Who Was Ramanujan?

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
 
 
"Hardy" and "Ramanujan" in The Man Who Knew Infinity
This essay is in Idea Makers: Personal Perspectives on the Lives & Ideas of Some Notable People »

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). 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:

The sum of all positive integers equal to -1/12

Surprising claims from Ramanujan's letter to Hardy (Reproduced by kind permission of the Syndics of Cambridge University Library)

Then there are statements that suggest a kind of experimental approach to mathematics:

Empirical math in Ramanujan's letter (Reproduced by kind permission of the Syndics of Cambridge University Library)

But some things get more exotic, with pages of formulas like this:

Remarkable formulas in Ramanujan's letter (Reproduced by kind permission of the Syndics of Cambridge University Library)

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:

Modern Mathematica reproduces Ramanujan's results

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?

One of the surviving math pages from Ramanujan's letter to Hardy (reproduced by kind permission of the Syndics of Cambridge University Library) One of the surviving math pages from Ramanujan's letter to Hardy (reproduced by kind permission of the Syndics of Cambridge University Library) One of the surviving math pages from Ramanujan's letter to Hardy (reproduced by kind permission of the Syndics of Cambridge University Library) One of the surviving math pages from Ramanujan's letter to Hardy (reproduced by kind permission of the Syndics of Cambridge University Library) One of the surviving math pages from Ramanujan's letter to Hardy (reproduced by kind permission of the Syndics of Cambridge University Library)

One of the surviving math pages from Ramanujan's letter to Hardy (reproduced by kind permission of the Syndics of Cambridge University Library) One of the surviving math pages from Ramanujan's letter to Hardy (reproduced by kind permission of the Syndics of Cambridge University Library) One of the surviving math pages from Ramanujan's letter to Hardy (reproduced by kind permission of the Syndics of Cambridge University Library) One of the surviving math pages from Ramanujan's letter to Hardy (reproduced by kind permission of the Syndics of Cambridge University Library)

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.

The math book Ramanujan got at age 16 The math book Ramanujan got at age 16 The math book Ramanujan got at age 16 The math book Ramanujan got at age 16 The math book Ramanujan got at age 16 The math book Ramanujan got at age 16

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:

18-year-old Ramanujan is missing!

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.

A page from Ramanujan's first published paper  A page from Ramanujan's first published paper  A page from Ramanujan's first published paper  A page from Ramanujan's first published paper

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 G. H. Hardy G. H. 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:

The modern version of integrals from Hardy's first research paper

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’”.

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 responds to Hardy (Reproduced by kind permission of the Syndics of Cambridge University Library) Ramanujan responds to Hardy(Reproduced by kind permission of the Syndics of Cambridge University Library) Ramanujan responds to Hardy (Reproduced by kind permission of the Syndics of Cambridge University Library) Ramanujan responds to Hardy (Reproduced by kind permission of the Syndics of Cambridge University Library)

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 out 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 zeros off the real axis, 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 1:1^s + 1:2^s + 1:3^s + ... 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]:

The sum of all positive integers

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.

 Pages from a paper of Ramanujan's   Pages from a paper of Ramanujan's   Pages from a paper of Ramanujan's

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: Sqrt[2]^(Sqrt[2]+Sqrt[3]) or 2^Sqrt[3]. 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 couldn’t 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):

Ramanujan discovers an "almost integer"

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 ≅):

Approximations to pi

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:

Ramanujan's series for pi

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:

Machin's series for pi

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:

A “curious” fact from Ramanujan's paper

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:

An approximate "squaring of the circle" by Ramanujan

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

 
 
Ramanujan in Cambridge with friends Ramanujan in Cambridge at graduation with friends
Ramanujan in Cambridge at graduation Ramanujan in Cambridge 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 “some tamarind (seeds being removed) and good cocoanut oil by parcel post”. But of more importance, as Ramanujan reported it, was that the “professors here… have lost their interest in mathematics owing to the present war”.

Ramanujan told a friend that he had “changed [his] plan of publishing [his] results”. He said that he would wait to publish any of the old results in his notebooks until the war was over. But he said that since coming to England he had learned “their methods”, and was “trying to get new results by their methods so that I can easily publish these results without delay”.

In 1915 Ramanujan published a long paper entitled “Highly Composite Numbers” about maxima of the function (DivisorSigma in the Wolfram Language) that counts the number of divisors of a given number. Hardy seems to have been quite involved in the preparation of this paper—and it served as the centerpiece of Ramanujan’s analog of a PhD thesis.

