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

Einstein was anxious to return to Germany. He said to Bohr: "I will propose to the Berlin Academy of Sciences that you get the opportunity to speak in Berlin as soon as possible, but you must practice more German."

Bohr said: "German? I can only try my best."

Einstein said to Li Yu again: "Mr. Li Yu, you must not forget to come to Berlin."

Laue also said: "Professor Planck has always thought of you."

Li Yu said: "It will definitely happen."

"I would like to study some more physics problems with you," Einstein continued. "I have discovered some very interesting phenomena that may be able to verify my theory. But this requires both understanding of physics and There is no one who knows astronomy, but you are the most suitable person. So I will wait for you in Berlin."

Li Yu said: "In this case, we will see you later."

Quantum mechanics and relativity were both on the right track, and the two most exciting physics buildings of the early 20th century were immediately erected.

Li Yu had no choice but to leave England for the time being, because the Royal Society meeting was about to be held in London. He was a foreign academician after all, and it would be a shame not to attend the meeting.

The meeting was relatively routine, but because Li Yu crossed disciplines, he had to attend meetings in mathematics, physics, and astronomy.

Physics has just met a group of great masters, and the Royal Society is instantly dwarfed; astronomy has not yet reached its full potential.

As for mathematics, Li Yu didn't dare to say much at all.

Fortunately, the advancement of mathematics was obvious. By the beginning of the 20th century, all categories had been developed, and each category had been studied to an extremely in-depth level. After Poincaré's death, no one could master all branches of mathematics.

So mathematicians get along very well. You study your topology, and I study my number theory. You definitely know what I don’t know, but I also know what you don’t.

Everyone helps each other and makes progress.

Even so, with Li Yu's mathematical level, he would not dare to make a mistake in front of mathematicians.

In physics, he dared to say that he was ahead of an era, but in mathematics, there was no way to brag, and he would be slapped in the face every minute.

So in the middle of the meeting, when a group of great masters were chatting happily, Li Yu sneaked out under the pretext of going to the toilet.

The mathematics conference was held in Cambridge. Li Yu came to the campus, sat under a tree, and said helplessly: "Are you kidding me? Riemann functions and combinatorial mathematics, are these things I should listen to?"

A student next to him suddenly raised his head: "What language is this?"

Li Yu said casually: "Chinese."

"Are you Chinese?" said the student with a tanned skin.

Li Yu looked at the other person. He took off his leather shoes and put them aside. He sat cross-legged under the shade of the tree and read a thick notebook.

"Indian?" Li Yu asked.

The Indian student said: "How did you guess it right?"

Li Yu wanted to say it was because of your easily identifiable Hinglish, but it was obviously too offensive, so he replied: "The color between your eyebrows is different from other places. It should be because you spit something frequently. It seems to be Brahmanical customs in India; and your hair and skin look neat, you should bathe frequently, which is completely different from European habits."

The Indian student said: "What a logical mind. Have you read Conan Doyle's books?"

Li Yu smiled and said, "I just saw it on the boat."

"Oh, talking about taking a boat, it's really disappointing," the Indian student said with a painful expression. "I just sailed six thousand miles on a boat a month ago. If I hadn't practiced swinging every day a month in advance, I would have vomited. Dark and dark."

Li Yu tentatively said: "Your Excellency, your surname is your great name."

"Ramanujan," said the Indian student.

Okay, it’s really this guy.

Li Yu shook hands with him and said, "Li Yu."

"Li Yu! I know you!" Ramanujan said loudly, "I read your Star Wars, "Game Theory" and "Fractals and Chaos" on the ship. I liked them very much."

“多谢。”李谕笑道，然后一眼瞟见了他那本赫赫有名的笔记，低头看去，没想到第一眼竟然是1+2+3+4+5+6+7+……=-1/12。

Li Yu was quite surprised that Ramanujan was the first to come up with this famous formula.

"This formula," Li Yu pointed and asked, "how did you derive it?"

"Derivation? Why derivation?" Ramanujan asked rhetorically.

Li Yu said: "Mathematics, especially advanced mathematics, doesn't require proof and derivation?"

Ramanujan said: "Every number is extremely familiar to me, so why should I spend energy to prove it?"

