each natural science discipline: biology and physics, chemistry and psychology - is determined by the material features of its subject, the specific features of the area of the real world that it studies. The object or phenomenon itself can be studied by different methods, including mathematical ones, but by changing the methods, we still remain within the boundaries of this discipline, since the content of this science is the real subject, and not the research method. For mathematics, the material subject of research is not of decisive importance; the applied method is important. For example, trigonometric functions can be used both to study oscillatory motion and to determine the height of an inaccessible object. And what phenomena of the real world can be investigated using the mathematical method? These phenomena are determined not by their material nature, but exclusively by formal structural properties, and above all by those quantitative relationships and spatial forms in which they exist.
So, mathematics does not study material objects, but research methods and structural properties of the object of study, which allow applying certain operations to it (summation, differentiation, etc.). However, a significant part of mathematical problems, concepts and theories has as its primary source real phenomena and processes. For example, arithmetic and number theory emerged from the primary practical task of counting objects. Elementary geometry had as its source problems associated with comparing distances, calculating the areas of plane figures or the volumes of spatial bodies. All this needed to be found, since it was necessary to redistribute land between users, calculate the size of granaries or the volume of earthworks during the construction of defense structures.
A mathematical result has the property that it can not only be used in the study of a particular phenomenon or process, but also be used to study other phenomena, the physical nature of which is fundamentally different from those previously considered. So, the rules of arithmetic are applicable in economic problems, and in technical issues, and in solving problems of agriculture, and in scientific research. The rules of arithmetic were developed millennia ago, but they retained their practical value forever. Arithmetic is an integral part of mathematics, its traditional part is no longer subject to creative development within the framework of mathematics, but it finds and will continue to find numerous new applications. These applications may be of great importance for mankind, but they will no longer contribute to mathematics proper.
Mathematics, as a creative force, has as its goal the development of general rules that should be used in numerous special cases. The one who creates these rules, creates something new, creates. The one who applies ready-made rules no longer creates in mathematics itself, but, quite possibly, creates new values in other areas of knowledge with the help of mathematical rules. For example, today the data from the interpretation of satellite images, as well as information about the composition and age of rocks, geochemical and geophysical anomalies are processed using computers. Undoubtedly, the use of a computer in geological research leaves this research geological. The principles of operation of computers and their software were developed without taking into account the possibility of their use in the interests of geological science. This possibility itself is determined by the fact that the structural properties of geological data are in accordance with the logic of certain computer programs.
Two definitions of mathematics have become widespread. The first of these was given by F. Engels in Anti-Dühring, the other by a group of French mathematicians known as Nicolas Bourbaki in the article The Architecture of Mathematics (1948).
"Pure mathematics has as its object the spatial forms and quantitative relations of the real world." This definition not only describes the object of study of mathematics, but also indicates its origin - the real world. However, this definition by F. Engels largely reflects the state of mathematics in the second half of the 19th century. and does not take into account those of its new areas that are not directly related to either quantitative relations or geometric forms. This is, first of all, mathematical logic and disciplines related to programming. Therefore, this definition needs some clarification. Perhaps it should be said that mathematics has as its object of study spatial forms, quantitative relations, and logical constructions.
The Bourbaki argue that "the only mathematical objects are, properly speaking, mathematical structures." In other words, mathematics should be defined as the science of mathematical structures. This definition is essentially a tautology, since it says only one thing: mathematics is concerned with the objects it studies. Another defect of this definition is that it does not clarify the relation of mathematics to the world around us. Moreover, Bourbaki emphasize that mathematical structures are created independently of the real world and its phenomena. That is why Bourbaki was forced to declare that “the main problem is the relationship between the experimental world and the mathematical world. That there is a close relationship between experimental phenomena and mathematical structures seems to have been confirmed in a completely unexpected way by the discoveries of modern physics, but we are completely unaware of the deep reasons for this ... and perhaps we will never know them.
Such a disappointing conclusion cannot arise from the definition of F. Engels, since it already contains the assertion that mathematical concepts are abstractions from certain relations and forms of the real world. These concepts are taken from the real world and are associated with it. In essence, this explains the amazing applicability of the results of mathematics to the phenomena of the world around us, and at the same time the success of the process of mathematization of knowledge.
Mathematics is not an exception from all areas of knowledge - it also forms concepts that arise from practical situations and subsequent abstractions; it allows one to study reality also approximately. But at the same time, it should be borne in mind that mathematics does not study things of the real world, but abstract concepts, and that its logical conclusions are absolutely strict and precise. Its proximity is not internal in nature, but is associated with the compilation of a mathematical model of the phenomenon. We also note that the rules of mathematics do not have absolute applicability, they also have a limited area of application, where they reign supreme. Let us explain the expressed idea with an example: it turns out that two and two are not always equal to four. It is known that when mixing 2 liters of alcohol and 2 liters of water, less than 4 liters of the mixture is obtained. In this mixture, the molecules are arranged more compactly, and the volume of the mixture is less than the sum of the volumes of the constituent components. The addition rule of arithmetic is violated. You can also give examples in which other truths of arithmetic are violated, for example, when adding some objects, it turns out that the sum depends on the order of summation.
Many mathematicians consider mathematical concepts not as a creation of pure reason, but as abstractions from really existing things, phenomena, processes, or abstractions from already established abstractions (abstractions of higher orders). In the Dialectic of Nature, F. Engels wrote that “... all so-called pure mathematics is engaged in abstractions ... all its quantities are, strictly speaking, imaginary quantities ...” These words quite clearly reflect the opinion of one of the founders of Marxist philosophy about the role of abstractions in mathematics. We should only add that all these "imaginary quantities" are taken from reality, and are not constructed arbitrarily, by a free flight of thought. This is how the concept of number came into general use. At first, these were numbers within units, and, moreover, only positive integers. Then the experience forced me to expand the arsenal of numbers to tens and hundreds. The concept of the unboundedness of a series of integers was born already in an era historically close to us: Archimedes in the book “Psammit” (“Calculation of grains of sand”) showed how it is possible to construct numbers even larger than given ones. At the same time, the concept of fractional numbers was born from practical needs. Calculations related to the simplest geometric figures have led mankind to new numbers - irrational ones. Thus, the idea of the set of all real numbers was gradually formed.
The same path can be followed for any other concepts of mathematics. All of them arose from practical needs and gradually formed into abstract concepts. One can again recall the words of F. Engels: “... pure mathematics has a meaning independent of the special experience of each individual ... But it is completely wrong that in pure mathematics the mind deals only with the products of its own creativity and imagination. The concepts of number and figure are not taken from anywhere, but only from the real world. The ten fingers on which people learned to count, that is, to perform the first arithmetic operation, are anything but the product of the free creativity of the mind. In order to count, it is necessary to have not only objects to be counted, but also to have the ability to be distracted when considering these objects from all other properties except number, and this ability is the result of a long historical development based on experience. Both the concept of a number and the concept of a figure are borrowed exclusively from the external world, and did not arise in the head from pure thinking. There had to be things that had a certain form, and these forms had to be compared before one could come to the concept of a figure.
