If you ask the question: "Who is the greatest physicist in modern times?" People with certain modern knowledge will blurt out: "Einstein!" If you ask again: "Who is the greatest physicist like Einstein?" The greatest mathematician of equal status?" The correct answer should be: "Hilbert!"
Hilbert has many similarities with Einstein. They all grew up in the German cultural tradition that is good at theoretical thinking, and they all have good philosophical cultivation and artistic temperament. They all made epoch-making contributions in several important research fields, had a huge impact on contemporary scientists, and still play a leading role today. In 1914, when the German government asked a group of the most famous German scientists and artists to publish a "Message to the Civilized World" to support the Kaiser's war actions, only two people did not sign it: one was Einstein, and the other was Hilbert.
At 1 o'clock in the afternoon on January 23, 1862, a child was born in K?nigsberg, the capital of East Prussia. He was a descendant of the Hilbert family, and his name was David. K?nigsberg, the birthplace of David Hilbert, is not far from the Baltic Sea. The Breegel River flows through the city and enters the sea 4 miles away. This is the birthplace of the Kingdom of Prussia. Its industry and commerce are very developed, and it has a famous university. The great philosopher Kant spent most of his life here. This is the sphere of influence of Protestants. People value life, reason, and "faith from the heart." The abstract and critical thinking abilities of the German people have always been well developed, and the general public is very interested in philosophy and natural science. It is said that after Kant's "Critique of Pure Reason" was published, it even became a decoration on the dressing tables of noble ladies and young ladies to show "knowledge". This kind of elegance is rare in other countries.
Having the honor to be the hometown of the philosopher Kant was a rare advantage for Hilbert. The people of K?nigsberg regarded Kant as the city's greatest resident. Every year on April 22, the anniversary of the philosopher's birth, the underground chapel near K?nigsberg Cathedral is open to the public. Hilbert's mother would always take the young Hilbert to pay homage to Kant's bust surrounded by laurel flowers, and spell out the motto on the church wall word for word:
" "There are two things which fill our souls with ever-increasing wonder and awe the more deeply and persistently we think about them: the starry sky above us and the moral law within us."
Hilbert's mother was an unusual woman, "a weirdo" in German terms. Not only was she interested in philosophy and astronomy, she was also fascinated by mathematics. His mother's influence naturally made Hilbert admire Kant's philosophy since he was a child. Until his later years, when he gave a speech on "Natural Cognition and Logic" at the K?nigsberg Congress of Natural Scientists, he said: "I think that in essence, the basic ideas of Kant's epistemology are also reflected in my research on mathematical causes. ”
Many mathematicians showed high mathematical talents when they were children. Pascal, Newton, Leibniz, Gauss, Abel, Galois... are all legendary mathematical prodigies. Hilbert did not have such outstanding performance when he was a child. In this regard, he is somewhat similar to Einstein. It is said that Einstein's intellectual performance was average when he was a child, he was taciturn, and he did not cope well with the school's syllabus and rarely attracted the attention of teachers. The same goes for Hilbert, who was not very quick at grasping new concepts and had a poor memory. He lacks interest in courses that require rote memorization, especially language courses, but he is quite diligent. Whenever he wants to understand something, he always has to figure it out thoroughly through his own digestion, otherwise he will never give up. One of the reasons why he became interested in mathematics is that mathematics does not require rote memorization, but can be deduced through logic, so it is easier to master. Hilbert's family thought he was a bit strange. His mother wants to help him write essays, but he can explain math problems to the teacher. No one in the family really understands him.
An important reason why Hilbert's talent was not revealed when he was a child was that the school environment at the beginning was not very suitable for him. The Royal Preparatory School, chosen by his parents for him, was of excellent reputation and Kant himself was a graduate. However, this school's curriculum is very conservative, with a large proportion of language courses, a small weight of mathematics courses, and no natural science. In school, there are few opportunities to think independently and express personal opinions. It was not until the beginning of the last semester of prep school that Hilbert transferred to William Preparatory School. The environment here has been greatly improved, not only focusing on mathematics, but even discussing new developments in geometry. Hilbert's academic performance has improved significantly, and he has received excellent grades in almost all courses. And the math score was "excellent". The character comments on the back of his diploma were: His diligence was “exemplary” and he “had a strong interest in mathematics.” “He showed a strong interest in mathematics and a deep understanding; Methods to master the courses taught by teachers and be able to use them correctly and flexibly."
At the age of 18, Hilbert entered the University of K?nigsberg. This is a university with an excellent scientific tradition, and the famous mathematician Jacobi once taught here. His successor was Rich Lauter, who not only made outstanding contributions in the field of polyperiodic functions, but also turned Weierstrass from an ordinary middle school teacher into a professional mathematician.
