David Hilbert obtained theKonesburg UniversityDoctor's degree, and then stay in the university to obtain the lecturer qualification and become an associate professor;In 1893, he was appointed full professor;In 1895, he was transferred to Gottingen University as a professor;On August 8, 1900ParisThe second sessionInternational Congress of MathematiciansHe proposed 23 mathematical problems that mathematicians should strive to solve in the 20th century[6]；Retired in 1930;In 1942, he became an honorary member of the Berlin Academy of Sciences;He died on February 14, 1943[7]。
David Hilbert's research fields include algebraic invariants, algebraic number fields, geometric foundations, variational methods, integral equations, infinite dimensional spaces, physics and mathematical foundations, etc[2]。
It puts forward 23 mathematical problems that mathematicians should strive to solve in the 20th century In 1942, he became an honorary member of the Berlin Academy of Sciences
On January 23, 1862, David Hilbert was born in Weilao, near Konisburg, East Prussia (now Kaliningrad, Russia). In middle school, he made friends with Herman Minkowski (German: Hermann Minkowski, Einstein's teacher) and entered Konisburg University together.
In 1880, despite his father's willingness to let him study law, he entered the University of Konesburg to study mathematics.
In 1884, he received a doctor's degree from the University of Konesburg, and later stayed in the university to obtain the qualification of lecturer and was promoted to associate professor.
In 1893, he was appointed full professor.
In 1895, he was transferred to Gottingen University as a professor. Since then, he has lived and worked in Gottingen, the hometown of mathematics.
On August 8, 1900, at the Second International Conference of Mathematicians in Paris, Hilbert proposed 23 mathematical problems that mathematicians in the new century should strive to solve, which is considered to be the highest point of mathematics in the 20th century.The research on these problems has strongly promoted the development of mathematics in the 20th century, and has had a far-reaching impact in the world.
In 1930, he retired.
In 1942, he became an honorary member of the Berlin Academy of Sciences.
He died on February 14, 1943[1]。
Key achievements
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Scientific research achievements
Overview of scientific research
In 1932, Hilbert was lecturing
David Hilbert's mathematical work can be divided into several different periods, in which he almost concentrates on one kind of problems.In chronological order, his main research contents include: invariant theory, algebraic number field theory, geometric basis, integral equation, physics, and general mathematical basis. His research topics interspersed with them include: Dirichlet's principle and variational method, Walling problem, eigenvalue problem, "Hilbert space", etc.
In 1900, at the International Congress of Mathematicians in Paris, Hilbert gave a speech entitled "Mathematical Problems".He put forward 23 most important mathematical problems according to the achievements and development trends of mathematical research in the past, especially in the 19th century.These 23 problems, collectively known as Hilbert problem, later became the difficulties that many mathematicians tried to overcome, and had a profound impact on the research and development of modern mathematics.Some of the Hilbert problems have been successfully solved, and some have not yet been solved.The belief that every mathematical problem can be solved explained in his speech is a great inspiration to mathematical workers.He said: "Among us, we often hear the voice: Here is a mathematical problem, find the answer! You can find it through pure thinking, because there is no unknowable in mathematics."
In 1930, in his speech accepting the title of Honorary Citizen of Gothenburg, he once again confidently declared that "we must know, we will know." After Hilbert died, this sentence was engraved on his tombstone.Hilbert's Geometric Basis (1899) is the representative work of axiomatic thought. In the book, Euclid's geometry is sorted into a pure deductive system based on a group of simple axioms, and the relationship between axioms and the logical structure of the whole deductive system are explored.In 1904, he began to study the basic problems of mathematics again. After years of deliberation, in the early 1920s, he proposed a plan on how to demonstrate the consistency of number theory, set theory or mathematical analysis.He suggested that mathematics should be formalized into a symbolic language system from several formal axioms, and the corresponding logical system should be established from the point of view of finiteness, which does not assume real infinity.Then the logical properties of this formal language system are studied, and meta mathematics and proof theory are established.Hilbert's purpose is to try to give an absolute proof of the non contradiction of a formal language system, so as to overcome the crisis caused by the paradox and eliminate the doubt on the reliability of mathematical foundation and mathematical reasoning methods once and for all.
