1900 On August 8th, at the Second International Congress of Mathematicians held in Paris, he put forward 23 mathematical problems that mathematicians should strive to solve in the new century, which were considered as the commanding heights of mathematics in the 20th century. The study of these problems strongly promoted the development of mathematics in the 20th century and had a far-reaching impact in the world. The school of mathematics led by Hilbert was a banner of mathematics at the end of 19 and the beginning of the 20th century. Hilbert is called "the uncrowned king of mathematics".
(The famous Goldbach conjecture is also one of the problems. The mathematicians in China, represented by Chen Jingrun, have made great breakthroughs, but they have not been completely solved. )
Hilbert was born in Lao Wei near Konigsberg in East Prussia (Kaliningrad, the former Soviet Union). He was a studious student in middle school. He showed great interest in science, especially mathematics, and was good at mastering and applying the contents of the teacher's lectures flexibly and profoundly. 1880, against his father's wishes, he asked him to study law, entered the university of konigsberg to study mathematics, and obtained his doctorate in 1884, then stayed in school to obtain the qualification of lecturer and was promoted to associate professor. 1893 was hired as a full professor, 1895 was transferred to the University of G? ttingen as a professor, and has been living and working in G? ttingen ever since. He retired on 1930. During this period, he became a member of the School of Communication of the Berlin Academy of Sciences, and won the Steiner Prize, the Lobachevsky Prize and the Boyle Prize. 1930 won the science prize of Swedish Academy in Mittag-Leffler, and 1942 became an honorary academician of Berlin Academy of Sciences. Hilbert is an upright scientist. On the eve of the First World War, he refused to sign the book To the Civilized World published by the German government for deceptive propaganda. During the war, he dared to publish an article in memory of "enemy mathematician" Dabu. After Hitler came to power, he resisted and wrote against the Nazi government's policy of excluding and persecuting Jewish scientists. Due to the increasingly reactionary policies of the Nazi government, many scientists were forced to emigrate, and the once-flourishing Gottingen School declined, and Hilbert died alone in 1943.
Hilbert is one of the mathematicians who had a profound influence on mathematics in the 20th century. He led the famous Gottingen School, made the University of Gottingen an important mathematical research center in the world at that time, trained a group of outstanding mathematicians and made great contributions to the development of modern mathematics. Hilbert's mathematical work can be divided into several different periods, and in each period he almost devoted himself to one kind of problems. In chronological order, his main research contents include: invariant theory, algebraic number field theory, geometric foundation, integral equation, physics and general mathematical foundation. His research topics are interspersed with Dirichlet principle and variational method, Welling problem, eigenvalue problem, "Hilbert space" and so on. In these fields, he has made great or pioneering contributions. Hilbert believes that science has its own problems in every era, and the solution of these problems is of far-reaching significance to the development of science. He pointed out: "As long as a branch of science can raise a large number of questions, it is full of vitality, and the lack of questions indicates the decline and termination of independent development."
At the 1900 International Congress of Mathematicians held in Paris, Hilbert gave a famous speech entitled "Mathematical Problems". According to the achievements and development trend of mathematical research in the past, especially in the 19th century, he put forward 23 most important mathematical problems. These 23 problems, collectively called Hilbert problems, later became the difficulties that many mathematicians tried to overcome, which had a far-reaching impact on the research and development of modern mathematics and played a positive role in promoting it. Some Hilbert problems have been satisfactorily solved, while others have not yet been solved. The belief that every mathematical problem can be solved in his speech is a great encouragement to mathematicians. He said: "among us, we often hear such a voice: here is a math problem, find out its answer!" " You can find it through pure thinking, because there is no unknowability in mathematics. Thirty years later, 1930, in his speech accepting the title of honorary citizen of Konigsberg, he declared confidently again: "We must know, and we will know. "
Hilbert's Fundamentals of Geometry (1899) is a masterpiece of axiomatic thought. Euclidean geometry is sorted out in the book and becomes a pure deductive system based on a set of simple axioms, and the relationship between axioms and the logical structure of the whole deductive system are discussed. 1904 began to study the basic problems of mathematics. After years of deliberation, in the early 1920s, he put forward a scheme on how to demonstrate the consistency of number theory, set theory or mathematical analysis. He suggested formalizing mathematics from several formal axioms into a symbolic language system, and establishing a corresponding logical system from the point of view of never assuming infinite reality. Then, the logical properties of this formal language system are studied, so as to establish meta-mathematics and proof theory. Hilbert's purpose is to try to give an absolute proof that the formal language system is not contradictory, so as to overcome the crisis caused by paradox and eliminate the doubt on the reliability of mathematical foundation and mathematical reasoning method once and for all. However, in 1930, the young Austrian mathematical logician Godel (K.G.? Del, 1906 ~ 1978) gets a negative result, which proves that Hilbert scheme is impossible to realize. However, as Godel said, Hilbert's scheme based on mathematics "still retains its importance and continues to arouse people's high interest". Hilbert's works include The Complete Works of Hilbert (three volumes, including his famous Report on Number Theory), Fundamentals of Geometry, and General Theoretical Foundations of Linear Integral Equations. He co-authored Methods of Mathematical Physics, Fundamentals of Theoretical Logic, Intuitive Geometry and Fundamentals of Mathematics.