"He is such an interesting person," Hardy walked over with a smile and blew his pipe. "I am more interested in the mysteries of the East now."

Li Yu greeted him: "Professor Hardy."

Hardy said: "If other mathematicians saw this formula, they would definitely think they had met a madman. Who would have thought that someone who studies the Riemann function would come up with the weird result that the sum of all natural numbers is -1/12?"

Ramanujan said: "The wonder of mathematics does not lie in the proof process, but in the interesting conclusions."

Hada knocked on his pipe: "This is why I asked you to come to Cambridge. If you stick to what you think, you will never publish a paper."

Ramanujan said anxiously: "Professor, you promised me! As long as I come to England, you will definitely help me publish the formulas in my notes."

Hada said: "If you really want to publish it and get the approval of the mathematicians in the meeting just now, you must do what I say."

Ramanujan did not understand the operating mechanism of modern mathematics at all and said: "Professor, you don't know how much I gave up before I was willing to leave India. According to the teachings of Brahmanism, anyone who leaves casually will be deprived of religious membership. If I cannot publish a paper , I can’t go back to India.”

"Of course it can be published," Hardy didn't even know how to explain it to him, "Forget it, come to my office first."

Hardy really valued Ramanujan very much. In just a few letters, it was clear that Ramanujan was a mathematical genius that was unique among all.

It's a pity that he had too little mathematical training and was completely self-taught.

Ramanujan's self-study process was very simple. He just relied on a "Collection of Mathematical Formulas" given by a college student to master number theory.

Simply unimaginable.

Ramanujan’s treasured notebook was filled with dense formulas, all of which were originally created by Ramanujan himself. Although 30 to 40% of the content in it was the result of previous work, Ramanujan did not know it in advance. , I figured it out by myself. And even so, most of them are brand new results.

It's hard not to be fascinated by this staggering number of new mathematical formulas.

Later generations estimated that Ramanujan's notebook, which was filled with mathematical formulas after nearly 10 years of hard work, contained three to four thousand theorems and formulas. They are page after page, with very few proofs or explanations. They are as concise and concise as aphorisms, compressing an extremely rich mathematical truth into one or two lines.

In later generations, publishing three papers a week would be enough for him to publish for more than ten years.

Ramanujan's notes troubled an entire generation of mathematicians, and by 1921, seven years after they had been made public, Hardy said there was "a great deal of unpublished material" waiting to be analysed.

Two years later, he wrote a paper devoted to Ramanujan's work on hypergeometric series in chapters 12 and 13 of the first notebook. Hardy had to declare: "It is only of these two chapters that I have so far been able to make a truly exploratory analysis."

Later, a Hungarian mathematician, Polya, visited Hardy and borrowed from him a copy of Ramanujan's unpublished notes.

A few days later, Polya almost went crazy and returned it to Hardy.

Because he could no longer watch, Polya said: "As long as he is caught in Ramanujan's magic net, he will definitely devote his whole life to proving these theorems and never discover anything that belongs to him again."

A few years later, Watson, a professor of pure mathematics at Cambridge University, began to study Ramanujan's notes again. After two years of research, he found that it was not easy. It would take several months to prove just a pair of modular equations.

Fortunately, the rewards were so great that he felt "it was worth spending a considerable part of his life editing these notebooks and making Ramanujan's early work known to the world."

Watson estimates the work will take five years. In fact, until his energy declined in the late 5s, he had spent nearly 20 years working on this, writing more than 30 papers and a lot of never-published notes, but he still could not finish studying Ramanu. Kim's Notes.

In short, although Ramanujan did not provide proofs, he gave thousands of correct formulas. His creativity is incredible in such a difficult field as mathematics.

If Ramanujan had been born 100 years earlier, his fame would definitely be greater.

Because his approach is more like that of eighteenth-century mathematicians: giving very clever and beautiful formulas, but without proof.

He could just write: "I don't want to prove them because my goddess told me so."

The effect may be comparable to Fermat's sentence that has deceived mathematicians for three hundred years: "I have obtained a beautiful proof, but unfortunately the space here is too small for me to write."

Several people came to Hardy's office. Hardy first said to Li Yu: "After you left just now, they were discussing the role of chaos theory in mathematical applications. Obviously, your achievements are the best in this era in applied mathematics. "

In terms of mathematics, what Li Yu could do was only applied mathematics, and he couldn't do any pure mathematics.

