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Intuiting the Beauty of the Infinite: Ramanujan and Hardy’s Friendship and Collaboration

By David Baggett 

The Man Who Knew Infinity, a recent movie based on a book of the same name by Robert Kanigel, recounts the short but remarkable life story of India’s great mathematical prodigy Srivivasa Ramanujan (henceforth SR). Although what follows is a response to the film, the book is well-worth reading, filled with luscious prose such as in this sample: “The Cauvery was a familiar, recurrent constant of Ramanujan’s life. At some places along its length, palm trees, their trunks heavy with fruit, leaned over the river at rakish angles. At others, leafy trees formed a canopy of green over it, their gnarled, knotted roots snaking along the riverbank.”

The movie begins by quoting Bertrand Russell (a character in the movie itself): “Mathematics, rightly viewed, possesses not only truth but supreme beauty.” It then shows SR in India, doing his mathematics (without much formal training) while trying to eke out a living for his family. His passion and talent for math are obvious; trying to describe maths (the preferred British abbreviation) to his wife, he says it’s like a painting, but with colors you can’t see. There are patterns everywhere in mathematics, he adds, revealed in the most incredible forms. Finding himself in need of someone who could understand and appreciate his ground-breaking work, SR wrote G. H. Hardy, legendary professor at Cambridge, and eventually Hardy invited SR to traverse the ocean and come work with him there.

This incredible opportunity required SR to leave his wife behind and endure the long journey and culture shock of moving to England, which contributes to a compelling narrative, with many twists and turns I’m not discussing but that make for a terrific, sometimes heart-wrenching tale. Despite the trials and challenges (including a war), what’s amazing was how much work SR and Hardy were able to do over the next five years—publishing dozens of groundbreaking articles.

The divergent worldviews of the two men make the dynamics of their friendship particularly fascinating to chronicle. SR was a devout Hindu whereas Hardy was a committed atheist—though the first time Hardy says this to SR in the movie (“I’m what’s called an ‘atheist’”), SR replies, “You believe in God. You just don’t think he likes you.” Incidentally, this is a key structuring question in C. S. Lewis’s moving novel Till We Have Faces: whereas both Psyche and Orual believe in the gods, Psyche believed they were marvelous and loving, but Orual thought they were only dark, unkind, and mysterious. In Rudolph Otto’s terminology, Orual was familiar with the tremendum aspect of the Numinous, but Psyche with both the tremendum (the awe-inspiring mystery) and the fascinans aspect of the Numinous. Fascinans is the aspect of the Divine involving consuming attraction, rapturous longing—and is often connected to the imagination, beauty, even poetry.

The diametric difference in SR’s and Hardy’s ultimate worldviews proves to be related to a central aspect of the plot. Hardy is adamant about the need to show step-by-step proofs of SR’s conclusions, while SR is depicted as functioning on a much more intuitive level. I’m not concerned for now what artistic liberties the moviemakers might have taken in this regard, but it is true that SR would often write down the conclusions of his work and not all the intervening steps. There may be at least a partial explanation of this which is fairly prosaic: paper tended to be in short supply for SR in India. But it’s at least intriguing to consider the explanation advanced in the movie: SR possessed incredibly strong intuitive skills. Mystifying Hardy, SR could just see things that few others could and felt little need to offer the proofs.

Hardy—though incredibly impressed with SR’s abilities, likening him to an artist like Mozart, who could write a whole symphony in his head—repeatedly says that intuition is not enough. Intuition must be “held accountable.” Proofs mattered, to avoid projecting the appearance of SR’s mathematical dance or art as on a par with conjuring.

It isn’t that SR’s intuitions were infallible. His theory of primes, however intuitively obvious, turned out to be wrong. Still, though, many of his intuitions were eventually vindicated and proved right. One among other interesting questions that SR’s reliance on intuitions raises is how much discursive analysis they involve. It’s a vexed question among epistemologists whether intuitions are a lightning quick series of inferences, or something more immediately and directly apprehended. The quickness with which they come naturally lends itself to the latter analysis, but perhaps there’s something to the former option—particularly if much of the analysis is done beneath the level of conscious awareness. In the Sherlock Holmes stories, for example, Sherlock’s inferences would come so quickly that Watson characterized them as resembling intuitions; likewise, realizing it’s sometimes easier to know something than to explain the justification for it, Sherlock himself recognized the way knowledge can have features that resemble more immediate apprehendings than just the deliverances of the discursive intellect. A couple of real-life Sherlocks, Al Plantinga and Phil Quinn had a dust up some years back on whether basic beliefs are formed inferentially or not.

