DOI: https://doi.org/10.62204/2336-498X-2025-1-15

AXIOMATIC METHOD AS A SCIENTIFIC BASIS

OF MODERN GEOMETRY COURSES

 

Natalia Shapovalova,

Ph.D. in Physico-mathematical Sciences, Associate Professor,
Mykhailo Dragomanov State University of Ukraine,
n.v.shapovalova@udu.edu.ua; ORCID: 0009-0000-7084-1460

Larisa Panchenko,

Ph.D. in Pedagogy, Associate Professor,
Mykhailo Dragomanov State University of Ukraine,
larpan97@gmail.com; ORCID: 0009-0001-8156-286X

Olga Mihulova,

Master of Mathematics,
Mykhailo Dragomanov State University of Ukraine,
rukovoditel10b@gmail.com; ORCID: 0009-0008-9916-0560

 

Annotation. The article reveals the essence and substantiates the expediency of using the axiomatic method of constructing geometry, both Euclidean and non-Euclidean, in institutions of higher education. The expediency of using the axiomatic method as a logical basis for constructing a geometry course is analyzed. Different approaches to building the axiomatics of Euclidean geometry are analyzed. Different systems of substantiation of Euclidean geometry are described. The problems of constructing a geometric theory based on the axiomatic method are studied. The specifics of the application of the axiomatic method in the study of Euclidean geometry in institutions of higher and secondary education are described. The main methodical aspects of these processes and the improved scientific principles of teaching geometry in institutions of higher and secondary education are revealed.

Keywords: geometry, basics of geometry, Euclidean geometry, axiomatic method of construction of geometry, educational process, learning, scientific approach, logical basis.

 

Problem statement. The purpose of the article is to, with the help of available sources, scientific and methodical literature, own experience, reveal the essence and justify the expediency of using the axiomatic method of constructing geometry, both Euclidean and non-Euclidean. To analyze different approaches to building the axiomatics of Euclidean geometry. To describe the specifics of the application of the axiomatic method in the study of Euclidean geometry in institutions of higher and secondary education. To reveal the main methodical aspects of these processes and to improve the scientific principles of teaching geometry in institutions of higher and secondary education.

Presenting main material. The axiomatic method is a way of building a scientific theory, where the basis of the theory is based on some initial propositions, which are called the axioms of the theory, and all other statements of the theory are obtained as logical consequences of the axioms.

The axiomatic or deductive method is one of the main methods of building a scientific theory. Its application assumes that a rigorous scientific construction of any mathematical theory must satisfy the following requirements:

  1. Any statement must be among the list of axioms or rigorously proven on the basis of axioms and previously formulated and proven theorems.
  2. Any concept must be either among the main ones or defined with the help of the main and previously defined concepts.

The axiomatic method is that:        

  1. The main concepts are listed and named.
  2. Certain laws are formulated that express the properties of these basic concepts (axioms).
  3. A number of concepts are formulated that are not included in the list of basic concepts, which we mean using basic concepts and axioms (definitions). A definition is a sentence in which the content of a new concept is revealed with the help of already known concepts and their properties [3].
  4. A number of statements are formulated, which we prove using the rules of logic and previously proven statements (theorems).

The interpretation of basic concepts is giving them a certain content, building models of a certain theory. In order for a system of axioms to serve as a scientific justification for a certain theory, three requirements must be met:

  1. Consistency or compatibility of the system of axioms.
  2. Independence or minimality of the axiom system.
  3. Completeness or categoricalness of the system of axioms.

A scientific theory that satisfies these requirements is a meaningful axiomatic theor [20].y.

A system of axioms is called consistent if there are no two contradictory statements in the list of its axioms and in the list of its consequences (theorems). The requirement of non-contradiction is proven by building its model on the basis of that scientific theory, the non-contradiction of which we have previously established.

There are arithmetic, algebraic, geometric, physical, chemical, economic, astronomical, architectural, optical, biological, space, virtual and real models. In geometry, we very often use the arithmetic model, taking advantage of the fact that the arithmetic of the real numbers is consistent. In particular, each geometric object is assigned an ordered set of real numbers called its coordinates. This makes it possible to determine the position of a point or body using numbers and to use the coordinate method when proving theorems, statements and solving problems.