Let us consider whether there are concepts in science that are created without connection with the past progress of science and the current progress of practice. We know very well that scientific mathematical creativity is preceded by the study of many subjects at school, university, reading books, articles, conversations with specialists both in their own field and in other fields of knowledge. A mathematician lives in a society, and from books, on the radio, from other sources, he learns about the problems that arise in science, engineering, and social life. In addition, the thinking of the researcher is influenced by the entire previous evolution of scientific thought. Therefore, it turns out to be prepared for the solution of certain problems necessary for the progress of science. That is why a scientist cannot put forward problems at will, on a whim, but must create mathematical concepts and theories that would be valuable for science, for other researchers, for humanity. But mathematical theories retain their significance in the conditions of various social formations and historical eras. In addition, often the same ideas arise from scientists who are not connected in any way. This is an additional argument against those who adhere to the concept of free creation of mathematical concepts.
So, we told what is included in the concept of "mathematics". But there is also such a thing as applied mathematics. It is understood as the totality of all mathematical methods and disciplines that find applications outside of mathematics. In ancient times, geometry and arithmetic represented all mathematics, and since both found numerous applications in trade exchanges, the measurement of areas and volumes, and in matters of navigation, all mathematics was not only theoretical, but also applied. Later, in ancient Greece, there was a division into mathematics and applied mathematics. However, all eminent mathematicians were also engaged in applications, and not only in purely theoretical research.
The further development of mathematics was continuously connected with the progress of natural science and technology, with the emergence of new social needs. By the end of the XVIII century. there was a need (primarily in connection with the problems of navigation and artillery) to create a mathematical theory of motion. This was done in their works by G. V. Leibniz and I. Newton. Applied mathematics has been replenished with a new very powerful research method - mathematical analysis. Almost simultaneously, the needs of demography and insurance led to the formation of the beginnings of probability theory (see Probability Theory). 18th and 19th centuries expanded the content of applied mathematics, adding to it the theory of ordinary and partial differential equations, equations of mathematical physics, elements of mathematical statistics, differential geometry. 20th century brought new methods of mathematical research of practical problems: the theory of random processes, graph theory, functional analysis, optimal control, linear and non-linear programming. Moreover, it turned out that number theory and abstract algebra found unexpected applications to the problems of physics. As a result, the conviction began to take shape that applied mathematics as a separate discipline does not exist and that all mathematics can be considered applied. Perhaps, it is necessary to say not that mathematics is applied and theoretical, but that mathematicians are divided into applied and theoreticians. For some, mathematics is a method of cognition of the surrounding world and the phenomena occurring in it, it is for this purpose that the scientist develops and expands mathematical knowledge. For others, mathematics itself represents a whole world worthy of study and development. For the progress of science, scientists of both types are needed.
Mathematics, before studying any phenomenon with its own methods, creates its mathematical model, i.e., lists all those features of the phenomenon that will be taken into account. The model forces the researcher to choose those mathematical tools that will allow to adequately convey the features of the phenomenon under study and its evolution. As an example, let's take a planetary system model: the Sun and the planets are considered as material points with the corresponding masses. The interaction of each two points is determined by the force of attraction between them
where m 1 and m 2 are the masses of the interacting points, r is the distance between them, and f is the gravitational constant. Despite the simplicity of this model, for the past three hundred years it has been transmitting with great accuracy the features of the motion of the planets of the solar system.
Of course, each model roughens reality, and the task of the researcher is, first of all, to propose a model that, on the one hand, most fully conveys the factual side of the matter (as they say, its physical features), and, on the other hand, gives a significant approximation to reality. Of course, several mathematical models can be proposed for the same phenomenon. All of them have the right to exist until a significant discrepancy between the model and reality begins to affect.
Mathematics 1. Where did the word mathematics come from 2. Who invented mathematics? 3. Main themes. 4. Definition 5. Etymology On the last slide.
Where did the word come from (go to the previous slide) Mathematics from Greek - study, science) - the science of structures, order and relationships, historically based on the operations of counting, measuring and describing the shape of objects. Mathematical objects are created by idealizing the properties of real or other mathematical objects and writing these properties in a formal language.
Who invented mathematics (go to the menu) The first mathematician is usually called Thales of Miletus, who lived in the VI century. BC e. , one of the so-called Seven Wise Men of Greece. Be that as it may, it was he who was the first to structure the entire knowledge base on this subject, which has long been formed within the world known to him. However, the author of the first treatise on mathematics that has come down to us was Euclid (III century BC). He, too, deservedly be considered the father of this science.
Main topics (go to the menu) The field of mathematics includes only those sciences in which either order or measure is considered, and it does not matter at all whether these are numbers, figures, stars, sounds, or anything else in which this measure is found . Thus, there must be some general science that explains everything pertaining to order and measure, without entering into the study of any particular subjects, and this science must be called not by the foreign, but by the old, already common name of General Mathematics.
Definition (go to the menu) Modern analysis is based on classical mathematical analysis, which is considered as one of the three main areas of mathematics (along with algebra and geometry). At the same time, the term "mathematical analysis" in the classical sense is used mainly in curricula and materials. In the Anglo-American tradition, classical mathematical analysis corresponds to the course programs with the name "calculus"
Etymology (go to the menu) The word "mathematics" comes from other Greek. , which means study, knowledge, science, etc. -Greek, originally meaning receptive, successful, later related to study, later related to mathematics. Specifically, in Latin, it means the art of mathematics. The term is other -Greek. in the modern sense of the word “mathematics” is already found in the works of Aristotle (4th century BC). in "The Book of Selected Briefly on the Nine Muses and on the Seven Free Arts" (1672)
In the 3rd century BC, a book of Euclid with the same name appeared in Alexandria, in the Russian translation of "Beginnings". From the Latin name "Beginnings" came the term "elementary geometry". Although the writings of Euclid's predecessors have not come down to us, we can form some opinion about these writings from Euclid's Elements. In the "Beginnings" there are sections that are logically very little connected with other sections. Their appearance is explained only by the fact that they were introduced according to tradition and copy the "Beginnings" of Euclid's predecessors.
Euclid's Elements consists of 13 books. Books 1 - 6 are devoted to planimetry, books 7 - 10 are about arithmetic and incommensurable quantities that can be built using a compass and straightedge. Books 11 to 13 were devoted to stereometry.
The "Beginnings" begin with a presentation of 23 definitions and 10 axioms. The first five axioms are "general concepts", the rest are called "postulates". The first two postulates determine actions with the help of an ideal ruler, the third - with the help of an ideal compass. The fourth, "all right angles are equal to each other," is redundant, since it can be deduced from the rest of the axioms. The last, fifth postulate read: "If a line falls on two lines and forms interior one-sided angles in the sum of less than two lines, then, with an unlimited continuation of these two lines, they will intersect on the side where the angles are less than two lines."
