David Hilbert (1862–1943) was one of the most prominent mathematicians of the late 19th and early 20th centuries. As a mathematician, he was energetic, thoughtful, and creative; as a mentor, he was tireless, seductive, and approachable. His great achievements in mathematics and science have had a profound impact, and Hilbert's 23 questions directly guided the direction of mathematics in the 20th century. It is rare in the world that the work of a mathematician does not originate in some way from Hilbert's work. Hilbert is like the Alexander of the mathematical world, leaving his great and illustrious name on the entire mathematical map.
Königsberg and Minkowski
Hilbert was born on 23 January 1862 in Königsberg, the capital of East Prussia (the birthplace of the Problem of the Seven Bridges of Euler), the son of a local judge and the daughter of a merchant. His enlightened parents taught him from an early age the virtues of Prussian honesty and hard work. As a child, Hilbert was a bit sluggish compared to his peers and didn't go to school until he was 8 years old. His parents sent him to frederick preparatory school, a prestigious local school, which was Kant's former alma mater. Hilbert, however, was not at all interested in the tedious curriculum of Greek and Latin, and the only thing that really fascinated him was mathematics, and he simply transferred to the More Emphasis on Mathematics Education At William Preparatory School. Here, Hilbert was finally stimulated by his talents, achieving excellent grades in almost all courses, and "superclass" in mathematics. The school wrote in Hilbert's graduation review:
"His diligence is exemplary, he has a strong interest in mathematics, he has a deep understanding, he grasps what the teacher teaches in a very good way, and can apply it with confidence and flexibility."
It was also in his hometown of Königsberg that Hilbert met a lifelong friend, Hermann Minkowski (1864-1909), also a master of mathematics. Interestingly, it was Minkowski, the head of the mathematics department, who chased Einstein in Switzerland to drop him out of school.
Minkowski
In 1872, Minkowski's family fled to Königsberg to settle down in Königsberg to escape the Tsar's relentless persecution of the Jews. Minkowski's eldest brother and father re-established himself as a wealthy businessman, and his second brother was Oscar Minkowski (1858–1931), the later famous "father of insulin." Minkowski completed eight years of preparatory school in just five and a half years, then entered the University of Königsberg, and two and a half years later transferred to the then German mathematical center berlin university. It happened that in the year of Minkowski's graduation, the Paris Academy of Sciences announced that the title of the 1883 Grand Prize was a representation of integers split into the sum of five squares. Encouraged by his eldest brother, Minkowski pondered carefully, and with his strong mathematical talent, he wrote a 140-page solution paper, which was finally awarded at the same time as henry Smith, a veteran British mathematician. At this time, Minkowski was not yet 19 years old. The British were furious, but Königsberg was sensational. The grand prize judge, Jordan, the authority of French mathematics at the time (1838-1922), was not afraid of pressure, insisted on awarding the prize to the young and very mathematically talented Minkowski, and wrote to him: "Do it! I beg you to be a great mathematician! ”
Jodang
News of Minkowski's award soon reached Hilbert, who, despite his father's admonition, happily went to visit Minkowski across the river. With the same love for mathematics, they soon became close friends, and a sincere friendship accompanied them throughout their lives.
On the back hill of Königsberg University, Hilbert and Minkowski, as well as Herwitz (1859-1919, German mathematician), often walked and discussed mathematics and exchanged ideas. In the future, all three of them will become important figures in the mathematical world. The University of Königsberg has a good tradition and excellent teachers, and Jacques Bie has taught here, and professor Richard Lauter, the discoverer of Weierstrass, has also worked here. The atmosphere here is relaxed, the teacher can teach what he wants, the students want to learn anything at will, there is usually no roll call and examination, and students can give full play to their interests and strengths. Hilbert was also able to study mathematics with all his might.
Invariants and number theory
A few years later, Hilbert completed his doctoral thesis on the invariant nature of certain algebraic forms, and obtained some wonderful results. On the advice of his mentor Lindemann (1852-1939, a German mathematician who proved that π is a transcendent number), Hilbert decided to go out of Königsberg and see the wider mathematical world outside.
