To The Question Of The Method Of Studying The Concept Of Proof And The Axiomatic Method By Bachelors And Masters Of Pedagogical Education
The decrease in the level of logical literacy of graduates of modern schools is due, in particular, to the fact that the school lacks a holistic course of logic and does not pay attention to the systematic education of the logical thinking of students. But to a large extent this is due to the fact that future teachers of mathematics studying in classical and pedagogical universities in the direction of “Pedagogical education” did not receive the corresponding logical knowledge and did not form logical competencies on the university bench, since in higher education institutions, the training of mathematics teachers is holistic the course of logic is also missing. The most important components of the logical and didactic competence of a future mathematics teacher are the ability to analyze the structure of mathematical sentences, knowledge of the concept of rigorous proof of a mathematical theorem, knowledge of the methods of proof of mathematical theorems, understanding of how mathematical theories are constructed and how. The purpose of the article is to show how, using the subject and methods of the course of mathematical logic, to form the concepts of rigorous mathematical proof, axiomatic method and axiomatic theory for future teachers of mathematics. A formalized calculus of statements can serve as an excellent methodological model for this. Using this formal axiomatic theory as an example, the future teacher of mathematics can demonstrate in detail and with all the proof the process of constructing such a theory, as well as the process of studying its properties (a metatheory of such a theory). Neither the geometry course, nor even the course of numerical systems possesses such methodological capabilities. The paper considers one aspect of such a study related to the construction of an axiomatic theory on the basis of two different axiom systems - the three-axiom Mendelssohn system and the thirteen-axiom Henzen – Kleene system. The equivalence of these two systems of axioms is established through formal proofs that each axiom of the first system is a theorem of the axiomatic theory built on the basis of the second system of axioms, on the contrary, each axiom of the second system of axioms is a theorem of the axiomatic theory built on the basis of the first system of axioms. In the process of these studies, the components of logical principles in teaching mathematics are vividly presented - the concepts of proof, the axiomatic method and axiomatic theory. In conclusion, it is recommended that the considered methodology for the formation of the concepts of rigorous mathematical proof, axiomatic method and axiomatic theory for future teachers of mathematics be applied using other systems of axioms on the basis of which a formal calculus of propositions can be constructed. Recommendations are given on selecting the content of a course in mathematical logic for undergraduate and graduate levels for future mathematics teachers.