This week’s release of the movie The Man Who Knew Infinity (which I saw in rough form last fall through its mathematican-producers Manjul Bhargava and Ken Ono) leads me to write about its subject, Srinivasa Ramanujan…
A Remarkable Letter
They used to come by physical mail. Now it’s usually email. From around the world, I have for many years received a steady trickle of messages that make bold claims—about prime numbers, relativity theory, AI, consciousness or a host of other things—but give little or no backup for what they say. I’m always so busy with my own ideas and projects that I invariably put off looking at these messages. But in the end I try to at least skim them—in large part because I remember the story of Ramanujan.
On about January 31, 1913 a mathematician named G. H. Hardy in Cambridge, England received a package of papers with a cover letter that began: “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. I am now about 23 years of age….” and went on to say that its author had made “startling” progress on a theory of divergent series in mathematics, and had all but solved the longstanding problem of the distribution of prime numbers. The cover letter ended: “Being poor, if you are convinced that there is anything of value I would like to have my theorems published…. Being inexperienced I would very highly value any advice you give me. Requesting to be excused for the trouble I give you. I remain, Dear Sir, Yours truly, S. Ramanujan”.
What followed were at least 11 pages of technical results from a range of areas of mathematics (at least 2 of the pages have now been lost). Early on, there are a few things that on first sight might seem absurd, like that the sum of all positive integers can be thought of as being equal to –1/12:
Then there are statements that suggest a kind of experimental approach to mathematics:
But then things get more exotic, with pages of formulas like this:
What are these? Where do they come from? Are they even correct?
The concepts are familiar from college-level calculus. But these are not just complicated college-level calculus exercises. Instead, when one looks closely, each one has something more exotic and surprising going on—and seems to involve a quite different level of mathematics.
Today we can use Mathematica or Wolfram|Alpha to check the results—at least numerically. And sometimes we can even just type in the question and immediately get out the answer:
And the first surprise—just as G. H. Hardy discovered back in 1913—is that, yes, the formulas are essentially all correct. But what kind of person would have made them? And how? And are they all part of some bigger picture—or in a sense just scattered random facts of mathematics?
The Beginning of the Story
Needless to say, there’s a human story behind this: the remarkable story of Srinivasa Ramanujan.
He was born in a smallish town in India on December 22, 1887 (which made him not “about 23”, but actually 25, when he wrote his letter to Hardy). His family was of the Brahmin (priests, teachers, …) caste but of modest means. The British colonial rulers of India had put in place a very structured system of schools, and by age 10 Ramanujan stood out by scoring top in his district in the standard exams. He also was known as having an exceptional memory, and being able to recite digits of numbers like pi as well as things like roots of Sanskrit words. When he graduated from high school at age 17 he was recognized for his mathematical prowess, and given a scholarship for college.
While in high school Ramanujan had started studying mathematics on his own—and doing his own research (notably on the numerical evaluation of Euler’s constant, and on properties of the Bernoulli numbers). He was fortunate at age 16 (in those days long before the web!) to get a copy of a remarkably good and comprehensive (at least as of 1886) 1055-page summary of high-end undergraduate mathematics, organized in the form of results numbered up to 6165. The book was written by a tutor for the ultra-competitive Mathematical Tripos exams in Cambridge—and its terse “just the facts” format was very similar to the one Ramanujan used in his letter to Hardy.
By the time Ramanujan got to college, all he wanted to do was mathematics—and he failed his other classes, and at one point ran away, causing his mother to send a missing-person letter to the newspaper:
Ramanujan moved to Madras (now Chennai), tried different colleges, had medical problems, and continued his independent math research. In 1909, when he was 21, his mother arranged (in keeping with customs of the time) for him to marry a then-10-year-old girl named Janaki, who started living with him a couple of years later.
Ramanujan seems to have supported himself by doing math tutoring—but soon became known around Madras as a math whiz, and began publishing in the recently launched Journal of the Indian Mathematical Society. His first paper—published in 1911—was on computational properties of Bernoulli numbers (the same Bernoulli numbers that Ada Lovelace had used in her 1843 paper on the Analytical Engine). Though his results weren’t spectacular, Ramanujan’s approach was an interesting and original one that combined continuous (“what’s the numerical value?”) and discrete (“what’s the prime factorization?”) mathematics.
