Chapter 21: The Bird and the Frog

"I believe mathematics should be beautiful. It is certainly never boring; it possesses a unique aesthetic."

"I don't particularly enjoy the media labeling me as a hermit. It just so happens that the first major paper I worked on concerned Fermat's Last Theorem; that doesn't mean I am only capable of tackling massive problems. Not everyone is as fortunate as I am to have successfully produced results on such grand topics.

I believe young scholars must still consider their survival. They need to start with easier projects to prove their value, which makes it easier to secure a good faculty position. Once they are settled, they can then attempt more difficult problems and take on larger projects. This approach allows for a better balance between life and academic aspirations."

"I am very fond of my professor's metaphor regarding mathematicians; he divides them into two categories: frogs and birds.

Birds soar high in the sky, overlooking a vast mathematical landscape that stretches to the distant horizon. They love those concepts that unify our thinking and integrate the many problems of different fields. Frogs live in the mud beneath the sky, seeing only the flowers growing around them. They enjoy exploring the details of specific problems, solving them one at a time."

"No, there is no hierarchy between birds and frogs; mathematics requires both.

Mathematics is rich and beautiful because birds provide it with broad, magnificent vistas, while frogs clarify its intricate details. Birds see further, but frogs see deeper.

The world of mathematics is both vast and profound; we need both birds and frogs to work in concert to explore it."

Most of Lin Ran’s interview revolved around mathematics itself. The definitions of birds and frogs, due to their profound implications, became widely circulated among the mathematical community after being translated into English.

When the news reached Europe, the nominal supervisor Horkheimer had found for Lin Ran had to face the questioning of his peers, who asked whether he was a bird or a frog, and why, if he had such deep insights, he had not shared them sooner.

Meanwhile, young mathematicians were all pondering whether they were frogs or birds, and whether they possessed the talent to be birds.

On the way back to Li Zhengdao’s residence, Yang Zhenning remarked with emotion, "It is so well said. Physicists can likewise be divided into birds and frogs. Those like Einstein point out our direction, define the scope, and tell everyone what can be studied, while physicists who work on concrete problems are like frogs, deeply buried in a field, constantly excavating its potential."

Li Zhengdao nodded, "Randolph does not seem like a young man in his early twenties at all. He gives me the impression that he is very clear about what he is doing and what he intends to do. Up until now, I have always felt pushed along by problems, pushed along by the constant surprises the physical world brings me.

But at his age, he already possesses a complete mathematical map, clearly delineating his understanding of the mathematical world. This is truly rare."

The two legends, who had both won the Nobel Prize in Physics in their thirties, actually felt a sense of being washed away by the tide in the face of this young rising star.

...

"Taniyama-kun, do you see? Our conjecture back then was indeed correct. All elliptic curves over Q are modular. As we expected, this conjecture truly plays a crucial role in the field of mathematics.

It is a pity you can no longer see it.

I truly cannot understand why you passed away so suddenly. If Lin-kun had been able to prove the Taniyama conjecture two years earlier, would you be here with me now in the seminar room at the University of Tokyo, discussing mathematical problems?

Lin-kun is truly a remarkable figure. The 'Randolph Program' he proposed has brought a shock to the entire mathematical world. Problems in many fields can be linked back to the program itself, and the significance behind completing the program makes all mathematicians feel excited.

I really wish you could see this scene as well."

In a temple five kilometers southwest of Kita-Saitama District in Saitama Prefecture, Japan, a young man in a suit and leather shoes stood before a grave, holding the latest issue of *Inventiones Mathematicae*, whispering to himself.

Standing before the grave was Goro Shimura, of the Taniyama-Shimura conjecture; lying within the grave was his dear friend, Yutaka Taniyama, also of the Taniyama-Shimura conjecture.

During Taniyama’s lifetime, both were teachers at the University of Tokyo—the former an associate professor, the latter a lecturer. Together, they completed the Taniyama-Shimura conjecture based on Taniyama’s initial hypothesis.

Because this conjecture was proposed by Japanese mathematicians in the 1950s, a time when Japanese mathematicians were unknown and held little status in the international community, the Taniyama-Shimura conjecture was buried in piles of old papers.

Aside from Taniyama and Shimura, no one thought the conjecture was anything special. It was destined to wait until the 1970s, when the great Andrew Weil unearthed it and deemed it important, leading to its promotion. By the 1980s, the German mathematician Gerhard Frey proposed that the Taniyama-Shimura conjecture might be equivalent to Fermat's Last Theorem to a certain extent.

Finally, Andrew Wiles, building on the work of his predecessors, completed the Taniyama-Shimura conjecture for a specific form, thereby proving Fermat's Last Theorem and making the Taniyama-Shimura conjecture famous alongside it.

Taniyama had committed suicide in 1958. Judging from his suicide note, he was exhausted and had lost faith in the future. In the environment of post-war Japan, Taniyama’s ideas were criticized as groundless and his behavior as eccentric, which in Japan meant being a social outcast.

To add a note: Taniyama’s fiancée also committed suicide after his death, leaving a note that read: "We promised that no matter where, we would be together, never to be separated. Since he is dead, I must follow him."

Wiles' proof of Fermat's Last Theorem had been something that could be traced—the result of several generations of mathematicians connecting the dots—and he was the one to finish it. But now, seeing it proven by Lin Ran felt like a bolt from the blue.

For everyone who had known or studied Fermat’s conjecture—no, it should be called Fermat's Last Theorem now—this was an incredible event.

The methods he used were ones that no mathematician had ever considered before. The Taniyama-Shimura conjecture? Many mathematicians had never even heard of it.

To Shimura, it felt like finding a kindred spirit in the high mountains and flowing waters. The conjecture he had made was actually used in the proof of Fermat's Last Theorem, which meant not only fame but also an improvement in his material circumstances. He had originally been unable to secure a faculty position at the University of Tokyo and was forced to move to Osaka University; later, dissatisfied with Osaka University, he went to Princeton.

Now, because the Taniyama-Shimura conjecture was at the heart of the Fermat theorem, the University of Tokyo had contacted him overnight, urging him to return and offering him a full professorship.

Beyond his joy, Taniyama felt a deep sense of melancholy; his dear friend had missed seeing this by only two years.

"The Department of Mathematics at the University of Tokyo has sent an invitation for a visiting professorship to Lin-kun, hoping that he will accept their guidance at the University of Tokyo this summer."