Миф о трех кризисах в основаниях математики Часть 2

Abstract

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. In the first part of this paper, it has been demonstrated that there is no evidence to recognize the so-called “first foundational crisis” in pre-Euclidean Greek mathematics caused by the discovery of incommensurable magnitudes. The second part is mainly devoted to the historical situation of the 18th century and the beginning of the 19th century, the alleged epoch of “the second foundational crisis” caused by the obscurity of the basic notions of the calculus. Despite the lack of clarity as far as the foundations of mathematical analysis are concerned, there are no signs of a foundational crisis in 18th-century mathematics. On the contrary, in the 19th century, at the time of “the second scientific revolution”, there are striking illustrations of crisis awareness. The cultural processes that engendered this awareness are shown to be identical for the 19th century and the beginning of the 20th century, hence the second and the third foundational crises are proved to be the different stages of the same unique crisis. Finally, we have only one foundational crisis instead of three.