When Ramanujan’s mathematical friends didn’t succeed in getting him a scholarship, Ramanujan started looking for jobs, and wound up in March 1912 as an accounting clerk—or effectively, a human calculator—for the Port of Madras (which was then, as now, a big shipping hub). His boss, the Chief Accountant, happened to be interested in academic mathematics, and became a lifelong supporter of his. The head of the Port of Madras was a rather distinguished British civil engineer, and partly through him, Ramanujan started interacting with a network of technically oriented British expatriates. They struggled to assess him, wondering whether “he has the stuff of great mathematicians” or whether “his brains are akin to those of the calculating boy”. They wrote to a certain Professor M. J. M. Hill in London, who looked at Ramanujan’s rather outlandish statements about divergent series and declared that “Mr. Ramanujan is evidently a man with a taste for Mathematics, and with some ability, but he has got on to wrong lines.” Hill suggested some books for Ramanujan to study.
Meanwhile, Ramanujan’s expat friends were continuing to look for support for him—and he decided to start writing to British mathematicians himself, though with some significant help at composing the English in his letters. We don’t know exactly who all he wrote to first—although Hardy’s long-time collaborator John Littlewood mentioned two names shortly before he died 64 years later: H. F. Baker and E. W. Hobson. Neither were particularly good choices: Baker worked on algebraic geometry and Hobson on mathematical analysis, both subjects fairly far from what Ramanujan was doing. But in any event, neither of them responded.
And so it was that on Thursday, January 16, 1913, Ramanujan sent his letter to G. H. Hardy.
Who Was Hardy?
G. H. Hardy was born in 1877 to schoolteacher parents based about 30 miles south of London. He was from the beginning a top student, particularly in mathematics. Even when I was growing up in England in the early 1970s, it was typical for such students to go to Winchester for high school and Cambridge for college. And that’s exactly what Hardy did. (The other, slightly more famous, track—less austere and less mathematically oriented—was Eton and Oxford, which happens to be where I went.)
Cambridge undergraduate mathematics was at the time very focused on solving ornately constructed calculus-related problems as a kind of competitive sport—with the final event being the Mathematical Tripos exams, which ranked everyone from the “Senior Wrangler” (top score) to the “Wooden Spoon” (lowest passing score). Hardy thought he should have been top, but actually came in 4th, and decided that what he really liked was the somewhat more rigorous and formal approach to mathematics that was then becoming popular in Continental Europe.
The way the British academic system worked at that time—and basically until the 1960s—was that as soon as they graduated, top students could be elected to “college fellowships” that could last the rest of their lives. Hardy was at Trinity College—the largest and most scientifically distinguished college at Cambridge University—and when he graduated in 1900, he was duly elected to a college fellowship.
Hardy’s first research paper was about doing integrals like these:
For a decade Hardy basically worked on the finer points of calculus, figuring out how to do different kinds of integrals and sums, and injecting greater rigor into issues like convergence and the interchange of limits.
His papers weren’t grand or visionary, but they were good examples of state-of-the-art mathematical craftsmanship. (As a colleague of Bertrand Russell’s, he dipped into the new area of transfinite numbers, but didn’t do much with them.) Then in 1908, he wrote a textbook entitled A Course of Pure Mathematics—which was a good book, and was very successful in its time, even if its preface began by explaining that it was for students “whose abilities reach or approach something like what is usually described as ‘scholarship standard’”.
But by 1910 or so, Hardy had pretty much settled into a routine of life as a Cambridge professor, pursuing a steady program of academic work. But then he met John Littlewood. Littlewood had grown up in South Africa and was eight years younger than Hardy, a recent Senior Wrangler, and in many ways much more adventurous. And in 1911 Hardy—who had previously always worked on his own—began a collaboration with Littlewood that ultimately lasted the rest of his life.
As a person, Hardy gives me the impression of a good schoolboy who never fully grew up. He seemed to like living in a structured environment, concentrating on his math exercises, and displaying cleverness whenever he could. He could be very nerdy—whether about cricket scores, proving the non-existence of God, or writing down rules for his collaboration with Littlewood. And in a quintessentially British way, he could express himself with wit and charm, but was personally stiff and distant—for example always theming himself as “G. H. Hardy”, with “Harold” basically used only by his mother and sister.
