On the problem of axiomatization of geometry

Abstract

An analysis of the foundations of geometry within the framework of the correct methodological basis – the unity of formal logic and rational dialectics – is proposed. The analysis leads to the following result: (1) geometry is an engineering science, but not a field of mathematics; (2) the essence of geometry is the construction of material figures (systems) and study of their properties; (3) the starting point of geometry is the following system principle: the properties of material figures (systems) determine the properties of the elements of figures; the properties of elements characterize the properties of figures (systems); (4) the axiomatization of geometry is a way of construction of the science as a set (system) of practical principles. Sets (systems) of practice principles can be complete or incomplete; (5) the book, “The Foundations of Geometry” by David Hilbert, represents a methodologically incorrect work. It does not satisfy the dialectical principle of cognition, “practice  theory practice,” because practice is not the starting point and final point in Hilbert’s theoretical approach (analysis). Hilbert did not understand that: (a) scientific intuition must be based on practical experience; intuition that is not based on practical experience is fantasy; (b) the correct science does not exist without definitions of concepts; the definitions of geometric concepts are the genetic (technological) definitions that shows how given material objects arise (i.e., how a person creates given material objects); (c) the theory must be constructed within the framework of the correct methodological basis: the unity of formal logic and rational dialectics. (d) the theory must satisfy the correct criterion of truth: the unity of formal logic and rational dialectics. Therefore, Hilbert cannot prove the theorem of trisection of angle and the theorem of sum of interior angles (concluded angles) of triangle on the basis of his axioms. This fact signifies that Hilbert’s system of axioms is incomplete. In essence, Hilbert’s work is a superficial, tautological and logically incorrect verbal description of Figures 1-52 in his work. 