Hilbert and Einstein have many similarities. 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, and had a great influence on contemporary scientists, and they still play a leading role today. 19 14, when the German government asked a group of the most famous German scientists and artists to publish this book to the civilized world to support the Kaiser's war action, only two people did not sign it: one was Einstein and the other was Hilbert.
1 86265438+1October 23rd1point, a child was born in konigsberg, the capital of East Prussia. He is descended from the Hilbert family. His name is David. David? Hilbert's birthplace, Konigsberg, is not far from the Baltic Sea. The Broig River runs through the city and flows into the sea four miles away. This is the birthplace of Prussia. Its industry and commerce are very developed, and there is a famous university where the great philosopher Kant spent most of his life. This is the sphere of influence of Protestants People value life, reason and "heartfelt belief". Germans' abstract and speculative abilities have always been developed, and the general public is interested in philosophy and natural science. It is said that when Kant's Critique of Pure Reason was published, it even became an ornament for noble ladies to display "knowledge" on their dressing tables, which is rare in other countries.
Hilbert is fortunate to be a countryman of the philosopher Kant, which is a rare advantage. People in Konigsberg regard Kant as the greatest resident of the city. April 22nd is the birthday anniversary of this philosopher, and the underground shrine near Konigsberg Cathedral is open to the public. Hilbert's mother always led the young Hilbert to pay tribute to Kant's bust surrounded by osmanthus in the moonlight, and read the motto on the temple wall word by word:
"There are two things. The deeper and longer we think about them, the more surprises and awe they arouse and fill our hearts. This is the starry sky above us and the moral law in our hearts. "
Hilbert's mother is an unusual woman. In German words, she is "a freak". She is not only interested in philosophy and astronomy, but also fascinated by mathematics. The influence of his mother naturally made Hilbert respect Kant's philosophy from an early age. Until his later years, when he delivered a speech on "Natural Cognition and Logic" at the conference of natural scientists in Konigsberg, he also said: "I think, in essence, the basic idea of Kant's epistemology is also reflected in my research on mathematical reasons."
Many mathematicians showed extraordinary mathematical talent when they were young. Pascal, Newton, Leibniz, Gauss, Abel and Galois are all legendary mathematical prodigies. Hilbert did not show such outstanding performance when he was a child. At this point, he is somewhat similar to Einstein. It is said that Einstein was average in intelligence and taciturn when he was a child. He didn't cope with the school syllabus well and seldom attracted the attention of teachers. The same is true of Hilbert, who doesn't understand new concepts quickly and has a poor memory. He lacks interest in rote learning courses, especially language courses, but he works hard. Whenever he wants to understand something, he always makes it clear through his own digestion, otherwise he will never stop. One of the reasons for his interest in mathematics is that mathematics can be deduced by logic instead of rote learning, so it is easier to master. Hilbert's family thought he was a little strange. His mother wants to help him with his composition, but he can explain the math problems to the teacher, and no one in his family really knows him.
An important reason why Hilbert's talent as a child was not exposed was that the original school environment was not suitable for him. The Royal Special Preparatory School chosen by his parents has an excellent reputation, and Kant himself is a graduate of the school. However, the curriculum of this school is rather old-fashioned, with a large proportion of language courses and a small weight of mathematics courses, and there is no natural science. In school, there are few opportunities to think and express one's opinions independently. It was not until the last semester of the preparatory school that Hilbert transferred to William Preparatory School. The environment here has been greatly improved, not only paying attention to mathematics, but even discussing the new development of geometry. Hilbert's academic performance has improved significantly, and almost all courses have achieved excellent results. Got an "A" in math. The moral evaluation behind his graduation certificate is: his diligence is "exemplary", "he has a strong interest in mathematics" and "he shows a very strong interest in mathematics and has a profound understanding; He can master the courses taught by the teacher well and use them correctly and flexibly. "
/kloc-at the age of 0/8, Hilbert entered the university of konigsberg. This is a university with a fine scientific tradition, and the famous mathematician jacoby once taught here. His successor is Ricky Laut, who has not only made outstanding contributions in the field of multi-periodic functions, but also changed Wilstrass from an ordinary middle school teacher to a professional mathematician. Known as the "father of modern analysis", Wilstrass made outstanding achievements in mathematics research in his early years, but he worked as a middle school teacher for more than ten years because he didn't have a degree. Richter found him and persuaded the University of Konigsberg to grant him an honorary doctorate. This important turning point fundamentally changed the fate of Wilstrass. There is also a versatile theoretical physicist Newman at the University of Konigsberg. He founded the first institute of theoretical physics in German universities and opened a course. This form of academic activities plays an important role in cultivating talents. The fine tradition of Konigsberg University in mathematics and theory has a far-reaching influence on Hilbert's later academic development.