So in early 1913 there was Hardy: a respectable and successful, if personally reserved, British mathematician, who had recently been energized by starting to collaborate with Littlewood—and was being pulled in the direction of number theory by Littlewood’s interests there. But then he received the letter from Ramanujan.
The Letter and Its Effects
Ramanujan’s letter began in a somewhat unpromising way, giving the impression that he thought he was describing for the first time the already fairly well-known technique of analytic continuation for generalizing things like the factorial function to non-integers. He made the statement that “My whole investigations are based upon this and I have been developing this to a remarkable extent so much so that the local mathematicians are not able to understand me in my higher flights.” But after the cover letter, there followed more than nine pages that listed over 120 different mathematical results.
Again, they began unpromisingly, with rather vague statements about having a method to count the number of primes up to a given size. But by page 3, there were definite formulas for sums and integrals and things. Some of them looked at least from a distance like the kinds of things that were, for example, in Hardy’s papers. But some were definitely more exotic. Their general texture, though, was typical of these types of math formulas. But many of the actual formulas were quite surprising—often claiming that things one wouldn’t expect to be related at all were actually mathematically equal.
At least two pages of the original letter have gone missing. But the last page we have again seems to end inauspiciously—with Ramanujan describing achievements of his theory of divergent series, including the seemingly absurd result about adding up all the positive integers, 1+2+3+4+…, and getting –1/12.
So what was Hardy’s reaction? First he consulted Littlewood. Was it perhaps a practical joke? Were these formulas all already known, or perhaps completely wrong? Some they recognized, and knew were correct. But many they did not. But as Hardy later said with characteristic clever gloss, they concluded that these too “must be true because, if they were not true, no one would have the imagination to invent them.”
Bertrand Russell wrote that by the next day he “found Hardy and Littlewood in a state of wild excitement because they believe they have found a second Newton, a Hindu clerk in Madras making 20 pounds a year.” Hardy showed Ramanujan’s letter to lots of people, and started making enquiries with the government department that handled India. It took him a week to actually reply to Ramanujan, opening with a certain measured and precisely expressed excitement: “I was exceedingly interested by your letter and by the theorems which you state.”
Then he went on: “You will however understand that, before I can judge properly of the value of what you have done, it is essential that I should see proofs of some of your assertions.” It was an interesting thing to say. To Hardy, it wasn’t enough to know what was true; he wanted to know the proof—the story—of why it was true. Of course, Hardy could have taken it upon himself to find his own proofs. But I think part of it was that he wanted to get an idea of how Ramanujan thought—and what level of mathematician he really was.
His letter went on—with characteristic precision—to group Ramanujan’s results into three classes: already known, new and interesting but probably not important, and new and potentially important. But the only things he immediately put in the third category were Ramanujan’s statements about counting primes, adding that “almost everything depends on the precise rigour of the methods of proof which you have used.”
Hardy had obviously done some background research on Ramanujan by this point, since in his letter he makes reference to Ramanujan’s paper on Bernoulli numbers. But in his letter he just says, “I hope very much that you will send me as quickly as possible… a few of your proofs,” then closes with, “Hoping to hear from you again as soon as possible.”
Ramanujan did indeed respond quickly to Hardy’s letter, and his response is fascinating. First, he says he was expecting the same kind of reply from Hardy as he had from the “Mathematics Professor at London”, who just told him “not [to] fall into the pitfalls of divergent series.” Then he reacts to Hardy’s desire for rigorous proofs by saying, “If I had given you my methods of proof I am sure you will follow the London Professor.” He mentions his result 1+2+3+4+…=–1/12 and says that “If I tell you this you will at once point out to me the lunatic asylum as my goal.” He goes on to say, “I dilate on this simply to convince you that you will not be able to follow my methods of proof… [based on] a single letter.” He says that his first goal is just to get someone like Hardy to verify his results—so he’ll be able to get a scholarship, since “I am already a half starving man. To preserve my brains I want food…”
Ramanujan makes a point of saying that it was Hardy’s first category of results—ones that were already known—that he’s most pleased about, “For my results are verified to be true even though I may take my stand upon slender basis.” In other words, Ramanujan himself wasn’t sure if the results were correct—and he’s excited that they actually are.