In 1930, the young Austrian mathematical logician K.G? Del (1906-1978) obtained a negative result, proving that the Hilbert scheme was impossible to realize.But just as Godel said, Hilbert's plan on mathematical basis "still has its importance and continues to arouse people's high interest." The academic treatise Hilbert's Complete Works (three volumes), including his famous Report on Number Theory, Geometric Basis, and General Theoretical Basis of Linear Integral Equations, was co authored with others, including Mathematical Physics MethodsFundamentals of Theoretical Logic, Intuitive Geometry, and Fundamentals of Mathematics.
Hilbert Problem At the 1900 Paris International Congress of Mathematicians, Hilbert delivered a famous speech entitled "Mathematical Problems".He put forward 23 most important mathematical problems according to the achievements and development trends of mathematical research in the past, especially in the 19th century.These 23 problems, commonly known as the Hilbert problem, later became the difficulties that many mathematicians tried to overcome, which had a profound impact on the research and development of modern mathematics and played an active role in promoting it.Some of the Hilbert problems have now been satisfactorily solved, while others remain unsolved.The belief that every mathematical problem can be solved explained in his speech is a great inspiration to mathematical workers.Hilbert's 23 questions are divided into four parts: the first to the sixth are basic mathematical problems;Questions 7 to 12 are number theory;Questions 13 to 18 are algebraic and geometric;Questions 19 to 23 belong to mathematical analysis.(1) Cantor's continuum cardinal number problem.In 1874, Cantor conjectured that there was no other cardinality between the cardinality of the countable set and the cardinality of the real number set, that is, the famous continuum hypothesis.In 1938, the Austrian mathematical logician Godel, who lived in the United States, proved that there was no contradiction between the continuum hypothesis and the axiom system of ZF set theory.In 1963, the American mathematician P. Choen proved that the continuum hypothesis and ZF axiom were independent of each other.Therefore, the continuum hypothesis cannot be proved by ZF axiom.In this sense, the problem has been solved.(2) There is no contradiction in the system of arithmetic axioms.The non contradiction of Euclidean geometry can be summed up as the non contradiction of arithmetic axioms.Hilbert once proposed to use the method of proof theory of formalism plan to prove it, and Godel published the incompleteness theorem in 1931 to deny it.In 1936, G. Gentaen (1909-1945) used transfinite induction to prove the non contradiction of the system of arithmetic axioms.(3) It is impossible to prove that two tetrahedrons with equal base and height have the same volume only according to the contract axiom.The meaning of the problem is that there are two tetrahedrons of equal height bottom, which cannot be decomposed into a finite number of small tetrahedrons, so that these two groups of tetrahedrons are congruent with each other.M. Dehn solved the problem in 1900.(4) The shortest distance between two points is a straight line.This question is generally raised.There are many geometries satisfying this property, so some restrictions need to be imposed.In 1973, the Soviet mathematician Pogleov announced that the problem was solved under the condition of symmetrical distance.(5) The condition that topology becomes a Lie group (topological group).This problem is referred to as the analyticity of continuous groups, that is, whether every local Euclidean group must be a Lie group.In 1952, it was jointly solved by Gleason, Montgomery and Zippin[4]。In 1953, Japan's Yoshihiko Yamamay had been fully affirmed.(6) The axiomatization of physics that plays an important role in mathematics.In 1933, Soviet mathematician Kolmogorov axiomatized the theory of probability.Later, he succeeded in quantum mechanics and quantum field theory.But many people doubt whether all branches of physics can be fully axiomatic.(7) The proof of the transcendence of some numbers.It should be proved that if α is an algebraic number and β is an algebraic number of irrational numbers, then α ^ β must be a transcendental number or at least an irrational number (for example, 2 ^ √ 2 and exp (π)).Gelfond of the Soviet Union in 1929, Schneider and Siegel of Germany in 1935 independently proved its correctness.But the transcendental number theory is far from complete.There is no unified method to determine whether the 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 here Riemann conjecture, Goldbach conjecture and twin prime number problem.Riemann conjecture has not been solved yet.The Goldbach conjecture and the twin prime problem have not been finally solved, and the best results belong to Chinese mathematicians Chen Jingrun and Zhang Yitang respectively.