Axiomatic method of constructing a scientific theory in mathematics. Axiomatic method of constructing a scientific theory Axiomatic method of constructing a theory

The axiomatic method is a way of constructing mathematical theory, in which the basis is based on some provisions that are accepted without proof (axioms), and all the rest are derived from them in a purely logical way. With a radical application of this approach, mathematics is reduced to pure logic, such things as intuition, visual geometric representations, inductive reasoning, and so on are expelled from it. What is the essence of mathematical creativity disappears. Why then was this method invented? To answer this question we need to go back to the very beginnings of mathematics.

1. Axioms: two understandings

As we remember from school, mathematical proofs, axioms and theorems appeared in Ancient Greece. The axiomatic construction of geometry was canonized in the book from which many generations were taught mathematics - in Euclid's Elements. However, in those days the concept of an axiom was understood differently than it is now. Until now, school textbooks sometimes say that axioms are obvious truths accepted without proof. In the 19th century, this concept changed a lot because the word “obvious” disappeared. Axioms are no longer obvious; they are still accepted without proof, but can, in principle, be completely arbitrary statements. Behind this small, at first glance, change is a rather radical change in philosophical position - a refusal to recognize the only possible mathematical reality. The main role in this change, of course, was played by the history of the emergence of non-Euclidean geometry, which occurred in the 19th century thanks to the work of such scientists as N. I. Lobachevsky and J. Bolyai.

2. The problem of the parallel lines axiom

The history of non-Euclidean geometry began with attempts to prove the so-called fifth postulate of Euclid - the famous axiom of parallels: through a point outside a line, no more than one line can be drawn parallel to the given one. This statement was noticeably different in nature from the rest of Euclid’s axioms. It seemed to many that it needed to be proven; it was not as obvious as the other axioms. These attempts were not successful for centuries; many mathematicians proposed their own “solutions”, in which other mathematicians subsequently found errors. (Now we know that these attempts were obviously doomed to failure; this was one of the first examples of unprovable mathematical statements).

3. Lobachevsky geometry

Only in the 19th century was it realized that perhaps this statement was in fact unprovable and that there was some other geometry, completely different from ours, in which this axiom was false. What did Lobachevsky do? He did what mathematicians often do when trying to prove a statement. A favorite technique is proof by contradiction: suppose that this statement wrong. What follows from this? To prove the theorem, mathematicians try to derive a contradiction from the assumption made. But in this case, Lobachevsky received more and more new mathematical, geometric consequences from the assumption made, but they lined up into a very beautiful, internally consistent system, which nevertheless differed from the Euclidean one we are used to. A new world of non-Euclidean geometry, unlike the one we are used to, was unfolding before his eyes. This led Lobachevsky to the realization that such geometry was possible. At the same time, the axiom of parallels in Lobachevsky’s geometry clearly contradicted our everyday geometric intuition: not only was it not intuitively obvious, but from the point of view of this intuition it was false.

However, it is one thing to imagine that this is possible in principle, and another to prove strictly mathematically that such a system of axioms for geometry is consistent. This was achieved several decades later in the works of other mathematicians - Beltrami, Klein and Poincaré, who proposed models of the axioms of non-Euclidean geometry within the framework of ordinary Euclidean geometry. They actually established that the inconsistency of Lobachevsky's geometry would entail the inconsistency of the Euclidean geometry familiar to us. The opposite is also true, that is, from the point of view of logic, both systems turn out to be completely equal.

Having said that, there is one caveat that needs to be made. The history of non-Euclidean geometry is well illustrated by another phenomenon observed more than once in the history of science. Sometimes the solution to a problem arises not after, but before the problem itself receives a precise formulation that is well understood by everyone. This was the case in this case: in the middle of the 19th century full list the axioms of elementary geometry did not yet exist. Euclid's Elements were not sufficiently consistent in terms of their implementation of the axiomatic method. Many of Euclid's arguments appealed to visual intuition; his axioms were clearly not enough even for a meaningful formulation of the problem of the unprovability of the parallel postulate. Lobachevsky with Bolyai, and Beltrami with Klein and Poincaré were in a similar position. Setting the problem of unprovability at the proper level of rigor required the development of a completely new apparatus of mathematical logic and that same axiomatic method.

