A Man of Science and Engineering in Troubled Times Wandering in the Late Qing Dynasty

After class, Li Yu found Hilbert and said with a smile: "Professor, listening to your lecture is worth ten years of reading."

Hilbert said: "I didn't expect you to come to listen. If I had known, I would have talked about game theory."

"It's worth looking forward to," Li Yu said, and then pulled out a manuscript. "It would be great if you can help me prove a few more mathematical theorems!"

"What theorem?" Hilbert asked.

Li Yu said: "It is a conjecture involving chess in game theory. For a two-person complete information game, there must be a strategy. Either the first player must win, or the second player must win, or both parties must draw."

Hilbert touched his beard: "You mean, starting from the first move of chess, even if the opponent has not yet made a move, the winner or loser can already be determined?"

Li Yu said: "Yes, game theory is mathematics. Mathematically speaking, the chessboard is limited, so the possible moves are also limited. There must be a winning strategy."

Hilbert, who played chess frequently, said: "But I have never heard of anyone who never lost at chess."

"Because the complexity of playing chess is exponential and cannot be proved exhaustively," Li Yu said. "Take chess as an example, all its positions are at least 10 to the 50th power."

Hilbert was a mathematician and he knew exactly how huge this number was.

Go is more complex than chess. Even if some repetitive situations are removed, the order of magnitude of all Go positions can reach 10 to the power of 170.

You must know that there are only 10^80 atoms in the entire universe. Even if one atom represents a Go situation, it is impossible to represent all Go situations by exhausting all the atoms in the universe.

If you use a computer to calculate, you need to draw the game tree, which is more complicated, at least 10 to the power of 360.

Even the fastest supercomputer in the world can perform 100 billion floating-point operations per second.If a path can be calculated with one floating point operation, then it would take 1^10 seconds to calculate all the possible situations of the Go game.

The age of the universe is only 138 billion years, which is approximately only 10 to the 17th power of seconds.

So it’s really hard to imagine the huge number of poems.

But this is mathematics. Just because something is physically impossible does not mean it is mathematically impossible.

From the perspective of game theory, optimal solutions must exist for all chess games.

But as for how to prove it, of course it cannot be exhaustive and can only use mathematical techniques.

Hilbert thought about it for a while and said: "Interesting! I like this conjecture, but regarding game theory, I am not the best at the University of Göttingen. There is a young professor named Zermelo who is simply obsessed with game theory. .”

Hilbert was very accurate in reading people. The conjecture Li Yu mentioned just now is actually Zermelo's theorem.

In fact, what Li Yu was thinking about was the theorem about equilibrium in game theory, which is the famous Nash equilibrium in later generations, of which Zermelo's theorem is a special case.

With the proof of Zermelo's theorem, it will be of great help to prove the Nash equilibrium.

Li Yu said: "Please also ask Professor Hilbert to help introduce you."

"Yes, but he may not be able to get away today, because tomorrow there will be two groups of people conducting mathematical discussions on set theory. Zermelo, as an important supporter of set theory, will debate with each other," Hilbert said, "Would you like to join in the fun tomorrow?"

"Of course I do," Li Yu said, "I am a fan of set theory."

"Okay, I have more power with you," Hilbert said, "But there are a lot of people coming from the other side. I want to find my good friends to help Zemelo stand together."

Li Yu asked: "You mean Professor Minkowski?"

"Yes, he happens to be in class. Let's go see if he has finished speaking." Hilbert said.

At present, class times in European universities are relatively random, often spanning noon.

Arriving outside Minkowski's classroom, Hilbert found him constantly calculating on the blackboard.

Hilbert took out his watch and said to his assistant Born beside him: "The get out of class is about to end, but it seems that he has no intention of stopping at all. Is Professor Minkowski still working on the four-color problem today?"

Born said: "Yes, Mr. Professor, if I remember correctly, he has been teaching for four weeks, but he has not yet completed the proof."

Li Yu was stunned and asked: "Professor Minkowski wants to prove the four-color theorem in class?!"

"That's right," Hilbert said. "Four weeks ago, when he was teaching topology, he happened to mention the four-color problem."

Li Yu asked: "It's normal for topology to talk about the four-color problem, but shouldn't Professor Minkowski be going to prove it right away, or in class?"

Hilbert said: "You have seen that what he calculated is the proof of the four-color problem."

The four-color problem is simple to describe: any map using only four colors can color countries with common borders in different colors.It is one of the three major mathematical conjectures along with Goldbach's conjecture and Fermat's conjecture.

