The discovery of irrational numbers led to the first mathematical crisis, and the solution of the crisis also promoted the development of logic and the systematization of geometry.

The second mathematical crisis is caused by infinitesimal contradiction, which reflects the contradiction between finiteness and infinity in mathematics. Mathematics has always been a contradiction between calculation method and analysis method, with clear concept and strict logic. In this respect, practical mathematicians pay more attention to blind application. And more mathematicians and philosophers who pay attention to it criticize it. Only when these two aspects are well coordinated can the contradiction be solved. Later, operator calculus and δ function also repeated this process, starting with formal calculus and arbitrary application, and it was not until Schwal that the strict system of generalized function theory was established.

For the third mathematical crisis, some people think that it is only the crisis of mathematical foundation and has nothing to do with mathematics. This view is one-sided. It is true that the problem involves mathematical logic and set theory, but it involves infinite set from the beginning, and if modern mathematics is separated from infinite set, it can be said that it is difficult to move. Because if only finite sets or countable sets are considered, most mathematics will no longer exist. And even with these limited mathematical contents, there are many problems involving infinite methods, such as using analytical methods to solve many problems in number theory. From this point of view, the third mathematical crisis is a profound mathematical crisis.