Axiomatic method: description, stages of development and examples.

12.9. Axiomatic method For the ancient Greeks, the objects of mathematics had a real existence in the “world of ideas.” Some properties of these objects seemed completely undeniable to the mind's eye and were declared axioms, others - non-obvious - should

(Greek axioma - significant, accepted position) - a way of constructing a theory in which some true statements are selected...

(Greek axioma - significant, accepted position) - a method of constructing a theory in which some true statements are selected as initial positions (axioms), from which the remaining true statements (theorems) of this theory are then logically deduced and proven. Scientific significance of A.M. was justified by Aristotle, who was the first to divide the entire set of true statements into basic (“principles”) and those requiring proof (“provable”). In its development A.M. went through three stages. At the first stage A.M. was meaningful, axioms were accepted on the basis of their obviousness. An example of such a deductive construction of a theory is Euclid’s “Elements.” At the second stage, D. Hilbert introduced a formal criterion for the application of A.M. - the requirement of consistency, independence and completeness of the axiom system. At the third stage A.M. becomes formalized. Accordingly, the concept of “axiom” has changed. If at the first stage of development of A.M. it was understood not only as the starting point of evidence, but also as a true position that does not need proof due to its obviousness, then at present the axiom is substantiated as a necessary element of the theory, when confirmation of the latter is considered at the same time as confirmation of its axiomatic foundations as the starting point of construction . In addition to the main and introductory statements in A.M. The level of special inference rules also began to stand out. Thus, along with axioms and theorems, as the set of all true statements of a given theory, axioms and theorems for the rules of inference are formulated - metaaxioms and metatheorems. In 1931, K. Gödel proved a theorem about the fundamental incompleteness of any formal system, because it contains undecidable propositions that are both unprovable and irrefutable. Taking into account the limitations imposed on it, AM is considered as one of the main methods for constructing a developed formalized (and not just substantive) theory, along with the hypothetico-deductive method (which is sometimes interpreted as “semi-axiomatic”) and the method of mathematical hypothesis. The hypothetico-deductive method, in contrast to A.M., involves the construction of a hierarchy of hypotheses, in which weaker hypotheses are derived from stronger ones within the framework of a single deductive system, where the strength of the hypothesis increases with distance from the empirical basis of science. This allows us to weaken the power of A.M. restrictions: to overcome the closedness of the axiomatic system due to the possibility of introducing additional hypotheses that are not strictly bound by the initial provisions of the theory; introduce abstract objects of different levels of organization of reality, i.e. remove the restriction on the validity of the axiomatics “in all worlds”; remove the requirement of equality of axioms. On the other hand, A.M., in contrast to the method of mathematical hypothesis, which focuses on the very rules for constructing mathematical hypotheses related to unstudied phenomena, allows one to appeal to certain content subject areas.

V.L. Abushenko

Axiomatic Method

One of the ways of deductively constructing scientific theories, in which: 1) a certain set of accepted ones is selected without...

One of the methods of deductively constructing scientific theories, in which: 1) a certain set of propositions of a certain theory (axioms) are accepted without proof; 2) the concepts included in them are not clearly defined within the framework of this theory; 3) the rules of definition and rules of inference of a given theory are fixed, allowing one to introduce new terms (concepts) into the theory and logically derive some propositions from others; 4) all other propositions of this theory (theorem) are derived from (1) on the basis of (3). The first ideas about A. m. arose in Ancient. Greece (Eleatics, Plato. Aristotle, Euclid). Subsequently, attempts were made to provide an axiomatic presentation of various sections of philosophy and science (Spinoza, Newton, etc.) These studies were characterized by a meaningful axiomatic construction of a certain theory (and only one), while the main attention was paid to the definition and selection of intuitively obvious axioms. Starting from the second half In the 19th century, in connection with the intensive development of problems of substantiation of mathematics and mathematical logic, axiomatic theory began to be considered as a formal (and from the 20-30s of the 20th century - as a formalized) system, establishing relationships between its elements (signs) and describing any sets of objects that satisfy it. At the same time, the main attention began to be paid to establishing the consistency of the system, its completeness, the independence of the system of axioms, etc. Due to the fact that sign systems can be considered either regardless of the content that can be represented in them, or taking it into account, syntactic and semantic ones are distinguished axiomatic systems (only the latter represent scientific knowledge itself) This distinction necessitated the formulation of the basic. requirements for them, on two levels, syntactic and semantic (syntactic and semantic consistency, completeness, independence of axioms, etc.) The analysis of formalized axiomatic systems led to the establishment of their fundamental limitations, chief among which is the impossibility of complete axiomatization of sufficiently developed systems proved by Gödel scientific theories (for example, arithmetic of natural numbers), which implies the impossibility of complete formalization of scientific knowledge. Axiomatization is only one of the methods for constructing scientific knowledge, but its use as a means of scientific discovery is very limited. Axiomatization is usually carried out after the theory has already been sufficiently constructed in its content, and serves the purpose of its more accurate representation, in particular, the strict derivation of all consequences from the accepted premises. In the last 30-40 years, much attention has been paid to the axiomatization of not only mathematical disciplines, but also certain sections of physics, biology, psychology, economics, linguistics, etc., including theories of the structure and dynamics of scientific knowledge. When studying natural science (in general, any non-mathematical) knowledge, mathematical methods appear in the form of a hypothetico-deductive method (see also Formalization)