4. Creation of an axiomatic method

The situation was comprehended after the publication of D. Hilbert’s book “Foundations of Geometry”; he proposed the concept of the axiomatic method with which we began. Hilbert realized that in order to understand the foundations of geometry, it was necessary to completely exclude from the axioms everything except logic. He colorfully expressed this idea as follows: “The validity of the axioms and theorems will not be shaken at all if we replace the usual terms “point, line, plane” with others, equally conventional: “chair, table, beer mug”!

It was Hilbert who constructed the first consistent and complete system of axioms for elementary geometry, this happened at the very end of the 19th century. Thus, the axiomatic method was actually created in order to prove the impossibility of proving certain, in this case geometric, statements.

Hilbert was proud of his discovery and thought that this method could be extended to all mathematics as a whole: not only to elementary geometry, but also to arithmetic, analysis, and set theory. He proclaimed the "Hilbert Program", the goal of which was to develop systems of axioms for all parts of mathematics (and even parts of physics) and then establish the consistency of mathematics by limited means. As soon as Hilbert realized the possibilities of the axiomatic method, it seemed that a direct path was open for such development. Hilbert even uttered a famous phrase in 1930, which translated into Russian sounds like “We must know, and we will know,” meaning that everything that mathematicians should know, they will sooner or later learn. This goal, however, turned out to be unrealistic, which became clear much later. What's most amazing is that the theorem that effectively refuted these hopes, Kurt Gödel's incompleteness theorem, was announced at the same conference in 1930 at which Hilbert gave his famous speech, exactly one day before this event.

5. Possibilities of the axiomatic method

Hilbert's axiomatic method allows one to build mathematical theories on clearly defined mathematical statements, from which others can be derived logically. Hilbert actually went further and decided that the reduction of mathematics to logic could be continued. You can further ask the question: “Is it possible to get rid of the explanation of the meaning of what a logical operation is?” Logic itself can be removed from the axiomatic method. From axiomatic theories we move on to formal axiomatic theories - these are theories written in symbolic form, while mathematics turns not just into a sequence of logical conclusions, but into some kind of game of rewriting formal expressions according to certain rules. It is this game, which makes absolutely no sense if you look at it naively, that provides the exact mathematical model of what a “proof” is. By analyzing this game, one can prove that mathematical theorems cannot be proven. But the main thing: as a result of formalization, mathematicians for the first time built fully formalized languages, which led to the creation of programming languages ​​and database languages. Modern development Computer technology is ultimately based on discoveries that were made in mathematics at the beginning of the 20th century.

6. Criticism of the axiomatic method

Many mathematicians criticize the axiomatic method for what it was created for: it takes the meaning out of mathematics. Because first we rid mathematics of various geometric concepts, of intuition. Moving on to a formal axiomatic theory, we, in general, banish logic from mathematics. And as a result, all that remains of the substantive proof is a skeleton consisting of formal symbols. The advantage of the latter is precisely that we do not know what “meaning” and “intuition” are, but we know exactly what manipulations with finite strings of characters are. This allows us to build an accurate mathematical model of a complex phenomenon - evidence - and subject it to mathematical analysis.

Mathematical proof was originally a psychological process of convincing an interlocutor of the correctness of a particular statement. In the formal system this is not the case: everything has been reduced to a purely mechanical process. This purely mechanical process can be performed by a computer. However, like any model, the mechanical process conveys only some of the features of real evidence. This model has its limits of applicability. It is wrong to think that formal proofs are “real” mathematical proofs or that mathematicians actually work within certain formal systems.