At Herwitz's suggestion, Hilbert first visited Felix Christian Klein (1849–1925), a legend and leader of German mathematics at the time. Klein was so impressed with him that after listening to a report from Hilbert, he predicted that he would become a great instrument in the future, and carefully collected his speeches. Unexpectedly, decades later, when celebrating Hilbert's 60th birthday, Klein, who was already in a wheelchair, actually took out the speech of that year and gave it to Hilbert, which was a big gift. After bidding farewell to Klein, he went to Paris, the center of mathematics at the time. At this time, the French mathematical community was no longer as cold and arrogant as in Cauchy's time, and the French mathematical colleagues led by Poincaré warmly welcomed Hilbert's arrival. In Paris, Hilbert had an in-depth and friendly exchange with Charles Hermite (1822-1901), especially on the "Göldán question" that both were concerned about. The trip to Paris was accompanied by many recent developments and achievements in mathematics. On his way back to Königsberg, he also made a special trip to Göttingen to visit the eccentric Leopold Kronecker (1823-1891). Kronecker was unusually thin, less than a meter and a half tall, and was ostracized by the mathematical community for his often snarky criticism of mathematics. To the surprise of the outside world, he received Hilbert warmly.
Klein
Poincaré
Kronecker
After submitting a paper on invariants, Hilbert successfully obtained the lecturer qualification, and invariant became the number one problem in Hilbert's mind at this time. The study of invariant problems was first pioneered by the British mathematicians Arthur Cayley (1821-1895) and James Joseph Sylvester (1814-1897), but soon the Germans caught up, and the "king of invariants" Gordan (1837-1912) was born. The problem of invariants was a big hit in mathematics at that time, and the "Gordan problem" was the focus of research, and the so-called "Gordan problem" said whether there was a set of bases (that is, a set of finite invariants) that could be represented by the rational integer form of this set of bases?
Gordan
After a long period of deliberation and a flash of inspiration, Hilbert proved that the "Göldan problem" was true in extremely concise mathematical language. Both Klein and Gloria were overjoyed to congratulate Hilbert. But Hilbert's proof eventually sparked a famous controversy in the history of mathematics. A group of mathematicians, represented by Crohneck and Gordan, believed that only existence without construction could not be regarded as a real proof, and a controversy was waged with Hilbert. Mere proof of existence is now commonplace in mathematics, but it was not acceptable to some old-school mathematicians at the time. Unable to be bothered, Hilbert finally constructed a solution to the "Gordan problem", and then Gordan closed his mouth. Minkowski was very pleased to send a congratulatory letter, "I have long known that it is only a matter of time before you solve this old problem of invariants—as if only that point was missing on the "i"; but it was so surprisingly solved at once that I was very happy, and let me congratulate you." Hilbert became famous in world war I, and his position in the mathematical community suddenly rose sharply.
In 1892, Hilbert succeeded Hvwitz as an associate professor in Königsberg. Having achieved fame in invariants, Hilbert's attention at this time had shifted to algebraic number theory. Just a year later, he gave a new and very concise proof of π and transcendence, which once again shocked the mathematical community. In the autumn of that year, Hilbert traveled to Munich to attend the annual meeting of the German Mathematical Society. At the meeting, Hilbert presented two new proofs for decomposing numbers in a domain into prime ideals, and his work once again impressed the other members. Excited, everyone hoped that Hilbert and Minkowski could write a report on the development of number theory, because the work of Kummer (1810-1893, German mathematicians) and Richard Dedekind (1831-1916) was too obscure.
Dedekin
Good things always happened to Hilbert, and soon after, due to the vacancy of a chair at the University of Göttingen, Klein extended an invitation to Hilbert, and Hilbert gladly accepted, coming to the holy land of mathematics a hundred years after Gauss's arrival. He will also create new glory here and spend the rest of his life.
In 1897, after a long period of hard work, the 400-page "Number Theory Report" was officially completed and published. It exceeded the initial expectations of the members of the German Mathematical Society in every respect. They had hoped only for an overview of the current state of the theory, but what they received was a truly masterpiece, integrating all the problems into a beautiful and complete theoretical system, combining previously isolated number theory with algebraic and function theory. Minkowski excitedly wrote after receiving the report:
"I have no doubt that Hilbert will be among the great scholars in the field of number theory."