The five "general concepts" of Euclid are the principles of measuring lengths, angles, areas, volumes: "equal to the same are equal to each other", "if equals are added to equals, the sums are equal to each other", "if equals are subtracted from equals, the remainders are equal among themselves", "combining with each other are equal to each other", "the whole is greater than the part".
Then came the criticism of Euclid's geometry. Euclid was criticized for three reasons: for the fact that he considered only such geometric quantities that can be constructed using a compass and straightedge; for breaking up geometry and arithmetic and proving for integers what he had already proved for geometric quantities, and, finally, for the axioms of Euclid. The fifth postulate, Euclid's most difficult postulate, has been most strongly criticized. Many considered it superfluous, and that it can and should be deduced from other axioms. Others believed that it should be replaced by a simpler and more illustrative one, equivalent to it: "Through a point outside a straight line, no more than one straight line can be drawn in their plane that does not intersect this straight line."
Criticism of the gap between geometry and arithmetic led to the extension of the concept of number to a real number. Disputes about the fifth postulate led to the fact that at the beginning of the 19th century N.I. Lobachevsky, J. Bolyai and K.F. Gauss built a new geometry in which all the axioms of Euclid's geometry were fulfilled, with the exception of the fifth postulate. It was replaced by the opposite statement: "In a plane through a point outside a line, more than one line can be drawn that does not intersect the given one." This geometry was as consistent as the geometry of Euclid.
The Lobachevsky planimetry model on the Euclidean plane was built by the French mathematician Henri Poincaré in 1882.
Draw a horizontal line on the Euclidean plane. This line is called the absolute (x). The points of the Euclidean plane lying above the absolute are the points of the Lobachevsky plane. The Lobachevsky plane is an open half-plane lying above the absolute. Non-Euclidean segments in the Poincaré model are arcs of circles centered on the absolute or line segments perpendicular to the absolute (AB, CD). The figure on the Lobachevsky plane is the figure of an open half-plane lying above the absolute (F). Non-Euclidean motion is a composition of a finite number of inversions centered on the absolute and axial symmetries whose axes are perpendicular to the absolute. Two non-Euclidean segments are equal if one of them can be translated into the other by a non-Euclidean movement. These are the basic concepts of the axiomatics of Lobachevsky's planimetry.
All axioms of Lobachevsky's planimetry are consistent. "A non-Euclidean line is a semicircle with ends on the absolute, or a ray with origin on the absolute and perpendicular to the absolute." Thus, the assertion of Lobachevsky's axiom of parallelism holds not only for some line a and a point A not lying on this line, but also for any line a and any point A not lying on it.
Behind Lobachevsky's geometry, other consistent geometries also arose: projective geometry separated from Euclidean, multidimensional Euclidean geometry developed, Riemannian geometry arose (a general theory of spaces with an arbitrary law of measuring lengths), etc. From the science of figures in one three-dimensional Euclidean space, geometry for 40 - 50 years has turned into a set of various theories, only somewhat similar to its progenitor - the geometry of Euclid.
The main stages of the formation of modern mathematics. Structure of modern mathematics
Academician A.N. Kolmogorov identifies four periods in the development of mathematics Kolmogorov A.N. - Mathematics, Mathematical Encyclopedic Dictionary, Moscow, Soviet Encyclopedia, 1988: the birth of mathematics, elementary mathematics, mathematics of variables, modern mathematics.
During the development of elementary mathematics, the theory of numbers gradually grows out of arithmetic. Algebra is created as a literal calculus. And the system of presentation of elementary geometry created by the ancient Greeks - the geometry of Euclid - for two millennia ahead became a model of the deductive construction of mathematical theory.
In the 17th century, the demands of natural science and technology led to the creation of methods that make it possible to mathematically study movement, the processes of changing quantities, and the transformation of geometric figures. With the use of variables in analytic geometry and the creation of differential and integral calculus, the period of mathematics of variables begins. The great discoveries of the 17th century are the concept of an infinitesimal quantity introduced by Newton and Leibniz, the creation of the foundations for the analysis of infinitesimal quantities (mathematical analysis).
The concept of a function comes to the fore. Function becomes the main subject of study. The study of a function leads to the basic concepts of mathematical analysis: limit, derivative, differential, integral.
The appearance of the brilliant idea of R. Descartes on the method of coordinates also belongs to this time. Analytical geometry is created, which allows studying geometric objects by methods of algebra and analysis. On the other hand, the coordinate method opened up the possibility of a geometric interpretation of algebraic and analytic facts.
Further development of mathematics led at the beginning of the 19th century to the formulation of the problem of studying possible types of quantitative relations and spatial forms from a fairly general point of view.
The connection between mathematics and natural science is becoming more and more complex. New theories arise and they arise not only as a result of the demands of natural science and technology, but also as a result of the inner need of mathematics. A remarkable example of such a theory is the imaginary geometry of N.I. Lobachevsky. The development of mathematics in the 19th and 20th centuries allows us to attribute it to the period of modern mathematics. The development of mathematics itself, the mathematization of various fields of science, the penetration of mathematical methods into many areas of practical activity, the progress of computer technology have led to the emergence of new mathematical disciplines, for example, operations research, game theory, mathematical economics, and others.
The main methods in mathematical research are mathematical proofs - rigorous logical reasoning. Mathematical thinking is not limited to logical reasoning. Mathematical intuition is necessary for the correct formulation of the problem, for evaluating the choice of the method for solving it.
In mathematics, mathematical models of objects are studied. The same mathematical model can describe the properties of real phenomena that are far from each other. So, the same differential equation can describe the processes of population growth and the decay of radioactive material. For a mathematician, it is not the nature of the objects under consideration that is important, but the relations existing between them.
There are two types of reasoning in mathematics: deduction and induction.
Induction is a research method in which a general conclusion is built on the basis of particular premises.
Deduction is a method of reasoning by means of which a conclusion of a particular nature follows from general premises.
Mathematics plays an important role in natural science, engineering and humanities research. The reason for the penetration of mathematics into various branches of knowledge is that it offers very clear models for studying the surrounding reality, in contrast to the less general and more vague models offered by other sciences. Without modern mathematics, with its developed logical and computing apparatus, progress in various areas of human activity would be impossible.
Mathematics is not only a powerful tool for solving applied problems and a universal language of science, but also an element of a common culture.
Basic features of mathematical thinking
On this issue, of particular interest is the characteristic of mathematical thinking given by A.Ya. Khinchin, or rather, its specific historical form - the style of mathematical thinking. Revealing the essence of the style of mathematical thinking, he singles out four features common to all eras that noticeably distinguish this style from the styles of thinking in other sciences.