Separately, it is worth mentioning the teaching of mathematics. There is nothing worse than basing schoolchildren’s education on performing mechanical actions (algorithms) or on constructing formal logical conclusions. This way you can ruin any creative beginning in a person. Accordingly, when teaching mathematics, you should not approach it from the position of a strict axiomatic method in the sense of Hilbert - that is not what it was created for.

The axiomatic method was first successfully applied by Euclid to construct elementary geometry. Since that time, this method has undergone significant evolution and has found numerous applications not only in mathematics, but also in many branches of exact natural science (mechanics, optics, electrodynamics, relativity theory, cosmology, etc.).

The development and improvement of the axiomatic method occurred along two main lines: firstly, the generalization of the method itself and, secondly, the development of logical techniques used in the process of deriving theorems from axioms. To more clearly imagine the nature of the changes that have taken place, let us turn to the original axiomatics of Euclid. As is known, the initial concepts and axioms of geometry are interpreted in one and only way. By point, line and plane, as the basic concepts of geometry, idealized spatial objects are meant, and geometry itself is considered as the study of the properties of physical space. It gradually became clear that Euclid's axioms turned out to be true not only for describing the properties of geometric, but also other mathematical and even physical objects. So, if by a point we mean a triple of real numbers, by a straight line or a plane - the corresponding linear equations, then the properties of all these non-geometric objects will satisfy the geometric axioms of Euclid. Even more interesting is the interpretation of these axioms with the help of physical objects, for example, the states of a mechanical and physicochemical system or the variety of color sensations. All this indicates that the axioms of geometry can be interpreted using objects of a very different nature.

This abstract approach to axiomatics was largely prepared by the discovery of non-Euclidean geometries by N. I. Lobachevsky, J. Bolyai, C. F. Gauss and B. Riemann. Most consistent expression A New Look on axioms as abstract forms that allow many different interpretations, found in the famous work of D. Hilbert “Foundations of Geometry” (1899). “We think,” he wrote in this book, “of three different systems of things: we call the things of the first system points and denote A, B, C,...; We call things of the second system direct and denote a, b, c,...; We call things of the third system planes and designate them as a, B, y,...". From this it is clear that by “point”, “straight line” and “plane” we can mean any system of objects. It is only important that their properties are described by the corresponding axioms. The next step on the path to abstraction from the content of axioms is associated with their symbolic representation in the form of formulas, as well as the precise specification of those rules of inference that describe how from some formulas (axioms) other formulas (theorems) are obtained. As a result of this, meaningful reasoning with concepts at this stage of research turns into some operations with formulas according to pre-prescribed rules. In other words, meaningful thinking is reflected here in calculus. Axiomatic systems of this kind are often called formalized syntactic systems, or calculi.

All three types of axiomatization considered are used in modern science. Formalized axiomatic systems are resorted to mainly when studying the logical foundations of a particular science. Such research has gained the greatest scope in mathematics in connection with the discovery of paradoxes in set theory. Formal systems play a significant role in the creation of special scientific languages, with the help of which it is possible to eliminate as much as possible the inaccuracies of ordinary, natural language.

Some scientists consider this point to be almost the main thing in the process of applying logical-mathematical methods in specific sciences. Thus, the English scientist I. Woodger, who is one of the pioneers of the use of the axiomatic method in biology, believes that the application of this method in biology and other branches of natural science consists in creating a scientifically perfect language in which calculus is possible. The basis for constructing such a language is an axiomatic method, expressed in the form of a formalized system, or calculus. The initial symbols of two types serve as the alphabet of a formalized language: logical and individual.

Logical symbols represent logical connections and relationships common to many or most theories. Individual symbols represent objects of the theory under study, such as mathematical, physical or biological. Just as a certain sequence of letters of the alphabet forms a word, so a finite collection of ordered symbols forms the formulas and expressions of a formalized language. To distinguish meaningful expressions of a language, the concept of a correctly constructed formula is introduced. To complete the process of constructing an artificial language, it is sufficient to clearly describe the rules for deriving or converting one formula to another and highlight some correctly constructed formulas as axioms. Thus, the construction of a formalized language occurs in the same way as the construction of a meaningful axiomatic system. Since meaningful reasoning with formulas is unacceptable in the first case, the logical derivation of consequences here comes down to performing precisely prescribed operations for handling symbols and their combinations.