(9) The proof of general reciprocity law in arbitrary number field.In 1921, it was basically settled by Japan's Shoji Takagi, and in 1927 by Germany's E. Artin.The domain theory is still developing.(10) Can we use finite steps to determine whether there is a rational integer solution to an indefinite equation?To find the integer root of an equation with integer coefficients, it is called Diophantine (ancient Greek mathematician, about 210-290) equation solvable.Around 1950, American mathematicians Davis, Putnan and Robinson made critical breakthroughs.In 1970, Baker and Philos reached a positive conclusion on the equation containing two unknowns.1970.Mathematician Matissevich of the Soviet Union finally proved that in general, the answer is no.Although negative results have been obtained, a series of valuable by-products have been produced, many of which are closely related to computer science.(11) Quadratic form theory in general algebraic number field.German mathematicians Hasse and Siegel obtained important results in the 1920s.In the 1960s, French mathematician A. Weil made new progress.(12) The composition of class domain.That is to say, the Kronecker theorem on Abelian fields is extended to arbitrary algebraic rational fields.There are only some sporadic results of this problem, which is far from being solved completely.(13) It is impossible to solve a general seventh order algebraic equation by a combination of two variable continuous functions.(14) Establish the foundation of algebraic geometry.From 1938 to 1940, Van der Walden, a Dutch mathematician, and Wei Yi, a French mathematician, solved the problem.Note 1 The strict basis of Schubert's counting calculus.A typical question is: there are four lines in 3D space. How many lines can intersect these four lines?Schubert gave an intuitive solution.Hilbert demanded to generalize the problem and give it a strict basis.There are some computable methods, which are closely related to algebraic geometry.But the strict foundation has not yet been established.(15) Topological study of algebraic curves and surfaces.The first half of this problem involves the maximum number of algebraic curves with closed branching curves.The second half requires to discuss the maximum number N (n) and relative position of the limit cycle with dx/dy=Y/X, where X and Y are n-degree polynomials of x and y.For the case of n=2 (i.e. secondary system), in 1934, Froscher obtained N (2) ≥ 1;In 1952, Bottin obtained N (2) ≥ 3;In 1955, the Soviet Union's Podrovski announced that N (2) ≤ 3, a result that had been shocking for a while, was questioned because some lemmas were denied.As for the relative position, Chinese mathematicians Dong Jinzhu and Ye Yanqian proved in 1957 that (E2) does not exceed two strings.In 1957, Chinese mathematicians Qin Yuanxun and Pu Fujin gave an example that the equation of n=2 has at least three limit cycles in a string.In 1978, under the guidance of Qin Yuanxun and Hua Luogeng, Shi Songling and Wang Mingshu of China respectively cited at least 4 specific examples of limit cycles.In 1983, Qin Yuanxun further proved that the quadratic system has at most four limit cycles and is (1,3) structure, thus finally solving the structure problem of the solution of the quadratic differential equation, and providing a new way to study the Hilbert (16) problem.(16) Construct space with congruent polyhedron.In 1910, German mathematician Bieberbach, and in 1928 Reinhart made a partial solution.(17) Is the solution of the regular variational problem always an analytic function?German mathematician Bernrtein (1929) and Soviet mathematician Petrovsky (1939) have solved the problem.(18) The general boundary value problem is studied.This problem has made rapid progress and has become a large branch of mathematics, which is still developing.(19) The existence of solutions for Fuchs class linear differential equations with given singular points and single valued groups.This problem belongs to the large-scale theory of linear ordinary differential equations.Hilbert in 1905 and H. Rohrl in 1957 reached important results respectively.In 1970, French mathematician Deligne made outstanding contributions.(20) The analytic function is monovalued by automorphic function.This problem involves the difficult Riemann surface theory. In 1907, P. Koebe solved a variable case, which made an important breakthrough in the research of the problem.Other aspects have not yet been resolved.(21) Develop the research of variational methods.This is not a clear mathematical problem.The variational method has developed greatly in the 20th century[3]。
Academic treatise
In 1899, David Hilbert published Geometric Basis, which became the representative work of modern axiomatic methods, and thus promoted the formation of "mathematical axiomatic chemistry school".