So how was he getting his results? I’ll say more about this later. But he was certainly doing all sorts of calculations with numbers and formulas—in effect doing experiments. And presumably he was looking at the results of these calculations to get an idea of what might be true. It’s not clear how he figured what was actually true—and indeed some of the results he quoted weren’t in the end true. But presumably he used some mixture of traditional mathematical proof, calculational evidence, and lots of intuition. But he didn’t explain any of this to Hardy.
Instead, he just started conducting a correspondence about the details of the results, and the fragments of proofs he was able to give. Hardy and Littlewood seemed intent on grading his efforts—with Littlewood writing about some result, for example, “(d) is still wrong, of course, rather a howler.” Still, they wondered if Ramanujan was “an Euler”, or merely “a Jacobi”. But Littlewood had to say, “The stuff about primes is wrong”—explaining that Ramanujan incorrectly assumed the Riemann zeta function didn’t have complex zeros, even though it actually has an infinite number of them, which are the subject of the whole Riemann hypothesis. (The Riemann hypothesis is still a famous unsolved math problem, even though an optimistic teacher suggested it to Littlewood as a project when he was an undergraduate…)
What about Ramanujan’s strange 1+2+3+4+… = –1/12? Well, that has to do with the Riemann zeta function as well. For positive integers, ζ(s) is defined as the sum And given those values, there’s a nice function—called Zeta[s] in the Wolfram Language—that can be obtained by continuing to all complex s. Now based on the formula for positive arguments, one can identify Zeta[–1] with 1+2+3+4+… But one can also just evaluate Zeta[–1]:
It’s a weird result, to be sure. But not as crazy as it might at first seem. And in fact it’s a result that’s nowadays considered perfectly sensible for purposes of certain calculations in quantum field theory (in which, to be fair, all actual infinities are intended to cancel out at the end).
But back to the story. Hardy and Littlewood didn’t really have a good mental model for Ramanujan. Littlewood speculated that Ramanujan might not be giving the proofs they assumed he had because he was afraid they’d steal his work. (Stealing was a major issue in academia then as it is now.) Ramanujan said he was “pained” by this speculation, and assured them that he was not “in the least apprehensive of my method being utilised by others.” He said that actually he’d invented the method eight years earlier, but hadn’t found anyone who could appreciate it, and now he was “willing to place unreservedly in your possession what little I have.”
Meanwhile, even before Hardy had responded to Ramanujan’s first letter, he’d been investigating with the government department responsible for Indian students how he could bring Ramanujan to Cambridge. It’s not quite clear quite what got communicated, but Ramanujan responded that he couldn’t go—perhaps because of his Brahmin beliefs, or his mother, or perhaps because he just didn’t think he’d fit in. But in any case, Ramanujan’s supporters started pushing instead for him to get a graduate scholarship at the University of Madras. More experts were consulted, who opined that “His results appear to be wonderful; but he is not, now, able to present any intelligible proof of some of them,” but “He has sufficient knowledge of English and is not too old to learn modern methods from books.”
The university administration said their regulations didn’t allow a graduate scholarship to be given to someone like Ramanujan who hadn’t finished an undergraduate degree. But they helpfully suggested that “Section XV of the Act of Incorporation and Section 3 of the Indian Universities Act, 1904, allow of the grant of such a scholarship [by the Government Educational Department], subject to the express consent of the Governor of Fort St George in Council.” And despite the seemingly arcane bureaucracy, things moved quickly, and within a few weeks Ramanujan was duly awarded a scholarship for two years, with the sole requirement that he provide quarterly reports.
A Way of Doing Mathematics
By the time he got his scholarship, Ramanujan had started writing more papers, and publishing them in the Journal of the Indian Mathematical Society. Compared to his big claims about primes and divergent series, the topics of these papers were quite tame. But the papers were remarkable nevertheless.
What’s immediately striking about them is how calculational they are—full of actual, complicated formulas. Most math papers aren’t that way. They may have complicated notation, but they don’t have big expressions containing complicated combinations of roots, or seemingly random long integers.