The main purpose of using formalized languages ​​in science is a critical analysis of the reasoning with the help of which new knowledge in science is obtained. Since formalized languages ​​reflect some aspects of meaningful reasoning, they can also be used to assess the possibilities of automating intellectual activity.

Abstract axiomatic systems are most widely used in modern mathematics, which is characterized by an extremely general approach to the subject of research. Instead of talking about concrete numbers, functions, lines, surfaces, vectors and the like, the modern mathematician considers various sets of abstract objects, the properties of which are precisely formulated by means of axioms. Such collections, or sets, together with the axioms that describe them, are now often called abstract mathematical structures.

What advantages will the axiomatic method give to mathematics? Firstly, it significantly expands the scope of application of mathematical methods and often facilitates the research process. When studying specific phenomena and processes in a particular area, a scientist can use abstract axiomatic systems as ready-made tools of analysis. Having made sure that the phenomena under consideration satisfy the axioms of some mathematical theory, the researcher can immediately use all the theorems that follow from the axioms without additional labor-intensive work. The axiomatic approach saves a specialist in a specific science from performing rather complex and difficult mathematical research.

For a mathematician, this method makes it possible to better understand the object of research, highlight the main directions in it, and understand the unity and connection of different methods and theories. The unity that is achieved with the help of the axiomatic method, in the figurative expression of N. Bourbaki, is not the unity “that gives a skeleton devoid of life. It is the nutritious juice of the body in full development, a malleable and fruitful research instrument...” Thanks to the axiomatic method, especially in its formalized form, it becomes possible to fully reveal the logical structure various theories. In its most perfect form, this applies to mathematical theories. In natural science knowledge we have to limit ourselves to axiomatizing the main core of theories. Further, the use of the axiomatic method makes it possible to better control the course of our reasoning, achieving the necessary logical rigor. However main value axiomatization, especially in mathematics, is that it acts as a method for studying new patterns, establishing connections between concepts and theories that previously seemed isolated from each other.

The limited use of the axiomatic method in natural science is explained primarily by the fact that its theories must constantly be monitored by experience.

Because of this, natural science theory never strives for complete completeness and isolation. Meanwhile, in mathematics they prefer to deal with systems of axioms that satisfy the requirement of completeness. But as K. Gödel showed, any consistent system of axioms of a non-trivial nature cannot be complete.

The requirement for consistency of a system of axioms is much more important than the requirement for their completeness. If a system of axioms is contradictory, it will not be of any value for knowledge. By limiting ourselves to incomplete systems, we can axiomatize only the main content naturally scientific theories, leaving the possibility for further development and refinement of the theory by experiment. Even such a limited goal in a number of cases turns out to be very useful, for example, for discovering some implicit premises and assumptions of the theory, monitoring the results obtained, their systematization, etc.

The most promising application of the axiomatic method is in those sciences where the concepts used have significant stability and where one can abstract from their change and development.

It is under these conditions that it becomes possible to identify formal-logical connections between the various components of the theory. Thus, the axiomatic method, to a greater extent than the hypothetico-deductive method, is adapted for the study of ready-made, achieved knowledge.

Analysis of the emergence of knowledge and the process of its formation requires turning to materialist dialectics, as the most profound and comprehensive doctrine of development.

Axiomatic method of constructing a scientific theory in mathematics

The axiomatic method appeared in Ancient Greece, and is now used in all theoretical sciences, primarily in mathematics.

The axiomatic method of constructing a scientific theory is as follows: basic concepts are identified, the axioms of the theory are formulated, and all other statements are deduced logically, based on them.

The main concepts are highlighted as follows. 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 come to elementary concepts that cannot be defined through 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 proven; they had to be justified. 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, based on it, further statements can be proven.

