On the last Tuesday of January 1913, on a clear, cold winter’s morning in Cambridge University in England, a 35 year old mathematician of rapidly growing fame and esteem opened a letter. The letter, thick and unwieldy, bore the grime and wear-and-tear of a long journey, and a ragged line of stamps, unfamiliar and exotic. It started with the words, “Dear Sir, I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras on a salary of only £20 per annum............”
The writer of the letter was a 25 year old chubby-cheeked, smallpox-scarred young man with large, shining eyes, a devoutly religious Brahmin from a humble family of modest means in the deep south of India. His name was Srinivasa Ramanujan. The recipient, living over 8,000 kilometers away was a cricket-loving atheist of strong likes and dislikes and wide-ranging interests, a man with patrician, delicately chiseled features who could not bear to look at himself in the mirror, a socially awkward and shy person who nevertheless possessed a fine wit, called Godfrey Harold Hardy.
They were both men from humble backgrounds. They were both brilliant mathematicians. And yet, they could not have been more different from each other. Hardy was a rigorously trained mathematician who believed strongly in mathematical fundamentals, in iron-clad proofs, in precise, step-by step notations when solving a problem. He was an avowed atheist. Ramanujan, on the other hand, worked through intuition and inspiration, and his mind skipped and leaped across entire mathematical landscapes, touching down only when he felt he needed to. His mathematical visions, profoundly original, even strange, were, he firmly believed, granted to him by Namagiri, the goddess of the temple near his home in Kumbakonam, the one his mother prayed to when she could not conceive a child. His religious faith was unquestioning, solid, and it was a source of solace to him, never a burden with all the rituals and prohibitions it entailed.
People often wonder how and why it is that so many South Indians excel in mathematics. As one not blessed with even a hint of mathematical ability, I wish I knew the answer. It is said that mathematics permeates so many aspects of life in the south; it is a natural part of life there, woven into the warp and weft of day-to-day activities, not set apart as a school subject that induces terror and paralysis in so many children. It is there, visible in the beautiful kolams that are drawn on the thresholds of most homes, as audible mathematics in the rhythmic intricacies of Carnatic music, palpable in the complex patterns of the korvais and theermanams of Bharata Natyam. And Indians were, a thousand years before the Europeans, path-breaking pioneers in mathematical concepts. The idea of zero, so taken for granted today, was an astonishing idea for its time (some time in the 2nd century BC), and opened the doors for many other mathematical innovations. Every Indian school child is taught about Aryabhatta, who, in the 5th century, came up with one of the earliest and most accurate values for π (pi), and Brahmagupta. There was Bhaskara in the 12th century, but, alas, after that, India fell off the mathematics radar, it was no longer at the forefront of anything mathematical. Throughout, Indian mathematicians stressed the results, not the method by which the results were obtained. Western mathematicians, following Euclid’s footsteps, were unwavering in their insistence on formal, step-by-step proof.
India fell behind; the West surged ahead. Mathematics grew by leaps and bounds, names like Newton, Gauss, Reimann, Leibniz, Euler, Poincare, Legrange, Fermat, Bernoulli, and Hardy’s own contemporaries like Littlewood and Barnes contributed to the great flowering of this science - or art.
For those who are privileged to see and understand its pure beauty, mathematics is an art. Hardy, he of the rigorously analytical approach, wrote: "The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours, or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics”.
So Ramanujan and Hardy were not so unalike at all. Beauty, as the saying goes, is in the eyes of the beholder. Both of them beheld the wondrous beauties of mathematics, the one, as the inspired dreamscape of a lovely goddess, the other, as an ideal form, shaped and defined by rules and proofs, flawless in its precision and perfection. One reality, two visions, both awe-inspiring.
Ramanujan wrote about the convergent series, which formed part of the infinite series that he so loved. In a convergent series : 1+ 1/2+1/4+1/8+1/16... the series converges - at infinity - to two. Only at infinity. And I wondered: was this how Ramanujan viewed infinity - as the place where God and man converge, their ultimate meeting place? Was that what the goddess Namagiri whispered to him in his dreams? How did Ramanujan look at divergent series, decomposition....what would I give to know!
Ramanujan was particularly dazzled by prime numbers, and he labored hard to find a mathematical pattern, a method to their apparent randomness. A prime number is one that is divisible by itself and one. 2,3,5,7,11.....the prime numbers come at you with no predictability, no particular sequence. Lay them out on a grid, and what emerges is the sheer randomness with which they appear. Yet, Ramanujan was convinced that there had to be a method in how they were distributed, that there had to be a formula that would predict when or how often they would appear.
With minimal mathematical training, most of it self-taught, Ramanujan came up with a formula that he claimed would give the number of prime numbers less than x. He wrote in his first letter to Hardy, “ I have found a function which exactly represents the number of prime numbers less than x”. He did not include that function in his letter, hoping, perhaps, that Hardy would be sufficiently intrigued to want to know more, and write back. Hardy was more than sufficiently intrigued. He was startled by the contents of the letter, some of which looked like sheer lunacy to his eyes, the rest, like sheer genius. A type of genius - wild, untamed, uncontrolled, glinting with madness - that Hardy had never encountered before.
After consulting his friend and colleague, John Littlewood, Hardy wrote back to Ramanujan. The rest is history.