Axiomatic Method

A method of constructing a theory in which it is based on certain initial provisions - axioms or postulates...

A method of constructing a theory in which it is based on certain initial provisions - axioms or postulates, from which all other statements of this theory must be deduced in a purely logical way.

Axiomatic Method

A method of constructing a scientific theory in which some provisions of the theory are chosen as initial ones, and all the rest...

A method of constructing a scientific theory in which some provisions of the theory are chosen as initial ones, and all its other provisions are deduced from them in a purely logical way, through evidence. Statements proven on the basis of axioms are called theorems.

A. m. is a special way of defining objects and relationships between them (see: Axiomatic definition). A. m. is used in mathematics, logic, as well as in certain branches of physics, biology, etc. A. m. originated in antiquity and gained great fame thanks to Euclid’s “Elements,” which appeared around 330–320. BC e. Euclid, however, failed to describe in his “axioms and postulates” all the properties of geometric objects that he actually used; his evidence was accompanied by numerous drawings. The “hidden” assumptions of Euclid’s geometry were revealed only in modern times by D. Hilbert (1862-1943), who considered axiomatic theory as a formal theory that establishes relationships between its elements (signs) and describes any sets of objects that satisfy it. Nowadays, axiomatic theories are often formulated as formalized systems containing a precise description of the logical means of deriving theorems from axioms. A proof in such a theory is a sequence of formulas, each of which is either an axiom or is obtained from previous formulas in the sequence according to one of the accepted rules of inference.

An axiomatic formal system is subject to the requirements of consistency, completeness, independence of the system of axioms, etc.

A.M. is only one of the methods for constructing scientific knowledge. It has limited application, since it requires a high level of development of an axiomatized substantive theory.

As the famous mathematician and logician K. Gödel showed, fairly rich scientific theories (for example, the arithmetic of natural numbers) do not allow complete axiomatization. This shows the limitations of A.M. and the impossibility of complete formalization of scientific knowledge (see: Gödel’s theorem).

An axiom is the starting point, original a position of a theory that is the basis of the proofs of other provisions (for example, theorems) of this theory, within which it accepted without evidence. In everyday consciousness and language, an axiom is a certain truth that is so indisputable that it does not require evidence.

So, axiomatic method- this is one of the methods of deductive construction of a scientific theory, in which a certain set of provisions accepted without proof, called “principles”, “postulates” or “axioms”, is selected, and all other proposals of the theory are obtained as logical consequence these axioms.

The axiomatic method in mathematics originates at least from Euclid, although the term “axiom” is often found in Aristotle: “... For proof is impossible for everything: after all, a proof must be given on the basis of something regarding something and to justify something . Thus, it turns out that everything that is proved must belong to the same genus, since all proving sciences use axioms in the same way.<…>An axiom has the highest degree of generality and is the essence of the beginning of everything.<…>I call the principles of proof the generally accepted provisions on the basis of which everyone builds their proofs.<…>There is no need to ask “why” about the principles of knowledge, and each of these principles in itself must be reliable. What is plausible is what seems right to all or most people, or to the wise, to all or most of them, or to the most famous and glorious.” (See, for example, Aristotle. Works in four volumes. T. 2. Topeka. M.: Mysl, 1978. P. 349).

As can be seen from the last fragment of Aristotle’s Topics, the basis for accepting an axiom is a certain “reliability” and even authority"famous and famous" people. But at present this is not considered a sufficient reason. Modern exact sciences, including mathematics itself, do not resort to obviousness as an argument of truth: the axiom is simply introduced and accepted without proof.