After the publication of the "Number Theory Report", Hilbert published a programmatic paper " Relative Abelian Domain Theory " , which pioneered the later well-known domain theory. If Hilbert's work on invariants was the end of a theory, his work on the field of algebraic numbers became the starting point for many mathematicians.
Geometric foundations, the resurrection of Dirichlet's principle, and Hilbert's 23 questions
As early as the time of Königsberg University, Hilbert began to consider the problem of geometric foundations. He believes that the definition of points, lines, and surfaces in Euclidean geometry is actually not too important, and it can be completely replaced by tables, chairs, and beer glasses. While what is really important is the adoption of the axiom system with regard to the axioms, hilbert believed that it must be complete, independent, and compatible with these three conditions. After thinking about this, Hilbert used analytic geometry to prove that any contradiction existing in Euclidean geometry can be equated with a contradiction in real arithmetic. This shows that both non-Euclidean and Euclidean geometries are at least as compatible as real arithmetic. Hilbert later compiled these results in his famous book, Fundamentals of Geometry. As soon as the book was published, it caused an immediate sensation, and no one expected that a mathematical master would pay attention to the basic field of mathematics and make it look new. Poincaré praised it in particular: "Some contemporary geometricians may feel that they have reached their limits in recognizing possible non-Euclidean geometries based on the denial of parallel postulates. If they read Professor Hilbert's work, this illusion will be eliminated. They will find in this work that their cocooning barrier has been completely broken. ”
We all know that Riemann established the classical theory of single-complex variable functions in his famous doctoral thesis, but Riemann's theory is based on a hypothesis, named Dirichlet's principle: there must be a solution to the edge value problem of Laplace's equations. Mathematicians at the time thought that this principle was correct, because the physical processes described by the equations must have a result. But Weierstrass, the "gossip terminator" who likes to "nitpick", does not think so, and after a long period of thinking, he is stunned to cite a counterexample that Dirichlet's principle does not hold, and the mathematical community is suddenly in an uproar. Years later, Hilbert once again focused on the "dead knot" that had been sealed for many years, because he did not want to write off the huge role played by Dirichlet's principle. He proved that by imposing certain restrictions on the properties of curves and boundary values, the shortcomings criticized by Vilstrass could be eliminated and the simplicity and elegance of Dilickré's principle restored. Klein excitedly said Hilbert had succeeded in "cutting the surface."
The turn of the century will soon come, and Hilbert will give a great gift to the development of mathematics in the 20th century, that is, the 23 profound questions he asked to guide the development of the 20th century. Regarding Hilbert's 23 question, I described it in detail in the previous article, so I will not repeat it here.
Integral equations, physics, and Waring problems
In the winter of 1900, the Swedish student Fred Holm (1866-1927, Swedish mathematician) brought a newly published paper on integral equations to Hilbert's seminar. FredHolm creatively gives solutions to some special integral equations, revealing some hidden similarities between the integral equations and the linear algebraic equations. Hilbert immediately realized the great significance of Fred Holm's paper. Without hesitation, he dropped the variational method he was studying and threw himself into the study of the integral equation. The development of integral equations was unusually slow and difficult before this, and Hilbert's addition ushered in a period of heyday in this field, directly giving birth to Hilbert's spatial and functional analysis.
In the autumn of 1902, with the tireless insistence and efforts of Hilbert and Klein, the German Ministry of Education agreed to Minkowski's appointment to Göttingen. Minkowski was overjoyed, saying: "Looking at the future of life and work, I see the best hope!" Even more excited was Hilbert, who was finally able to go back to Königsberg when he was walking and mingling. Göttingen, which was already a mecca of mathematics, was even more brilliant with the arrival of Minkowski.
The energetic Hilbert, under the influence of Minkowski, fell in love with physics again, and fell into the study of physics again. In a seminar, he and Minkowski studied electrodynamics, and this seminar produced the future master of physics, nobel laureate Born. Many of the results of the seminar coincided with Einstein's at the time, and Minkowski proposed his theory of space during this time, laying the foundation for the mathematical foundation of relativity.