For the next couple of years, Ramanujan prolifically wrote papers—and despite the war, they were published. A notable paper he wrote with Hardy concerns the partition function (PartitionsP in the Wolfram Language) that counts the number of ways an integer can be written as a sum of positive integers. The paper is a classic example of mixing the approximate with the exact. The paper begins with the result for large n:

Approximating the number of partitions for large n

But then, using ideas Ramanujan developed back in India, it progressively improves the estimate, to the point where the exact integer result can be obtained. In Ramanujan’s day, computing the exact value of PartitionsP[200] was a big deal—and the climax of his paper. But now, thanks to Ramanujan’s method, it’s instantaneous:

Thanks to Ramanujan it now takes almost no time to compute this

Cambridge was dispirited by the war—with an appalling number of its finest students dying, often within weeks, at the front lines. Trinity College’s big quad had become a war hospital. But through all of this, Ramanujan continued to do his mathematics—and with Hardy’s help continued to build his reputation.

But then in May 1917, there was another problem: Ramanujan got sick. From what we know now, it’s likely that what he had was a parasitic liver infection picked up in India. But back then nobody could diagnose it. Ramanujan went from doctor to doctor, and nursing home to nursing home. He didn’t believe much of what he was told, and nothing that was done seemed to help much. Some months he would be well enough to do a significant amount of mathematics; others not. He became depressed, and at one point apparently suicidal. It didn’t help that his mother had prevented his wife back in India from communicating with him, presumably fearing it would distract him.

Hardy tried to help—sometimes by interacting with doctors, sometimes by providing mathematical input. One doctor told Hardy he suspected “some obscure Oriental germ trouble imperfectly studied at present”. Hardy wrote, “Like all Indians, [Ramanujan] is fatalistic, and it is terribly hard to get him to take care of himself.” Hardy later told the now-famous story that he once visited Ramanujan at a nursing home, telling him that he came in a taxicab with number 1729, and saying that it seemed to him a rather dull number—to which Ramanujan replied: “No, it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways”: 1729=1^3+12^3=9^3+10^3. (Wolfram|Alpha now reports some other properties too.)

But through all of this, Ramanujan’s mathematical reputation continued to grow. He was elected a Fellow of the Royal Society (with his supporters including Hobson and Baker, both of whom had failed to respond to his original letter)—and in October 1918 he was elected a fellow of Trinity College, assuring him financial support. A month later World War I was over—and the threat of U-boat attacks, which had made travel to India dangerous, was gone.

And so on March 13, 1919, Ramanujan returned to India—now very famous and respected, but also very ill. Through it all, he continued to do mathematics, writing a notable letter to Hardy about “mock” theta functions on January 12, 1920. He chose to live humbly, and largely ignored what little medicine could do for him. And on April 26, 1920, at the age of 32, and three days after the last entry in his notebook, he died.

Notifying Hardy of Ramanujan's death (image courtesy of the Master and Fellows of Trinity College, Cambridge)

The Aftermath

From when he first started doing mathematics research, Ramanujan had recorded his results in a series of hardcover notebooks—publishing only a very small fraction of them. When Ramanujan died, Hardy began to organize an effort to study and publish all 3000 or so results in Ramanujan’s notebooks. Several people were involved in the 1920s and 1930s, and quite a few publications were generated. But through various misadventures the project was not completed—to be taken up again only in the 1970s.

Pages from Ramanujan's notebooks Pages from Ramanujan's notebooks Pages from Ramanujan's notebooks
Pages from Ramanujan's notebooks Pages from Ramanujan's notebooks Pages from Ramanujan's notebooks

In 1940, Hardy gave all the letters he had from Ramanujan to the Cambridge University Library, but the original cover letter for what Ramanujan sent in 1913 was not among them—so now the only record we have of that is the transcription Hardy later published. Ramanujan’s three main notebooks sat for many years on top of a cabinet in the librarian’s office at the University of Madras, where they suffered damage from insects, but were never lost. His other mathematical documents passed through several hands, and some of them wound up in the incredibly messy office of a Cambridge mathematician—but when he died in 1965 they were noticed and sent to a library, where they languished until they were “rediscovered” with great excitement as Ramanujan’s lost notebook in 1976.

When Ramanujan died, it took only days for his various relatives to start asking for financial support. There were large medical bills from England, and there was talk of selling Ramanujan’s papers to raise money.

