There is an old Legend in Europe: a famous chariot was firmly bound by a rope woven from the bark of a dogwood tree. Do you want to take the throne of the world? Then the knot must be untied. Countless clever and strong warriors came with hope and went away in despair, because the knots of the ropes were coiled and entangled, and the rope heads were hidden and difficult to find. One day, Alexander also came here, and after a little thought, he decisively drew his sword and cut the rope into two pieces. The intractable knot was thus easily "untied". Alexander thus enjoyed the right to rule the entire world.
On September 6, 1888, people were pleasantly surprised to learn that the famous problem that many mathematicians had struggled for more than a decade, the Golden Problem, had finally been overcome by a young man who was unknown at the time. The methods and methods he employed were so surprising and convincing, just as Alexander untied the knot; and just as the illustrious monarch left a great battle mark on the vast continent of Eurasia, the young man who devoted his life's efforts and talents to the vast field of mathematics, throughout almost all the frontiers of modern mathematics, and his glorious name was inscribed everywhere on the entire territory of mathematics. He is the Alexander of the mathematical world, David Hilbert! In the mathematical world, there is such a great god who has influenced the entire course of mathematical history in the twentieth century, and he is David Hilbert.
01 The problems of the new century indicate the direction of mathematics throughout the century
On August 8, 1900, the first summer of the new century, the Second Congress of International Mathematicians was being held in the lecture hall of the University of Paris. Standing at the podium was a medium-sized bearded scholar, who looked less than 40 years old, and his sonorous voice echoed throughout the hall, "At the dawn of the new century, who among us does not want to open the curtain of the future and see the prospects and mysteries of the development of our discipline?" What will the mathematical trends that will lead the next generation pursue?
In the vast and rich field of mathematical thought, what new methods and new achievements will the new century bring? The speaker's three questions and the powerful answers that followed shook the hearts of the participants, namely David Hilbert, the world's most famous German mathematician.
After half a year of careful thinking and meticulous preparation, Hilbert's 40-page speech is the famous lecture of "Mathematical Problems", and he put forward 23 problems that mathematicians in the new century should strive to solve based on the achievements of mathematics in the 19th century and the trend of future development.
These 23 problems include four types: mathematical foundation problems, number theory problems, algebraic and geometric problems, and mathematical analysis problems. Covers almost the entire field of modern mathematics. As he often says to his students, "A perfect presentation of a problem means that it is half solved." "
It has been proved afterwards that these problems are not only groundbreaking, but also the study of them has also promoted the development of mathematics worldwide, and has become the "commanding heights" of mathematical development in the new century.
At the end of his speech, Hilbert said with passion and infinite hope: "Mathematics is the basis of all precise knowledge of natural science, and in order to achieve our noble goal, let the new century bring to this discipline the masters of genius and countless enthusiastic believers." "
Hilbert's historic speech has stimulated the incomparable imagination of the world mathematical community, and it is also a great encouragement and encouragement to the mathematical community. It is widely believed that a mathematician who can solve any of the 23 problems in Hilbert's Paris Lecture is a major contribution to the mathematical sciences, and with the solution of the problem, it will certainly promote the development of mathematics in the 20th century.
Some people call him "the last all-rounder in mathematics", and his vision of mathematics is also quite profound. As Hilbert's friend and famous mathematician Minkowski put it, Hilbert's speech "provides a navigation map for the development of mathematics in the new century." In 1950, half a century after Hilbert's speech, Herman Weier, a famous mathematician at the Princeton Institute for Advanced Study in the United States and a student of Hilbert, summed up the history of mathematics in the first half of the 20th century at a conference of the American Mathematical Society, saying: "In the past 50 years, we mathematicians have measured our progress according to this navigation map. "
02 Hilbert's path to mathematics was natural
Hilbert was born in 1862 in Königsberg, East Prussia, the birthplace of many German celebrities, the German philosopher Kant, the Nobel Prize in chemistry Lavach, the famous mathematical "conjecture" Goldbach and the famous mathematician and Einstein's teacher Minkowski were born in this small town. It has been said that Königsberg, located on the Pregel River, has a mathematical charm, and the two tributaries have a total of 7 bridges, which makes it the famous "Königsberg Seven Bridges Problem". This is one of the four famous mathematical problems, which was later solved by the Swiss mathematician Euler, and thus pioneered the study of topology.
The influence of the environment, the influence of his family, and his own endowments made Hilbert a keen interest in science, especially mathematics, from the time he was in middle school. At the age of 17, he entered the University of Königsberg to study mathematics, where he developed a lifelong friendship with like-minded Minkowski.
03 Super great teacher
In 1895, a crucial year in his life, he came to Göttingen, germany's mathematical capital, and became a professor of mathematics at the University of Göttingen until his retirement.
Between the rolling green hills, there is a quaint town. Its small size, population of less than 120,000, rugged and narrow cobblestone roads, low old red-roofed houses, quiet and rustic style, make it superficially look like it is no different from other small towns in Germany, but if you stop, you will soon find that it is full of strong academic atmosphere, it is the world-famous German academic town - Göttingen. At the entrance to the Old Town Hall, there stands a plaque inscribed with an old Latin phrase that points out the characteristics of the small town of Göttingen: "There is no life outside Göttingen, and even if there is life, it is not such a life." "What should life be like in Göttingen?"