Weierstrass, known as the "Father of Modern Analysis", despite his outstanding achievements in mathematical research in his early years, did not have a degree and had been a middle school teacher for more than ten years. Richster discovered him and persuaded his brother to join him. The University of Nisborg awarded him an honorary doctorate. This important turning point fundamentally changed Weierstrass's destiny. There is also a versatile theoretical physicist Newman at the University of K?nigsberg. He founded the first Institute of Theoretical Physics at a German university and initiated study classes. This form of academic activity plays an important role in cultivating talents. The excellent tradition of mathematics and theory at the University of K?nigsberg had a profound impact on Hilbert's later academic development.
University life is simply too free for Hilbert. Professors can teach whatever courses they want, and students can take whatever courses they want. There are no minimum required courses here. The number of classes is not called, and there are no exams during normal times. The exam is only taken once to obtain a degree. Unexpected freedom caused many college students to spend their first year drinking and fighting swords. Weierstrass was a good drinker and sword-fighter when he was young, so he neglected his studies for a time. The mellow aroma of German beer and German drinking are world-famous. Fencing, which symbolizes youthful vitality and strong physique, has also become a traditional activity that college students are obsessed with. But none of this aroused Hilbert's enthusiasm. He devoted himself wholeheartedly to the kingdom of mathematics and discovered a new world where he could develop freely spiritually. Not following the crowd was a key factor in Hilbert's growth. He followed his own path and pursued truth tirelessly. This persistence lasted throughout his life.
After graduating from college, Hilbert went to teach at the university in Leipzig. He taught while conducting mathematical research. The Goldin question established his position in academia.
Goldan was a well-known mathematician at the time, 25 years older than Hilbert. Goldin's academic focus is on the study of invariants. Goldin's question is: Is there a set of basis (that is, a set of finite number of invariants) such that all other invariants (even though there are an infinite number of them) ) can be expressed in the form of organized functions of this set of basis.
Hilbert returned to K?nigsberg. This problem occupied his entire body and mind. He was thinking about it whether at work or in entertainment, and even while dancing. On September 6, 1888, he sent a short note from Lauching to the "Newsletter" of the G?ttingen Scientific Society. In this commentary, he completely unexpectedly opened up a new path, showing how to use a unified method to establish the algebraic form of any number of variables.
“Suppose an infinite set of algebraic formal systems containing a finite number of variables is given. Under what conditions, there exists a finite set of algebraic formal systems such that all other forms can be expressed as The coefficients of their linear combinations are the organized functions of the original variables!"
The answer he finally got was: Such a form always exists.
This sensational proof of the finite existence of the invariant system is based on a lemma, or an auxiliary theorem, which is about the existence of finite bases of modules. "Module" is a mathematical concept that Hilbert got while studying Clonic's work. This lemma is so simple that it seems extremely trivial, and the proof of Goldan's general theorem can be derived directly from it. This work was the first example of the spiritual essence of Hilbert's ideas - what one of his students described as "a natural simplicity of thought, not derived from authority or past experience."
In the following years, Hilbert's status in academia rose. He did everything that most young people do at this age: get married, have children, and accept professors. At the same time, he also decided to explore new research areas.
From 1898 to 1899, Hilbert taught geometry at the University of G?ttingen. He came to a new conclusion: theorems derived from axioms can be established for any explanation of basic concepts and basic relationships. , as long as these concepts and relationships satisfy the axioms. On this basis, a set of simple, complete, and independent axioms is established. With this set of axioms it is possible to prove all the well-known theorems of Euclidean geometry.
Hilbert made important achievements in the field of number theory, and also put forward many insights in physics and logic. 1941 was Hilbert’s 80th birthday. The Berlin Academy of Sciences voted to commemorate this birthday: to give special honor to the 92-page book on the fundamentals of geometry. Of all Hilbert's influential works, it had the most profound impact on the advancement of mathematics.
On the day he went to the Academy of Sciences to make this decision, Hilbert fell on the street in G?ttingen and broke his arm. This unfortunate accident left him unable to move his body and caused various complications. A little more than a year later - on February 14, 1943, he passed away.
Only a few close friends attended the funeral held at his home that morning. Arnold Sowerfeld, one of Hilbert's earliest students, came from Munich to stand beside the coffin and talk about Hilbert's work.
What was his greatest mathematical achievement? "Is it invariants? Is it the number theory that he loved so much? Is it the foundation of geometry? - That was the first breakthrough in this field since Euclidean geometry." The greatest achievement.
In terms of the basics of function theory and variational calculations, Hilbert's proof established the correctness of the conjectures of Riemann and Dirichlet. The research on the theory of integral equations also reached its peak... Soon, in the new physics... they bore the most beautiful fruits. His theory of gases had a fundamental effect on new experimental knowledge and remains obsolete to this day. Also, his contribution to the general theory of relativity is of eternal value. As for his last efforts to explore the true knowledge of mathematics, the jury is still out, but when it is possible to further develop this field, it will not be bypassed and must be continued by Hilbert.
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