Before the physical waste heat had dissipated, Hilbert set out to tackle the Waring problem: each positive integer must be represented as the sum of 4 squares, the sum of 9 cubic numbers, the sum of 19 quaternions, and so on; in general, each nth power has a finite number. This puzzle has plagued the mathematical community for more than a hundred years without anyone solving it. The creative Hilbert followed Herwitz's approach and ultimately proved the existence of the problem.
Minkowski's misfortune and the clouds of war
Just as he was excitedly preparing to tell Minkowski, Minkowski collapsed due to a sudden acute appendicitis, and soon after died of malignant complications! The bad news made Hilbert miserable, losing his best friend, and mathematics losing a sincere heart.
A few years later, World War I broke out suddenly. The rational Hilbert refused to sign the war manifesto, which was criticized by the authorities and even the school's teachers and students. As the war intensified, a large number of teachers and students were conscripted into the army, and the usually crowded Göttingen suddenly became lonely. Göttingen's academic exchanges with the outside world also came to an abrupt end, which worried Hilbert. Worse still, after the war, German mathematics has been ostracized by the international mathematical community and appears to be isolated. It was not until the International Congress of Mathematicians in 1928 that the impasse was broken. In spite of the obstruction of domestic and foreign forces, Hilbert resolutely led a 67-member German mathematical delegation to the meeting, and Hilbert was particularly warmly welcomed. He made a generous statement:
"I am overjoyed that, after a long and difficult period, mathematicians from all over the world are gathered here again. For the sake of the prosperity of this science, which we love so much, we should do and can only do so. It should be noted that, as mathematicians, we stand on the top of the mountain of precision scientific research. We have no choice but to take up this noble duty incumbent upon us. Restrictions of any kind, especially of nationalities, are incompatible with the nature of mathematics. Artificially creating ethnic or racial differences in scientific research is an expression of extreme ignorance of science, and its reasons are not worth refuting. Mathematics does not distinguish between races... For mathematics, the whole civilized world is a country! ”
Hilbert again spoke at the congress about his views on mathematical foundations, especially axioms. But then Gödel's principle of incompleteness directly shattered Hilbert's illusion that the narrow compatibility of arithmetic systems was unprovable. Weil vividly portrays it as: "God exists because there is no contradiction in mathematics; the devil also exists because we cannot prove that there is no contradiction." ”
Stopped thinking...
On Hilbert's 70th birthday, Klein's new building was built, but he was not at all happy. The rise to power of the Nazis in Germany caused large numbers of Jews to flee Göttingen, and the entire campus became as lonely as it was during the war, with a hint of death. In the last few years of his life, Hilbert constantly watched his colleagues and students flee due to persecution, and the old man's heart was filled with infinite sadness. Soon after, the elderly Hilbert fell down the street and could barely get out of bed. To make matters worse, the wound also triggered a series of complications. On February 14, 1943, David Hilbert passed away and great souls stopped thinking. He was buried on the other side of the river, where Klein was buried. The tombstones in the meadow are inscribed only with their respective names and dates.
epilogue
For Hilbert's lifelong work and contributions, we borrow from the evaluation of his earliest student, Sommerfeld:
What was Hilbert's greatest mathematical achievement? Is it a variable? Is it his favorite number theory? Is it the basis of geometry?—— that is the greatest achievement in the field after Euclidean geometry and non-Euclidean geometry. In terms of the fundamentals of function theory and the calculation of variations, Hilbert proved the correctness of the Riemann and Dirichlet conjectures. The study of integral equations has also reached its peak... Soon, in the new physics... They bear fruit again. His theory of gases, which had a fundamental effect on new experimental knowledge, has not yet become obsolete. Moreover, his contribution to general relativity is of eternal value. As for his last effort to explore the truth of mathematics, the jury is still out. However, when it is possible to develop further in this area, it will not bypass it and will have to move forward through Hilbert.
Weyl also commented on his teacher, saying:
Hilbert was like a bagpiper in a variegated dress, and the sweet sound of the flute seduced many rats and jumped into the abyss of mathematics with him.
"We must know, we must know", this is Hilbert's lifelong belief, but also the driving force that guides countless latecomers forward. As a mathematical giant, Hilbert's evaluation of him need not be repeated, and the real greatness need not be said!