During my years as a mathematician, not one film-maker has tried to teach me how to write better articles. So I’m not going to tell Matt Brown, the writer/director of “The Man Who Knew Infinity”, what he should have done differently in a movie that, as the fine print on the poster reminds us, is merely based on the life of Ramanujan. If I knew as much about movie-making as Matt Brown does, I probably would have made the same choices he did.
But I am going to tell you, fellow-members of the movie-going public, what characteristics of the math life are conveyed by the film, and what characteristics aren’t. I’m not saying that the film in and of itself is inaccurate, but it does recycle some tropes about mathematics that you’ve probably seen in other movies about mathematicians and that give an inaccurate picture of mathematics. Along the way, you’ll meet the surprising base-ten expansion of the infinite product .9 × .99 × .999 × .9999 × … and learn what it has to do with Ramanujan’s story. (I’m going to assume that you’ve read my blog essay Sri Ramanujan and the Secrets of Lakshmi from last month, or that you already know something about the life and work of Ramanujan.)
Here are some characteristics of mathematics that you’ll learn about from the movie:
1. Math is a creative endeavor that can evoke esthetic delight.
Actor Dev Patel, bringing Matt Brown’s script to vibrant life, conveys the sense of beauty that Ramanujan finds in his pursuit of mathematics. Early in the film, in talking to his wife about the artistic side of math, Ramanujan resorts to metaphor, speaking to her of his formulas as paintings composed of “colors you cannot see”. His desire to find others who see those colors impels him to leave her and all the other people he knows and loves, to seek his destiny in England in 1914.
2. Doing research in math isn’t easy, even for great mathematicians.
The film shows Ramanujan’s struggle to find proofs of his insights that will satisfy his collaborator Hardy and the rest of the mathematical establishment. Such struggle, less anguished but just as strenuous, is I think the norm for productive mathematicians. Of course there are some active researchers who find a narrow vein of expertise and mine it for years in a low-key, whistle-while-you-work way, but most of us frequently step out of our comfort-zones. Once we get really good at doing something, we set our sights a bit higher and try to do something a bit harder but still (we hope) within our powers.
3. Math is often collaborative.
In the film we see two mathematicians working together, an image often missing from popular portrayals of mathematical research. People have different strengths, and research can benefit from multiple talents and perspectives. A recent paper of mine took over ten years to come to fruition; from the start my original collaborators and I knew in a heuristic sense why a certain result ought to be true, and we had computer evidence that it probably was true, but we didn’t have an airtight proof that it had to be true. A younger mathematician (from India, as it happens) was the one who did most of the hard work of designing a proof that would do the job. It was an intricate proof, too; we talked about it for several days, and I still didn’t understand all the details. It certainly wasn’t the kind of thing I would have been able to come up with on my own.
4. Genius can come from anywhere, and it doesn’t always arrive equipped with credentials.
Hardy wasn’t the first English mathematician with whom Ramanujan attempted to establish a connection; Hardy was merely the first to take him seriously. It’s easy to see why other English mathematicians would have succumbed to the temptation to dismiss Ramanujan. After all, Ramanujan had no academic credentials beyond an unfinished college degree. Racism no doubt played a role too; Ramanujan didn’t look like English mathematicians’ internalized picture of what a mathematician was. And yet his turned out to be one of the greatest mathematical minds of the 20th century.
5. Genius isn’t about always being right.
As the film makes clear, Ramanujan’s intuition didn’t give him 20/20 vision into the mists of mathematical reality; in particular, his work on prime numbers contained errors. Genius is about incisive originality, not infallibility. It’s almost inevitable that a researcher who goes into territory where no one has gone before will misread certain signs, jump to wrong conclusions, and take some mis-steps. (That’s one reason why Ramanujan was so grateful to have so much paper when he arrived in England; a mathematician needs a lot of it, and much of it gets thrown away!) The question is, can one detect one’s own mistakes, recalibrate one’s intuition, and continue to move forward?
6. Rigor and inspiration are complementary.
In the movie, Ramanujan and Hardy respectively embody the opposed yet cooperating principles of intuition and rigor. When Ramanujan’s work on primes turns out to be flawed, Hardy uses the occasion to reiterate to Ramanujan the importance of not just intuiting that a result is true but also subjecting it to rigorous scrutiny, and constructing a proof that supports insight with ironclad, step-by-step logic. His insistence on this point borders on the condescending. The impression of arrogance is leavened by the fact that Hardy not only thinks that Ramanujan is a better mathematician than he is, but he doesn’t mind if Ramanujan knows it. Late in life, struggling with depression, Hardy famously wrote: “I still say to myself when I am depressed and find myself forced to listen to pompous and tiresome people ‘Well, I have done one thing you could never have done, and that is to have collaborated with Littlewood and Ramanujan on something like equal terms.'”
