It was not until the 1920s of 19 that some mathematicians paid more attention to the strict foundation of calculus. It took more than half a century from the beginning of the work of Porzano, Abel, Cauchy, De Ricchelli and others to the end of the work of Wilstras, Dedekind and Cantor, which basically solved the contradiction and laid a strict foundation for mathematical analysis.

Porzano gave a correct definition of continuity; Abel pointed out that it is necessary to strictly limit the abuse of series expansion and summation; Cauchy started with the definition of variables in the algebra analysis course of 182 1, and realized that functions don't have to have analytic expressions. He grasped the concept of limit and pointed out that infinitesimal and infinitesimal are not fixed quantities but variables, and infinitesimal is a variable with zero as the limit.

Derivative and integral are defined. De Reichley gave a modern definition of function. On the basis of these works, Wilstrass eliminated the inaccuracies, gave the current universal definition of limit and continuous definition, and strictly established derivatives and integrals on the basis of limit.

From 65438 to the early 1970s, Wilstrass, Dedekind, Cantor and others independently established the real number theory, and established the basic theorem of the limit theory on the basis of the real number theory, thus making the mathematical analysis based on the strict basis of the real number theory.

Extended data:

From the outbreak of the second mathematical crisis to 2 1 century, there have been different opinions. Euler, a famous mathematician, insisted that the result should be 0/0 in the calculation of derivative. For example, he said that if the value of the earth is calculated, the error of a dust or even thousands of dust can be ignored.

But in the operation of calculus, "the strictness of geometry requires that even such a small error cannot exist." Marx put forward in Mathematical Manuscripts that the result of derivative operation should be strictly concrete 0/0, and criticized the so-called "infinite approximation".

This crisis not only did not hinder the rapid development and wide application of calculus, but also made calculus gallop in various scientific and technological fields, solved a large number of physical problems, astronomical problems and mathematical problems, and greatly promoted the development of industrial revolution.

As far as calculus itself is concerned, after the baptism of this crisis, it has been systematized and integrated and expanded to different branches, becoming the "overlord" of mathematics in the18th century. At the same time, the second mathematical crisis also promoted the process of strict analysis, algebraic abstraction and non-Europeanization of geometry in19th century.

Baidu Encyclopedia-The Second Mathematical Crisis