First, the mathematician is characterized by the dominance of the logical scheme of reasoning brought to the limit. A mathematician who loses sight of this scheme, at least temporarily, loses the ability to think scientifically altogether. This peculiar feature of the style of mathematical thinking has a lot of value in itself. Obviously, to the maximum extent it allows you to monitor the correctness of the flow of thought and guarantees against errors; on the other hand, it forces the thinker to have before his eyes the totality of available possibilities during analysis and obliges him to take into account each of them without missing a single one (such omissions are quite possible and, in fact, are often observed in other styles of thinking).
Secondly, conciseness, i.e. the conscious desire to always find the shortest logical path leading to a given goal, the merciless rejection of everything that is absolutely necessary for the impeccable validity of the argument. A mathematical essay of good style, does not tolerate any "water", no embellishing, weakening the logical tension of ranting, distraction to the side; extreme stinginess, severe strictness of thought and its presentation are an integral feature of mathematical thinking. This feature is of great value not only for mathematical, but also for any other serious reasoning. Laconism, the desire not to allow anything superfluous, helps both the thinker and his reader or listener to fully concentrate on a given train of thought, without being distracted by secondary ideas and without losing direct contact with the main line of reasoning.
The luminaries of science, as a rule, think and express themselves succinctly in all fields of knowledge, even when their thought creates and sets out fundamentally new ideas. What a majestic impression, for example, the noble stinginess of thought and speech of the greatest creators of physics: Newton, Einstein, Niels Bohr! Perhaps it is difficult to find a more striking example of what a profound effect the style of thinking of its creators can have on the development of science.
For mathematics, the conciseness of thought is an indisputable law, canonized for centuries. Any attempt to burden the presentation with not necessarily necessary (even if pleasant and exciting for the listeners) pictures, distractions, oratory is placed under legitimate suspicion in advance and automatically causes critical alertness.
Thirdly, a clear dissection of the course of reasoning. If, for example, when proving a proposition, we must consider four possible cases, each of which can be broken down into one or another number of subcases, then at each moment of reasoning, the mathematician must clearly remember in which case and subcase his thought is now being acquired and which cases and subcases he still has to consider. With all sorts of branched enumerations, the mathematician must at every moment be aware of the generic concept for which he enumerates its component species concepts. In ordinary, non-scientific thinking, we very often observe confusion and jumps in such cases, leading to confusion and errors in reasoning. It often happens that a person begins to enumerate the species of one of the genus, and then imperceptibly to the listeners (and often to himself), using the insufficient logical distinctness of the reasoning, jumped to another genus and ends with the statement that both genera are now classified; and listeners or readers do not know where the boundary lies between species of the first and second kind.
In order to make such confusions and jumps impossible, mathematicians have long made extensive use of simple external methods of numbering concepts and judgments, sometimes (but much less often) used in other sciences. Those possible cases or those generic concepts that should be considered in this reasoning are renumbered in advance; within each such case, those subcases to be considered that it contains are also renumbered (sometimes, for distinction, using some other numbering system). Before each paragraph, where the consideration of a new subcase begins, the designation accepted for this subcase is put (for example: II 3 - this means that the consideration of the third subcase of the second case begins here, or the description of the third type of the second kind, if we are talking about classification). And the reader knows that until he comes across a new numerical rubric, everything that is presented applies only to this case and subcase. It goes without saying that such numbering is only an external device, very useful, but by no means obligatory, and that the essence of the matter is not in it, but in that distinct division of argumentation or classification, which it both stimulates and marks by itself.
Fourthly, scrupulous accuracy of symbols, formulas, equations. That is, “each mathematical symbol has a strictly defined meaning: replacing it with another symbol or rearranging it to another place, as a rule, entails a distortion, and sometimes complete destruction of the meaning of this statement.”
Having singled out the main features of the mathematical style of thinking, A.Ya. Khinchin notes that mathematics (especially the mathematics of variables) by its nature has a dialectical character, and therefore contributes to the development of dialectical thinking. Indeed, in the process of mathematical thinking there is an interaction between visual (concrete) and conceptual (abstract). “We cannot think of lines,” Kant wrote, “without drawing it mentally, we cannot think of three dimensions for ourselves without drawing three lines perpendicular to each other from one point.”
The interaction of concrete and abstract “led” mathematical thinking to the development of new and new concepts and philosophical categories. In ancient mathematics (mathematics of constants), these were “number” and “space”, which were originally reflected in arithmetic and Euclidean geometry, and later in algebra and various geometric systems. The mathematics of variables was “based” on the concepts that reflected the motion of matter - “finite”, “infinite”, “continuity”, “discrete”, “infinitely small”, “derivative”, etc.
If we talk about the current historical stage in the development of mathematical knowledge, then it goes in line with the further development of philosophical categories: the theory of probability “masters” the categories of the possible and the random; topology - categories of relation and continuity; catastrophe theory - jump category; group theory - categories of symmetry and harmony, etc.
In mathematical thinking, the main patterns of constructing logical connections similar in form are expressed. With its help, the transition from the singular (say, from certain mathematical methods - axiomatic, algorithmic, constructive, set-theoretic and others) to the special and general, to generalized deductive constructions is carried out. The unity of the methods and the subject of mathematics determines the specifics of mathematical thinking, allows us to speak of a special mathematical language that not only reflects reality, but also synthesizes, generalizes, and predicts scientific knowledge. The power and beauty of mathematical thought lies in the utmost clarity of its logic, the elegance of constructions, and the skilful construction of abstractions.
Fundamentally new possibilities of mental activity opened up with the invention of the computer, with the creation of machine mathematics. Significant changes have taken place in the language of mathematics. If the language of classical computational mathematics consisted of formulas of algebra, geometry and analysis, focused on the description of the continuous processes of nature, studied primarily in mechanics, astronomy, physics, then its modern language is the language of algorithms and programs, including the old language of formulas as a particular case.
The language of modern computational mathematics is becoming more and more universal, capable of describing complex (multi-parameter) systems. At the same time, I would like to emphasize that no matter how perfect the mathematical language, enhanced by electronic computing technology, it does not break ties with the diverse “living”, natural language. Moreover, spoken language is the basis of an artificial language. In this regard, the recent discovery of scientists is of interest. The point is that the ancient language of the Aymara Indians, which is spoken by about 2.5 million people in Bolivia and Peru, turned out to be extremely convenient for computer technology. As early as 1610, the Italian Jesuit missionary Ludovico Bertoni, who compiled the first Aymara dictionary, noted the genius of its creators, who achieved high logical purity. In Aymara, for example, there are no irregular verbs and no exceptions to the few clear grammatical rules. These features of the Aymara language allowed the Bolivian mathematician Ivan Guzmán de Rojas to create a system of simultaneous computer translation from any of the five European languages included in the program, the “bridge” between which is the Aymara language. The computer "Aymara", created by a Bolivian scientist, was highly appreciated by specialists. Summarizing this part of the question about the essence of the mathematical style of thinking, it should be noted that its main content is the understanding of nature.
Axiomatic Method
Axiomatics is the main way of constructing a theory, from antiquity to the present day, confirming its universality and all applicability.