But it was not until 1976 that a mathematician completed the proof of the four-color theorem using a computer.

Strictly speaking, the proof is completed through exhaustive method.

From the perspective of mathematicians, the proof method is not very beautiful and not very mathematical, so it has been criticized by many mathematicians.

Hilbert said: "At that time, Minkowski declared to the students in class, 'This theorem has not been proved because until now, only some third-rate mathematicians have studied it specifically,' and then he The student confidently said, ‘I believe I can prove it!’ But he obviously didn’t finish that day’s lesson, so he spent the next four weeks working on the proof.”

Li Yu looked at the calculations on the blackboard and said, "It doesn't seem to be going well."

Hilbert watched it with great interest: "I must see when he gives in."

After another half hour, Minkowski had filled the blackboard with writing. He picked up the blackboard eraser and wanted to erase a piece of it, but suddenly there was a thunder in the sky.

Minkowski's hand trembled, and after a pause of more than ten seconds, he put down the blackboard eraser helplessly, and then said: "God seems to be angered by my pride... I admit that my proof of the four-color theorem is incomplete. .”

Hearing what he said, all the students below the podium breathed a sigh of relief. God knows what the hell they have been listening to this month!

Minkowski asked again: "Where did I go before I talked about the four-color theorem?"

The student replied: "It's topology, professor."

Minkowski said: "Okay, we will continue to talk about topology in the next class. But it has been delayed for a month, and the progress will be accelerated after that."

The faces of the students all turned dark.

Damn, this is topology!It’s also a super difficult class in the math department!They even want to compress the course!
But Minkowski couldn't care less, so he clapped his hands and walked out of the classroom.

When Hilbert saw him, he laughed loudly: "I knew you wouldn't last long. I made a bet with Director Klein, and it looks like I won!"

Minkowski said nonchalantly: "If I can't do it, you definitely can't do it."

Hilbert said proudly: "But I already knew that I was not good, so I didn't go to the head-on challenge, and I also won an extra 100 marks."

Minkowski said speechlessly: "It's really yours!"

Hilbert and Minkowski have been friends for 40 to [-] years. He smiled and said, "Okay, I'll treat you to dinner today, and Mr. Li Yu just arrived."

Minkowski's expression improved: "That's pretty much it!"

Several people came to Hilbert's home. After entering, Li Yu could guess just by looking at the layout of the room that Hilbert often held banquets at home.

Minkowski saw a newly purchased telescope and said to Hilbert: "Don't tell me you want to study astronomy again? I guess you study astrology at most."

"Tch!" Hilbert snorted, "If you gather the ten smartest people in the world and ask them what is the stupidest thing in the world? They will definitely tell you: There is nothing stupider than astrology. !”

Once he got home his accent changed to an East Prussian accent, which was very interesting.

Hilbert's wife brought out tea and said to him: "It's best not to let the church people hear it. They can't even rehabilitate Galileo who made the telescope. Oh, by the way, Galileo is facing religion. At trial, he did not seem to stand up for his beliefs."

Hilbert immediately objected: "Galileo was not a fool. Only a fool believes that scientific truth requires religious martyrdom, and scientific achievements rely on time to prove their correctness."

During dinner, Li Yu clearly felt that Hilbert and Minkovsky were two very active thinkers. Minkovsky always liked to quote Goethe's poems, especially "Faust" during the conversation.

Hilbert suddenly said to Li Yu: "In my opinion, you are a very talented person, and your knowledge involves many fields of physics, mathematics, astronomy, mechanics, communications, and science fiction literature. So I really want to Let me ask you a question, what is the most important technology in the future?"

Li Yu put down the knife and fork in his hand, thought for a while and said, "Professor, do you prefer a more serious answer or a more relaxed answer?"

Hilbert said: "Of course it's relaxed, this is not a university."

Li Yu said: "Go for a ride on the moon."

Hilbert almost spit out the red wine when he reached his mouth: "Go to the moon and go for a drive?! It sounds like something you would say to a girl, like reaching for the stars and the moon. Why do you say that?"

Li Yu said with a smile: "Because to achieve this goal, a lot of additional technical problems must be solved, which means that humans have to face a lot of scientific and technological difficulties, from mathematics to physics, aerodynamics, mechanics, materials science, etc. There are countless .”

"Excellent answer!" Minkowski couldn't help but applaud him, and then asked Hilbert, "What do you think is the most important mathematical problem in this? You can only choose one. Don't say it is the four-color problem. "

Hilbert blurted out: "Of course it is the zero point problem of the Riemann zeta function, and it is absolutely important!"