No mathematician should ever allow himself to forget that mathematics, more than any art or science, is a young man's game," the British mathematician G.H. Hardy wrote in A Mathematician's Apology. But the older guys are now catching up.
Since Hardy wrote those lines in 1940, it has been conventional wisdom that mathematical breakthroughs are most often made in a moment of brilliance by a born genius at a young age, rather than an experienced practitioner after decades of work.
Last month, however, Zhang Yitang, a 50-year-old lecturer in mathematics at the University of New Hampshire, defied Hardy's glib assertion. Zhang, who had not published any original work since 2001, submitted a paper to the peer-reviewed Annals of Mathematics in which he solved one of the most longstanding and difficult problems in pure maths. His proof - that there are an infinite number of consecutive pairs of prime numbers (those that are divisible only by 1 and themselves such as 3, 5, 7, 11) separated by less than 70 million - may be meaningless to the layperson, but to number theorists it is earth-shaking.
The fact that Zhang is well into middle age gives hope to legions of mid-career mathematicians oppressed by Hardy's dictum that groundbreaking work in their field should be left to the young.
Of course Hardy could point to many examples in the history of mathematics to support his assertion. The French mathematician Evariste Galois laid the foundations for modern algebra in the 1800s while he was still a teenager and died at the age of 21. During the same era, the Norwegian Niels Abel, aged 19, independently came up with group theory, which is invaluable in many areas of mathematics and physics. Srinivasa Ramanujan, the Indian maths prodigy mentored by Hardy at Cambridge University, compiled 3,900 results in identity and equations before he died at age 32 in 1920.
In more recent times, there's Terence Tao, whose parents emigrated to Australia from Hong Kong. Tao is a polymath who does brilliant work across many mathematical disciplines such as number theory, harmonic analysis and combinatorics. He received his PhD in mathematics from Princeton University at 20 years old, was at 24 appointed the youngest ever full professor at the University of California at Los Angeles, and at 30, in 2006, received the Fields Medal, the highest honour in mathematics.
The media reinforces the stereotype of youthful mathematical creativity. In the movie A Beautiful Mind John Nash, who as a graduate student in his early twenties did pioneering work in game theory, is depicted hanging out at a bar in Princeton when a sudden insight leads him to the concept that became known as the Nash equilibrium, which is today widely applied in economics and conflict analysis.
While such young guns make romantic figures for feel-good movies, Zhang's story may be even more inspirational for being the achievement of age, experience, persistence and sheer hard slog. It took him over three years of intensive, single-minded research in his late forties to solve the prime numbers problem.
He is not the only late bloomer. At age 41, Andrew Wiles, a Princeton and Oxford University mathematician, cracked Fermat's Last Theorem, which had vexed mathematicians for 358 years since Pierre de Fermat came up with it in 1637. Wiles had pondered the problem since he discovered it in a library when he was a 10-year-old student in Scotland. After seven years of intense and solitary work, he presented his results at Cambridge University in 1993, and, like Zhang, stunned his fellow mathematicians. It took another year for him to correct an error in his first proof in collaboration with his former student Richard Taylor.
An equally difficult problem, the Poincaré conjecture, was also solved by a mature thinker. In 1904 Henri Poincare, one of the most creative mathematicians of all time, made his conjecture about the topology, or shape and space, of a three-dimensional sphere. The Clay Mathematics Institute in the United States offered US$1 million to the person who could prove the conjecture. In 2006, the Russian mathematician Grigori Perelman did so. He was 40 years old. Offered the cash award as well as the Fields Medal, Perelman turned both down. Declaring, "I am not interested in money or fame", he was the first and only person to decline the prestigious medal. He said his contribution was no more significant than that of an American mathematician, Richard Hamilton, who devised the technique that allowed him to prove the conjecture.
Why have recent mathematical breakthroughs been made by older brains? There is just much more mathematics requiring more time to master than during Hardy's day a century ago. As Jordan Ellenberg, an expert in algebraic geometry at the University of Wisconsin, has noted, today there are no whiz kids like Galois and Abel. It simply takes them longer to learn from many more intellects.
Wiles tapped into the work of the Japanese mathematicians Yutaka Taniyama and Goro Shimura in two distinct branches of maths to figure out Fermat's Last Theorem. Perelman's proof of the Poincaré conjecture was aided by Hamilton's work in differential geometry at Columbia University. Zhang's breakthrough in prime numbers built on the work of Dan Goldston at San Jose State University in the United States, Janos Pintz of the Renyi Institute of Mathematics in Budapest and Cem Yalcin Yildirim of Bogazici University in Istanbul.
As for whether the frontiers of mathematics are best advanced by youthful flashes of intuition or long years of logical deduction, Poincare provided an answer. He wrote: "Logic and intuition have their necessary role. Each is indispensable."
He should know. He came up with his famous conjecture when he was 50 years old.
Tom Yam is a Hong Kong-based management consultant. He holds a PhD in electrical engineering and an MBA from the Wharton School of the University of Pennsylvania