Now here are some characteristics of mathematics that you won’t learn from the movie, or for that matter from various other movies about mathematicians, but which you should still keep in mind, especially if you teach mathematics and therefore communicate your beliefs about mathematics to many other people:
1. Women do math, and do it well.
I don’t fault Matt Brown for making a movie about a male mathematician. Ramanujan’s story is one of the great stories from the world of mathematics, and it’s a story about a male mathematician.
I also don’t mean to insult your intelligence by pointing out that many mathematicians are women. Actually, I take that back: I do mean to insult your intelligence! And everyone else’s intelligence too, including mine. That’s because people, with their fallible brains, are not very good at making unbiased judgments of other people. Our brains tend to sacrifice accuracy for the sake of quick decision-making. Faced with the complex task of forming judgments about people, we fall back on cognitive heuristics, namely stereotypes. We replace a complex social construct like the notion of a “mathematician” by an inner catalogue of exemplars, rather than a checklist of attributes. And some of those exemplars don’t even come from real life, but are creations of the entertainment industry.
So it’s very natural that if I ask you to picture a mathematician, you’ll picture someone who looks like Hardy, or maybe Ramanujan, and not someone who looks like Emmy Noether, say (see Endnote #4 for more about her). It’s natural, but insidious, because then if we encounter someone who might or might not be a mathematician, we unconsciously ask the wrong question: not “Does she teach math or write about math or create new math?” but “Does she look like Jeremy Irons or Russell Crowe or Matt Damon?” And part of what makes the problem so hard to uproot is that it’s a chicken-and-egg problem that we’re all complicit in it, in a big-ole cultural feedback loop. Should we blame screenwriters for not writing more scripts that show us women as creative geniuses? Studio executives, for not green-lighting the few such scripts that cross their desks? Movie-goers, for not going to such movies, or not signalling their interest in going? Mathematically talented girls who, lacking visible role models, abandon math at one point or another in their education, thereby failing to inspire the next generation of mathematicians and screenwriters?
It’s a tough problem. One recent study found that kids absorb societal messages about math and gender by the time they turn 10. But there are some solutions you can adopt on an individual level, even if they only affect your own consciousness. For one thing, when you see a movie, you can try to evaluate it in a broader context that takes into account all the movies that you’re not seeing because they’re not being made. Likewise for other sorts of stories, like the ones found in books and plays and radio programs. You can try to consciously monitor and subvert your own unconcious biases with questions like “Would I have the same reaction if X were a man rather than a woman?”. It’s not a solution, but it’s a start. (See Endnote #3 for more on this.)
2. People who weren’t child prodigies do math, and do it well.
Earlier, I praised the film’s analogy between the beauty of visual art and the beauty of mathematics. But Ramanujan’s speech to his wife, telling her that theorems are made of “colors you cannot see”, if the emphasis is on the word you and the word cannot, worries me; it sounds as if Ramanujan might be saying that he sees colors that most other people can’t, no matter how hard they try, because they just weren’t born with the right kind of eyes. I don’t deny the existence of genius, but I maintain that huge swaths of mathematical knowledge that may at first seem daunting and bizarre and unseeable can eventually seem natural and straightforward, if you are persistent. Mathematical vision is a faculty that can be trained, not an organ one is either born with or born without. The real handicap faced by students who say they “just can’t do math” is not a lack of ability but a mistaken belief in their own limitations.
Just as there are movies about mathematicians we never get to see, there are people who could become mathematicians and make important contributions but don’t, all because of the “prodigy myth”. It’s a hidden problem, because we never see the papers they don’t write or the theorems they don’t prove. But as a teacher, I’ve repeatedly been reminded that the precocious students aren’t always the most insightful ones. Sometimes, it’s actually “slow” students, the ones who aren’t satisfied with an explanation that satisfies their classmates, who are the deepest thinkers.
Related to the prodigy myth is the “genius myth”, which would have us think that in order to be an effective mathematician you need to have some sort of superhuman proficiency. It’s true that some people are born with more natural facility than others, but just because something is hard for you doesn’t mean you can’t excel at it. The lesson of Wilma Rudolph is one that applies in all fields of endeavor.
