1. The discovery of irrational numbers in three crises in the history of mathematical development: the first mathematical crisis: in the 5th century BC, the discovery of incommensurable measures led to the Pythagorean paradox.

2. This paradox directly violated the fundamental creed of Pythagoras school, which led to the "crisis" of cognition at that time, thus resulting in the first mathematical crisis.

3. The second mathematical crisis:18th century, differential method and integral method have been widely and successfully applied in production and practice, and most mathematicians have no doubt about the reliability of this theory.

4. 1734 British philosopher and archbishop Becker published "An analyst or mathematician who doesn't believe in orthodoxy", which puts forward the so-called Becker paradox for the infinitesimal problem, the basis of calculus.

This has caused a debate in the field of mathematics and even philosophy for a century and a half.

6. It led to the second mathematical crisis in the history of mathematics.

7. The third mathematical crisis: The third crisis in the history of mathematics was caused by the sudden impact of 1897. This crisis was caused by the discovery of the marginal paradox of Cantor's general set theory.