Hilbert problem
At the 1900 International Congress of Mathematicians held in Paris, Hilbert gave a famous speech entitled "Mathematical Problems". According to the achievements and development trend of mathematical research in the past, especially in the 19th century, he put forward 23 most important mathematical problems. These 23 problems, collectively called Hilbert problems, later became the difficulties that many mathematicians tried to overcome, which had a far-reaching impact on the research and development of modern mathematics and played a positive role in promoting it. Some Hilbert problems have been satisfactorily solved, while others have not yet been solved. The belief that every mathematical problem can be solved in his speech is a great encouragement to mathematicians.
Hilbert's 23 problems belong to four blocks: 1 to 6 are basic mathematical problems; Questions 7 to 12 are number theory problems; Problems 13 to 18 belong to algebraic and geometric problems; 19 to 23 belong to mathematical analysis.
(1) Cardinality of Cantor continuum.
1874, Cantor speculated that there was no other cardinality between countable set cardinality and real set cardinality, that is, the famous continuum hypothesis. 1938, Austrian mathematical logician Godel living in the United States proved that there is no contradiction between the continuum hypothesis and the axiomatic system of ZF set theory. 1963, American mathematician P.Choen proved that the continuum hypothesis and ZF axiom are independent of each other. Therefore, the continuum hypothesis cannot be proved by ZF axiom. In this sense, the problem has been solved.
(2) Arithmetic axiom system is not contradictory.
The contradiction of Euclidean geometry can be summed up as the contradiction of arithmetic axioms. Hilbert once put forward the method of proving formalism plan, but Godel's incompleteness theorem published in 193 1 denied it. Gnc(G. genta en,1909-1945)1936 proved the non-contradiction of the arithmetic axiomatic system by means of transfinite induction.
(3) It is impossible to prove that two tetrahedrons with equal base and equal height are equal in volume only according to the contract axiom.
The significance of the problem is that there are two tetrahedrons with equal height, which cannot be decomposed into finite small tetrahedrons, so that the congruence of the two tetrahedrons (M. DEHN) has been solved in 1900.
(4) Take a straight line as the shortest distance between two points.
This question is rather general. There are many geometries that satisfy this property, so some restrictions are required. 1973, the Soviet mathematician Bo gref announced that this problem was solved under the condition of symmetrical distance.
(5) Conditions for topology to be a Lie group (topological group).
This problem is simply called the analytic property of continuous groups, that is, whether every local Euclidean group must be a Lie group. 1952 was solved by Gleason, Montgomery and Zipin. 1953, Hidehiko Yamanaka of Japan got a completely positive result.
(6) Axiomatization of physics, which plays an important role in mathematics.
1933, the Soviet mathematician Andrei Andrey Kolmogorov axiomatized probability theory. Later, he succeeded in quantum mechanics and quantum field theory. However, many people have doubts about whether all branches of physics can be fully axiomatized.
(7) Proof of transcendence of some numbers.
It is proved that if α is algebraic number and β is algebraic number of irrational number, then α β must be transcendental number or at least irrational number (such as 2√2 and eπ). Gelfond of the Soviet Union (1929) and Schneider and Siegel of Germany (1935) independently proved its correctness. But the theory of transcendental number is far from complete. At present, there is no unified method to determine whether a given number exceeds the number.
(8) The distribution of prime numbers, especially for Riemann conjecture, Goldbach conjecture and twin prime numbers.
Prime number is a very old research field. Hilbert mentioned Riemann conjecture, Goldbach conjecture and twin prime numbers here. Riemann conjecture is still unsolved. Goldbach conjecture and twin prime numbers have not been finally solved, and the best result belongs to China mathematician Chen Jingrun.
(9) Proof of the general law of reciprocity in arbitrary number field.
192 1 was basically solved by Kenji Takagi of Japan, and 1927 was basically solved by E.Artin of Germany. However, category theory is still developing.
(10) Can we judge whether an indefinite equation has a rational integer solution by finite steps?
Finding the integer root of the integral coefficient equation is called Diophantine (about 2 10-290, an ancient Greek mathematician) equation solvable. Around 1950, American mathematicians such as Davis, Putnam and Robinson made key breakthroughs. In 1970, Baker and Feros made positive conclusions about the equation with two unknowns. 1970. The Soviet mathematician Marty Sevic finally proved that, on the whole, the answer is negative. Although the result is negative, it has produced a series of valuable by-products, many of which are closely related to computer science.
Quadratic theory in (1 1) algebraic number field.