In modern times, we’re used to seeing incredibly complicated formulas routinely generated by Mathematica. But usually they’re just intermediate steps, and aren’t what papers explicitly talk much about. For Ramanujan, though, complicated formulas were often what really told the story. And of course it’s incredibly impressive that he could derive them without computers and modern tools.
(As an aside, back in the late 1970s I started writing papers that involved formulas generated by computer. And in one particular paper, the formulas happened to have lots of occurrences of the number 9. But the experienced typist who typed the paper—yes, from a manuscript—replaced every “9” with a “g”. When I asked her why, she said, “Well, there are never explicit 9’s in papers!”)
Looking at Ramanujan’s papers, another striking feature is the frequent use of numerical approximations in arguments leading to exact results. People tend to think of working with algebraic formulas as an exact process—generating, for example, coefficients that are exactly 16, not just roughly 15.99999. But for Ramanujan, approximations were routinely part of the story, even when the final results were exact.
In some sense it’s not surprising that approximations to numbers are useful. Let’s say we want to know which is larger: or . We can start doing all sorts of transformations among square roots, and trying to derive theorems from them. Or we can just evaluate each expression numerically, and find that the first one (2.9755…) is obviously smaller than the second (3.322…). In the mathematical tradition of someone like Hardy—or, for that matter, in a typical modern calculus course—such a direct calculational way of answering the question seems somehow inappropriate and improper.
And of course if the numbers are very close one has to be careful about numerical round-off and so on. But for example in Mathematica and the Wolfram Language today—particularly with their built-in precision tracking for numbers—we often use numerical approximations internally as part of deriving exact results, much like Ramanujan did.
When Hardy asked Ramanujan for proofs, part of what he wanted was to get a kind of story for each result that explained why it was true. But in a sense Ramanujan’s methods didn’t lend themselves to that. Because part of the “story” would have to be that there’s this complicated expression, and it happens to be numerically greater than this other expression. It’s easy to see it’s true—but there’s no real story of why it’s true.
And the same happens whenever a key part of a result comes from pure computation of complicated formulas, or in modern times, from automated theorem proving. Yes, one can trace the steps and see that they’re correct. But there’s no bigger story that gives one any particular understanding of the results.
For most people it’d be bad news to end up with some complicated expression or long seemingly random number—because it wouldn’t tell them anything. But Ramanujan was different. Littlewood once said of Ramanujan that “every positive integer was one of his personal friends.” And between a good memory and good ability to notice patterns, I suspect Ramanujan could conclude a lot from a complicated expression or a long number. For him, just the object itself would tell a story.
Ramanujan was of course generating all these things by his own calculational efforts. But back in the late 1970s and early 1980s I had the experience of starting to generate lots of complicated results automatically by computer. And after I’d been doing it awhile, something interesting happened: I started being able to quickly recognize the “texture” of results—and often immediately see what might be likely be true. If I was dealing, say, with some complicated integral, it wasn’t that I knew any theorems about it. I just had an intuition about, for example, what functions might appear in the result. And given this, I could then get the computer to go in and fill in the details—and check that the result was correct. But I could derive why the result was true, or tell a story about it; it was just something that intuition and calculation gave me.
Now of course there’s a fair amount of pure mathematics where one can’t (yet) just routinely go in and do an explicit computation to check whether or not some result is correct. And this often happens for example when there are infinite or infinitesimal quantities or limits involved. And one of the things Hardy had specialized in was giving proofs that were careful in handling such things. In 1910 he’d even written a book called Orders of Infinity that was about subtle issues that come up in taking infinite limits. (In particular, in a kind of algebraic analog of the theory of transfinite numbers, he talked about comparing growth rates of things like nested exponential functions—and we even make some use of what are now called Hardy fields in dealing with generalizations of power series in the Wolfram Language.)
So when Hardy saw Ramanujan’s “fast and loose” handling of infinite limits and the like, it wasn’t surprising that he reacted negatively—and thought he would need to “tame” Ramanujan, and educate him in the finer European ways of doing such things, if Ramanujan was actually going to reliably get correct answers.
Seeing What’s Important
Ramanujan was surely a great human calculator, and impressive at knowing whether a particular mathematical fact or relation was actually true. But his greatest skill was, I think, something in a sense more mysterious: an uncanny ability to tell what was significant, and what might be deduced from it.