Ramanujan’s wife was 21 when he died, but as was the custom, she never remarried. She lived very modestly, making her living mostly from tailoring. In 1950 she adopted the son of a friend of hers who had died. By the 1960s, Ramanujan was becoming something of a general Indian hero, and she started receiving various honors and pensions. Over the years, quite a few mathematicians had come to visit her—and she had supplied them for example with the passport photo that has become the most famous picture of Ramanujan.

She lived a long life, dying in 1994 at the age of 95, having outlived Ramanujan by 73 years.

What Became of Hardy?

Hardy was 35 when Ramanujan’s letter arrived, and was 43 when Ramanujan died. Hardy viewed his “discovery” of Ramanujan as his greatest achievement, and described his association with Ramanujan as the “one romantic incident of [his] life”. After Ramanujan died, Hardy put some of his efforts into continuing to decode and develop Ramanujan’s results, but for the most part he returned to his previous mathematical trajectory. His collected works fill seven large volumes (while Ramanujan’s publications make up just one fairly slim volume). The word clouds of the titles of his papers show only a few changes from before he met Ramanujan to after:

Word clouds of the titles of Hardy's papers, before Ramanujan Word clouds of the titles of Hardy's papers, after Ramanujan

Shortly before Ramanujan entered his life, Hardy had started to collaborate with John Littlewood, who he would later say was an even more important influence on his life than Ramanujan. After Ramanujan died, Hardy moved to what seemed like a better job in Oxford, and ended up staying there for 11 years before returning to Cambridge. His absence didn’t affect his collaboration with Littlewood, though—since they worked mostly by exchanging written messages, even when their rooms were less than a hundred feet apart. After 1911 Hardy rarely did mathematics without a collaborator; he worked especially with Littlewood, publishing 95 papers with him over the course of 38 years.

Hardy’s mathematics was always of the finest quality. He dreamed of doing something like solving the Riemann hypothesis—but in reality never did anything truly spectacular. He wrote two books, though, that continue to be read today: An Introduction to the Theory of Numbers, with E. M. Wright; and Inequalities, with Littlewood and G. Pólya.

Hardy lived his life in the stratum of the intellectual elite. In the 1920s he displayed a picture of Lenin in his apartment, and was briefly president of the “scientific workers” trade union. He always wrote elegantly, mostly about mathematics, and sometimes about Ramanujan. He eschewed gadgets and always lived along with students and other professors in his college. He never married, though near the end of his life his younger sister joined him in Cambridge (she also had never married, and had spent most of her life teaching at the girls’ school where she went as a child).

In 1940 Hardy wrote a small book called A Mathematician’s Apology. I remember when I was about 12 being given a copy of this book. I think many people viewed it as a kind of manifesto or advertisement for pure mathematics. But I must say it didn’t resonate with me at all. It felt to me at once sanctimonious and austere, and I wasn’t impressed by its attempt to describe the aesthetics and pleasures of mathematics, or by the pride with which its author said that “nothing I have ever done is of the slightest practical use” (actually, he co-invented the Hardy-Weinberg law used in genetics). I doubt I would have chosen the path of a pure mathematician anyway, but Hardy’s book helped make certain of it.

The beginning of Hardy's A Mathematician's Apology The beginning of Hardy's A Mathematician's Apology

To be fair, however, Hardy wrote the book at a low point in his own life, when he was concerned about his health and the loss of his mathematical faculties. And perhaps that explains why he made a point of explaining that “mathematics… is a young man’s game”. (And in an article about Ramanujan, he wrote that “a mathematician is often comparatively old at 30, and his death may be less of a catastrophe than it seems.”) I don’t know if the sentiment had been expressed before—but by the 1970s it was taken as an established fact, extending to science as well as mathematics. Kids I knew would tell me I’d better get on with things, because it’d be all over by age 30.

Hardy explains that "mathematics... is a young man's game" Hardy explains that "mathematics... is a young man's game"

Is that actually true? I don’t think so. It’s hard to get clear evidence, but as one example I took the data we have on notable mathematical theorems in Wolfram|Alpha and the Wolfram Language, and make a histogram of the ages of people who proved them. It’s not a completely uniform distribution (though the peak just before 40 is probably just a theorem-selection effect associated with Fields Medals), but particularly if one corrects for life expectancies now and in the past it’s a far cry from showing that mathematical productivity has all but dried up by age 30.

