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October 12, 2024 - February 10, 2025
always the nagging question: What might have been, had he been discovered a few years earlier, or lived a few years longer?
Ramanujan was a man who grew up praying to stone deities; who for most of his life took counsel from a family goddess, declaring it was she to whom his mathematical insights were owed; whose theorems would, at intellectually backbreaking cost, be proved true—yet leave mathematicians baffled that anyone could divine them in the first place.
So specialized is mathematics today, I am told, that most mathematical papers appearing in most mathematics journals are indecipherable even to most mathematicians.
In the West, all through the centuries, artists have sought to give expression to religious feeling, creating Bach fugues and Gothic cathedrals in thanks and tribute to their gods. In South India today, such religious feeling hangs heavy in the air, and to discern a spiritual resonance in Ramanujan’s mathematics seems more natural by far than it does in the secular West.
“An equation for me has no meaning,” he once said, “unless it expresses a thought of God.”
“Ramanujan”—pronounced Rah-MAH-na-jun, with only light stress on the second syllable,
Like the American city of Des Moines, with its similar relationship to the corn-rich countryside of Iowa, Kumbakonam was more cosmopolitan than its surroundings,
their family deity, the Goddess Namagiri of Namakkal,
From his mother, Ramanujan absorbed tradition, mastered the doctrines of caste, learned the puranas. He learned to sing religious songs, to attend pujas, or devotions, at the temple, to eat the right foods and forswear the wrong ones—learned, in short, what he must do, and what he must never do, in order to be a good Brahmin boy.
Brahmins with heads so shaved in front that they looked prematurely bald, prominent caste marks of dried, colored paste upon their foreheads, locks of hair in the back like little ponytails, and thin, white, knotted threads worn diagonally across their bare chests,
Four percent of the South Indian population, Brahmins
The first three castes were entitled to wear the sacred thread that affirmed them “twice-born.”
Ramanujan wore a caste mark on his forehead—the namam, a broad white “U” intersected by a red vertical slash—wholly distinct from the three white horizontal stripes worn by Shaivites.
He came to understand trigonometric functions not as the ratios of the sides in a right triangle, as usually taught in school, but as far more sophisticated concepts involving infinite series. He’d rattle off the numerical values of π
drawn into conversation, and soon was expatiating on the ties he saw between God, zero, and infinity—keeping
So special and distinct was the South in the minds of its inhabitants that in writing overseas they were apt to make “South India” part of the return address.
There were gods for every purpose, to suit any frame of mind, any mood, any psyche, any stage or station of life. In taking on different forms, God became formless; in different names, nameless.
Later, in high school, Ramanujan saw how trigonometric functions could be expressed in a form unrelated to the right triangles in which, superficially, they were rooted. It was a stunning discovery. But it turned out that the great Swiss mathematician Leonhard Euler had anticipated it by 150 years. When Ramanujan found out, he was so mortified that he secreted the papers on which he had recorded the results in the roof of his house.
Everyone was struck by Ramanujan’s gifts; but there was nothing new in that. Nor was there anything new in that nothing tangible came of it.
Ramanujan’s world was one in which numbers had properties built into them.
There was nothing “wrong” in what Ramanujan did; it was just weird. Ramanujan was not in contact with other mathematicians.
Once, while a student at Pachaiyappa’s College, he is said to have warned the parents of a sick child to move him away; “the death of a person,” he told them, “can occur only in a certain space-time junction point.” Another time, in a dream, he saw a hand write across a screen made red by flowing blood, tracing out elliptic integrals.
the quantity 2n − 1. That, a friend remembered him explaining, stood for “the primordial God and several divinities.
Later, in England, Ramanujan would build a theory of reality around Zero and Infinity, though his friends never quite figured out what he was getting at.
“An equation for me has no meaning unless it expresses a thought of God.”
“Infinite series,” one mathematician has written, “were Ramanujan’s first love.”
trigonometric functions, historically rooted in right triangles and ratios, can be evaluated in a way seemingly unrelated—as the sums of infinite series.
the editor’s work in connection with Ramanujan’s contributions was by no means light,”
Carr, writing a synopsis of results rather than making original contributions, had given proofs, where he did so at all, only in bare outline. Now Ramanujan, who was making original contributions, clung to the pattern.
The marvel of British rule was that so few administered it.
In The Case of the Philosophers’ Ring, a Sherlock Holmes mystery written half a century after the death of Arthur Conan Doyle, the characters include Ramanujan and Hardy.
in no photograph of Hardy does he ever look relaxed.
He knew what privilege meant, and he knew that he had possessed it.”
“privilege” it was, it was of a rare and perhaps ideal sort—in circumstances at once economically modest and culturally enriched, like the immigrant Jews of two generations ago, say, or the immigrant Asians of today.
Winchester was still reckoned among the most brutal of the public schools.
Of twenty-six class hours each week, five each went to Greek, Latin, and history, three to French, two to divinity, two to science, and four to mathematics.
He wore his honours as he had worn his misfortunes, with becoming modesty,
The word is pronounced try-poss,
Hardy felt he could do better, and in September 1908 completed A Course of Pure Mathematics, the first rigorous exposition in English of mathematical concepts other texts sloughed over in their rush to get to practical applications or cover broad expanses of mathematical ground.
Hardy held to a regular routine. He read the London Times over breakfast, especially the cricket scores. He worked for four hours or so in the morning, then had a light lunch in Hall, perhaps played a little tennis in the afternoon. His career was well in place, his life comfortable, his future secure. Then the letter came from India.
Years later, most of the formulas in Ramanujan’s letter would become the subjects of papers in the Journal of the London Mathematical Society and other mathematical journals.
Roughly speaking, it was as if, instead of stating the aphorism “penny-wise, pound-foolish,” Ramanujan had for his own reasons expressed it as “two pennies wise, seven-and-a-half pounds foolish”—leaving the listener, distracted by the particulars, harder pressed to extract its meaning.
Orders of Infinity, the book that had caught Ramanujan’s eye in India.
(The gamma function is like the more familiar “factorial”—4!, read “four factorial,” = 4 × 3 × 2 × 1—except that it extends the idea to numbers other than integers.)
Ramanujan’s theorems which, he would write, “defeated me completely; I had never seen anything in the least like them before. A single look at them is enough to show that they could only be written down by a mathematician of the highest class.” And then, in a classic Hardy flourish, he added: “They must be true because, if they were not true, no one would have the imagination to invent them.”
The notebooks would frustrate whole generations of mathematicians, who were forever underestimating the sheer density of mathematical riches they contained.
It was Euler who, in his 1748 book, Introductio in analysin infinitorum, gave the trigonometric functions the form they have today.
what did modern mathematics look like to someone who had the deepest insight, but who had literally never heard of most of it?”
Euler’s elegant relationship, eiπ = − 1 which in a single, strange, beautiful statement of mathematical truth ties trigonometry and geometry to natural logarithms and thence to the whole world of “imaginary” numbers.