Having identified the basic concepts and formulated axioms, we then derive theorems and other concepts in a logical way. This is the logical structure of geometry. Axioms and basic concepts constitute 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 the 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 the formulation and proof of 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 in geometry based on definitions and axioms. This course and its subsequent treatments were called "Elements".

Then, in the 3rd century. BC, a book of Euclid with the same name appeared in Alexandria, in the Russian translation of “Beginnings”. The term “elementary geometry” comes from the Latin name “Beginnings”. Despite the fact that the works of Euclid's predecessors have not reached us, we can form some opinion about these works based on Euclid's Elements. In the "Principles" there are sections that are logically very little connected with other sections. Their appearance can only be explained by the fact that they were introduced according to tradition and copy the “Elements” 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 constructed using a compass and ruler. Books 11 to 13 were devoted to stereometry.

The Principia begins 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 using an ideal ruler, the third - using an ideal compass. The fourth, “all right angles are equal to each other,” is redundant, since it can be deduced from the remaining axioms. The last, fifth postulate read: “If a straight line falls on two straight lines and forms internal one-sided angles in the sum of less than two straight lines, then, with an unlimited extension of these two straight lines, they will intersect on the side where the angles are less than two straight lines.”

Five " general concepts“Euclidean principles of measuring lengths, angles, areas, volumes: “equals are equal to each other”, “if equals are added to equals, the sums are equal”, “if equals are subtracted from equals, the remainders are equal”, “those combined with each other are equal to each other,” “the whole is greater than the part.”

Next began criticism of Euclid's geometry. Euclid was criticized for three reasons: because he considered only those geometric quantities that can be constructed using a compass and ruler; for the fact that he separated geometry and arithmetic and proved for integers what he had already proved for geometric quantities, and, finally, for the axioms of Euclid. The most heavily criticized postulate was the fifth, Euclid's most complex postulate. Many considered it superfluous, and that it could and should be deduced from other axioms. Others believed that it should be replaced by a simpler and more obvious one, equivalent to it: “Through a point outside a line, no more than one straight line can be drawn in their plane that does not intersect the given line.”

Criticism of the gap between geometry and arithmetic led to the expansion of the concept of number to real number. Disputes about the fifth postulate led to the fact that at the beginning of the 19th century N.I. Lobaczewski, J. Bolyai and K.F. Gauss constructed 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 Euclid's geometry.

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

Let's draw a horizontal line on the Euclidean plane (see Figure 1). This line is called absolute (x). Points of the Euclidean plane lying above the absolute are 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 segments of straight lines perpendicular to the absolute (AB, CD). A figure on the Lobachevsky plane is a 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 transferred to the other by a non-Euclidean motion. These are the basic concepts of the axiomatics of Lobachevsky planimetry.

All axioms of Lobachevsky planimetry are consistent. The definition of a straight line is as follows: “A non-Euclidean straight line is a semicircle with ends at the absolute or a ray with a beginning at the absolute and perpendicular to the absolute.” Thus, the statement of Lobachevsky’s parallelism axiom is satisfied 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 (see Figure 2).

After Lobachevsky’s geometry, other consistent geometries arose: projective geometry separated from Euclidean, multidimensional Euclidean geometry emerged, Riemannian geometry arose (the general theory of spaces with an arbitrary law for 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 ancestor - Euclidian geometry. 60,896.

This method is used to construct theories of mathematics and exact science. The advantages of this method were realized back in the third century by Euclid when constructing a system of knowledge on elementary geometry. In the axiomatic construction of theories, a minimum number of initial concepts and statements are precisely distinguished from the rest. An axiomatic theory is understood as a scientific system, all provisions of which are derived purely logically from a certain set of provisions accepted in this system without proof and called axioms, and all concepts are reduced to a certain fixed class of concepts called indefinable. The theory is defined if the system of axioms and the set of logical means used - the rules of inference - are specified. Derived concepts in axiomatic theory are abbreviations for combinations of basic ones. The admissibility of combinations is determined by axioms and rules of inference. In other words, definitions in axiomatic theories are nominal.