Honor recognition
time
Honor recognition
Awarding unit
1930
Mittag Leffler Award
Swedish Academy of Sciences
1942
Honorary academician of the Berlin Academy of Sciences
Berlin Academy
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Communication academician of Berlin Academy of Sciences
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Steiner Award
Lobachevsky Award
Poyoy Award
Personal life
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marriage and family
Hilbert's Former Residence in Gottingen
In 1892, she got married.
Anecdotes of characters
1. There are so many mathematical nouns named after Hilbert that Hilbert doesn't even know them.For example, Hilbert once asked his colleagues in the department, "What is Hilbert space?"
2. In 1916, Amy Nott, a talented young woman, came to the University of Gottingen.Hilbert greatly appreciated her knowledge and immediately decided to let her stay as a lecturer to assist in the research of relativity.However, discrimination against women was quite serious at that time, and Hilbert's proposal was strongly opposed by professors of linguistics and history.Hilbert stood up and shouted: "Gentlemen, this is the school, not the bathhouse!" So he angered his opponent. Hilbert was unmoved by this and decided to let Nott take the place of the class in his own name.
3. One of his students bought a car and died in a car accident.At the funeral, the family of the deceased asked Mr. Hilbert to say a few words, so he said: "Little Klaus was the best among my students. He had an extraordinary talent in mathematics. He covered a wide range of mathematical problems, such as..." He paused for a moment, then said: "Consider a group of differentiable functions on the unit interval, and then take their closures..."
Character evaluation
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Hilbert in 1912
Hilbert was one of the mathematicians who had a profound influence on the mathematics of the twentieth century. He led the famous Gottingen School, made Gottingen University an important center of mathematical research in the world at that time, and cultivated a group of outstanding mathematicians who made great contributions to the development of modern mathematics.
Hilbert's research in geometry has had the greatest impact on this field after Euclid.Through the systematic study of Euclid's geometric axioms, Hilbert put forward 21 such axioms and analyzed their significance.He has made contributions in many areas of mathematics and physics.[1](ScotlandSt Andrews UniversitySchool of Mathematics and Statistics)
Hilbert's Tomb.[3]
Hilbert is one of the greatest mathematicians in Germany and even in the world in the first half of the twentieth century.In his 60 years of research career spanning two centuries, he has almost walked all the frontiers of modern mathematics, thus deeply penetrating his thoughts into the whole modern mathematics.Hilbert is the core of the Gottingen School of Mathematics. He has attracted young scholars from all over the world with his hard work and sincere personal qualities, making Gottingen's tradition have an impact in the world.When Hilbert died, the German magazine Nature published the view that it is rare for a mathematician in the world to derive his work from Hilbert's work in some way.He is like Alexander of the mathematical world, leaving his famous name on the whole mathematical map.
The mathematical school led by Hilbert is a banner of the mathematical world at the end of the 19th century and the beginning of the 20th century. Hilbert is known as "the uncrowned king of the mathematical world". He is a genius among talents[5]。