The requirement of independence is that the list of axioms does not include such a statement that is a consequence of others. Let us have a system of axioms ∑a1,a2,…aᵢ,…,aₖ. Let’s choose axiom аᵢ and prove that it is not a consequence of all other axioms. We will reject this axiom of аᵢ and introduce a statement that is opposite in content to аᵢ. We will get a new system: ∑’a1,a2,…a̅ᵢ,…,aₖ. And we will prove that it is consistent. If as a result we got that the system of axioms ∑’ is consistent, it means that axiom аᵢ is not a consequence of all other axioms, because if axiom аᵢ was a consequence, then in the theory that was built on the system of axioms ∑’, two contradictory statements would hold and it would not be consistent. From here we can conclude that axiom аᵢ is not a consequence of all other axioms of the system of axioms Σ.

The requirement of completeness is that, given a certain system of axioms, we must say precisely about any statement whether it is true or false. The requirement of completeness is proved by establishing an isomorphism between two different models of the corresponding axiom system. A system of axioms will be categorical if all its interpretations are isomorphic. That is, the categorical requirement of the axiom system is stronger. The categorical nature of the axiom system implies its completeness, but not vice versa [20].

In contrast to substantive axiomatic theories, there are formal axiomatic theories in which the rules of logical deduction are introduced.

Consider Gödel’s incompleteness theorem (1931). Let us have a consistent or compatible system of axioms ∑a1,a2,…aᵢ,…,aₖ., on which the formal theory T(Σ) is built.

Gödel’s incompleteness theorem. In any non-contradictory (compatible) formal system with minimal arithmetic possibilities (addition, multiplication, generality quantifier, existence quantifier), there will be a formally undecidable statement.

That is, a true statement formulated in terms of the theory under consideration, which cannot be proven on the basis of the formal system under consideration. Thus, it cannot be deduced as a logical consequence of the system of axioms Σ under consideration, nor can its negative statement be derived on the basis of this system of axioms Σ.

A consistent system of axioms Σ is said to be complete if, in the theory T(Σ) built on the basis of this system of axioms Σ, any statement formulated in terms of the considered theory T(Σ) can either be proved or disproved, that is, a negative statement can be proved.

Thus, the completeness requirement is that, given the complete system of axioms Σ, we can prove the truth or falsity of any statement formulated in terms of the theory T(Σ). Therefore, true statements are fulfilled in every model (in all models) of a given system of axioms Σ, false statements are not fulfilled in any of its models, and formally unsolvable statements are fulfilled only in some models of the system of axioms Σ, but there are such models of this system axioms of Σ in which they do not hold. If a theory consists only of true and false statements, then it is complete.

In our time, axiomatic theories, which are based on set-theoretic concepts, which allow the basic concepts of the axiomatic theory to be given a certain set-theoretic interpretation in the form of sets and some relations between their elements, are strongly developing. For this interpretation of geometric objects, it is necessary to use the concept of mathematical structure.

A mathematical structure is an axiomatic theory whose axioms are expressed in terms of set theory. A mathematical structure can be defined by the assignment of one or more sets of relations, assigned to them, and a certain system of axioms that express the properties of these relations.

Therefore, in the axiomatic presentation of a certain geometry, the concept of mathematical structure and theory is used quite widely and requires detailed study.

A mathematical structure can be defined by specifying one or more basic sets: M1, M2,…,Mₚ, the elements of which are connected by basic relations: ρ1, ρ2, …,ρₙ. The properties of these relations are expressed in axioms: a1, a2, …,aₖ. The set of all axioms ∑a1,a2,…aᵢ,…,aₖ is called a system of axioms of a mathematical structure.

Mathematical structures are denoted as follows:

S = (M1, M2,…, Mₚ, ρ1, ρ2,…,ρₙ) +  ∑a1,a2,…aᵢ,…,aₖ

Among the basic sets, some are basic, the other part of the sets, which is not included in the selected list, is called auxiliary.

Examples of reference notes on this topic can be found in the publication [17]

To understand the construction of geometry using the concept of mathematical structure, it is advisable to give an example of the construction of Euclidean geometry based on Hermann Weyl’s axiomatics. This axiomatics is also called point-vector axiomatics, since the main undefined concepts in it are points and vectors.

In the system of G. Weil’s axioms, there are two main concepts: a point and a vector. The relations “adding vectors”, “multiplying a vector by a number”, “scalar product of vectors” and “subtracting a vector from a point” are called basic relations. The sets of all points and vectors are denoted by the symbols T and V, respectively. Weyl’s axiomatics consists of five groups.

The axioms of the first and second groups allow us to define the concept of a vector space. A vector space over the field of real numbers is the set V, for the elements (vectors) of which the operations of adding vectors and multiplying a vector by a real number are defined so that the requirements of the first and second groups of axioms are met.

