Chapter 22: Birds and Frogs

I believe mathematics should be beautiful; it is certainly not boring—it possesses a unique kind of beauty.

I don’t like how the media calls me a recluse; I just happened to write my first major paper on Fermat’s Last Theorem, but that doesn’t mean I can only tackle big problems. Not everyone is as lucky as I am, able to produce results on such grand topics.

I think young scholars should consider survival first—they should start with easier problems to prove their worth, so they can secure good teaching positions, and only then attempt difficult, large-scale problems, better balancing life and academic ideals.

I greatly admire my professor’s analogy of mathematicians: he divides them into two types, frogs and birds.

Birds soar high in the sky, surveying vast mathematical landscapes stretching to the distant horizon. They love concepts that unify our thinking and integrate problems across different fields. Frogs live in the mud beneath the sky, seeing only the flowers growing around them. They delight in exploring the details of specific problems, solving one at a time.

No, birds and frogs are not superior or inferior to each other; mathematics needs both.

Mathematics is rich and beautiful because birds give it sweeping, majestic vistas, while frogs clarify its intricate details. Birds see farther; frogs see deeper.

The world of mathematics is both vast and profound—we need birds and frogs working together to explore it.

Lin Ran’s interview focused largely on mathematics itself; his definition of birds and frogs, due to its profound symbolism, became widely circulated among mathematicians after being translated into English.

After the news reached Europe, Horkheimer’s nominal advisor for Lin Ran had to face questions from colleagues: “Are you a bird or a frog? If you understood this so deeply, why didn’t you tell us?”

Young mathematicians everywhere began pondering whether they were frogs or birds—and whether they had the innate talent to be birds.

On the way back to Lee Chengdao’s residence, Yang Zhenning remarked, “That’s beautifully said. Physicists can be divided into birds and frogs too. Einstein, for instance, showed us the direction, defined the scope, and told us what to study; physicists tackling specific problems are like frogs, deeply buried in one field, endlessly excavating its potential.”

Lee Chengdao nodded: “Randolph doesn’t seem like someone barely in his twenties. He gives me the impression that he knows exactly what he’s doing and what he wants to do. I’ve always felt pushed forward by problems, driven by the constant surprises the physical world presents me.”

And at his age, he already possesses a complete map of mathematics, clearly outlining his understanding of the mathematical world—it’s truly extraordinary.

Two giants who won the Nobel Prize in Physics in their thirties felt, before this young wave, as if they’d been washed ashore.

“Taniyama-kun, did you see? Our old conjecture was right—all elliptic curves over Q are modular. This conjecture, as we predicted back then, plays a crucial role in mathematics.”

Too bad you can’t see it now.

I truly can’t understand why you suddenly passed away. If Lin Jun had proven the Taniyama conjecture two years earlier, wouldn’t you still be sitting with me in Tokyo University’s seminar room, discussing mathematics?

Lin Jun is truly an extraordinary figure. His Randolph Program shook the entire mathematical world. Problems across many fields can now be linked to the program itself, and the deeper meaning behind completing it has thrilled every mathematician.

I truly wish you could have seen this moment.

At a temple five kilometers southwest of North Saitama District, Saitama Prefecture, a young man in a suit stood before a grave, murmuring while holding the latest issue of “New Advances in Mathematics.”

Standing before the grave was Goro Shimura, of the Taniyama-Shimura conjecture; inside the grave lay his closest friend, Yutaka Taniyama, the other half of the Taniyama-Shimura conjecture.

Before Taniyama’s death, both were professors at Tokyo University—Shimura an associate professor, Taniyama a lecturer. Together, they built upon Taniyama’s conjecture to complete the Taniyama-Shimura conjecture.

Because this conjecture was proposed by Japanese mathematicians, and in the 1950s Japanese mathematicians had no international standing, the Taniyama-Shimura conjecture was buried in forgotten archives.

Aside from Taniyama and Shimura, no one thought the conjecture was significant. It would have remained obscure until the 1970s, when the great André Weil rediscovered it and declared it important, and then in the 1980s, the German mathematician Gerhard Frey proposed that the Taniyama-Shimura conjecture was equivalent to Fermat’s conjecture in some way.

Finally, Andrew Wiles completed a special case of the Taniyama-Shimura conjecture on the foundation of prior work, thereby proving Fermat’s conjecture, and in doing so, brought the Taniyama-Shimura conjecture to global fame alongside Fermat’s Last Theorem.

Taniyama committed suicide in 1958. From his suicide note, it was clear he died from exhaustion and loss of faith in the future. In postwar Japan, his ideas were criticized as baseless, and sometimes even labeled eccentric—meaning he was seen as socially alienated.

By the way, Taniyama’s fiancée also committed suicide after his death. Her suicide note read: “We promised each other that wherever we were, we would always be together, never apart. Since he is gone, I must follow him.”

Wiles’s proof of Fermat’s Last Theorem was, originally, a natural culmination of generations of work, completed by him at the end. Now, proven by Lin Ran, it felt like a thunderclap from a clear sky.

To anyone who had studied or researched Fermat’s conjecture—now properly called Fermat’s Last Theorem—it was an unimaginable event.

Because the method he used had never been conceived by any mathematician before; the Taniyama-Shimura conjecture? These two mathematicians had never even been heard of.

To Shimura, it felt like finding a soulmate in the mountains and rivers—his conjecture had been used to prove Fermat’s Last Theorem, bringing not only fame but also improved material circumstances. He had been denied a position at Tokyo University and was forced to move to Osaka University; dissatisfied there, he later went to Princeton.

Now, because the Taniyama-Shimura conjecture is central to Fermat’s Last Theorem, Tokyo University contacted him overnight, urging him to return immediately and offering him a full professorship.

Shimura felt both joy and sorrow—just two years too late, his closest friend could not witness this moment.

“The Department of Mathematics at Tokyo University has sent Lin Jun an invitation as a visiting professor, hoping he will accept instruction there this summer.”