German mathematician david hilbert (1862 ~ 1943) is one of the greatest mathematicians in the 20th century. His contribution to mathematics is enormous and multifaceted, and his research fields involve algebraic invariants, algebraic number fields, geometric foundations, variational methods, integral equations, infinite dimensional spaces, physics and mathematical foundations. His "Fundamentals of Geometry" published in 1899 became the representative work of modern axiomatic method, which promoted the formation of "axiomatic chemistry school of mathematics". It can be said that Hilbert is the founder of modern formal axiom school. 1900 Hilbert gave a famous speech entitled "Mathematical Problems" at the second international congress of mathematicians held in Paris at the age of 38. In his speech, according to the achievements and development trend of mathematics research in the19th century, he put forward 23 questions facing the new century with outstanding foresight. These 23 questions involve the most important fields of modern mathematics (the famous Goldbach conjecture is part of the eighth question), and the study of these questions has strongly promoted the development of various branches of mathematics in the 20th century.

This paper introduces two short stories about Hilbert's youth.

First, the teacher wants to push the class now.

1In the autumn of 880, 18-year-old Hilbert entered his hometown University of Konigsberg. Although my father wanted to study law very much, he did not hesitate to enter the philosophy department to study mathematics (the university was still located in the philosophy department at that time). Hilbert discovered how free college life was at that time. Unexpected freedom makes many young people spend their precious first year in college on traditional activities of students' mutual aid associations, such as drinking and sword fighting. However, for Hilbert, the more fascinating thing about college life is that he can finally devote all his energy to mathematics freely.

In the first semester of college, Hilbert took three courses: integral calculus, matrix theory and surface curvature theory. According to the regulations. In the second semester, he can transfer to another university to attend classes. Hilbert chose Heidelberg University, which was the most attractive and romantic school among all universities in Germany at that time. Hilbert took lazarus fuchs's class at the University of Heidelberg. Fuchs is a famous expert in differential equations, and his name is almost synonymous with linear differential equations. His lecture is really different and impressive. He didn't prepare much before class, and now he wants to push what he wants to say in class. So there is often a problem that can't be pushed down on the blackboard. At this time, he thought of another method, sometimes he had to change several methods in succession, but in the end he could always deduce the result. That's what he is. He is used to putting himself in danger in class. What do students think of such a class? One of his students later recalled that such a course gave students "an opportunity to see the actual process of mathematical thinking." We can imagine that Hilbert, who is good at thinking and learning, will certainly understand how a mathematician thinks. This exploration process, which involves hitting a wall and finally finding a solution, is invisible in textbooks anyway. It is actually very enlightening to show students the actual process of thinking. Professor Xu Lizhi, a famous expert in mathematical methodology in China, believes that this has definitely played a very good role in Hilbert's growth. I think this is also very enlightening for us today. Learning mathematics should not only learn the solution of this problem, but also learn how to find it. Learn to think.

Second, the routine steps under the apple tree

Hilbert stayed in Heidelberg for one semester, and could have been allowed to transfer to Berlin for lectures next semester, but he was reluctant to part with his hometown, so he returned to the University of Konigsberg. 1in the spring of 882, Hilbert still decided to stay in konigsberg.

At this time, Herman Jokowski returned to the University of Konigsberg after studying in Berlin for three semesters. Minkowski has been excellent in mathematics since he was a child. It is said that once in a math class, the teacher "hung up the blackboard" because he misunderstood the question, and the students shouted in unison: "Shut up and help!" When he was studying in Berlin, he won an award for his excellent math work. At this time, Jokowski, who is only 17 years old, is immersed in a very profound research-solving a problem on a list published by the Paris Academy of Sciences: turning a table into the sum of five squares. A year later, in the spring of 1883, 18-year-old Jokowski and the famous British mathematician Smith * * * won the grand prize of the Paris Academy of Sciences. This incident caused a sensation throughout Konigsberg. Hilbert's father sued for this.

Tell your son not to make friends with such a famous person rashly. However, because of their love for mathematics and common belief, Hilbert and Minkowski, two years younger than him, soon became good friends.

1884 In the spring, Adolf leonid hurwicz, a young mathematician, came to Koenigsberg from G? ttingen as an associate professor. Before he was 25 years old, he made outstanding research achievements in function theory. Hilbert and Minkowski soon established a close relationship with their new teacher. They three young people must meet at 5 o'clock every afternoon on time, and then go for a walk under the apple tree. Hilbert later recalled: "in the day-to-day walk, we were all immersed in discussing the practical problems of current mathematics;" Exchange our newly acquired understanding of the problem, exchange ideas and research plans with each other. "Among the three of them, Leonid Hurvich" has a solid basic knowledge and has always been in good order ",so he is a natural leader and convinces the other two. At that time, Hilbert found that this learning method was many times better than boring in a dark classroom or library. This routine walk lasted for eight and a half years. With this most leisurely and interesting way of learning, they explored "every corner" of mathematics and inspected every kingdom in the mathematical world. Hilbert later recalled: "At that time, I never thought that we would take ourselves so far!" " In this way, the three people "formed a lifelong friendship." "

As Professor Xu Lizhi pointed out, discussions between tutors and friends also played a very important role in Hilbert's growth and development. It is conceivable that that period was an important stage for Hilbert's rapid growth in talent, knowledge and knowledge. Without this experience, it is inconceivable that Hilbert could raise so many famous questions in so many important fields at once in 1900.

This short story about Hilbert's walk tells us that in addition to classroom activities, extracurricular communication between teachers and students and between classmates is also an important way of learning, which is very beneficial to mathematics learning. Moreover, because there are no books, papers and pens for communication and no complicated deduction and calculation, we can only say something that can be "said" in words, that is, to understand and analyze problems and tap the soul of the commander-in-chief in the form of deduction ... and these are very important for learning mathematics well. Students might as well often invite a few good students to walk and chat together, which will definitely be fun.