Using the concept of mathematical structure, the following definition of vector space can be given. A vector space is a mathematical structure (V, φ1, φ2) with a basis set V and operations φ1, φ2, for which the requirements of the first and second groups of axioms are fulfilled.

A vector space in which the operation of the scalar product of vectors is defined in such a way that the requirements of the axioms of the first, second, and fourth groups of axioms are met is called a Euclidean space. Or the Euclidean space is a mathematical structure (V, φ1,, φ2, φ3)) with a basis set V and operations φ1,, φ2, φ3, for which the requirements of the first, second, and fourth groups of axioms are fulfilled.

Two axiom systems Σ and Σ´ are called equivalent if the theories built on the basis of these axiom systems coincide, i.e. Т(Σ)=Т(Σ´).

The formation of students’ skills in building models of various axiomatics, checking the fulfillment of requirements for the system of axioms contributes and sets them up for understanding the scientific construction of geometric knowledge and facts. The need not only to state or verify, but strictly scientific proof of the facts of both Euclidean and non-Euclidean geometries, requires students to have a thorough theoretical and practical basis for further professional activity.

Therefore, the creation and use of reference notes, where the information necessary for learning the educational material is concisely presented in the form of diagrams, figures, tables, diagrams, color correspondences, etc., is urgent and expedient, especially in the conditions of distance learning [17, p.83].

The reference outline is a visual structural and logical diagram, which is used to present the educational material in a condensed form.

The pedagogical feature of the reference synopsis is that the educational material is offered in the form of a compact structural and logical scheme that is quickly remembered, has the form of a system of didactic blocks with the content of the educational material encoded in it.

The didactic essence of the reference synopsis is determined with the help of keywords or phrases, abbreviations, pictures, graphs, formulas, conventional signs or other means of coding, which allow you to quickly learn and reproduce the content of the studied material.

The psychological essence of the reference synopsis consists in the intensification of the educational and cognitive activity of students by creating favorable conditions for the effective course of the processes of perception, memorization and reproduction of large in volume and integral in nature arrays of educational information.

The reference synopsis is easily reproduced, which allows you to create a situation of success in learning, in addition, such material reflects the connections not only between course topics, but also between different educational disciplines, that is, it provides interdisciplinary connections, which greatly contributes to the development of the individual’s thinking and its comprehensive development.

The significance of reference notes, which are both a means of visualization and a means of systematizing acquired knowledge, and a means of intensification of the educational process, is determined by time.

An example of one of the supporting notes in teaching the axiomatic method to students of mathematical specialties of the Faculty of Mathematics, Informatics and Physics of Mykhailo Dragomanov State University of Ukraine, namely, when checking the fulfillment of three requirements for the axiom system, can be found in our publication [17, p.83-86]. Reference notes as one of the means of learning the system of axioms D. Hilbert of Euclidean geometry can be seen in our publication [18].

Problems of constructing a geometric theory based on the axiomatic method.

Questions arise in geometry:

– can we build the same geometric theory, having different initial, basic, undefined concepts and relations?

− can we build different geometric theories, having different initial, basic, undefined concepts and relations?

– can we build the same geometric theory, having the same initial, basic, undefined concepts and relations, but formulating different systems of axioms?

In order to answer these questions, we will consider various axiomatic approaches to the construction of Euclidean geometry.

The axiomatic justification of geometry was first given by D. Hilbert in 1899, already after non-Euclidean geometry, namely hyperbolic geometry, was discovered.

Hilbert’s axiomatics contains 20 axioms, they describe 8 main objects: 3 main concepts (point, line, plane) and 5 main relations (incidence, belonging or combination, the relation “lie between” or the relation of order, congruence, continuity, parallelism).

Hilbert’s system of axioms of Euclidean geometry consists of 5 groups that describe basic relations.

The first group: axioms of incidence, belonging or conjunction, which describe the relation of incidence of points and a line, a line and a plane, points and a plane.

Group II: axioms of order, which describe the basic relation “lie between”” associated with points incident to a line.

Group III: axioms of congruence, which describe congruence relations for segments, angles, triangles.

IV group: axioms of continuity, which describe the property of continuity of the location of points on a straight line.

Group V consists of only one axiom, namely, the axiom of parallelism: through point A, which does not belong to line a, in the plane defined by point A and line a, no more than one line can be drawn that does not intersect line a.

It is interesting that neither Hilbert’s axiom of parallelism nor Lobachevsky’s axiom of parallelism contains the word parallel. They only indicate the number of straight lines that pass through a given point and do not cross a given straight line.