Distribution of the ages at which mathematicians proved "notable" theorems

My own feeling—as someone who’s getting older myself—is that at least up to my age, many aspects of scientific and technical productivity actually steadily increase. For a start, it really helps to know more—and certainly a lot of my best ideas have come from making connections between things I’ve learned decades apart. It also helps to have more experience and intuition about how things will work out. And if one has earlier successes, those can help provide the confidence to move forward more definitively, without second guessing. Of course, one must maintain the constitution to focus with enough intensity—and be able to concentrate for long enough—to think through complex things. I think in some ways I’ve gotten slower over the years, and in some ways faster. I’m slower because I know more about mistakes I make, and try to do things carefully enough to avoid them. But I’m faster because I know more and can shortcut many more things. Of course, for me in particular, it also helps that over the years I’ve built all sorts of automation that I’ve been able to make use of.

A quite different point is that while making specific contributions to an existing area (as Hardy did) is something that can potentially be done by the young, creating a whole new structure tends to require the broader knowledge and experience that comes with age.

But back to Hardy. I suspect it was a lack of motivation rather than ability, but in his last years, he became quite dispirited and all but dropped mathematics. He died in 1947 at the age of 70.

Littlewood, who was a decade younger than Hardy, lived on until 1977. Littlewood was always a little more adventurous than Hardy, a little less austere, and a little less august. Like Hardy, he never married—though he did have a daughter (with the wife of the couple who shared his vacation home) whom he described as his “niece” until she was in her forties. And—giving a lie to Hardy’s claim about math being a young man’s game—Littlewood (helped by getting early antidepressant drugs at the age of 72) had remarkably productive years of mathematics in his 80s.

Ramanujan’s Mathematics

What became of Ramanujan’s mathematics? For many years, not too much. Hardy pursued it some, but the whole field of number theory—which was where the majority of Ramanujan’s work was concentrated—was out of fashion. Here’s a plot of the fraction of all math papers tagged as “number theory” as a function of time in the Zentralblatt database:

Fraction of mathematics papers tagged as "number theory" vs. time

Ramanujan’s interest may have been to some extent driven by the peak in the early 1900s (which would probably go even higher with earlier data). But by the 1930s, the emphasis of mathematics had shifted away from what seemed like particular results in areas like number theory and calculus, towards the greater generality and formality that seemed to exist in more algebraic areas.

In the 1970s, though, number theory suddenly became more popular again, driven by advances in algebraic number theory. (Other subcategories showing substantial increases at that time include automorphic forms, elementary number theory and sequences.)

Back in the late 1970s, I had certainly heard of Ramanujan—though more in the context of his story than his mathematics. And I was pleased in 1982, when I was writing about the vacuum in quantum field theory, that I could use results of Ramanujan’s to give closed forms for particular cases (of infinite sums in various dimensions of modes of a quantum field—corresponding to Epstein zeta functions):

Using results from Ramanujan in a 1982 physics paper

Starting in the 1970s, there was a big effort—still not entirely complete—to prove results Ramanujan had given in his notebooks. And there were increasing connections being found between the particular results he’d got, and general themes emerging in number theory.

A significant part of what Ramanujan did was to study so-called special functions—and to invent some new ones. Special functions—like the zeta function, elliptic functions, theta functions, and so on—can be thought of as defining convenient “packets” of mathematics. There are an infinite number of possible functions one can define, but what get called “special functions” are ones whose definitions survive because they turn out to be repeatedly useful.

And today, for example, in Mathematica and the Wolfram Language we have RamanujanTau, RamanujanTauL, RamanujanTauTheta and RamanujanTauZ as special functions. I don’t doubt that in the future we’ll have more Ramanujan-inspired functions. In the last year of his life, Ramanujan defined some particularly ambitious special functions that he called “mock theta functions”—and that are still in the process of being made concrete enough to routinely compute.

If one looks at the definition of Ramanujan’s tau function it seems quite bizarre (notice the “24”):

The seemingly arbitrary definition of Ramanujan's tau function

And to my mind, the most remarkable thing about Ramanujan is that he could define something as seemingly arbitrary as this, and have it turn out to be useful a century later.

Are They Random Facts?

In antiquity, the Pythagoreans made much of the fact that 1+2+3+4=10. But to us today, this just seems like a random fact of mathematics, not of any particular significance. When I look at Ramanujan’s results, many of them also seem like random facts of mathematics. But the amazing thing that’s emerged over the past century, and particularly over the past few decades, is that they’re not. Instead, more and more of them are being found to be connected to deep, elegant mathematical principles.

