The three mathematical crises in the history of mathematics occurred in the 5th century BC,17th century and19th century respectively, all during the great development of western culture. Therefore, the occurrence of mathematical crisis has its certain cultural background.

These three mathematical crises are:

The first time: in ancient Greece, it was triggered by the discovery of an incommensurable line segment-irrational number and some intuitive experience;

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

The third time: Russell discovered the paradox in set theory and endangered the foundation of the whole 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.