The Background and Significance of 'Crises' in the Foundations of Mathematics
Languages of publication
The watersheds in the development of mathematics that lay bare the ambiguities and contradictions among its very basic notions and conceptions are commonly described as 'crises' in mathematics. They coincide with periods of intense mathematical research, often inspired by philosophical doctrines which help to clarify the notions and methods with the use of which many fundamental theorems are proved. The paper presents three best known and most widely discussed crises in the foundations of mathematics, reviews the causes of their appearance and discusses the difficulties that they have led to as well as possible ways of their resolution. The three crises arose in connection with the discovery of the incommensurable line segments in the Pythagorean School, in connection with the operations on the infinitesimals by the creators of the differential calculus (17th-18th centuries) and with respect to the foundations of mathematics connected with operations on the actually infinite sets (19th and 20th centuries).
Publication order reference
CEJSH db identifier