Chapter 21: Birds And Frogs
“I believe mathematics should be beautiful. It certainly won’t be boring; it contains a unique sense of beauty.”
“I don’t really like the media calling me a hermit-like character. I just happened to make my first big paper on Fermat’s Last Theorem, which doesn’t mean I can only work on big problems. Not everyone can be as lucky as me, able to produce results on big topics.
I believe young scholars still need to consider survival. They should first work on some easy topics to prove their value, so they can find good teaching positions. Once stable, they can try tackling difficult problems and big topics, which better balances life and academic ideals.”
“I really like my professor’s metaphor about mathematicians. He compared mathematicians to two types: one is frogs, the other is birds.
Birds soar in the high sky, overlooking the vast mathematical vision extending to the distant horizon. They like those concepts that unify our thoughts and integrate many problems from different fields. Frogs live in the muddy ground under the sky, seeing only the flowers growing around them. They enjoy exploring the details of specific problems, solving one at a time.”
“No, there is no hierarchy between birds and frogs. Mathematics needs both birds and frogs.
Mathematics is rich and beautiful because birds give it a vast and spectacular vision, while frogs clarify its intricate details. Birds see farther, and frogs see deeper.
The world of mathematics is both vast and profound. We need birds and frogs to work together to explore it.”
Lin Ran’s interview mostly revolved around mathematics itself. The definition of birds and frogs in the interview, due to its profound implication, became widely circulated among mathematicians after being translated into English.
After the news reached Europe, Horkheimer, Lin Ran’s nominal mentor, had to face questions from his colleagues, asking if he was a bird or a frog, and why he hadn’t told them about such a profound understanding before.
Young mathematicians were all thinking about whether they were frogs or birds, and whether they had the talent to be birds.
On the way back to Tsung-Dao Lee’s residence, Yang Zhenning sighed: “Well said. Physicists can also be divided into birds and frogs. Ones like Einstein point out directions for us, delineate scopes, and tell everyone what can be researched, while physicists working on specific problems are like frogs, deeply digging into the potential of one field.”
Tsung-Dao Lee nodded: “Randolph doesn’t seem like a young man in his early twenties at all. He gives me the feeling that he is very clear about what he is doing and what he wants to do. Even now, I still feel like I’m being pushed by problems, pushed by the constant surprises from the physical world.
And at his age, he already has a complete mathematical map, clearly depicting his understanding of the mathematical world. This is truly rare.”
The two big shots who won the Nobel Prize in Physics in their thirties actually had the illusion of being washed up on the beach in front of the young new generation.
“Yoshiyuki Goro, did you see? Our conjecture from back then was indeed correct. All elliptic curves over Q are modular. This conjecture really has a crucial role in the field of mathematics, just as we expected back then.
It’s a pity you can’t see it.
I really can’t understand why you suddenly passed away. If Lin Jun had proven the Taniyama-Shimura conjecture two years earlier, could you still be chatting with me about mathematical problems in the seminar room at the University of Tokyo now?
Lin Jun is truly a remarkable character. The Randolph Program he proposed brought a shock to the entire mathematics community. Problems in many fields can be connected to the program itself, and the significance contained behind completing the program excites all mathematicians even more.
I really wish you could see this scene too.”
At a temple five kilometers southwest of North Saitama District, Saitama Prefecture, Japan, a young man in a suit holding the latest issue of New Progress in Mathematics magazine stood in front of the graveyard and murmured.
Standing in front of the graveyard was Goro Shimura from the Taniyama-Shimura conjecture; lying in the graveyard was his close friend, Yutaka Taniyama from the Taniyama-Shimura conjecture.
During Taniyama’s lifetime, both were teachers at the University of Tokyo; the former was an associate professor, the latter a lecturer. Together, they completed the Taniyama-Shimura conjecture based on Taniyama’s conjecture.
Because this conjecture was proposed by Japanese mathematicians, and Japanese mathematicians in the 1950s were obscure with no status in the international mathematics community, the Taniyama-Shimura conjecture was also buried in the pile of old papers.
Besides Taniyama and Shimura, no one thought this conjecture was anything special. It was originally going to wait until the 1970s to be dug up by the big shot Andrew Wiles, who said it was important and then promoted it. In the 1980s, German mathematician Gerhard Frey proposed that the Taniyama-Shimura conjecture should be equated with Fermat’s Conjecture to some extent.
Finally, Andrew Wiles completed the special case of the Taniyama-Shimura conjecture based on predecessors’ work, thereby completing Fermat’s Conjecture, making the Taniyama-Shimura conjecture famous along with Fermat’s Last Theorem.
Taniyama committed suicide in 1958. From the content of Taniyama’s suicide note, he killed himself due to exhaustion and loss of confidence in the future. In postwar Japan, Taniyama’s ideas were criticized as baseless and sometimes even for his eccentric behavior, which in Japan meant being out of step.
As an aside, Taniyama’s fiancée also committed suicide after Taniyama did. Her suicide note said: “We promised to be together no matter where, never to separate. Since he died, I must follow him.”
Originally, Wiles’s proof of Fermat’s Last Theorem was traceable, the result of generations of mathematicians’ efforts, finally completed by him. But now proven in Lin Ran’s hands, it feels like a bolt from the blue.
For everyone who knew or studied Fermat’s Conjecture—no, now it should be called Fermat’s Last Theorem—it was utterly unbelievable.
Because the method he used had never been thought of by any mathematician before. Taniyama-Shimura conjecture? They’d never even heard of these two mathematicians.
And to Shimura, it felt like a kindred spirit encounter across mountains and flowing water. His own conjecture was actually used in the proof of Fermat’s Last Theorem, which not only meant fame but also improved real treatment. Originally, he couldn’t get a teaching position at the University of Tokyo and had to move to Osaka University. Later dissatisfied with Osaka University, he went to Princeton.
Now, because the Taniyama-Shimura conjecture is core to Fermat’s Theorem, the University of Tokyo contacted him overnight, asking him to come work immediately and offering him a full professor position.
Besides being happy, Shimura felt sorrowful. Just two years late, and his close friend couldn’t see this scene.
“The University of Tokyo Mathematics Department has sent an invitation letter for a visiting professor position to Lin Jun, hoping to receive his guidance at the University of Tokyo this summer.”
