G. H. Hardy

Godfrey Harold Hardy (1877–1947) was the leading British pure mathematician of his generation and, through his thirty-five-year partnership with J. E. Littlewood, half of the most famous collaboration in mathematics. He was a purist who prized mathematics for its beauty, a champion of rigour who helped drag English analysis into the modern age — and, almost against his will, the man who discovered Ramanujan.

The telegram from Madras

In 1913 Hardy received a letter from an unknown clerk in Madras, crammed with strange and beautiful formulae stated without proof. Most mathematicians who had received earlier versions dismissed it as a crank's. Hardy, after an evening puzzling over the pages with Littlewood, realised the writer was a genius — the results "must be true, because, if they were not true, no one would have had the imagination to invent them." He brought Srinivasa Ramanujan to Cambridge, and their collaboration produced deep results in the theory of numbers and partitions. Hardy called discovering Ramanujan the "one romantic incident" of his life.

The most retold moment is the taxicab. Visiting the ailing Ramanujan, Hardy remarked that his cab's number, 1729, seemed rather dull. "No," Ramanujan replied at once, "it is the smallest number expressible as a sum of two cubes in two different ways" — 1729 = 1^3 + 12^3 = 9^3 + 10^3. Such numbers are "taxicab numbers" to this day.

Hardy–Littlewood

With Littlewood, Hardy wrote around a hundred papers on the analytic theory of numbers — pioneering the circle method for additive problems, and the prime k-tuple conjectures that still guide research on the distribution of the primes. He also, with Ramanujan, found the astonishing asymptotic formula for the number of partitions of an integer. His motto for how to do mathematics was simple and severe: get the hard, quantitative heart of the matter, and prove it.

In 1908 Hardy dashed off a short note settling a question in genetics: in a large randomly-mating population, allele frequencies stay constant across generations unless something disturbs them. This Hardy–Weinberg principle is a cornerstone of population genetics, taught to biology students worldwide — and Hardy thought it so obvious he was almost embarrassed by it. It is a delicious irony: the proudly "useless" pure mathematician's most applied and widely-used result was a throwaway he considered beneath him. In his memoir A Mathematician's Apology he insisted that "real" mathematics is useless and the better for it — a claim the twentieth century, with number theory at the heart of cryptography, would gently refute.

The Apology, and a life of the mind

Hardy's A Mathematician's Apology (1940) is one of the most widely read books ever written about what it is like to be a mathematician — an elegant, melancholy defence of mathematics as a creative art, written as his own powers waned. He loved cricket almost as much as mathematics, kept up a lifelong rivalry with the idea of God (whom he treated as a personal adversary), and mentored a generation. His verdict on his own career was that his greatest contribution was not a theorem at all, but the discovery of Ramanujan.