To enunciate these principles in a direct and formal way requires layers of abstract mathematical concepts and language which have taken decades to develop. But somehow, through his experiments and intuition, Ramanujan managed to find concrete examples of these principles. Often his examples look quite arbitrary—full of seemingly random definitions and numbers. But perhaps it’s not surprising that that’s what it takes to express modern abstract principles in terms of the concrete mathematical constructs of the early twentieth century. It’s a bit like a poet trying to express deep general ideas—but being forced to use only the imperfect medium of human natural language.

It’s turned out to be very challenging to prove many of Ramanujan’s results. And part of the reason seems to be that to do so—and to create the kind of narrative needed for a good proof—one actually has no choice but to build up much more abstract and conceptually complex structures, often in many steps.

So how is it that Ramanujan managed in effect to predict all these deep principles of later mathematics? I think there are two basic logical possibilities. The first is that if one drills down from any sufficiently surprising result, say in number theory, one will eventually reach a deep principle in the effort to explain it. And the second possibility is that while Ramanujan did not have the wherewithal to express it directly, he had what amounts to an aesthetic sense of which seemingly random facts would turn out to fit together and have deeper significance.

I’m not sure which of these possibilities is correct, and perhaps it’s a combination. But to understand this a little more, we should talk about the overall structure of mathematics. In a sense mathematics as it’s practiced is strangely perched between the trivial and the impossible. At an underlying level, mathematics is based on simple axioms. And it could be—as it is, say, for the specific case of Boolean algebra—that given the axioms there’s a straightforward procedure to figure out whether any particular result is true. But ever since Gödel’s theorem in 1931 (which Hardy must have been aware of, but apparently never commented on) it’s been known that for an area like number theory the situation is quite different: there are statements one can give within the context of the theory whose truth or falsity is undecidable from the axioms.

It was proved in the early 1960s that there are polynomial equations involving integers where it’s undecidable from the axioms of arithmetic—or in effect from the formal methods of number theory—whether or not the equations have solutions. The particular examples of classes of equations where it’s known that this happens are extremely complex. But from my investigations in the computational universe, I’ve long suspected that there are vastly simpler equations where it happens too. Over the past several decades, I’ve had the opportunity to poll some of the world’s leading number theorists on where they think the boundary of undecidability lies. Opinions differ, but it’s certainly within the realm of possibility that for example cubic equations with three variables could exhibit undecidability.

So the question then is, why should the truth of what seem like random facts of number theory even be decidable? In other words, it’s perfectly possible that Ramanujan could have stated a result that simply can’t be proved true or false from the axioms of arithmetic. Conceivably the Goldbach conjecture will turn out to be an example. And so could many of Ramanujan’s results.

Some of Ramanujan’s results have taken decades to prove—but the fact that they’re provable at all is already important information. For it suggests that in a sense they’re not just random facts; they’re actually facts that can somehow be connected by proofs back to the underlying axioms.

And I must say that to me this tends to support the idea that Ramanujan had intuition and aesthetic criteria that in some sense captured some of the deeper principles we now know, even if he couldn’t express them directly.

Automating Ramanujan

It’s pretty easy to start picking mathematical statements, say at random, and then getting empirical evidence for whether they’re true or not. Gödel’s theorem effectively implies that you’ll never know how far you’ll have to go to be certain of any particular result. Sometimes it won’t be far, but sometimes it may in a sense be arbitrarily far.

Ramanujan no doubt convinced himself of many of his results by what amount to empirical methods—and often it worked well. In the case of the counting of primes, however, as Hardy pointed out, things turn out to be more subtle, and results that might work up to very large numbers can eventually fail.

So let’s say one looks at the space of possible mathematical statements, and picks statements that appear empirically at least to some level to be true. Now the next question: are these statements connected in any way?

Imagine one could find proofs of the statements that are true. These proofs effectively correspond to paths through a directed graph that starts with the axioms, and leads to the true results. One possibility is then that the graph is like a star—with every result being independently proved from the axioms. But another possibility is that there are many common “waypoints” in getting from the axioms to the results. And it’s these waypoints that in effect represent general principles.

If there’s a certain sparsity to true results, then it may be inevitable that many of them are connected through a small number of general principles. It might also be that there are results that aren’t connected in this way, but these results, perhaps just because of their lack of connections, aren’t considered “interesting”—and so are effectively dropped when one thinks about a particular subject.