An axiom must be logically stronger than other statements that are derived from it as consequences. The system of axioms of a theory potentially contains all the consequences, or theorems, that can be proven with their help. Thus, all the essential content of the theory is concentrated in it. Depending on the nature of the axioms and means of logical inference, the following are distinguished:

• 1) formalized axiomatic systems, in which axioms are initial formulas, and theorems are obtained from them according to certain and precisely listed transformation rules, as a result of which the construction of a system turns into a kind of manipulation with formulas. Appeal to such systems is necessary in order to present the initial premises of the theory and logical means of conclusion as accurately as possible. axioms. The failure of Lobachevsky's attempts to prove Euclid's parallel axiom led him to the conviction that another geometry was possible. If the doctrine of axiomatics and mathematical logic had existed at that time, then erroneous proofs could have easily been avoided;
• 2) semi-formalized or abstract axiomatic systems, in which the means of logical inference are not considered, but are assumed to be known, and the axioms themselves, although they allow many interpretations, do not act as formulas. Such systems are usually dealt with in mathematics;
• 3) meaningful axiomatic systems assume a single interpretation, and the means of logical inference are known; are used to systematize scientific knowledge in exact natural sciences and other developed empirical sciences.

A significant difference between mathematical axioms and empirical ones is also that they have relative stability, while in empirical theories their content changes with the discovery of new important results of experimental research. It is with them that we constantly have to take into account when developing theories, therefore axiomatic systems in such sciences can never be either complete or closed for derivation.

The axiomatic method is one of the ways of deductively constructing scientific theories, in which:
1. a certain set of propositions of a certain theory (axioms) accepted without proof is selected;
2. the concepts included in them are not clearly defined within the framework of this theory;
3. the rules of definition and rules for choosing a given theory are fixed, allowing one to introduce new terms (concepts) into the theory and logically deduce some proposals from others;
4. all other propositions of this theory (theorem) are derived from 1 on the basis of 3.

In mathematics, AM originated in the works of ancient Greek geometers. Brilliant, remaining the only one until the 19th century. The model for using AM was geometric. system known as Euclid's "Beginnings" (c. 300 BC). Although at that time the question of describing the logic did not yet arise. means used to extract meaningful consequences from axioms, in the Euclidian system the idea of ​​​​obtaining the entire basic content of geometrics is already quite clearly carried out. theories by a purely deductive method from a certain, relatively small number of statements - axioms, the truth of which seemed clearly obvious.

Opening at the beginning 19th century non-Euclidean geometry by N. I. Lobachevsky and J. Bolyai was the impetus for the further development of AM. They established that, replacing the usual and, it would seem, the only “objectively true” V postulate of Euclid about parallels with its negation, You can develop purely logical. by geometric a theory as harmonious and rich in content as Euclid’s geometry. This fact forced mathematicians of the 19th century. pay special attention to the deductive method of constructing mathematical. theories, which led to the emergence of new problems associated with the very concept of mathematical mathematics, and formal (axiomatic) mathematical. theories. As axiomatic experience accumulated. presentation of mathematical theories - here it is necessary to note, first of all, the completion of a logically impeccable (in contrast to Euclid's Elements) construction of elementary geometry [M. Pash (M. Pasch), J. Peano (G. Peano), D. Hilbert (D. Hilbert)] and the first attempts to axiomatize arithmetic (J. Peano), - the concept of formal axiomatic was clarified. systems (see below); a specific feature arose. problems on the basis of which the so-called evidence theory as the main section of modern mathematics. logic.