College life is simply too free for Hilbert. Professors can teach whatever courses they want, and students can choose whatever courses they want. There are no minimum required courses, no roll call, and no exams at ordinary times until you take the exam and get your degree. Unexpected freedom has made many college students spend their first year drinking and fighting swords. When he was young, Wilstrass was addicted to alcohol and a swordsman, so he was once lazy in his studies. Full-bodied German beer and German booze are famous all over the world. Fencing, a symbol of youthful vitality and strong physique, has also become a traditional activity that college students are obsessed with. But all this did not arouse Hilbert's enthusiasm. He devoted himself to the kingdom of mathematics and found a new world that can develop freely in spirit. Not going with the flow is the key factor for Hilbert's growth. He went his own way and pursued the truth tirelessly. This persistent spirit runs through his life.
After graduating from college, Hilbert went to a university in Leipzig to teach. He is doing math research while teaching. Gordan's problems established his position in academic circles.
Gordan was a famous mathematician at that time, 25 years older than Hilbert. Gordan's academic focus is to study invariants. Goldan's question is whether there is a set of bases (that is, a set of finite invariants) so that all other invariants (although their number is infinite) can be expressed in the form of rational integral functions of this set of bases.
Hilbert returned to Konigsberg, and this problem occupied his whole body and mind. He was thinking about it when he was working, playing and even dancing. On September 6th, 1888, he sent a short comment to the communication of Gottingen Science Society from Laoxing Town. In this note, he completely unexpectedly opened up a new road, showing how to establish gordan's theorem for the algebraic form of any variable in a unified way.
"Given an infinite set of algebraic formal systems with finite variables, under what conditions there exists a set of algebraic formal systems with a finite number, so that all other forms can be expressed as their linear combinations, and the coefficients are rational integral functions of those variables!"
He finally got the answer: this form has always existed.
The sensational proof of the finite existence of this invariant system is based on a lemma, or an auxiliary theorem, that is, the existence of finite bases of modules. "Module" is a mathematical concept that Hilbert got when he studied the work of cloning Nick. This lemma is so simple that it seems extremely common that the proof of gordan's general theorem can be directly derived from it. This work is the first example of the spiritual essence of Hilbert's thought-one of his students described it as "a natural and simple thought, not from authority or past experience".
In the following years, Hilbert's position in the academic circle rose. He did everything that most young people should do at this age: get married, have children, accept the letter of appointment of professors, and at the same time he decided to open up new research fields.
From 1898 to 1899, Hilbert taught geometry at the University of G? ttingen, and he came to a new conclusion: the theorems derived from axioms are valid for the interpretation of any basic concepts and relationships, as long as these concepts and relationships meet the axioms. On this basis, a set of simple, complete and independent axioms is established. All well-known theorems in Euclidean geometry can be proved by this set of axioms.
Hilbert has made great achievements in the field of number theory and put forward many insights in physics and logic. 194 1 is Hilbert's 80th birthday. The Berlin Academy of Sciences voted to commemorate this birthday: to give this 92-page Basic Geometry a special honor. Among all Hilbert's influential works, it has had the most profound influence on the progress of mathematics.
On the day when he went to the Academy of Sciences to make this decision, Hilbert fell in the street of G? ttingen and broke his arm. The unfortunate accident caused his body to be unable to move, which led to various complications. A little over a year later-1943 February 14, he passed away.
Only a few close friends attended the funeral held at his home that morning. Arnold. Thornfield is one of Hilbert's earliest students, from Munich. He stood by the coffin and told the story of Hilbert's work.
"Is it an invariant? Is it his favorite number theory? Is it the basis of geometry? That is the greatest achievement in this field since Euclidean geometry. Hilbert's proof proves the correctness of riemann sum's Dirichlet conjecture on the basis of function theory and variational calculation. The study of integral equation theory also reached its peak ... Soon, in the new physics ... they produced the most beautiful fruit. His gas theory has had a fundamental impact on new experimental knowledge, and it is still not out of date. Moreover, his contribution to general relativity also has eternal value. As for his last effort to explore the true knowledge of mathematics, it is still inconclusive, but when this field is likely to develop further, it will not be bypassed and must be pushed forward by Hilbert. "