Three Universes of Ramanujan: Infinity, Tamil Nadu and Cambridge
I first read the story of Hardy and Ramanujan nearly twenty years ago, in Robert Kanigel’s superb biography, The Man Who Knew Infinity. I wept when I read the story then, and I wept again, when I re-read it a few years ago. It is an extraordinary story, a heartbreakingly tragic one of loneliness, exile, sickness, war and death, but it is also a profoundly uplifting one of warmth and friendship, perseverance and hard work, with a glimmer of infinite beauty.
More recently, I read another book, historical fiction, of the Hardy-Ramanujan collaboration, The Indian Clerk, by David Leavitt. Where Kanigel’s book brings South India alive, Leavitt’s book is a window into the Cambridge University and England of the early 20th century. You are there, along with Russell, D.H. Lawrence, Wittgenstein and Keynes, Littlewood and Hardy, at the cusp of the first world war, as these men, the intellectual giants of their day, banter and debate the issues, big and small, that were relevant to their lives. Homosexuality was rampant - or so Leavitt’s book leads us to believe - as were extra-marital affairs, and this book turns a mirror westward to capture a vivid picture of Hardy’s life and work in Cambridge, and of Ramanujan’s days there. The two books - one partly fictional, the other, a non-fiction biography - complement each other well to breathe life into the story and spirit of Ramanujan and Hardy.
What a delicate balancing act it must have been, for both of them. Ramanujan, homesick, cold, hungry, despondent because his wife never replied to his letters (his mother, strong-willed, controlling, with scant regard for the young girl who was Ramanujan’s wife, ensured that Janaki, the wife, never saw Ramanujan’s letters), yet bewitched by the dazzling universe of numbers that Namagiri gave him a glimpse of, that made him push himself hard, eager to prove himself and share his vision with the world. And Hardy, jolted out of his steady, stable existence, trying to grasp Ramanujan’s shadowy proofs and visions, aware that he should not constrain him too much and break Ramanujan’s spell of inspiration. Hardy was awed enough, and man enough, to admit, “I learned from him much more than he learned from me”.
And yesterday, I saw a live movie screening of a play, A Disappearing Number, that was based on - yes, the lives of Hardy and Ramanujan. Any telling of their story is moving and very sad, and this one was no less. Woven along with the Ramanujan-Hardy story were the tale of a modern couple - a mathematician, in thrall with the work of Ramanujan, and her mathematically illiterate hedge-fund husband - a call center worker, Barbara/Lakshmi, and a string theory physicist. It sounds like an incompatible mess of story lines, but it was so brilliantly done, and Ramanujan’s theories and visions were so beautifully integrated into the lives of the characters and the meaning and significance they sought for the ups and downs in their existence, that I was moved to tears yet again.
While I am on the topic of prime numbers and books, I would like to briefly mention a fascinating chapter I read in Dr. Oliver Sacks’s terrific book, “The Man who Mistook his Wife for a Hat”. Dr. Sacks, a well-known psychiatrist, writes about a pair of autistic, idiot-savant twins, John and Michael, whom he had met in a state hospital in 1966. The twins were severely mentally retarded, and Dr. Sacks presents them as “a sort of grotesque Tweedledum and Tweedledee, indistinguishable, mirror images, identical in face, in body movements, in personality, in mind, identical too in their stigmata of brain and tissue damage”. They were capable of all manner of number-related feats, like Dustin Hoffman’s character in the movie Rainman. They had a strange passion and vision for numbers, and Dr. Sacks relates a series of absolutely fascinating encounters with them. He finds them sitting in a corner of the hospital one day, reciting numbers, with a “mysterious, secret smile on their faces.....they seemed to be locked in a singular, purely numerical, converse”. In this numerical conversation, Dr Sacks writes, John would recite a 6-digit number, and after a short pause, during which he savored, with a smile, the number his twin had just recited, Michael would recite another 6-digit number. They carried on in this fashion for a while, while Dr. Sacks watched, bewildered and mesmerized. What Dr. Sacks found out, after digging into the matter, was that these highly retarded, autistic, “non-functioning” twins, were reciting prime numbers, many of them higher than anything the best computing powers of the day could come up with. Another day, armed with a “cheat sheet” of prime numbers, Dr.Sacks joined the twins in their prime number reciting game, reciting an eight-digit prime number. After a moment’s intense concentration, when the twins appeared to be weighing Dr. Sacks's contribution, a delighted grin spread across their faces, and, by making space for him on the floor, invited him to join them in their “game”. This I found utterly fascinating. In Dr. Sacks’s words, “ they summon up, they dwell among, strange scenes of numbers; they wander freely in great landscapes of numbers; they create, dramaturgically, a whole world made of numbers....they do not seem to “operate” with numbers.....they “see” them, directly, as a vast, natural scene”.
Perhaps this was how Ramanujan viewed the universe of numbers.
What I found heart-breaking about this story was that eventually, the twins were separated and moved into homes where they were taught to be more self-sufficient and "useful". And with this, they lost, forever, that vision of the landscape of numbers that they had had - never again did they, or could they, recite the prime numbers. Something incalculably precious, of infinite beauty, was lost to them. How sad.
Do read the books on Ramanujan. Even the most avid math-phobe will enjoy it - I guarantee it. So often I wonder, what if Hardy had not replied to Ramanujan? And I remember these beautiful lines from Thomas Gray's Elegy Written in a Country Churchyard:
Full many a gem of purest ray serene
The dark unfathom'd caves of ocean bear:
Full many a flower is born to blush unseen,
And waste its sweetness on the desert air.
(C) Kamini Dandapani