At present, in secondary education institutions, the congruence relation is not considered, but the equality relation is introduced, so let’s pay attention to the question of the equivalence of the congruence and equality relations. To do this, we will consider the concept of motion and show the connection between the relations of motion and congruence. The transformation of a set of points in space is called a movement if it translates two arbitrary points A and B into points A’ and B’ so that A’B’=AB. Considering the axioms of motion and studying the properties of equal figures, we can come to the conclusion that the relation of motion and congruence are equivalent. That is, in Hilbert’s axiomatics, the third group, that is, the axioms of congruence, can be replaced by the axioms of motion.

The first Saccheri-Legendre theorem is very interesting: The sum of the interior angles of a triangle cannot be greater than the sum of two right angles. and the second Saccheri-Legendre theorem: If there is at least one triangle on a plane whose sum of interior angles is equal to two straight lines, then every triangle of this plane has a sum of interior angles equal to two straight lines. If there is a triangle on the plane, the sum of the interior angles of which is less than two straight lines, then every triangle of this plane has the sum of the interior angles, less than two straight lines.

The introduction of the definition of the defect of a triangle as a value equal to the difference between 1800 and the sum of the internal angles of a triangle is a kind of marker for determining the type of geometry, namely: if the defect of a triangle is zero, then we have Euclidean geometry, if it is greater than zero, we have hyperbolic geometry.

The system of consequences arising only from axioms I-IV of groups of Hilbert’s system of axioms is called absolute geometry. And it is a common part of Euclidean and hyperbolic geometries.

Having a system of axioms D. Hilbert, Euclidean geometry can be strictly scientifically constructed.

The work of D. Hilbert’s Fundamentals of Geometry played an important role in the development of geometry. The modern axiomatic method and the theory of mathematical structures in the modern sense originate from it.

Friedrich Schur replaced David Hilbert’s axioms of congruence with axioms of motion, which describe the properties of motion — the mapping of points, lines, and planes into points, lines, and planes, respectively. Both groups of axioms of both systems perform the same task, defining the same concepts in different ways.

Hilbert’s axioms of congruence determine congruence relations directly, axioms of motion — through consequences.

In 1904, the American mathematician O. Veblen gives the construction of Euclidean geometry on the basis of “metric” axiomatics. His research was continued by another American mathematician R. L. Moore in 1908. But the most widespread of the “metric” axiomatics was the system of axioms of the American mathematician J. Birkhoff, who presented a system of planimetry axioms based on the use of a scale ruler and a protractor. The main concepts in the system are the concepts of “distance”, “angle measure”, “point”, “line”. This is a system of four axioms of Euclidean geometry. The concept of a real number is used in the formulation of the axioms. Therefore, Birkhoff’s axiomatics resembles the introduction of Euclidean geometry using a model.

George Birkhoff was involved in writing a school textbook using this system of axioms. This system influenced the system of axioms developed by the School Mathematics Study Group for the American school.

In the textbook O. IN. Pogorelov adopted a modification of Birkhoff’s axiomatics. The initial concepts in the system of axioms are: “point”, “line”, “plane”, “belonging”, “lie between”, “length of a segment”, “degree measure of an angle”. The axioms of planimetry are divided into six groups: axioms of belonging; axioms of mutual location of points on a straight line and on a plane; axioms of measuring segments and angles; axioms of setting segments and angles; axioms of equality of triangles; axiom of parallels. This system of axioms, with some changes, became the basis of a school geometry textbook.

Conclusions. In institutions of higher education, both Euclidean and non-Euclidean geometries are constructed using the axiomatic method. The logical foundations of geometry are the foundations of geometry, which must meet the requirements of logic.

Thus, we can build the same geometric theory, having different starting, basic, undefined concepts and relations. Such a theory is Euclidean geometry, examples of various axiomatics of which are considered in the work.

We can also build different geometric theories, having different initial, basic, undefined concepts and relations. Examples of such theories are spherical geometry, elliptic geometry, Galilean geometry.

We cannot build the same geometric theory, having the same initial, basic, undefined concepts and relations, but having formulated different systems of axioms. Such an example is Euclidean geometry and hyperbolic geometry, projective geometry.

In order not to violate the principle of continuity and consistency in education, we consider it expedient to build a geometry course in secondary education institutions using the axiomatic method. And from the very beginning of teaching geometry, use the axiomatic method as the basis for structuring students’ thinking. The use of the principles of continuity and systematicity in education gradually gives good results not only when studying geometry, but also for studying other disciplines, as it teaches to structure the material, to see the logical structure of the relevant discipline. Otherwise, it will not be a construction of geometry, but a story about geometry.

 

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