The construction of a mathematical theory is based on the axiomatic method. The scientific theory is based on some initial provisions, called axioms, and all other provisions of the theory are obtained as logical consequences of the axioms.
The axiomatic method appeared in ancient Greece, and is currently used in almost all theoretical sciences, and, above all, in mathematics.
Comparing three, in a certain respect, complementary geometries: Euclidean (parabolic), Lobachevsky (hyperbolic), and Riemannian (elliptic), it should be noted that, along with some similarities, there is a big difference between spherical geometry, on the one hand, and the geometries of Euclid and Lobachevsky - on the other.
The fundamental difference between modern geometry is that it now embraces the "geometries" of an infinite number of different imaginary spaces. However, it should be noted that all these geometries are interpretations of Euclidean geometry and are based on the axiomatic method first used by Euclid.
On the basis of research, the axiomatic method has been developed and widely used. As a special case of applying this method is the method of traces in stereometry, which allows solving problems on the construction of sections in polyhedra and some other positional problems.
The axiomatic method, first developed in geometry, has now become an important tool of study in other branches of mathematics, physics, and mechanics. Currently, work is underway to improve and study the axiomatic method of constructing a theory in more depth.
The axiomatic method of constructing a scientific theory consists in highlighting the basic concepts, formulating the axioms of the theories, and all other statements are derived in a logical way, relying on them. It is known that one concept must be explained with the help of others, which, in turn, are also defined with the help of some well-known concepts. Thus, we arrive at elementary concepts that cannot be defined in terms of others. These concepts are called basic.
When we prove a statement, a theorem, we rely on premises that are considered already proven. But these premises were also proved, they had to be substantiated. In the end, we come to unprovable statements and accept them without proof. These statements are called axioms. The set of axioms must be such that, relying on it, one can prove further statements.
Having singled out the main concepts and formulated the axioms, then we derive theorems and other concepts in a logical way. This is the logical structure of geometry. Axioms and basic concepts form the foundations of planimetry.
Since it is impossible to give a single definition of the basic concepts for all geometries, the basic concepts of geometry should be defined as objects of any nature that satisfy the axioms of this geometry. Thus, in the axiomatic construction of a geometric system, we start from a certain system of axioms, or axiomatics. These axioms describe the properties of the basic concepts of a geometric system, and we can represent the basic concepts in the form of objects of any nature that have the properties specified in the axioms.
After formulating and proving the first geometric statements, it becomes possible to prove some statements (theorems) with the help of others. The proofs of many theorems are attributed to Pythagoras and Democritus.
Hippocrates of Chios is credited with compiling the first systematic course of geometry based on definitions and axioms. This course and its subsequent processings were called "Elements".
Axiomatic method of constructing a scientific theory
The creation of a deductive or axiomatic method of constructing science is one of the greatest achievements of mathematical thought. It required the work of many generations of scientists.
A remarkable feature of the deductive system of presentation is the simplicity of this construction, which makes it possible to describe it in a few words.
The deductive system of presentation is reduced to:
1) to the list of basic concepts,
2) to the presentation of definitions,
3) to the presentation of the axioms,
4) to the presentation of theorems,
5) to the proof of these theorems.
An axiom is a statement accepted without proof.
A theorem is a statement that follows from axioms.
Proof is an integral part of the deductive system, it is reasoning that shows that the truth of a statement follows logically from the truth of previous theorems or axioms.
Within a deductive system, two questions cannot be resolved: 1) about the meaning of the basic concepts, 2) about the truth of the axioms. But this does not mean that these questions are generally unsolvable.
The history of natural science shows that the possibility of an axiomatic construction of a particular science appears only at a fairly high level of development of this science, on the basis of a large amount of factual material, which makes it possible to clearly identify the main connections and relationships that exist between the objects studied by this science.
An example of the axiomatic construction of mathematical science is elementary geometry. The system of axioms of geometry was expounded by Euclid (about 300 BC) in the work "Beginnings" unsurpassed in its significance. This system has largely survived to this day.
Basic concepts: point, line, plane basic images; lie between, belong, move.
Elementary geometry has 13 axioms, which are divided into five groups. In the fifth group, there is one axiom about parallels (Euclid's V postulate): through a point on a plane, only one straight line can be drawn that does not intersect this straight line. This is the only axiom that caused the need for proof. Attempts to prove the fifth postulate occupied mathematicians for more than 2 millennia, up to the first half of the 19th century, i.e. until the moment when Nikolai Ivanovich Lobachevsky proved in his writings the complete hopelessness of these attempts. At present, the unprovability of the fifth postulate is a strictly proven mathematical fact.
Axiom about parallel N.I. Lobachevsky replaced the axiom: Let a straight line and a point lying outside the straight line be given in a given plane. Through this point, at least two parallel lines can be drawn to the given line.
From the new system of axioms N.I. Lobachevsky, with impeccable logical rigor, deduced a coherent system of theorems that constitute the content of non-Euclidean geometry. Both geometries of Euclid and Lobachevsky are equal as logical systems.
Three great mathematicians in the 19th century almost simultaneously, independently of each other, came to the same results of the unprovability of the fifth postulate and to the creation of non-Euclidean geometry.
Nikolai Ivanovich Lobachevsky (1792-1856)
Carl Friedrich Gauss (1777-1855)
Janos Bolyai (1802-1860)
Mathematical proof
The main method in mathematical research is mathematical proof - rigorous logical reasoning. By virtue of objective necessity, points out Corresponding Member of the Russian Academy of Sciences L.D. Kudryavtsev Kudryavtsev L.D. - Modern mathematics and its teaching, Moscow, Nauka, 1985, logical reasoning (which by its nature, if correct, is also rigorous) is a method of mathematics, mathematics is unthinkable without them. It should be noted that mathematical thinking is not limited to logical reasoning. For the correct formulation of the problem, for the evaluation of its data, for the selection of significant ones from them and for the choice of a method for solving it, mathematical intuition is also necessary, which makes it possible to foresee the desired result before it is obtained, to outline the path of research with the help of plausible reasoning. But the validity of the fact under consideration is proved not by checking it on a number of examples, not by conducting a number of experiments (which in itself plays a big role in mathematical research), but in a purely logical way, according to the laws of formal logic.
It is believed that mathematical proof is the ultimate truth. A decision that is based on pure logic simply cannot be wrong. But with the development of science and the tasks before mathematicians are put more and more complex.
“We have entered an era when the mathematical apparatus has become so complex and cumbersome that at first glance it is no longer possible to say whether the problem encountered is true or not,” believes Keith Devlin from Stanford University, California, USA. He cites as an example the “classification of simple finite groups”, which was formulated back in 1980, but a complete exact proof has not yet been imparted. Most likely, the theorem is true, but it is impossible to say for sure about this.