Take for example his paper “Modular Equations and Approximations to π”, published in 1914, in which he calculates (without a computer of course):
Most mathematicians would say, “It’s an amusing coincidence that that’s so close to an integer—but so what?” But Ramanujan realized there was more to it. He found other relations (those “=” should really be ≅):
Then he began to build a theory—that involves elliptic functions, though Ramanujan didn’t know that name yet—and started coming up with new series approximations for π:
Previous approximations to π had in a sense been much more sober, though the best one before Ramanujan’s (Machin’s series from 1706) did involve the seemingly random number 239:
But Ramanujan’s series—bizarre and arbitrary as they might appear—had an important feature: they took far fewer terms to compute π to a given accuracy. In 1977, Bill Gosper—himself a rather Ramanujan-like figure, whom I’ve had the pleasure of knowing for more than 35 years—took the last of Ramanujan’s series from the list above, and used it to compute a record number of digits of π. There soon followed other computations, all based directly on Ramanujan’s idea—as is the method we use for computing π in Mathematica and the Wolfram Language.
It’s interesting to see in Ramanujan’s paper that even he occasionally didn’t know what was and wasn’t significant. For example, he noted:
And then—in pretty much his only published example of geometry—he gave a peculiar geometric construction for approximately “squaring the circle” based on this formula:
Truth versus Narrative
To Hardy, Ramanujan’s way of working must have seemed quite alien. For Ramanujan was in some fundamental sense an experimental mathematician: going out into the universe of mathematical possibilities and doing calculations to find interesting and significant facts—and only then building theories based on them.
Hardy on the other hand worked like a traditional mathematician, progressively extending the narrative of existing mathematics. Most of his papers begin—explicitly or implicitly—by quoting some result from the mathematical literature, and then proceed by telling the story of how this result can be extended by a series of rigorous steps. There are no sudden empirical discoveries—and no seemingly inexplicable jumps based on intuition from them. It’s mathematics carefully argued, and built, in a sense, brick by brick.
A century later this is still the way almost all pure mathematics is done. And even if it’s discussing the same subject matter, perhaps anything else shouldn’t be called “mathematics”, because its methods are too different. In my own efforts to explore the computational universe of simple programs, I’ve certainly done a fair amount that could be called “mathematical” in the sense that it, for example, explores systems based on numbers.
Over the years, I’ve found all sorts of results that seem interesting. Strange structures that arise when one successively adds numbers to their digit reversals. Bizarre nested recurrence relations that generate primes. Peculiar representations of integers using trees of bitwise xors. But they’re empirical facts—demonstrably true, yet not part of the tradition and narrative of existing mathematics.
For many mathematicians—like Hardy—the process of proof is the core of mathematical activity. It’s not particularly significant to come up with a conjecture about what’s true; what’s significant is to create a proof that explains why something is true, constructing a narrative that other mathematicians can understand.
Particularly today, as we start to be able to automate more and more proofs, they can seem a bit like mundane manual labor, where the outcome may be interesting but the process of getting there is not. But proofs can also be illuminating. They can in effect be stories that introduce new abstract concepts that transcend the particulars of a given proof, and provide raw material to understand many other mathematical results.
For Ramanujan, though, I suspect it was facts and results that were the center of his mathematical thinking, and proofs felt a bit like some strange European custom necessary to take his results out of his particular context, and convince European mathematicians that they were correct.
Going to Cambridge
But let’s return to the story of Ramanujan and Hardy.
In the early part of 1913, Hardy and Ramanujan continued to exchange letters. Ramanujan described results; Hardy critiqued what Ramanujan said, and pushed for proofs and traditional mathematical presentation. Then there was a long gap, but finally in December 1913, Hardy wrote again, explaining that Ramanujan’s most ambitious results—about the distribution of primes—were definitely incorrect, commenting that “…the theory of primes is full of pitfalls, to surmount which requires the fullest of trainings in modern rigorous methods.” He also said that if Ramanujan had been able to prove his results it would have been “about the most remarkable mathematical feat in the whole history of mathematics.”