The difference in Hardy’s and SR’s styles, we come to see, is related to their divergent worldviews. Exasperated at Hardy’s recurring disparagement of intuition as lacking in substance, SR finally blurts out, “You say this word as if it is nothing. Is that all it is to you? All that I am? You’ve never even seen me. You are a man of no faith. . . . Who are you, Mr. Hardy?” The underlying dynamic that brought this exchange to a head was the way SR connected his own identity to those intuitions. Hardy had asked SR before how he got his ideas. Now SR gives his answer: “By my god. She speaks to me, puts formulas on my tongue when I sleep, sometimes when I pray.”

SR asks Hardy if he believes him, and adds, “Because if you are my friend, you will know that I am telling you the truth. If you are truly my friend.”

Even more than the math, this is a movie about men and their remarkable friendship and fertile partnership across radically divergent and conflicting paradigms. The humanity of the film is its best feature of all.
In Till We Have Faces, we find a similar scene. Orual can’t see the gold-and-amber castle that Psyche tells her of, but Orual also knows that Psyche had never told her a lie. One issue here is testimony, and the conditions that need to be in place to take it as reliable. Of course someone could be telling the truth, the best they understand it, and still be unreliable—for perhaps they’ve unwittingly made a mistake, or they’re delusional or confused.

At any rate, Hardy’s reply is transparent: “But I don’t believe in God. I don’t believe in anything I can’t prove.”

“Then you don’t believe in me,” SR responded. “Now do you see? An equation has no meaning to me unless it expresses the thought of God.”

Hardy remained skeptical of SR’s theology, but couldn’t dispute with the results. He would go to bat for SR to get him a fellowship at Cambridge, and in his impassioned defense of SR’s accomplishments he extolled his incredible originality, by which SR could apprehend so much truth otherwise missed. On Hardy’s view, the creativity and originality, though they provided SR a lens through which to see, didn’t subjectivize SR’s findings; rather, they were a tool for seeing farther and seeing more.

This contrasts with, say, Simon Critchley’s interpretation of the poetry of Wallace Stevens. On (Critchley’s) Stevens’s view, the only reality we experience is mediated through categories furnished by the poetic imagination, rendering our perspectives products of the imagination and, thus, subjective—yet still able to be believed despite their fictive nature. This is what some might call a more “postmodern” perspective than Hardy’s more traditional view that there’s an objective reality we’re able to discern, however imperfectly and through a glass darkly.

In real life, when Hardy died, one mourner spoke of his “profound conviction that the truths of mathematics described a bright and clear universe, exquisite and beautiful in its structure, in comparison with which the physical world was turbid and confused. It was this which made his friends . . . think that in his attitude to mathematics there was something which, being essentially spiritual, was near to religion.”

Hardy didn’t believe in God, but he did believe in SR and in the objectivity of mathematical truth. He wrote of his Platonism in his Mathematician’s Apology, and the movie captures this too. In one of his defenses of SR, he related the story of the way SR said mathematical truths are thoughts of God—a view parallel to, say, Plantinga’s view that modal and necessary moral truths are also thoughts in the mind of God. Then Hardy added, “Despite everything in my being set to the contrary, perhaps he’s right. For isn’t this exactly our justification for pure mathematics? We are merely the explorers of infinity in the pursuit of absolute perfection. We do not invent these formulae—they already exist and lie in wait for only the brightest minds to divine and prove. In the end, who are we to question Ramanujan—let alone God?”

Though math, on Hardy’s view, is discovered, not invented, it may take those with prodigious talents to uncover its deepest truths. Speaking of which, near the start of the film Hardy had said, “I didn’t invent Ramanujan. I discovered him.” Even more than the math, this is a movie about men and their remarkable friendship and fertile partnership across radically divergent and conflicting paradigms. The humanity of the film is its best feature of all.

After five years of collaboration between these unlikely friends, SR returned to India, having contracted a fatal disease—likely tuberculosis. Within a year he died, at the age of just 32. Hardy was crestfallen when he heard the news, and grieved the loss deeply. Near the end of the movie, he reflected on his collaboration with both SR and another colleague, Littlewood, saying he’d done something special indeed: “I have collaborated with both Littlewood and Ramanujan on something like equal terms.”

Paraphrasing Hardy, he once commented that out of 100 points, he would give himself 30 as a mathematician, 45 to Littlewood, 70 to Hilbert. And 100 to Ramanujan. In the year SR spent in India before his death, he poured his brilliant findings into another notebook. It was lost for a while, but when found, the importance of its discovery was likened to that of Beethoven’s “10th Symphony.” A century later, these formulas are being used to understand the behavior of black holes.

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