**Author:**

Kalanov, Temur Z.

**Category:**

Research Papers

**Sub-Category:**

Mathematics and Applied Mathematics

**Date Published:**

April 25, 2013

**Keywords:**

Pythagorean theorem, geometry, trigonometry, theory of number, mathematics, physics, philosophy of science

**Abstract:**

The critical analysis of the Pythagorean theorem and of the problem of irrational numbers is proposed. Methodological basis for the analysis is the unity of formal logic and of rational dialectics. It is shown that: 1) the Pythagorean theorem represents a conventional (conditional) theoretical proposition because, in some cases, the theorem contradicts the formal-logical laws and leads to the appearance of irrational numbers; 2) the standard theoretical proposition on the existence of incommensurable segments is a mathematical fiction, a consequence of violation of the two formal-logical laws: the law of identity of geometrical forms and the law of lack of contradiction of geometrical forms; 3) the concept of irrational numbers is the result of violation of the dialectical unity of the qualitative aspect (i.e. form) and quantitative aspect (i.e. content: length, area) of geometric objects. Irrational numbers represent a calculation process and, therefore, do not exist on the number scale. There are only rational numbers.

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