A computer solution cannot be called exact either, because such calculations always have an error. In 1998, Hales proposed a computer-assisted solution to Kepler's theorem, formulated back in 1611. This theorem describes the densest packing of balls in space. The proof was presented on 300 pages and contained 40,000 lines of machine code. 12 reviewers checked the solution for a year, but they never achieved 100% confidence in the correctness of the proof, and the study was sent for revision. As a result, it was published only after four years and without full certification of the reviewers.
All the latest calculations for applied problems are made on a computer, but scientists believe that for greater reliability, mathematical calculations should be presented without errors.
The theory of proof is developed in logic and includes three structural components: thesis (what is supposed to be proved), arguments (a set of facts, generally accepted concepts, laws, etc. of the relevant science) and demonstration (the procedure for deploying evidence itself; a consistent chain of inferences when The nth inference becomes one of the premises of the n+1th inference). The rules of proof are distinguished, possible logical errors are indicated.
Mathematical proof has much in common with the principles established by formal logic. Moreover, the mathematical rules of reasoning and operations obviously served as one of the foundations in the development of the proof procedure in logic. In particular, researchers of the history of the formation of formal logic believe that at one time, when Aristotle took the first steps to create laws and rules of logic, he turned to mathematics and to the practice of legal activity. In these sources, he found material for the logical constructions of the conceived theory.
In the 20th century, the concept of proof lost its strict meaning, which happened in connection with the discovery of logical paradoxes hidden in set theory and especially in connection with the results that K. Gödel's theorems on the incompleteness of formalization brought.
First of all, this affected mathematics itself, in connection with which it was believed that the term "proof" does not have a precise definition. But if such an opinion (which still holds today) affects mathematics itself, then they come to the conclusion that the proof should be accepted not in the logico-mathematical, but in the psychological sense. Moreover, a similar view is found in Aristotle himself, who believed that to prove means to conduct a reasoning that would convince us to such an extent that, using it, we convince others of the correctness of something. We find a certain shade of the psychological approach in A.E. Yesenin-Volpin. He sharply opposes the acceptance of truth without proof, linking it with an act of faith, and further writes: "I call the proof of a judgment an honest method that makes this judgment undeniable." Yesenin-Volpin reports that his definition still needs to be clarified. At the same time, does not the very characterization of evidence as an "honest method" betray an appeal to a moral-psychological assessment?
At the same time, the discovery of set-theoretic paradoxes and the appearance of Godel's theorems just contributed to the development of the theory of mathematical proof undertaken by intuitionists, especially the constructivist direction, and D. Hilbert.
Sometimes it is believed that mathematical proof is universal and represents an ideal version of scientific proof. However, it is not the only method; there are other methods of evidence-based procedures and operations. It is only true that the mathematical proof has a lot in common with the formal-logical proof implemented in natural science, and that the mathematical proof has certain specifics, as well as the set of techniques-operations. This is where we will stop, omitting the general thing that makes it related to other forms of evidence, that is, without expanding the algorithm, rules, errors, etc. in all steps (even the main ones). proof process.
Mathematical proof is a reasoning that has the task of substantiating the truth (of course, in the mathematical, that is, as deducibility, sense) of a statement.
The set of rules used in the proof was formed along with the advent of axiomatic constructions of mathematical theory. This was realized most clearly and completely in the geometry of Euclid. His "Principles" became a kind of model standard for the axiomatic organization of mathematical knowledge, and for a long time remained such for mathematicians.
Statements presented in the form of a certain sequence must guarantee a conclusion, which, subject to the rules of logical operation, is considered proven. It must be emphasized that a certain reasoning is a proof only with respect to some axiomatic system.
When characterizing a mathematical proof, two main features are distinguished. First of all, the fact that mathematical proof excludes any reference to empirical evidence. The entire procedure for substantiating the truth of the conclusion is carried out within the framework of the accepted axiomatics. Academician A.D. Aleksandrov emphasizes in this regard. You can measure the angles of a triangle thousands of times and make sure that they are equal to 2d. But math doesn't prove anything. You will prove it to him if you deduce the above statement from the axioms. Let's repeat. Here mathematics is close to the methods of scholasticism, which also fundamentally rejects argumentation by experimentally given facts.
For example, when the incommensurability of segments was discovered, when proving this theorem, an appeal to a physical experiment was excluded, since, firstly, the very concept of "incommensurability" is devoid of physical meaning, and, secondly, mathematicians could not, when dealing with abstraction, to bring to the aid material-concrete extensions, measurable by a sensory-visual device. The incommensurability, in particular, of the side and diagonal of a square, is proved based on the property of integers using the Pythagorean theorem on the equality of the square of the hypotenuse (respectively, the diagonal) to the sum of the squares of the legs (two sides of a right triangle). Or when Lobachevsky was looking for confirmation for his geometry, referring to the results of astronomical observations, then this confirmation was carried out by him by means of a purely speculative nature. Cayley-Klein and Beltrami's interpretations of non-Euclidean geometry also featured typically mathematical rather than physical objects.
The second feature of mathematical proof is its highest abstractness, in which it differs from proof procedures in other sciences. And again, as in the case of the concept of a mathematical object, it is not just about the degree of abstraction, but about its nature. The fact is that proof reaches a high level of abstraction in a number of other sciences, for example, in physics, cosmology and, of course, in philosophy, since the ultimate problems of being and thinking become the subject of the latter. Mathematics, on the other hand, is distinguished by the fact that variables function here, the meaning of which is in abstraction from any specific properties. Recall that, by definition, variables are signs that in themselves have no meanings and acquire the latter only when the names of certain objects are substituted for them (individual variables) or when specific properties and relations are indicated (predicate variables), or, finally, in cases of replacing a variable with a meaningful statement (propositional variable).
The noted feature determines the nature of the extreme abstractness of the signs used in the mathematical proof, as well as statements, which, due to the inclusion of variables in their structure, turn into statements.
The very procedure of proof, defined in logic as a demonstration, proceeds on the basis of the rules of inference, based on which the transition from one proven statement to another is carried out, forming a consistent chain of inferences. The most common are the two rules (substitution and derivation of conclusions) and the deduction theorem.
substitution rule. In mathematics, substitution is defined as the replacement of each of the elements a of a given set by some other element F(a) from the same set. In mathematical logic, the substitution rule is formulated as follows. If a true formula M in the propositional calculus contains a letter, say A, then by replacing it wherever it occurs with an arbitrary letter D, we get a formula that is also true as the original one. This is possible, and admissible precisely because in the calculus of propositions one abstracts from the meaning of propositions (formulas)... Only the values "true" or "false" are taken into account. For example, in the formula M: A--> (BUA) we substitute the expression (AUB) in place of A, as a result we get a new formula (AUB) -->[(BU(AUB) ].
The rule for inferring conclusions corresponds to the structure of the conditionally categorical syllogism modus ponens (affirmative mode) in formal logic. It looks like this:
a .
Given a proposition (a-> b) and also given a. It follows b.