In January 1914 a young Cambridge mathematician named E. H. Neville came to give lectures in Madras, and relayed the message that Hardy was (in Ramanujan’s words) “anxious to get [Ramanujan] to Cambridge”. Ramanujan responded that back in February 1913 he’d had a meeting, along with his “superior officer”, with the Secretary to the Students Advisory Committee of Madras, who had asked whether he was prepared to go to England. Ramanujan wrote that he assumed he’d have to take exams like the other Indian students he’d seen go to England, which he didn’t think he’d do well enough in—and also that his superior officer, a “very orthodox Brahman having scruples to go to foreign land replied at once that I could not go”.
But then he said that Neville had “cleared [his] doubts”, explaining that there wouldn’t be an issue with his expenses, that his English would do, that he wouldn’t have to take exams, and that he could remain a vegetarian in England. He ended by saying that he hoped Hardy and Littlewood would “be good enough to take the trouble of getting me [to England] within a very few months.”
Hardy had assumed it would be bureaucratically trivial to get Ramanujan to England, but actually it wasn’t. Hardy’s own Trinity College wasn’t prepared to contribute any real funding. Hardy and Littlewood offered to put up some of the money themselves. But Neville wrote to the registrar of the University of Madras saying that “the discovery of the genius of S. Ramanujan of Madras promises to be the most interesting event of our time in the mathematical world”—and suggested the university come up with the money. Ramanujan’s expat supporters swung into action, with the matter eventually reaching the Governor of Madras—and a solution was found that involved taking money from a grant that had been given by the government five years earlier for “establishing University vacation lectures”, but that was actually, in the bureaucratic language of “Document No. 182 of the Educational Department”, “not being utilised for any immediate purpose”.
There are strange little notes in the bureaucratic record, like on February 12: “What caste is he? Treat as urgent.” But eventually everything was sorted out, and on March 17, 1914, after a send-off featuring local dignitaries, Ramanujan boarded a ship for England, sailing up through the Suez Canal, and arriving in London on April 14. Before leaving India, Ramanujan had prepared for European life by getting Western clothes, and learning things like how to eat with a knife and fork, and how to tie a tie. Many Indian students had come to England before, and there was a whole procedure for them. But after a few days in London, Ramanujan arrived in Cambridge—with the Indian newspapers proudly reporting that “Mr. S. Ramanujan, of Madras, whose work in the higher mathematics has excited the wonder of Cambridge, is now in residence at Trinity.”
(In addition to Hardy and Littlewood, two other names that appear in connection with Ramanujan’s early days in Cambridge are Neville and Barnes. They’re not especially famous in the overall history of mathematics, but it so happens that in the Wolfram Language they’re both commemorated by built-in functions: NevilleThetaS and BarnesG.)
Ramanujan in Cambridge
What was the Ramanujan who arrived in Cambridge like? He was described as enthusiastic and eager, though diffident. He made jokes, sometimes at his own expense. He could talk about politics and philosophy as well as mathematics. He was never particularly introspective. In official settings he was polite and deferential and tried to follow local customs. His native language was Tamil, and earlier in his life he had failed English exams, but by the time he arrived in England, his English was excellent. He liked to hang out with other Indian students, sometimes going to musical events, or boating on the river. Physically, he was described as short and stout—with his main notable feature being the brightness of his eyes. He worked hard, chasing one mathematical problem after another. He kept his living space sparse, with only a few books and papers. He was sensible about practical things, for example in figuring out issues with cooking and vegetarian ingredients. And from what one can tell, he was happy to be in Cambridge.
But then on June 28, 1914—two and a half months after Ramanujan arrived in England—Archduke Ferdinand was assassinated, and on July 28, World War I began. There was an immediate effect on Cambridge. Many students were called up for military duty. Littlewood joined the war effort and ended up developing ways to compute range tables for anti-aircraft guns. Hardy wasn’t a big supporter of the war—not least because he liked German mathematics—but he volunteered for duty too, though was rejected on medical grounds.
Ramanujan described the war in a letter to his mother, saying for example, “They fly in aeroplanes at great heights, bomb the cities and ruin them. As soon as enemy planes are sighted in the sky, the planes resting on the ground take off and fly at great speeds and dash against them resulting in destruction and death.”
Ramanujan nevertheless continued to pursue mathematics, explaining to his mother that “war is waged in a country that is as far as Rangoon is away from [Madras]”. There were practical difficulties, like a lack of vegetables, which caused Ramanujan to ask a friend in India to send him