Understanding of the need to substantiate mathematics and specific tasks in this area arose in a more or less clear form already in the 19th century. At the same time, on the one hand, clarification of basic concepts and reduction of more complex concepts to the simplest on a precise and logically more and more strict basis were carried out by Ch. arr. in the field of analysis [A. Cauchy, functional-theoretic concepts of B. Bolzano and K. Weierstrass, continuum of G. Cantor and R. Dedekind (R .Dedekind)]; on the other hand, the discovery of non-Euclidean geometries stimulated the development of mathematical mathematics, the emergence of new ideas and the formulation of problems of more general metamathematics. character, first of all, problems associated with the concept of arbitrary axiomatic. theories, such as problems of consistency, completeness and independence of a particular system of axioms. The first results in this area were brought by the method of interpretation, which can roughly be described as follows. Let each initial concept and relation of a given axiomatic. theory T is put in correspondence with a certain concrete mathematical theory. an object. The collection of such objects is called. field of interpretation. Every statement of theory T is now naturally associated with a certain statement about the elements of the field of interpretation, which can be true or false. Then the statement of theory T is said to be true or false, respectively, under that interpretation. The field of interpretation and its properties themselves are usually the object of consideration of a mathematical theory, generally speaking another, mathematical one. theory T 1, in particular, can also be axiomatic. The method of interpretation allows us to establish the fact of relative consistency in the following way, that is, to prove propositions like: “if theory T 1 is consistent, then theory T is also consistent.” Let theory T be interpreted in theory T 1 in such a way that all axioms of theory T are interpreted by true judgments of theory T 1 . Then every theorem of the theory T, i.e., every statement A logically deduced from the axioms in T, is interpreted in T 1 by a certain statement deduced in T 1 from the interpretations of the axioms A i, and therefore true. The last statement is based on another assumption that we implicitly make of a certain similarity of logical. means of theories T and T 1, but in practice this condition is usually met. (At the dawn of the application of the method of interpretation, this assumption was not even specifically thought about: it was taken for granted; in fact, in the case of the first experiments, the proofs of theorems on the relative consistency of the logical means of theories T and T 1 simply coincided - this was the classical logic of predicates. ) Now let theory T be contradictory, that is, some assertion A of this theory can be deduced in it along with its negation. Then from the above it follows that the statements and will at the same time be true statements of the theory T 1, i.e., that the theory T 1 is contradictory. This method was, for example, proven [F. Klein (F. Klein), A. Poincare (N. Poincare)] consistency of non-Euclidean Lobachevsky geometry under the assumption that Euclidean geometry is consistent; and the question of the consistency of the Hilbert axiomatization of Euclidean geometry was reduced (D. Hilbert) to the problem of the consistency of arithmetic. The method of interpretation also allows us to solve the question of the independence of systems of axioms: to prove that the axiom of Atheory T does not depend on the other axioms of this theory, that is, it is not deducible from them, and, therefore, is essential to obtain the entire scope of this theory, it is enough construct such an interpretation of theory T, in which the Axiom Abyl would be false, and all other axioms of this theory would be true. Another form of this method of proving independence is the establishment of the consistency of the theory, which is obtained if in a given theory TaxiomA is replaced by its negation. The above-mentioned reduction of the problem of the consistency of Lobachevsky's geometry to the problem of the consistency of Euclidean geometry, and this latter - to the question of the consistency of arithmetic, has as its consequence the statement that Euclid's postulate is not deducible from the other axioms of geometry, unless arithmetic is consistent natural numbers. The weakness of the interpretation method is that in matters of consistency and independence of axiom systems, it makes it possible to obtain results that are inevitably only relative in nature. But an important achievement of this method was the fact that with its help the special role of arithmetic as such a mathematical science was revealed on a fairly accurate basis. theories, a similar question for a number of other theories is reduced to the question of consistency.