For example: If it is raining, then the pavement is wet, it is raining (a), therefore, the pavement is wet (b). In mathematical logic, this syllogism is written as follows (a-> b) a-> b.
The inference is determined, as a rule, by separating for implication. If an implication (a-> b) and its antecedent (a) are given, then we have the right to add to the reasoning (proof) also the consequent of this implication (b). Syllogism is coercive, constituting an arsenal of deductive means of proof, that is, absolutely meeting the requirements of mathematical reasoning.
An important role in mathematical proof is played by the deduction theorem - the general name for a number of theorems, the procedure of which makes it possible to establish the provability of the implication: A-> B, when there is a logical derivation of the formula B from the formula A. In the most common version of the propositional calculus (in the classical, intuitionistic and other types of mathematics), the deduction theorem states the following. If a system of premises G and a premise A are given, from which, according to the rules, B G, A B (- sign of derivability) can be deduced, then it follows that only from the premises of G can one obtain the sentence A --> B.
We have considered the type, which is a direct proof. At the same time, so-called indirect evidence is also used in logic; there are non-direct proofs that are deployed according to the following scheme. Not having, due to a number of reasons (inaccessibility of the object of study, the loss of the reality of its existence, etc.) the opportunity to conduct a direct proof of the truth of any statement, thesis, they build an antithesis. They are convinced that the antithesis leads to contradictions, and, therefore, is false. Then from the fact of the falsity of the antithesis one draws - on the basis of the law of the excluded middle (a v) - the conclusion about the truth of the thesis.
In mathematics, one of the forms of indirect proof is widely used - proof by contradiction. It is especially valuable and, in fact, indispensable in the acceptance of fundamental concepts and provisions of mathematics, for example, the concept of actual infinity, which cannot be introduced in any other way.
The operation of proof by contradiction is represented in mathematical logic as follows. Given a sequence of formulas G and the negation of A (G , A). If this implies B and its negation (G , A B, non-B), then we can conclude that the truth of A follows from the sequence of formulas G. In other words, the truth of the thesis follows from the falsity of the antithesis.
References:
1. N. Sh. Kremer, B. A. Putko, I. M. Trishin, M. N. Fridman, Higher Mathematics for Economists, textbook, Moscow, 2002;
2. L.D. Kudryavtsev, Modern mathematics and its teaching, Moscow, Nauka, 1985;
3. O. I. Larichev, Objective models and subjective decisions, Moscow, Nauka, 1987;
4. A.Ya.Halamizer, “Mathematics? - It's funny! ”, Author's edition, 1989;
5. P.K. Rashevsky, Riemannian geometry and tensor analysis, Moscow, 3rd edition, 1967;
6. V.E. Gmurman, Probability Theory and Mathematical Statistics, Moscow, Higher School, 1977;
7. World wide network Enternet.
Mathematics is the science of quantitative relations and spatial forms of the real world. In close connection with the demands of science and technology, the stock of quantitative relations and spatial forms studied by mathematics is constantly expanding, so that the above definition must be understood in the most general sense.
The purpose of studying mathematics is to increase the general outlook, the culture of thinking, the formation of a scientific worldview.
Understanding the independent position of mathematics as a special science became possible after the accumulation of a sufficiently large amount of factual material and arose for the first time in Ancient Greece in the 6th-5th centuries BC. This was the beginning of the period of elementary mathematics.
During this period, mathematical research dealt only with a rather limited stock of basic concepts that arose with the simplest demands of economic life. At the same time, a qualitative improvement of mathematics as a science is already taking place.
Modern mathematics is often compared to a big city. This is an excellent comparison, because in mathematics, as in a big city, there is a continuous process of growth and improvement. New areas are emerging in mathematics, elegant and deep new theories are being built, like the construction of new neighborhoods and buildings. But the progress of mathematics is not limited to changing the face of the city due to the construction of a new one. We have to change the old. Old theories are included in new, more general ones; there is a need to strengthen the foundations of old buildings. New streets have to be laid in order to establish connections between the distant quarters of the mathematical city. But this is not enough - architectural design requires considerable effort, since the diversity of styles in different areas of mathematics not only spoils the overall impression of science, but also interferes with understanding science as a whole, establishing links between its various parts.
Another comparison is often used: mathematics is likened to a large branched tree, which, systematically, gives new shoots. Each branch of the tree is one or another area of mathematics. The number of branches does not remain unchanged, as new branches grow, grow together at first growing separately, some of the branches dry up, deprived of nourishing juices. Both comparisons are successful and very well convey the actual state of affairs.
Undoubtedly, the demand for beauty plays an important role in the construction of mathematical theories. It goes without saying that the perception of beauty is very subjective and there are often quite ugly ideas about this. And yet one has to be surprised at the unanimity that mathematicians put into the concept of "beauty": the result is considered beautiful if from a small number of conditions it is possible to obtain a general conclusion relating to a wide range of objects. A mathematical derivation is considered beautiful if it is possible to prove a significant mathematical fact in it by simple and short reasoning. The maturity of a mathematician, his talent is guessed by how developed his sense of beauty is. Aesthetically complete and mathematically perfect results are easier to understand, remember and use; it is easier to identify their relationship with other areas of knowledge.
Mathematics in our time has become a scientific discipline with many areas of research, a huge number of results and methods. Mathematics is now so great that it is not possible for one person to cover it in all its parts, there is no possibility of being a universal specialist in it. The loss of connections between its separate directions is certainly a negative consequence of the rapid development of this science. However, at the basis of the development of all branches of mathematics there is a common thing - the origins of development, the roots of the tree of mathematics.
Euclid's geometry as the first natural science theory
Mathematics as a science of quantitative relations and spatial forms of reality studies the world around us, natural and social phenomena. But unlike other sciences, mathematics studies their special properties, abstracting from others. So, geometry studies the shape and size of objects, without taking into account their other properties: color, mass, hardness, etc. In general, mathematical objects (geometric figure, number, value) are created by the human mind and exist only in human thinking, in signs and symbols that form the mathematical language.
The abstractness of mathematics allows it to be applied in a variety of areas, it is a powerful tool for understanding nature.
Forms of knowledge are divided into two groups.
first group constitute forms of sensory cognition, carried out with the help of various sense organs: sight, hearing, smell, touch, taste.
Co. second group include forms of abstract thinking, primarily concepts, statements and inferences.
The forms of sensory cognition are Feel, perception and representation.
Each object has not one, but many properties, and we know them with the help of sensations.
Feeling- this is a reflection of individual properties of objects or phenomena of the material world, which directly (i.e. now, at the moment) affect our senses. These are sensations of red, warm, round, green, sweet, smooth and other individual properties of objects [Getmanova, p. 7].
From individual sensations, the perception of the whole object is formed. For example, the perception of an apple is made up of such sensations: spherical, red, sweet and sour, fragrant, etc.