David Hilbert (1862-1943), German mathematician and physicist, pointed out that the term axiomatic sometimes used in a broader and sometimes in a narrower sense of the word. With the broadest understanding of this term, we call the construction of a theory “axiomatic.” In this regard, D. Gilbert distinguishes content axiomatics and formal axiomatics.

The first “... introduces its basic concepts with reference to our experience, and either considers its main provisions to be obvious facts that can be directly verified, or formulates them as the result of a certain experience and thereby expresses our confidence that we succeeded in attacking on the trail of the laws of nature, and at the same time our intention to support this confidence with the success of the theory being developed. Formal axiomatics also needs to recognize the evidence of things of a certain kind - this is necessary both for the implementation of deduction and for establishing the consistency of the axiomatics itself - however, with the significant difference that this type of evidence is not based on any special epistemological relationship to the specific area under consideration science, but remains the same in the case of any axiomatics: we mean here such an elementary way of knowledge that it is generally a precondition for any accurate theoretical research.<…>Formal axiomatization necessarily needs a substantive one as its complement, since it is this latter that first guides us in the process of choosing appropriate formalisms, and then, when the formal theory is already at our disposal, it tells us how this theory should be applied to the field under consideration reality. On the other hand, we cannot limit ourselves to meaningful axiomatics for the simple reason that in science - if not always, then mostly - we are dealing with theories that do not completely reproduce the actual state of affairs, but are only simplifying idealization this position, which is their significance. This kind of theory, of course, cannot be justified by reference to the evidence of its axioms or experience. Moreover, its justification can be carried out only in the sense that the consistency of the idealization produced in it will be established, i.e. that extrapolation, as a result of which the concepts introduced in this theory and its main provisions exceed the boundaries of the visually obvious or the data of experience"(italics mine - Yu.E.). (Hilbert D., Bernays P. Foundations of Mathematics. M.: Nauka, 1979. P. 23.)

Thus, the modernly understood axiomatic method comes down to the following: a) select a set accepted without evidence axioms; b) the concepts included in them are not clearly defined within the framework of this theory; c) the rules of definition and rules of inference of a given theory are fixed, allowing one to logically deduce some assumptions from others; d) all other theorems are deduced from “a” on the basis of “c”. Various sections are currently built using this method. mathematicians(geometry, probability theory, algebra, etc.), physicists(mechanics, thermodynamics); attempts are being made to axiomatize chemistry And biology. Gödel proved the impossibility of complete axiomatization of sufficiently developed scientific theories (for example, the arithmetic of natural numbers), which implies the impossibility of complete formalization of scientific knowledge. When studying natural science knowledge, the axiomatic method appears in the form hypothetico-deductive method. The use of the concept “axiom” in everyday speech as a kind of a priori obviousness no longer reflects the essence of this concept. This Aristotelian understanding of this term in mathematics and natural science has now been overcome. It is appropriate to accompany the discussion of axiomatics with a fragment of the classic work of Karl Raymund Popper:

“A theoretical system can be called axiomatized if a set of axiom statements is formulated that satisfies the following four fundamental requirements: (a) the axiom system must be consistent(that is, it should not contain either self-contradictory axioms or contradictions between axioms). This is equivalent to the requirement that not every arbitrary statement is deducible in such a system. (b) The axioms of a given system must be independent, that is, the system must not contain axioms that can be derived from other axioms. (In other words, a certain statement can be called an axiom only if it is not deducible in the part of the system remaining after its removal). These two conditions relate to the axiom system itself. As for the relationship of the system of axioms to the main part of the theory, the axioms must be: (c) sufficient for the deduction of all statements belonging to the axiomatized theory, and d) necessary in the sense that the system should not contain unnecessary assumptions.<…>I consider two different interpretations of any system of axioms acceptable. The axioms can be considered either (1) as convention, either (2) as empirical, or scientific hypotheses"(Popper K.R. Logic of scientific research. M.: Respublika, 2005. P. 65).

In the history of science one can find a number of examples of the transition to an axiomatic way of presenting a theory. Moreover, the consistent application of this method to the logic of proving theorems in geometry made it possible to rethink this ancient science, opening the world of “non-Euclidean geometries” (A. I. Lobachevsky, J. Bolyai, K. Gauss, G. F. B. Riemann, etc. ). This method turned out to be convenient and productive, allowing one to build a scientific theory literally as a single crystal (this is how, in particular, theoretical mechanics and classical thermodynamics are now presented). Somewhat later, already in the 30s of the 20th century, the domestic mathematician Andrei Nikolaevich Kolmogorov (1903-1987) gave an axiomatic justification for the theory of probability, which, as historians of science confidently believe, was previously based on empirical images of gambling (“toss”, dice, cards). In this regard, it makes sense to offer the reader two fragments from the texts of the classics of science and pedagogy, who knew how to write, as Berdyaev said, not only “about something,” but also “something.”