The names of the streets here are silently explained to you, they are named after scholars who have studied and worked here for a long time, Gauss, Weber, Hahn, Riemann, Bunsen, Plant, Planck, Laue, Wien, Born, Klein, Hilbert, Oppenheimer... These names proclaim an achievement, a spirit, a demeanor, a personality, and it is no exaggeration to say that any one of these people, if a city or a university possesses one of them, is enough to make them an indelible page in the history of human civilization, and here they appear in brilliant succession.
The Göttingen spirit focused on "succession of literati", Klein's collaboration with Hilbert, Hilbert's collaboration with Minkowski, Gauss's collaboration with Weber, the founder of electromagnetism, Born and Frank's long-term partnership in theory and experimentation, and Pauli's and Heisenberg's brotherly help are all fruitful examples. In 1886 the mathematician Phyllis Christian Klein came to Göttingen, and after unremitting efforts, he hired Hilbert, who was 13 years younger than him and had already shown great genius. In Germany at that time, there was a self-evident "unspoken rule" that "there can be no more than two Jewish scholars in a place in a field", but Hilbert overcame many difficulties and hired Hermann Minkowski, a young Jewish man who won the All-German Prize in Mathematics at the age of 17. It was the spirit of "succession of literati" that brought Göttingen's school of mathematics into its heyday.
Since Hilbert came here as a professor in March 1895, Göttingen's mathematical prowess has attracted the attention of the world mathematical community. Hilbert pioneered a mathematical seminar that attracted mathematicians from all over the world. His deep understanding of mathematics, high-level teaching content, and rigorous lectures made him a true mathematics educator. When he lectured, hundreds of people often gathered in the lecture hall, and some even sat on the windowsill. Famous mathematicians and physicists of the 20th century, such as Wiener, von Neumann, Bohr, Born, and others, listened to his teachings several times. Göttingen became a place of mathematical pilgrimage admired by all of Germany, even in Europe and the world, and the undisputed center of mathematics in the world. At that time, almost all mathematics students in the world had a dream of "packing up their backpacks and going to Göttingen!" "Because there's Hilbert there.
In Göttingen, Hilbert worked and lived for 48 years. His invariant theory, algebraic field theory, integral equations, gravitational theory tensor theory, integral equation variational method, Waring problem, eigenvalue problem, Hilbert space, etc., can be called a pioneering contribution to mathematics. Among them, there are 40 theories, concepts and methods named after him alone, such as Hilbert space, Hilbert modulo forms, Hilbert rings, Hilbert transforms, Hilbert theorem, Hilbert procedures, etc. His books include The Complete Works of Hilbert, including Three Volumes of Reports on Number Theory, Fundamentals of Geometry, and General Theory of Linear Integral Equations, as well as Methods of Mathematical Physics, Fundamentals of Mathematical Logic, Intuitive Geometry, and Fundamentals of Mathematics. In his monumental work Fundamentals of Geometry, he sorted out Euclidean geometry and, on this basis, proposed a more rigorous and complete axiomatic system of geometry.
His achievements established a solid foundation for various branches of mathematics in the early 20th century, which set off the "axiomatic movement" of the mathematical community, leading a school of mathematics and becoming a banner of the mathematical community in the late 19th and early 20th centuries. Hilbert was a "genius of mathematical geniuses" and was revered as the "uncrowned king" of mathematics. However, Hilbert, who stands at the top of mathematics, does not regard mathematics as a cold holy place of "not eating incense in the world", and he once said to his students repeatedly: "For any mathematical theory, you must be able to face the first person who comes on the street and explain it to him, then you really understand it." "
04 Uncrowned King of Mathematics
Hilbert grew up in a religious environment and was baptized in Protestantism, but later he turned away from the church, firmly believing that mathematical truths exist independently of the assumptions of God or any prophet. Hilbert officially retired in 1930, and on September 8 of that year, the German Society of Science and Physics held a retirement ceremony for him. In thanksgiving, Hilbert first quoted a Famous Latin phrase: "We don't know, we can't know," and then he modified the quote and gave a sonorous and forceful response, saying that we should say, "We must know, we will certainly know." Today, this sentence has become a well-known saying among mathematicians around the world, and it is an encouragement and encouragement to those who come after.
A particular era doomed Hilbert to the bitterness of his later years. In 1933, after Hitler came to power, he fully implemented the fascist dictatorship. Under duress from sticks and muzzles, the Jews were expelled from Germany. As Hilbert watched his colleagues and students being forced to leave, looking back on the past, how could Hilbert treat him? The old man spent his old age in depression and loneliness. Hilbert died, but in almost all areas of mathematical development, he stood majestically; Hilbert's spirit still inspired future generations to continue to succeed. "We have to know, we have to know." This sonorous and heroic statement is not only deeply engraved on Hilbert's tombstone, but also permanently echoes in the hearts of scientists all over the world!
When Hilbert died in 1943, the German magazine Nature gave Hilbert a high opinion: "It is rare in the world today that the work of a mathematician is not derived from Hilbert's work in some way. He is like the Alexander of the mathematical world, leaving his illustrious name on the entire mathematical map.