Perception is a holistic reflection of an external material object that directly affects our senses [Getmanova, p. eight]. For example, the image of a plate, cup, spoon, other utensils; the image of the river, if we are now sailing along it or are on its banks; the image of the forest, if we have now come to the forest, etc.
Perceptions, although they are a sensory reflection of reality in our minds, are largely dependent on human experience. For example, a biologist will perceive a meadow in one way (he will see different types of plants), but a tourist or an artist will perceive it in a completely different way.
Performance- this is a sensual image of an object that is not currently perceived by us, but which was previously perceived by us in one form or another [Getmanova, p. ten]. For example, we can visually imagine the faces of acquaintances, our room in the house, a birch tree or a mushroom. These are examples reproducing representations, as we have seen these objects.
The presentation can be creative, including fantastic. We present the beautiful Princess Swan, or Tsar Saltan, or the Golden Cockerel, and many other characters from the fairy tales of A.S. Pushkin, whom we have never seen and never will see. These are examples of creative presentation over verbal description. We also imagine the Snow Maiden, Santa Claus, a mermaid, etc.
So, the forms of sensory knowledge are sensations, perceptions and representations. With their help, we learn the external aspects of the object (its features, including properties).
Forms of abstract thinking are concepts, statements and conclusions.
Concepts. Scope and content of concepts
The term "concept" is usually used to refer to a whole class of objects of an arbitrary nature that have a certain characteristic (distinctive, essential) property or a whole set of such properties, i.e. properties that are unique to members of that class.
From the point of view of logic, the concept is a special form of thinking, which is characterized by the following: 1) the concept is a product of highly organized matter; 2) the concept reflects the material world; 3) the concept appears in consciousness as a means of generalization; 4) the concept means specifically human activity; 5) the formation of a concept in the mind of a person is inseparable from its expression through speech, writing or symbol.
How does the concept of any object of reality arise in our minds?
The process of forming a certain concept is a gradual process in which several successive stages can be seen. Consider this process using the simplest example - the formation of the concept of the number 3 in children.
1. At the first stage of cognition, children get acquainted with various specific sets, using subject pictures and showing various sets of three elements (three apples, three books, three pencils, etc.). Children not only see each of these sets, but they can also touch (touch) the objects that make up these sets. This process of "seeing" creates in the mind of the child a special form of reflection of reality, which is called perception (feeling).
2. Let's remove the objects (objects) that make up each set, and invite the children to determine whether there was something in common that characterizes each set. The number of objects in each set was to be imprinted in the minds of the children, that there were “three” everywhere. If this is so, then a new form has been created in the minds of children - idea of the number three.
3. At the next stage, on the basis of a thought experiment, children should see that the property expressed in the word "three" characterizes any set of different elements of the form (a; b; c). Thus, an essential common feature of such sets will be singled out: "to have three elements". Now we can say that in the minds of children formed concept of number 3.
concept- this is a special form of thinking, which reflects the essential (distinctive) properties of objects or objects of study.
The linguistic form of a concept is a word or a group of words. For example, “triangle”, “number three”, “point”, “straight line”, “isosceles triangle”, “plant”, “coniferous tree”, “Yenisei River”, “table”, etc.
Mathematical concepts have a number of features. The main one is that the mathematical objects about which it is necessary to form a concept do not exist in reality. Mathematical objects are created by the human mind. These are ideal objects that reflect real objects or phenomena. For example, in geometry, the shape and size of objects are studied, without taking into account their other properties: color, mass, hardness, etc. From all this they are distracted, abstracted. Therefore, in geometry, instead of the word "object" they say "geometric figure". The result of abstraction are also such mathematical concepts as "number" and "value".
Main Features any concepts are the following: 1) volume; 2) content; 3) relationships between concepts.
When they talk about a mathematical concept, they usually mean the whole set (set) of objects denoted by one term (word or group of words). So, speaking of a square, they mean all geometric shapes that are squares. It is believed that the set of all squares is the scope of the concept of "square".
The scope of the concept the set of objects or objects to which this concept is applicable is called.
For example, 1) the scope of the concept of "parallelogram" is the set of such quadrangles as parallelograms proper, rhombuses, rectangles and squares; 2) the scope of the concept of "one-digit natural number" will be the set - (1, 2, 3, 4, 5, 6, 7, 8, 9).
Any mathematical object has certain properties. For example, a square has four sides, four right angles equal to the diagonals, the diagonals are bisected by the intersection point. You can specify its other properties, but among the properties of an object there are essential (distinctive) and non-essential.
The property is called significant (distinctive) for an object if it is inherent in this object and without it it cannot exist; property is called insignificant for an object if it can exist without it.
For example, for a square, all the properties listed above are essential. The property “side AD is horizontal” will be irrelevant for the square ABCD (Fig. 1). If this square is rotated, then side AD will be vertical.
Consider an example for preschoolers using visual material (Fig. 2):
Describe the figure.
Small black triangle. Rice. 2
Big white triangle.
How are the figures similar?
How are the figures different?
Color, size.
What does a triangle have?
3 sides, 3 corners.
Thus, children find out the essential and non-essential properties of the concept of "triangle". Essential properties - "have three sides and three angles", non-essential properties - color and size.
The totality of all essential (distinctive) properties of an object or object reflected in this concept is called the content of the concept .
For example, for the concept of "parallelogram" the content is a set of properties: it has four sides, it has four corners, opposite sides are pairwise parallel, opposite sides are equal, opposite angles are equal, the diagonals at the intersection points are divided in half.
There is a connection between the volume of a concept and its content: if the volume of a concept increases, then its content decreases, and vice versa. So, for example, the scope of the concept "isosceles triangle" is part of the scope of the concept "triangle", and the content of the concept "isosceles triangle" includes more properties than the content of the concept "triangle", because an isosceles triangle has not only all the properties of a triangle, but also others inherent only in isosceles triangles (“two sides are equal”, “two angles are equal”, “two medians are equal”, etc.).
Concepts are divided into single, common and categories.
A concept whose volume is equal to 1 is called single concept .
For example, the concepts: "Yenisei River", "Republic of Tuva", "city of Moscow".
Concepts whose volume is greater than 1 are called general .
For example, the concepts: "city", "river", "quadrilateral", "number", "polygon", "equation".
In the process of studying the foundations of any science, children generally form general concepts. For example, in the elementary grades, students get acquainted with such concepts as “number”, “number”, “single-digit numbers”, “two-digit numbers”, “multi-digit numbers”, “fraction”, “share”, “addition”, “term” , "sum", "subtraction", "subtracted", "reduced", "difference", "multiplication", "multiplier", "product", "division", "divisible", "divisor", "quotient", " ball, cylinder, cone, cube, parallelepiped, pyramid, angle, triangle, quadrilateral, square, rectangle, polygon, circle , "circle", "curve", "polyline", "segment", "length of segment", "ray", "straight line", "point", "length", "width", "height", "perimeter", "figure area", "volume", "time", "speed", "mass", "price", "cost" and many others. All these concepts are general concepts.