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

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.

The myth of the three crises in foundations of mathematics Part 1

The story of “the three crises in foundations of mathematics” is widely popular in Russian publications on the philosophy of mathematics. This paper aims at evaluating this story against the background of the contemporary scholarship in the history of mathematics. The conclusion is that it should be considered as a specimen of modern myth-making activity brought to the fore by an unconscious tendency to model the whole history of mathematics on the pattern of the foundational crisis of the first decades of the 20th century. What is more, the consideration of the specific role and character of the foundations in both early Greek mathematics and 18th-century mathematics gives an occasion to raise a more general question regarding the true meaning of the historicity of mathematics. The first part of this paper deals with the point whether there was a foundational crisis in pre-Euclidean Greek mathematics caused by the discovery of incommensurable magnitudes and Zeno’s paradoxes. The result is negative: we have no direct historical evidence of such a crisis; as for secondary considerations, they also mainly count against it. The idea of the first crisis in foundations of mathematics has emerged as a result of the unjustified transference of the modern grasp of foundational issues and the modern “mentalité de crise” to the ancient past.

Three Notes on the Method of Analysis and Synthesis in its Ancient and (Arabic) Medieval Contexts

Most historians and philosophers of philosophy and history of mathematics hold one interpretation or the other of the nature of method of analysis and synthesis in itself and in its historical development. In this paper, I am trying to prove – through three points – that, in fact, there were two understandings of that method in Greek mathematics and philosophy, and which were reflected in Arabic mathematical science and philosophy; this reflection is considered as proof also of this double nature of that method. Thus, we have to rethink the nature of Arabic philosophy systems.