R. Courant and G. Robbins: “There is one axiom in the Euclidian system, regarding which - on the basis of comparison with empirical data, using taut threads or light rays - it is impossible to say whether it is “true.” This is famous postulate about parallel, which states that through a given point located outside a given line one can draw one and only one line parallel to this one. A peculiar feature of this axiom is that the statement contained in it concerns the properties of the straight line along its entire length, and the line is assumed to be extended indefinitely in both directions: to say that two lines are parallel means to say that they cannot find a common point, no matter how far they are extended, It is quite obvious that within a certain limited part of the plane, no matter how extensive this part is, on the contrary, it is possible to draw through a given point many straight lines that do not intersect with a given straight line. Since the maximum possible length of a ruler, a thread, even a light beam traced with a telescope is certainly finite, and since inside a circle of finite radius there are many straight lines passing through a given point and not meeting a given straight line within the circle, it follows that Euclid's postulate can never be verified experimentally.<…>The Hungarian mathematician Bolyai and the Russian mathematician Lobachevsky put an end to doubts by constructing in all details a geometric system in which the axiom of parallelism was rejected. When Bolyai sent his work to the “king of mathematics” Gauss, from whom he was eagerly awaiting support, he received in response a notification that Gauss himself had made the discovery earlier, but he had refrained from publishing the results at the time, fearing too noisy discussions.

Let's see what the independence of the parallelism axiom means. This independence should be understood in the sense that it is possible to construct “geometric” sentences about points, lines, etc., free from internal contradictions, based on a system of axioms in which the parallelism axiom is replaced by its opposite. This construction is called non-Euclidean geometry(italics mine - Yu.E.). It took the intellectual fearlessness of Gauss, Bolyai and Lobachevsky to realize that geometry, not based on the Euclidean system of axioms, may be completely consistent(italics mine - Yu.E.).<…>We are now able to build simple “models” of such geometry that satisfy all the axioms of Euclid, except the axiom of parallelism” (R. Kurant, G. Robbins. What is mathematics? M.: Prosveshchenie, 1967. P. 250).

Various versions of non-Euclidean geometries (for example, Riemann geometry, as well as geometry in space of more than three dimensions) later found application in the construction of theories related to the microworld (relativistic quantum mechanics, particle physics) and, conversely, to the megaworld (general relativity) .

Finally, the opinion of Russian mathematician Andrei Nikolaevich Kolmogorov: “Probability theory or a mathematical discipline can and should be axiomatized in exactly the same sense as geometry or algebra. This means that, after the names of the objects under study and their basic relations are given, as well as the axioms to which these relations must obey, all further presentation should be based exclusively on these axioms, without relying on the usual concrete meaning of these objects and their relationships(italics mine - Yu.E.).<…>Any axiomatic (abstract) theory allows, as is known, an infinite number of specific interpretations. Thus, the mathematical theory of probability allows, along with those interpretations from which it arose, also many others.<…>Axiomatization of probability theory can be carried out in various ways, both in relation to the choice of axioms and the choice of basic concepts and basic relations. If we pursue the goal of possible simplicity of both the axiom system itself and the construction of a further theory from it, then it seems most appropriate to axiomatize the concepts of a random event and its probability. There are also other systems for the axiomatic construction of probability theory, namely those in which the concept of probability is not one of the basic concepts, but is itself expressed through other concepts [footnote: Cf., for example, von Mises R. Wahrscheinlichkeitsrechnung, Leipzig u. Wien, France Deuticke, 1931; Bernstein S.N. Theory of Probability, 2nd ed., Moscow, GTTI, 1934]. At the same time, they strive, however, for another goal, namely, as close as possible to the closest connection between mathematical theory and the empirical emergence of the concept of probability” (Kolmogorov A.N. Basic concepts of probability theory. M.: Nauka, 1974. P. 9).

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, and by a straight line and 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. The most consistent expression of the new view of axioms as abstract forms that allow many different interpretations was 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 of 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, the main value of axiomatization, especially in mathematics, is that it acts as a method for exploring 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, it is possible to axiomatize only the main content of natural science theories, leaving the possibility for further development and refinement of the theory through 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.