I have to say that these considerations lead to an important question for me. I have spent many years studying what amounts to a generalization of mathematics: the behavior of arbitrary simple programs in the computational universe. And I’ve found that there’s a huge richness of complex behavior to be seen in such programs. But I have also found evidence—not least through my Principle of Computational Equivalence—that undecidability is rife there.

But now the question is, when one looks at all that rich and complex behavior, are there in effect Ramanujan-like facts to be found there? Ultimately there will be much that can’t readily be reasoned about in axiom systems like the ones in mathematics. But perhaps there are networks of facts that can be reasoned about—and that all connect to deeper principles of some kind.

We know from the idea around the Principle of Computational Equivalence that there will always be pockets of “computational reducibility”: places where one will be able to identify abstract patterns and make abstract conclusions without running into undecidability. Repetitive behavior and nested behavior are two almost trivial examples. But now the question is whether among all the specific details of particular programs there are other general forms of organization to be found.

Of course, whereas repetition and nesting are seen in a great many systems, it could be that another form of organization would be seen only much more narrowly. But we don’t know. And as of now, we don’t really have much of a handle on finding out—at least until or unless there’s a Ramanujan-like figure not for traditional mathematics but for the computational universe.

Modern Ramanujans?

Will there ever be another Ramanujan? I don’t know if it’s the legend of Ramanujan or just a natural feature of the way the world is set up, but for at least 30 years I’ve received a steady stream of letters that read a bit like the one Hardy got from Ramanujan back in 1913. Just a few months ago, for example, I received an email (from India, as it happens) with an image of a notebook listing various mathematical expressions that are numerically almost integers—very much like Ramanujan’s exp-pi-sq-rt.

Mathematical results received in the mail a few months ago

Are these numerical facts significant? I don’t know. Wolfram|Alpha can certainly generate lots of similar facts, but without Ramanujan-like insight, it’s hard to tell which, if any, are significant.

Wolfram|Alpha finds possible formulas for a number

Over the years I’ve received countless communications a bit like this one. Number theory is a common topic. So are relativity and gravitation theory. And particularly in recent years, AI and consciousness have been popular too. The nice thing about letters related to math is that there’s typically something immediately concrete in them: some specific formula, or fact, or theorem. In Hardy’s day it was hard to check such things; today it’s a lot easier. But—as in the case of the almost integer above—there’s then the question of whether what’s being said is somehow “interesting”, or whether it’s just a “random uninteresting fact”.

Needless to say, the definition of “interesting” isn’t an easy or objective one. And in fact the issues are very much the same as Hardy faced with Ramanujan’s letter. If one can see how what’s being presented fits into some bigger picture—some narrative—that one understands, then one can tell whether, at least within that framework, something is “interesting”. But if one doesn’t have the bigger picture—or if what’s being presented is just “too far out”—then one really has no way to tell if it should be considered interesting or not.

When I first started studying the behavior of simple programs, there really wasn’t a context for understanding what was going on in them. The pictures I got certainly seemed visually interesting. But it wasn’t clear what the bigger intellectual story was. And it took quite a few years before I’d accumulated enough empirical data to formulate hypotheses and develop principles that let one go back and see what was and wasn’t interesting about the behavior I’d observed.

I’ve put a few decades into developing a science of the computational universe. But it’s still young, and there is much left to discover—and it’s a highly accessible area, with no threshold of elaborate technical knowledge. And one consequence of this is that I frequently get letters that show remarkable behavior in some particular cellular automaton or other simple program. Often I recognize the general form of the behavior, because it relates to things I’ve seen before, but sometimes I don’t—and so I can’t be sure what will or won’t end up being interesting.

Back in Ramanujan’s day, mathematics was a younger field—not quite as easy to enter as the study of the computational universe, but much closer than modern academic mathematics. And there were plenty of “random facts” being published: a particular type of integral done for the first time, or a new class of equations that could be solved. Many years later we would collect as many of these as we could to build them into the algorithms and knowledgebase of Mathematica and the Wolfram Language. But at the time probably the most significant aspect of their publication was the proofs that were given: the stories that explained why the results were true. Because in these proofs, there was at least the potential that concepts were introduced that could be reused elsewhere, and build up part of the fabric of mathematics.

It would take us too far afield to discuss this at length here, but there is a kind of analog in the study of the computational universe: the methodology for computer experiments. Just as a proof can contain elements that define a general methodology for getting a mathematical result, so the particular methods of search, visualization or analysis can define something in computer experiments that is general and reusable, and can potentially give an indication of some underlying idea or principle.

