Monday 22 December 2014

Hardy's Taxi

Srinivasa Ramanujan was born 22 December 1887 to a pair ordinary people in India and the combination of their genes generated a flaming genius of mathematics. Providentially the same random selection from the available genetic variants conferred resistance to smallpox. Without that, he would have succumbed with hundreds of others to a smallpox epidemic that swept through the region just as the boy turned two in December 1889.  Three of his brothers were scythed out of the gene-pool by the diseases which routinely kill small children in the Third World.  It was clear before he had reached double digits in age that Ramanujan had an extraordinary intuitive feel for numbers and their relationships.  Note also that Ramanujan is his given name, his father being Srinivasa Iyengar; this name ordering is common in India and shared my sub-continental friend who is treated every day with casual racism in Ireland. It is due to him that we have Ramanujan Primes, Ramanujan's Conjecture, Ramanujan's Sum, the Ramanujan-Soldner Constant and at least a dozen other substantive bricks in the wall of mathematics. It is weirdly patronising that we use his given name in giving him credit: we don't after all refer to Isaac's Laws of Universal Gravitation or Albert's e = mc2 formula; but they were white.

A lot of time Ramanujan's head was swirling with numbers and from an early age he took to writing down the results of his deliberations in notebooks and on the rare scraps of paper that he was able to lay his hands on.  Paper was sufficiently precious that the boy only wrote down the results and not the derivations. When his genius started to be noticed and then noised abroad, the lack of explicit proofs led to his work being dismissed by some established mathematicians. These professors must have thought that the mathematical insights were not due to a humble self-taught clerk called Srinivasa Ramanujan but to wholly other chap with the same name. Having established his credentials locally, his sponsors tried to alert the wider mathematics world to the genius on the fringes of their world. Ramanujan summarised some of his niftiest results and sent them off to three mathematicians at Cambridge University.  H.E. Baker and E.W. Hobson couldn't be bothered to deal with unsolicited letters from the colonies and returned their copies of the treasure trove unopened, but G.H. Hardy, although initially skeptical, became intrigued and eventually arranged for Ramanujan to be brought to Cambridge to be mined of his insights.

The diagram at the head of this piece is copied from Ramanujan's 1st note-book and shows how he could visualize rather complex geometrical theorems in his head.  A lot of his conjectures, infinite series and functions involve π and the trigonomical functions cos, sin and it seems that, like the Greeks before him, Ramanujan preferred to think in pictures although the results were written in algebra:
how do you start thinking up those sorts of relationships??  For Ramanujan, numbers were his pals, and for each he knew their names and antecedents, and their position in a  hierarchical criss-cross of series and relationships.

This is nowhere better illustrated than the famous story of Hardy going to visit Ramanujan in SW London, when the latter was coughing up tubercular blood again, and so couldn't come in to work. In Hardy's words "I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No", he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."  And begob, it's true:
1729 = 13 + 123 = 93 + 103
The tuberculosis caught up with Ramanujan and he died in 1920 absurdly young at 32. His widow donated his note-books and papers to Madras University.  A couple of years later, like the Elgin Marbles getting looted from Athens for "safe-keeping" in the British Museum, most of the notebooks were sent to Hardy in Cambridge who passed them on to another colleague who stored them in one of many tottering heaps of papers in his office. The "lost notebook" was saved from incineration when these papers were sorted in 1976 and caused a sensation in the mathematical world of the time.  In contrast to the Elgin Marbles, the value of the note-books lies not in the artifacts themselves but in the information and that can be transcribed and shared. Nearly 100 years after his death, quants are still working through the work and finding that another of Ramanujan'as bald statements is indeed provably true.  Brilliant!

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