Technology invades the modern world

Chapter 21 Birds and Frogs

“I think mathematics should be beautiful. It is certainly not boring. It contains a unique beauty.”

"I don't like the media calling me a hermit. It just happened that my first major paper was about Fermat's Last Theorem. It doesn't mean I only work on big problems. Not everyone is as lucky as me and can produce results on big projects.

I think young scholars still need to consider survival. They should first do some easy projects to prove their value, so that they can find good teaching positions. Once they are stable, they can try to solve some difficult problems and do some large projects. This way, they can better balance their life and academic ideals.

“I really like my professor’s metaphor about mathematicians, where he compares mathematicians to two types: frogs and birds.

Birds soar high in the sky, surveying vast mathematical vistas stretching to distant horizons. They relish concepts that unify our thinking and integrate diverse problems from diverse fields. Frogs live in the mud beneath the sky, seeing only the flowers growing around them. They delight in exploring the details of a particular problem, solving one at a time.

“No, there is no difference between birds and frogs; mathematics requires both birds and frogs.

Mathematics is rich and beautiful because the bird gives it a vast and magnificent perspective, while the frog clarifies its intricate details. The bird sees farther and the frog sees deeper.

The world of mathematics is vast and profound, and we need birds and frogs to work together to explore it."

Most of Lin Ran's interviews revolved around mathematics itself. The definitions of birds and frogs in the interview were widely circulated among mathematicians after being translated into English because of their profound meaning.

After the news reached Europe, the nominal mentor Horkheimer found for Lin Ran had to face questions from his colleagues, asking him whether he was a bird or a frog, and why he didn't tell them if he had such a profound understanding before.

Young mathematicians are all thinking about whether they are frogs or birds, and whether they have the talent to be birds.

On the way back to Tsung-Dao Lee's residence, Yang Zhenning said with emotion, "That's a very good point. Physicists can also be divided into birds and frogs. People like Einstein point out the direction for us, define the scope, and tell us what we can study. Physicists who work on specific problems are like frogs, burying themselves deeply in a field and constantly exploring its potential."

Li Zhengdao nodded and said, "Randolph doesn't seem like a young man in his early twenties at all. He gives me the feeling that he knows exactly what he is doing and what he wants to do. Until now, I have been driven by problems and the constant surprises that the world of physics brings me.

It is really rare that he has a complete mathematical map at this age, which clearly depicts his understanding of the mathematical world."

These two big men who won the Nobel Prize in Physics in their thirties actually had the illusion of being left on the beach in front of the young generation.

"Taniyama-kun, did you see that? Our conjecture back then was indeed correct. All elliptic curves on Q are modular. This conjecture really plays a crucial role in mathematics, just as we predicted.

Too bad you can't see it anymore.

I really can't understand why you passed away suddenly. If Mr. Lin could have proved the Taniyama conjecture two years earlier, would you still be discussing mathematical problems with me in the seminar room at the University of Tokyo?

Lin Jun is truly an extraordinary figure. His Randolph Program has shaken up the entire mathematical community. Problems in many fields can be linked to the program itself, and the implications of completing the program have excited all mathematicians. I really hope you can see this moment too.

In a temple five kilometers southwest of Kitasaitama County, Saitama Prefecture, a young man in a suit and tie stood in front of a cemetery, holding the latest "New Advances in Mathematics" magazine and murmured.

Standing in front of the cemetery is Goro Shimura, who was hypothesized by Taniyama-Shimura, and lying in the cemetery is his best friend, Taniyama Yutaka, who was also hypothesized by Taniyama-Shimura.

During Taniyama's lifetime, both of them were teachers at the University of Tokyo, the former as an associate professor and the latter as a lecturer. Based on Taniyama's conjecture, both of them jointly completed the Taniyama-Shimura conjecture.

Because this conjecture was proposed by Japanese mathematicians, who were relatively unknown in the 50s and had no status in the international mathematics community, the Taniyama-Shimura conjecture was also buried in the pile of old papers.

Except for Taniyama and Shimura, no one thought that this conjecture was anything special. It was not until the 70s that the great mathematician Andrew Weil discovered it and said it was very important, and then it was promoted. It was not until the 80s that the German mathematician Gerhard Frey proposed that the Taniyama-Shimura conjecture should be equivalent to the Fermat conjecture to some extent.

Finally, Andrew Wiles completed the Taniyama-Shimura conjecture in a special form based on the work of his predecessors and then completed the Fermat conjecture, making the Taniyama-Shimura conjecture famous along with Fermat's Last Theorem.

Among them, Taniyama committed suicide in 1958. Judging from the content of Taniyama's suicide note, he committed suicide because of fatigue and loss of confidence in the future. In the post-war Japanese environment, Taniyama's ideas were criticized as groundless, and sometimes he was even criticized for his peculiar behavior, which meant being out of place in Japan.

By the way, Taniyama's fiancée also committed suicide after Taniyama. Her suicide note read: "We agreed that no matter where we are, we will always be together and never separate. Now that he is dead, I will follow him."

Wiles's proof of Fermat's Last Theorem had a well-established history, the product of generations of mathematicians, finally completed by him. But now that it had been proven in Lin Ran's hands, it felt like a bolt from the blue.

This is an incredible thing for everyone who has understood and studied Fermat's conjecture, no, it should be called Fermat's Last Theorem.

Because the method he used was something no mathematician had ever thought about before. The Taniyama-Shimura conjecture? These two mathematicians had never heard of it.

Shimura, on the other hand, felt like meeting a kindred spirit. His conjecture, unexpectedly used in the proof of Fermat's Last Theorem, meant not only fame but also a real improvement in his salary. He had been unable to secure a teaching position at the University of Tokyo and was forced to move to Osaka University. Later, dissatisfied with Osaka University, he moved to Princeton.

Now, because the Taniyama-Shimura conjecture is the core of Fermat's theorem, the University of Tokyo contacted him overnight and asked him to come to work as soon as possible to give him a full professor position.

Besides being happy, Shimura was also sad. His best friend would not be able to see this scene just two years late.

"The Department of Mathematics at the University of Tokyo has sent Lin Jun an invitation letter as a visiting professor. I hope to receive his instruction at the University of Tokyo this summer."

(End of this chapter)