And so, a bit like many of the mathematics journals of Ramanujan’s day, I’ve tried to provide a journal and a forum where specific results about the computational universe can be reported—though there is much more that could be done along these lines.

When a letter one receives contains definite mathematics, in mathematical notation, there is at least something concrete one can understand in it. But plenty of things can’t usefully be formulated in mathematical notation. And too often, unfortunately, letters are in plain English (or worse, for me, other languages) and it’s almost impossible for me to tell what they’re trying to say. But now there’s something much better that people increasingly do: formulate things in the Wolfram Language. And in that form, I’m always able to tell what someone is trying to say—although I still may not know if it’s significant or not.

Over the years, I’ve been introduced to many interesting people through letters they’ve sent. Often they’ll come to our Summer School, or publish something in one of our various channels. I have no story (yet) as dramatic as Hardy and Ramanujan. But it’s wonderful that it’s possible to connect with people in this way, particularly in their formative years. And I can’t forget that a long time ago, I was a 14-year-old who mailed papers about the research I’d done to physicists around the world…

What If Ramanujan Had Mathematica?

Ramanujan did his calculations by hand—with chalk on slate, or later pencil on paper. Today with Mathematica and the Wolfram Language we have immensely more powerful tools with which to do experiments and make discoveries in mathematics (not to mention the computational universe in general).

It’s fun to imagine what Ramanujan would have done with these modern tools. I rather think he would have been quite an adventurer—going out into the mathematical universe and finding all sorts of strange and wonderful things, then using his intuition and aesthetic sense to see what fits together and what to study further.

Ramanujan unquestionably had remarkable skills. But I think the first step to following in his footsteps is just to be adventurous: not to stay in the comfort of well-established mathematical theories, but instead to go out into the wider mathematical universe and start finding—experimentally—what’s true.

It’s taken the better part of a century for many of Ramanujan’s discoveries to be fitted into a broader and more abstract context. But one of the great inspirations that Ramanujan gives us is that it’s possible with the right sense to make great progress even before the broader context has been understood. And I for one hope that many more people will take advantage of the tools we have today to follow Ramanujan’s lead and make great discoveries in experimental mathematics—whether they announce them in unexpected letters or not.

30 comments. Show all »

  1.  

    Always been a fan of your software but this is the first time I’m reading your blog. Definitely one of the most comprehensive, exhaustively fact-based and interesting blogs I’ve read. The casual reader may find it a bit lengthy, but your personal inputs and speculations are amazing. The unique perspective of not being a mathematician yourself but being extensively involved with the subject really makes this a unique read. Best two hours I’ve spent this week reading this and related material. Thank you, sir!

    Shah Megh
  2.  

    “Hardy’s mathematics was always of the finest quality. He dreamed of doing something like solving the Riemann hypothesis—but in reality never did anything truly spectacular.” Hardy was a top class mathematician. I don’t understand this statement.

    Llotrecaf
  3.  

    Interesting, I had no idea this was such a big deal. I’m pretty sure I have seen Ramanujan’s notes in my great uncle’s papers. Made no sense to me. If they are potentially interesting, I could raid some attics.

  4.  

    Very nice article. “A Mathematician’s Apology” was one of those things that induced me to major in Math some 40 years ago, and I always enjoy coming across articles about Hardy. I look forward to seeing this movie.

    Rick Jones
  5.  

    Kudos – As an avid Mathematica user, this is an excellent read!

    Satish
  6.  

    Thank you!, Brilliant. If experimental mathematics becomes more default way of exploring first , then I am sure we will make faster progress.

    Nikhil
  7.  

    Wonderful article, thank you for this!

    Vishvas Vasuki विश्वासो वासुकिजः
  8.  

    Wow great stuff. My Math teacher in college was Indian and boy did he know his stuff. It’s amazing how someone back then would master so much with no leads to go off from, pure genius.

    Arthur Segawa
  9.  

    Wonderful article. Complements Kanigel’s account by focusing on the math.

    Kalavai Venkat
  10.  

    Fascinating and brilliant article, carefully constructed and very well documented in its essence. Thank you. Wishing for the best for ‘The Man Who Knew Infinity’ which will be bound to be well received!

    Rekha Valliappan
  11.  

    Nice read sir. You forgot to mention his near misses on Fermat’s last theorem, I think that was realized only few months ago after almost a century :-)

    Prateek Varshney
  12.  

    I am just a simple Dutch math-teacher, but I have read your article with great pleasure. It is very inspiring and I hope to share it with some of my best pupils.

