1900, Hilbert was invited to attend the International Congress of Mathematicians held in Paris and delivered an important speech entitled "Mathematical Problems". In this historic speech, first of all, he put forward many important points:

Just like every career of human beings pursues a certain goal, mathematical research also needs its own problems. It is through the solution of these problems that researchers have exercised their iron will, discovered new ideas and reached a broader realm of freedom.

Hilbert particularly emphasized the role of major problems in the development of mathematics. He pointed out: "If we want to have an idea of the possible development of mathematical knowledge in the near future, we must review the problems raised by science today and hope to solve them in the future." At the same time, it is pointed out: "The profound significance of some problems to the general mathematical process and their important role in the personal work of researchers are undeniable. As long as a branch of science can ask a lot of questions, it is full of vitality, and no problem indicates the decline or suspension of independent development. "

He expounded the characteristics of major issues. A good issue should have the following three characteristics:

1 clear and easy to understand;

Although difficult, it gives people hope;

3 profound significance.

At the same time, he analyzed the difficulties often encountered in learning mathematics problems and some methods to overcome them. It was at this meeting that he put forward 23 problems that mathematicians should try to solve in the new century, namely the famous "Hilbert 23 problem".

The situation of solving the numbering problem in the field of promoting development

1 continuum hypothesis axiom set theory 1963 and Paul J.Cohen proved that the first problem is insoluble in the following sense. That is to say, in the zermelo-Frankel axiom system, the truth of the continuum hypothesis cannot be judged.

2 Compatibility of Arithmetic Axioms Mathematical Basis Hilbert's idea of proving the compatibility of arithmetic axioms later developed into a systematic Hilbert plan ("meta-mathematics" or "proof theory"), but Godel pointed out in the "incomplete theorem" of 193 1 year that it is impossible to prove the compatibility of arithmetic axioms with "meta-mathematics". The compatibility problem of mathematics has not been solved so far.

The geometric basis of equal volume of two kinds of high-base tetrahedrons is very fast (1900), and M.Dehn, a student of Hilbert, definitely answered it.

The problem that a straight line is the geometric basis of the shortest distance between two points is too general. After Hilbert, many mathematicians devoted themselves to constructing and exploring all kinds of special metric geometry, and made great progress in studying the fourth problem, but the problem was not completely solved.

5 The concept of Lie group does not define the topological group theory of the assumption of differentiability of group functions. After a long period of efforts, this problem was finally solved by Gleason, Montqomery, Zipping and others in 1952, and the answer was yes.

6 Mathematic treatment of physical axioms Mathematical physics has achieved great success in quantum mechanics, thermodynamics and other fields, but in general, what axiomatic physics means is still a problem that needs to be discussed. Axiomatization of probability theory was established by A.H.Konmoropob and others.

7 Irrational Numbers of Some Numbers and Transcendence Transcendental Number Theory 1934 A.O.temohm and Schneieder solved the second half of this problem independently.

Riemann conjecture is still a conjecture in the general case of number theory of 8 prime numbers. The Goldbach problem contained in the eighth question has not been solved so far. Mathematicians in China have done a series of excellent work in this field.

Proof of the most general reciprocal law in arbitrary number field. The field-like theory has been solved by Takagi Sadako (192 1) and E.Artin( 1927).

10 discriminant uncertainty analysis of the solvability of Diophantine equation 1970 mathematicians in the Soviet Union and the United States proved that the general algorithm expected by Hilbert did not exist.

The quadratic quadratic theory H. Hasse (1929) and C. L. Siegel (1936, 195 1) with arbitrary algebraic coefficients have obtained important results on this issue.

The Kroneker theorem on 12 Abel field is extended to any algebraic rational number field. The theory of complex multiplication has not been solved.

13 It is impossible to solve the ordinary seventh-order equation with a function with only two variables. The continuous function of equation theory and real function theory was denied by Soviet mathematicians in 1957. If you need to parse the function, the problem remains unsolved.

14 proves that the finite algebraic invariant theory of a complete function system gives a negative solution.

15 strict basic algebraic geometry of Schubert counting calculus Due to the efforts of many mathematicians, it is possible to treat the foundation of Schubert calculus purely by algebraic method, but the rationality of Schubert calculus remains to be solved. As for the foundation of algebraic geometry, it has been established by B.L. Vander Waals Deng (1938-40) and A.Weil (1950).

Topological curves and surfaces of 16 algebra, surface topology and the first half of qualitative theoretical problems of ordinary differential equations have obtained important results in recent years.

The theory of square expression domain (real domain) in positive definite form was solved by Artin in 1926.

18 is partially solved by the theory of space crystal group of congruent polyhedron.

Whether the solution of 19 regular variational problem must analyze the theory of elliptic partial differential equations has been solved in a sense.

20 general boundary value problems elliptic partial differential equation theory The research on boundary value problems of partial differential equations is developing vigorously.

2 1 Existence of linear partial differential equations with given value groups The large-scale theory of linear ordinary differential equations has been solved by Hilbert himself (1905) and H.Rohrl (Germany, 1957).

P.Koebe (Germany, 1907) has solved the case of univalent Riemannian surfaces with one variable of analytic relation.

23 Further Development of Variational Method Hilbert himself and many mathematicians have made important contributions to the development of Variational Method.