Li Yu said with a smile: "Mathematics is the language that describes the world. It is only a matter of time before mathematics is applied to all other fields."

Ramanujan seemed to disagree: "I think mathematics is mathematics, especially number theory. Whether it can be applied to other fields should not be considered by mathematicians. For example, the Riemann function is a theory about the distribution of prime numbers. In addition to mathematics Home, who cares? No one cares about number theory if they only think about applications."

"Your words make some sense," Li Yu said. "After all, it is a fact that mathematics precedes other disciplines. It may be hundreds of years or even thousands of years. There will always be other disciplines that use certain results of mathematics. Even if it is just pure In number theory, even if we don’t look at the results, the mathematical ideas born in the process can affect many fields.”

Li Yu's words were very flattering, and Ramanujan felt more comfortable after hearing them. He said proudly: "I don't believe that these wonderful formulas will be used in any discipline."

Li Yu could only say: "It will definitely happen."

Ramanujan certainly could not have imagined that many of his seemingly unfounded formulas would be used in quantum mechanics, black hole theory, and artificial intelligence in the future.

The key point is that Ramanujan himself didn’t know anything about quantum mechanics, black holes and artificial intelligence.

The magic is here.

Hardy's friend Littlewood also laughed: "Ramanujan not only dislikes mathematical training, but also does not like mathematical applications. All his body and mind belong to pure mathematical research. I have only seen this kind of person in books. .”

Ramanujan retorted again: "No, my body and mind belong to the goddess!"

Li Yu, Hardy and Littlewood didn't know much about Brahmanism, and they weren't very good at intervening in Ramanujan's religious issues - even after leading the British to rule India for so many years, they couldn't do anything about it.

It is more difficult to promote Christianity in India than in China.

But when you think about it, it seems even more incredible that it is difficult to spread in China... After all, India has a broad-based local religion, and the easiest place to spread it is probably in areas where no one believes in religion.

Hardy changed the subject and asked: "Are you still adapting to life in England?"

Ramanujan said: "It can only be said that it is okay. There are too few vegetables here, and there are some that I have never seen before. According to the teachings, they cannot be eaten. And everyone here has to wear shoes every day, which is really a torture."

Hardy said: "I have applied for a separate dormitory for you. You can stick to your vegetarian diet and pray in the room. However, during class, it is best to follow Cambridge rules."

Ramanujan said: "Thank you Professor."

Ramanujan was very pious. He always hung a statue of an Indian god in his room and performed Brahmin rituals every morning. He put on a white sacrificial loincloth, put a Brahmin caste mark on his forehead, wiped it off after praying, and only put on a suit when going out.

Hardy said: "But you have to take math classes well anyway and develop real mathematical literacy. I think you can do it."

Although Ramanujan didn't like mathematical proofs, he had no choice but to agree in order to publish his results.

Littlewood looked at his pocket watch and said to him: "Let's go. Since you like number theory, we will specially select some courses that interest you."

After Ramanujan and Littlewood walked out, Hardy said: "Mr. Li Yu, have you also noticed that Ramanujan is extraordinary?"

Li Yu nodded: "Geniuses always like to be unique."

“Ramanujan was truly amazing,” Hardy said. “The limitations and depth of his knowledge were equally astonishing.”

Li Yu said: "It is enough if there is something profound."

"It's hard to imagine that he can solve modular equations and complex multiplication theorems with an unheard-of order!" Hardy sighed. "His mastery of continued fractions, at least in terms of form, surpasses all mathematicians in the world. He can himself Finding the equation of the Riemann zeta function leads to many of the most famous functions in analytic number theory."

Li Yu said: "Isn't this great?"

Hardy said: "But he had never heard of the doubly periodic function, or even Cauchy's theorem; he could only give the vaguest explanation of what a mathematical proof was. All his results were obtained by such a method, - —miscellaneous arguments and intuitions.”

Hardy knew Ramanujan best and his evaluation was the most representative.

Years later, Hardy, like the Landau of physics, devised an informal rating scale for mathematician talent.

He rated himself 25 points and Littlewood 30 points. Hilbert, the greatest mathematician of his generation, rated 80 points.

He rated Ramanujan 100 points!