Further development- and in a certain sense this was the peak - AM received in the works of D. Hilbert and his school in the form of the so-called. method formalism in the foundations of mathematics. Within the framework of this direction, the next stage of clarifying the concept of axiomatic was developed. theories, namely the concept formal system. As a result of this clarification, it became possible to represent the mathematical ones themselves. theories as exact mathematical objects and build a general theory, or metatheory, such theories. At the same time, the prospect seemed tempting (and D. Hilbert was at one time fascinated by it) to solve all the main questions of the foundation of mathematics along this path. The main concept of this direction is the concept of a formal system. Any formal system is constructed as a precisely defined class of expressions - formulas, in which a subclass of formulas, called formulas, is distinguished in a certain precise way. theorems of this formal system. At the same time, the formulas of a formal system do not directly carry any meaningful meaning, and they can be constructed from arbitrary, generally speaking, icons or elementary symbols, guided only by considerations of technical convenience. In fact, the method of constructing formulas and the concept of a theorem of a particular formal system are chosen in such a way that this entire formal apparatus can be used to express, perhaps more adequately and completely, a particular mathematical (and non-mathematical) theory, more precisely, as its factual content and its deductive structure. The general scheme for constructing (specifying) an arbitrary formal system S is as follows.

I. System S language:

a) alphabet - a list of elementary symbols of the system;

b) rules of formation (syntax) - rules according to which formulas of the system S are constructed from elementary symbols; in this case, a sequence of elementary symbols is considered a formula if and only if it can be constructed using the rules of formation.

II. Axioms of the system S. A certain set of formulas (usually finite or enumerable) is identified, which are called. axioms of the system S.

III. System withdrawal rules S. A (usually finite) set of predicates is fixed on the set of all formulas of the system S. Let - k.-l. of these predicates, if the statement is true for these formulas, then they say that the formula follows directly from the formulas according to the rule

7. Probability theory:

Probability theory – a mathematical science that studies patterns in random phenomena. One of the basic concepts of probability theory is the concept random event (or simply events ).

Event is any fact that may or may not happen as a result of experience. Examples of random events: a six falling out when throwing a dice, a failure of a technical device, a distortion of a message when transmitting it over a communication channel. Some events are associated with numbers , characterizing the degree of objective possibility of the occurrence of these events, called probabilities of events .

There are several approaches to the concept of “probability”.

The modern construction of probability theory is based on axiomatic approach and is based on elementary concepts of set theory. This approach is called set-theoretic.

Let some experiment be carried out with a random outcome. Consider the set W of all possible outcomes of the experiment; we will call each of its elements elementary event and the set Ω is space of elementary events. Any event A in the set-theoretic interpretation there is a certain subset of the set Ω: .

Reliable is called the event W that occurs in each experiment.

Impossible is called an event Æ, which cannot occur as a result of experiment.

Incompatible are events that cannot occur simultaneously in the same experience.

Amount(combination) of two events A And B(denoted A+B, AÈ B) is an event that consists in the fact that at least one of the events occurs, i.e. A or B, or both at the same time.

The work(intersection) of two events A And B(denoted A× B, AÇ B) is an event where both events occur A And B together.

Opposite to the event A such an event is called, which is that the event A not happening.

Events A k(k=1, 2, …, n) form full group , if they are pairwise incompatible and in total form a reliable event.

Probability of the eventA they call the ratio of the number of outcomes favorable to this event to the total number of all equally possible incompatible elementary outcomes that form the complete group. So, the probability of event A is determined by the formula

where m is the number of elementary outcomes favorable to A; n is the number of all possible elementary test outcomes.

Here it is assumed that the elementary outcomes are incompatible, equally possible and form a complete group. The following properties follow from the definition of probability:
Its own article 1. The probability of a reliable event is equal to one. Indeed, if the event is reliable, then every elementary outcome of the test favors the event. In this case m = n, therefore,

P (A) = m / n = n / n = 1.

S in about with t in about 2. The probability of an impossible event is zero. Indeed, if an event is impossible, then none of the elementary outcomes of the test favor the event. In this case m = 0, therefore,

P (A) = m / n = 0 / n = 0.

With in about with t in about 3. The probability of a random event is a positive number between zero and one.Indeed, a random event favors only part of total number elementary test outcomes. In this case 0< m < n, значит, 0 < m / n < 1, следовательно,

0 <Р (А) < 1

So, the probability of any event satisfies the double inequality