First time: In ancient Greece, due to incommensurable line segments-

unreasonable

This discovery is caused by the conflict with some intuitive experience.

The second time: After Newton and Leibniz established the calculus theory, their understanding of infinitesimal was not profound.

The third time: when Russell found out.

set theory

The paradox in the book endangers the whole foundation of mathematics.

Although the three mathematical crises had a great influence on mathematics and philosophy at that time and caused some difficulties at that time, they never hindered the development and application of mathematics. On the contrary, after the predicament, it brought new vitality to the development of mathematics.

Extended data:

The formative period of mathematics is the period when human beings establish the most basic mathematical concepts. Since counting, human beings have gradually established the concept of natural numbers, simple calculation methods, and recognized the most basic and simplest geometric forms. Arithmetic and geometry are not separated.

Elementary mathematics is a constant period of mathematics. The most basic and simple achievements in this period constitute the main content of middle school mathematics. This period began in the 5th century BC, maybe earlier, and lasted for about two thousand years until17th century. This period gradually formed the main branches of elementary mathematics: arithmetic, geometry and algebra.

Modern mathematics. The period of modern mathematics began at the beginning of19th century. The beginning of the modern stage of mathematics development is characterized by profound changes in all foundations-algebra, geometry and analysis.

Baidu Encyclopedia-Three Crises of Mathematics