German mathematicians Hassel and Siegel made important achievements in the 1920s. In 1960s, French mathematician A.Weil made new progress.
Composition of (12) class domain.
That is, Kroneck's theorem on Abelian field is extended to any algebraic rational field. This problem has only some sporadic results and is far from being completely solved.
The impossibility of (13) combination of binary continuous functions to solve the seventh general algebraic equation.
The root of equation x7+ax3+bx2+cx+ 1=0 depends on three parameters A, B and C; X=x(a, b, c). Can this function be represented by a binary function? This problem is about to be solved. 1957 Arnold, a Soviet mathematician, proved that any continuous real function f(x 1, x2, x3) on [0, 1] can be written in the form of ∑ hi (ξi (x 1, x2), x3) (i. X3) can be written as ∑ hi (ξ i 1 (x 1) in 1964. Vituskin is extended to continuously differentiable, but the analytic function is not solved.
The finite proof of (14) some complete function systems.
That is, the polynomial fi (I = 1, ..., Xn), where r is the negative solution of this problem related to algebraic invariants by the rational function F(X 1, ..., Xm) and F. Japanese mathematician Masayoshi Nagata in 1959.
(15) Establish the foundation of algebraic geometry.
Dutch mathematicians Vander Waals Deng 1938 to 1940 and Wei Yi 1950 have solved the problem.
Note 1 Schubert's strict foundation of counting calculus.
A typical problem is that there are four straight lines in three-dimensional space. How many straight lines can intersect all four? Schubert gave an intuitive solution. Hilbert asked to generalize the problem and give a strict basis. Now there are some computable methods, which are closely related to algebraic geometry. But the strict foundation has not been established.
Topological research on (16) algebraic curves and surfaces.
The first half of this problem involves the maximum number of closed bifurcation curves in algebraic curves. In the second half, it is required to discuss the maximum number N(n) and relative position of limit cycles of dx/dy=Y/X, where x and y are polynomials of degree n of x and y. For the case of n=2 (i.e. quadratic system), 1934, Froxianer obtains n (2) ≥1; 1952, Bao Ting got n (2) ≥ 3; 1955, Podlovschi of the Soviet Union declared that n(2)≤3, which was the result of a shock for a while, but was questioned because some lemmas were rejected. Regarding the relative position, China mathematician and Ye proved in 1957 that (E2) does not exceed two strings. In 1957, China mathematicians Qin Yuanxun and Pu Fujin gave a concrete example. The equation with n = 2 has at least three series limit cycles. In 1978, under the guidance of Qin Yuanxun and Hua, Shi Songling and Wang of China respectively gave at least four concrete examples of limit cycles. In 1983, Qin Yuanxun further proved that the quadratic system has at most four limit cycles, and the structure is (1 3), thus finally solving the structural problem of the solution of the quadratic differential equation and providing a new way for studying the Hilbert problem (16).
The square sum representation of (17) semi-positive definite form.
The rational function f (x 1, ..., xn) is for any array (x 1, ..., xn). Are you sure that f can be written as the sum of squares of rational functions? 1927 Atin has been definitely solved.
(18) Construct space with congruent polyhedron.
German mathematicians Bieber Bach (19 10) and Reinhardt (1928) gave some answers.
(19) Is the solution of the regular variational problem always an analytic function?
German mathematician Berndt (1929) and Soviet mathematician Petrovsky (1939) have solved this problem.
(20) Study the general boundary value problem.
This problem is progressing rapidly and has become a major branch of mathematics. I was still researching and developing a few days ago.
(2 1) Proof of the existence of solutions for Fuchs-like linear differential equations with given singularities and single-valued groups.
This problem belongs to the large-scale theory of linear ordinary differential equations. Hilbert himself obtained important results in 1905 and H.Rohrl in 1957 respectively. Deligne, a French mathematician from 65438 to 0970, made outstanding contributions.
(22) Automorphic single-valued analytic function.
This problem involves the difficult Riemann surface theory. In 1907, P.Koebe solved a variant and made an important breakthrough in the study of this problem. Other aspects have not been solved.
(23) Carry out the research of variational method.
This is not a clear mathematical problem. Variational method has made great progress in the 20th century.
Hilbert's anecdote
1. There are so many mathematical terms named after Hilbert that some of them are unknown to Hilbert himself. For example, once Hilbert asked a colleague in the department, "What is Hilbert space?"
Emmy noether, a talented young woman, came to the University of G? ttingen. Hilbert appreciated her knowledge very much and immediately decided to let her stay as a lecturer to assist in the study of relativity. However, the discrimination against women was quite serious at that time, and Hilbert's suggestion was strongly opposed by professors such as linguistics and history. Hilbert was angry and shouted, "gentlemen, this is a school, not a bathhouse!" " "So he angered his opponent, and Hilbert was unmoved and decided to replace Jeannotte with his own name.