On a winter morning nearly 100 years ago, GH Hardy, 35, Cambridge’s rising star in mathematics, received a 10-page ‘Dear Sir’ letter from an Indian clerk, filled with theorems. Some of them suggested that what Hardy had asserted about the orders of infinity, and what the German mathematician Johann Gauss had asserted about prime numbers, was inadequate.
The letter-writer, S Ramanujan, said he could do better.
Dismissing it as a practical joke, Hardy picked up the London Times to follow England’s moves in the world. Politically, England was in a time of turmoil. Mathematically, it was in deep sulk. Isaac Newton’s Principia was still its Bible; it had no time for the Continent’s mathematical breakthroughs. The university system that had produced Newton produced GS Hardy — his mathematics insisted on proof.
Ramanujan, a self-taught 23-year-old whose education and culture was routed through his Tamil-Brahmin way of life, worked by intuition; his theorems, which mathematicians still call ‘good guesses’, bypassed proof. He often credited his results to the family deity, Namagiri.
Nonetheless, Hardy and Ramanujan began to collaborate, with the former facilitating Ramanujan’s tenure in Cambridge. Their collaboration is now the subject of a film, The Man Who Knew Infinity, starring Bafta awardee Dev Patel and Oscar winner Jeremy Irons. The film is based on the 1991 Robert Kanigel book of the same name.
Ramanujan, unsurprisingly, is the reference point among Indian mathematicians for talking up India’s mathematical tradition and why, like him, their personal lives feed off mathematics and vice-versa.
The Indian math tradition seems to be more about pure mathematics than applied. It’s a forced hierarchy, but mathematicians preserve their idealism about the former, says physicist Ranjit Nair.
Ramanujan, physicists almost seem to suggest, could have been one of them. ‘Guessing’, it turns out is important to both disciplines.
“First you guess. Don’t laugh, this is the most important step,” said physicist and Nobel laureate Richard Feynman, best known for his work in quantum mechanics.
“Even if you don’t solve a math problem, it can give you new ideas, allow you to see new connections,” says Shanta Laishram a mathematician with the Indian Statistical Institute (ISI), Delhi. “So when other people rediscovered the proof of Ramanujan’s formulas and identities, it became breakthroughs. Robert Langlands, a famous mathematician at IAS, Princeton, made a number of interesting conjectures; his ideas gave new insights for other mathematicians.”
This is perhaps why, when Ramanujan called out Hardy’s ‘mistakes’ in his letter, instead of suppressing it, the latter made it the subject of academic enquiry. He showed it to his colleagues, threw Ramanujan’s questions open in a seminar.
“Mathematics is a democratic space,” says Laishram. “Our job is to ask questions and seek out truth.” Asking questions and having an argument is not the same thing and mathematicians understand that, he adds. “I can tell my guide directly ‘You are wrong for X or Y reason’.”
He doesn’t risk applying the same approach in his private life. “Before I got married, I have had girls cool off me… ‘You ask too many questions!’ my other friends cautioned,” he recalls with a laugh.
If mathematics is the language scientists use to talk to each other, how do they talk to other people? Self-absorption is a common crime, says Antar Bandopadhyay, another ISI mathematician, with a poker face.
Outside of the world of mathematics, mathematicians, says professor A Raghuram of the Indian Institute of Science Education and Research, Pune, are no different from other people. He, too, put on his trousers one leg at a time. Friends, however, titter, when he passes a comment peppered with the linguistic idiosyncrasies common to mathematicians, such as ‘That movie was totally trivial’ or ‘That whiskey has a highly nontrivial taste’, he adds.
(Note: An elementary math equation such as 2 + 2 = 4 can be dismissed as ‘trivial’; new results, whose proof is not immediately evident, are ‘non-trivial’.)
Ramanujan clearly did not make proof the basis of all aspects of his life. Social theorist Ashis Nandy, who in his book, Alternative Sciences (1980), discussed the ‘method’ of two pre-colonial figures of science — Jagadish Chandra Bose and Ramanujan — says Bose tried to join the Vedantic point of view with research findings, to validate his culture through his science. But Ramanujan “did not try to convince himself or others of this connection, even though he was ‘possessed’ by mathematics”.
The highest form of truth and beauty is to be found in great theorems, and you may call this God, says professor Raghuram. “My guess is that is what Ramanujan meant by his references to Namagiri. There was no question of her handing out things to him,” agrees professor Ram Murty of Queen’s University, Ontario.
Does that mean God cannot be part of any equation for most mathematicians? In brief: Yes. But Bandopadhyay has his own take on the matter: “Suppose I say God is Green and non-God is Red and I say I believe in Red. Where does that leave me? Neither do I believe nor do I disbelieve. I cannot count out the existence of God nor his non-existence. This is not new in mathematics. In maths, we can have a theory with a postulate and another one with the negation of the postulate.”
Bandopadhyay, however, says there is no problem in upholding religious beliefs, and they do not harm or hamper one’s mathematics. “I have a mathematician friend who is an orthodox Hindu. He performs all the rituals, but his religion makes him humble, opens his mind to other things,” he says.
In these times of religious obscurantism gaining a foothold in the sciences, can mathematics retain its purity? Raghuram, who was one of the resource persons for the documentary The Genius of Srinavasa Ramanujan, says those who talk highly of Vedic Mathematics are unable to see that it was “a bag of tricks to solve a few numerical problems. If studied as an aspect of history, sociology and the culture of a different time, it is fine, but it is no substitute for modern mathematical curricula”.
Nair suggests that when seeking an ‘Indian tradition,’ it would in fact be better to refer to the Jyotisha Vedanga rather than Vedic Maths, which is of much recent origin and does not cite any sources. “In the Vedanga, the Jyotisha meant astronomy/mathematics. In the Sulva Sutras, the Pythagoras theorem in the Indian tradition makes its appearance, at least three or four centuries prior to Pythagoras,” he says.
Real mathematicians, says Raghuram, have a social obligation to talk about the dangers of ideas such as “interesting mathematics came out of Sanskrit aphorisms”. He adds: “The ratio of the circumference of a circle to its diameter is always the same universal constant, independent of which language it is expressed in, independent of which country the circle is drawn in, independent of the caste, sex, age and religion of the person drawing the circle.”
Mathematics cannot be a political project, though politicians are free to make it their business, say mathematicians. Comic relief, of course, cannot be ruled out from public life.