The problem of incommensurability and the crisis of foundations of the Ancient Greek mathematics

Abstract

The subject of this research is the problem of incommensurability and the crisis of foundations of the Ancient Greek mathematics. The article describes that the crisis of foundations was caused by the discovery of irrationality by the Pythagorean Hippasus of Metapontum, which resulted in the theoretical instability of mathematics of the Pythagoreans, who believed that everything could be expressed through numbers. The discovery of incommensurable line segments demonstrated that the relations between rational numbers cannot express any variable, for example the diagonal of a square with one side equal to one. Analysis is conducted on the achievements of the Pythagorean School in the field of mathematics. Special attention is given to the role of a number in the philosophy of this school. The article explores the main ways for overcoming this crisis, philosophical explanation of the unfolded situation, based on which the Pythagoreans formulate the methodological ways out of the discovered problem of incommensurability. It is noted that the Pythagoreans were actively elaborating on their philosophy and mathematical apparatus intending to find the answer to the discovery of incommensurability. The author’s special contribution lies in the statement that the discovery of irrationality was not critical for the Pythagoreans: they continued working towards the answer to the problem of incommensurability, as well as refined the mathematical theory of proportions, reconsidered the representation of infiniteness as a certain numerical characteristic of the things and processes. This article is first to advance a hypothesis on the possibility of dividing the object into an infinitely large number of infinitely small parts, which is now understood as the limit of function, which contributes to the development and application of dialectics. The problem of incommensurability led to the creation of new, complex theories in the history of science, culture, architecture, and art.