    Wim Haazen
  13.  

    Wonderful article. Thank you.

    Colin W. O'Sullivan
  14.  

    Mr Mark Littlewood will do a great service if he could raid the attics and unearth whatever papers of his own great uncle and Ramanujan and he could lay his hands on. The world of mathematics (and science) will be grateful.

    Subbiah Arunachalam
  15.  

    Fascinating read! I’m anticipating breakthroughs from Experimental Mathematics in the coming years…

    Paul Conroy
  16.  

    What’s the big deal with calculating P(200)?
    Just use Euler’s pentagonal number recursion.
    This should be a two day paper-and pencil job for a grad student in 1914.

    Kurt Shatov
  17.  

    I actually went to college because of Ramanujan.
    I read the book “The Man Who Knew Infinity”

    …and I couldn’t understand some of the advanced math in the book so I went to college to learn and I became side-tracked with computers and physics and I came up with the complete theory of everything….

    Everyone will realize I am correct in 40 or 50 years — we will all be long dead or too old to care– that’s the way the cookie crumbles.

  18.  

    You were my idol while I was doing my PhD and stuff like TeX, LaTeX and Mathematica were just evolving. This article is wonderful and thank you for adding your thoughts on (arguably) one of the greatest mathematicians the wold has ever seen.

  19.  

    Thank you for this very impressive article, Sir ! “Ramanujan’s brief life and death are symbolic of conditions in India. Of our millions how few get any education at all; how many live on the verge of starvation” (Jawaharlal Nehru)

    Dr Prakash Arumugam
  20.  

    I know next to nothing about mathematics but I was intrigued by the movie and wanted to find out more about Ramanujan. I really enjoyed the article even though a lot of it was way over my head! Your writing certainly kept my interest. Thanks

    Mary Ganger
  21.  

    Thanks sir for this post. It is very very interesting. I like it. And perhaps first post in my internet-life, I read a post all at a time.

    Sanjeev Kushwaha
  22.  

    This article would have been so much better with no mentions of Hardy. Hardy clearly attached his name to many of the Ramanujan’s works and hence the apology which only came later after the damage is done. Ramanujan is one of the, sorry the best Mathematician of the entire world, no western mathematician can even touch the dust of his feet, such was his skills. Apparently India failed to properly recognize Ramanujan as they are caught up in their own turmoil by using English as their language for education. Anyway, I am still waiting for an article which lists ALL the Ramanujan’s works without attaching westerners to HIS works

    Kalyan
  23.  

    I did empirical studies like Ramujan to fill in time while studying at university. As stated, most findings are mere curiosities. The real highly-advanced skill is to recognize the useful ones.

    Michael Whalan
  24.  

    FROM COCHIN,INDIA

    It is pretty hot here in India and therefore we tend to avoid physical work .
    We like to sit in the shade and brood about the nature of things. Sometimes
    we do accomplish much with a paper and pencil and of course knowledge and
    ideas. Remember Bose_Einstein Statistics, Remember SubramaniaChandrasekhar of
    Black-Holes fame. . He was nearly zeroised by Eddington.
    But we are a patient breed. We still learn our multiplication tables. And we do respect scholarship.
    Do read how the Russian Perelman did it solo.
    Hats off to all the mathematicians who struggling it alone. Einstein did it solo.

    jose francis v a
  25.  

    he really knows how mathematics works.
    my inspiring person. so love him

  26.  

    I want to thank you for the clarity and passion of your article. I found you online after seeing the movie (Dev Patel, etc.) on the plane earlier today. I am definitely not mathematically experienced – I love math, but I’m a humanities guy by training – but I am inspired and lifted emotionally both by Ramanujan’s life and your clear, comprehensive writing.

  27.  

    Wonderful blog!! Love the detail, the facts, and the love for mathematics. Thank you, thank you, thank you!

    Niraj Jain
  28.  

    Thank you for the revealing exposition. Ramanujan had a short and brilliant life. It is a shame he died so early, much like Mozart. I wonder what got him started into mathematical explorations.

    Dean Mindock
  29.  

    Lovely, inspriring post. Ramanujan has such a profound, unshakeable, intuitive grasp of mathematics – yet his fascination for it was deeper still “An equation means nothing to me unless it expresses a thought of God” – beautiful!

  30.  

    A very scholarly and engaging article on Ramanujan and Ramanujan-inspired perspective on mathematics and computation. I thoroughly enjoyed reading it. …. Interesting term “experimental mathematics”.

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