December 22 is Srinivasan Ramanujan’s birthday and is celebrated as National Mathematics Day. Readers may be familiar with the tremendous odds he overcame — crushing financial hardship, almost no college education, a long and inhospitable stay in England — to become one of the greatest mathematicians the world has ever seen. A compelling reason to celebrate this day is Ramanujan’s uncanny vision. His 1913 letter to GH Hardy, asking for help in publishing his findings, contained page after page of equations so outlandish that the Cambridge mathematician finally concluded they had to be true, because if not, “no one would have the imagination to invent them”. Even now, most of the 3,000-plus results Ramanujan produced look just as startling: Disparate mathematical terms so unusually juxtaposed that they might have been run through a kaleidoscope; each formula still coming together to express a perfect truth. Think of a savant chef combining unlikely lists of far-flung ingredients — chickpeas and jamun, say, seasoned with feni and soy — to not only create exquisite dishes but even launch new culinary movements. Indeed, Ramanujan’s discoveries are so profound and multi-layered that they continue to stimulate exciting new avenues of mathematical research, over 100 years after his death. His formulas are like X-rays that allow us to gaze deep into a hidden world of numbers and identify mesmerising underlying patterns at play.
What’s more, some of the patterns he discovered also show up in applied areas such as signal processing, astrophysics and quantum mechanics. This has happened before, where abstract theories can end up, centuries later, explaining how physical reality works (for example, the exotic “curved” geometries discovered in the 19th century were later adapted by Albert Einstein to model relativistic spacetime).
How did Ramanujan, unschooled as he was in higher maths, make such discoveries? Certainly, he did not use the route of formal logic — in fact, much of his time in Cambridge was spent composing verifiable proofs for the formulas he had written down. When asked, he credited the goddess Namagiri for his talents. Whether the divine played a role or not, Ramanujan’s extraordinary mathematical intuition certainly did. As did his formidable adeptness at computation — the algebraic formulas he arrived at were distilled from multitudes of arithmetic calculations he performed first to gain insight. This is similar to how centuries of measuring and experimenting with physical right triangles in Babylon, Mesopotamia, India and China (among other places) finally led to the theorem we now ascribe to Pythagoras.
Such contributions by the east have historically been downplayed in the mathematical narrative popular in the West, which deservedly lauds the Greeks for their insistence on rigorous proof, but often fails to adequately acknowledge the rich alternative mathematical traditions of non-European cultures. Ramanujan, who was always ambivalent about proofs, epitomises this divide, emphatically underlining for us the importance of both intuition and experimentation in mathematics.But the symbolism goes deeper. Ramanujan is one of the few non-White persons ever admitted to the worldwide pantheon of mathematical greats. The fact that he towers over so many of his fellow honorees sends a powerful message of hope, inspiration and self-affirmation to millions of students in India and beyond.
How relevant will Ramanujan remain, as we advance into the future? One could argue that his pencil-and-paper brand of computational dexterity is by now quite obsolete, given the advent of modern computing power. But such enhanced capability has actually fuelled even more interest in probing deeper into Ramanujan’s beloved world of numbers. Experimental mathematics is by now a respectable, well-established field, with journals devoted specifically to the kinds of investigations he engaged in.
Another changing dynamic is that we have realised how extensively mathematics is embedded in the DNA of our universe, and consequently, the subject has become much more interdisciplinary. In hindsight, we can now see that the links between his abstract theorems and physical reality, which once seemed coincidental, were really a harbinger of this symbiotic exchange – one which will only strengthen with time.
Perhaps the most dramatic impending change involves proofs. These have traditionally reigned supreme in mathematics. And yet, Ramanujan never regarded them as anything more than an inconvenient necessity. Imagine, then, how delighted he would feel to stand with us now, at the threshold of a brave new world of computer-assisted theorem proving.
The principle is simple — programme the foundational assumptions used in mathematics along with a compendium of basic rules, and let the computer do the deducing. An early demonstration came in 1994, when several results from Ramanujan’s notebooks were proven automatically by a code developed at Carnegie Mellon University. Since then, such techniques have advanced significantly, and a version called Lean has been proposed as a tool to help teach university maths students how to compose proofs.
So far, none of these programmes can run autonomously to prove anything too complicated – they are more like proof assistants, needing human oversight. But what if, as Fields medalists Paul Cohen and Timothy Gowers predicted, computers replace humans, and simply take over every aspect of mathematics? Surely Ramanujan would be opposed to so drastic an advance, since it would take away his very meaning in life.
Most of us can’t hope to come close to him in ability. What we can share with him is the joy of doing mathematics — be it a schoolchild exulting at having mastered multiplication, or a researcher finally cracking a difficult proof, or even a train commuter completing a newspaper sudoku puzzle. This, then, is what we have to do on National Mathematics Day for Ramanujan — simply enjoy some maths!
Manil Suri is a distinguished mathematics professor at the University of Maryland, Baltimore County, and author of The Big Bang of Numbers: How to Build the Universe Using Only MathsThe views expressed are personal