3. People with multiple interests and rich lives do mathematics, and do it well.
In pursuing his dream, Ramanujan left behind India (and, with it, everyone he knew and loved), and focused exclusively on mathematics for years. He also made huge contributions to math prior to his tragically early death. Coincidence? Of course not. In most fields of endeavor, there’ll be obsessives (think: Erdős) who achieve more because they devote themselves to their goals with monastic discipline. (Recall the way Isaac Newton explained how he made breakthroughs on so many problems: by always thinking about them. Also remember that he died a virgin.) These people may end up logging more breakthroughs than people with more balanced lives, but they end up just as dead. You need to decide for yourself what you want to pack on the bookshelf of your life between the bookends of Birth and Death, and I’m guessing it’s more than just books — maybe some photographs too? My point is that you can have both a math life and a non-math life, even though movies about mathematicians often focus on people who, through their own choice or untimely death, don’t.
4. Math isn’t just about formulas.
Of course formulas figure prominently in the movie; after all, most of Ramanunan’s work consisted of beautiful formulas. But there’s more to math than formulas. Take, for instance, Smale’s Theorem about it being possible to evert a sphere (to name just one piece of mathematics that I don’t understand but hope to understand someday). Stating Smale’s Theorem carefully may require formulas, and demonstrating it with computer graphics may require other sorts of formulas, but in and of itself, it’s about the ins and outs of self-penetrating surfaces, and formulas are incidental. I hope that as the art of illustrating mathematics advances, more and more non-mathematicians will come to appreciate the non-formulaic side of math.
5. Most mathematicians aren’t good at mental arithmetic.
I suppose it’s an exaggeration to claim, as present-day mathematician Robin Wilson has done, that “pure mathematicians don’t work with numbers”. But this exaggeration does have the virtue of being truer to life than the false image Wilson was trying to rebut: the mathematician as lightning-fast calculator. Yes, there are people like “mathemagician” Art Benjamin who can multiply two four-digit numbers as quickly as you can say them; but far more numerous are pure mathematicians who have trouble splitting a bill at restaurants. It appears that both Ramanujan and his “rival” Percy MacMahon (of whom more shortly) were enthusiastic mental calculators, but both of them knew the difference between this sort of mental exercise and the practice of actual mathematics. It made no sense to me that MacMahon would try to evaluate Ramanujan’s mathematical mettle by subjecting him to a test of mental arithmetic, as he does in the movie.
Let’s talk about MacMahon, and get to the mathematical meat of this essay. You may recall from last month the definition of p(n) as the number of partitions of the number n: p(0) = 1, p(1) = 1, p(2) = 2, p(3) = 3, p(4) = 5, p(5) = 7, etc. Ramanujan and Hardy came up with a general formula for p(n) and used this formula to compute p(200); they then checked their result against MacMahon’s combinatorial method of calculating p(200), and found that their method gave the right answer.
How did MacMahon do it? He used a different formula for p(n) that had been discovered by Euler over a century earlier:
p(n) = p(n−1) + p(n−2) − p(n−5) − p(n−7) + p(n−12) + p(n−15) − …
Some explanation is in order. The sum on the right appears to be an infinite sum, but for any specific value of n, all terms are equal to 0 (and hence can be ignored) from some point on, since p(−1), p(−2), etc. all equal 0. For instance, if you already know p(0) through p(9), you can compute p(10) by plugging into Euler’s formula: p(10) = p(9) + p(8) – p(5) – p(3) + p(−2) + p(−5) − … = 30 + 22 − 7 − 3 + 0 + 0 − … = 42. If you can follow that, you can see how MacMahon could have continued using this formula to derive p(11) = 56, p(12) = 77, …, p(50) = 204226, p(100) = 190569292, all the way up to p(200) = 3972999029388 (and no, he did not do it all in his head).
But what’s the infinitely long right-hand side of Euler’s equation?
The numbers 1, 2, 5, 7, 12, 15, … that get subtracted from n are the nonzero generalized pentagonal numbers; that is, numbers of the form k(3k+1)/2 where k can be any positive or negative integer, and the term p(n − k(3k+1)/2) in the right-hand side gets added or subtracted according to whether k is odd or even.
Euler’s equation is a consequence of his pentagonal number theorem. Another consequence is the beautiful structure for the digits of a number that I’ll call r, given by the infinite product .9 × .99 × .999 × .9999 × … If you write out r in decimal as
r = 0.8900100999989990000001000…
you begin to see that it’s a very different creature from rational numbers like
22/7 = 3.142857142857142857142857…
whose digits from some point onward just repeat and familiar irrational numbers like
π = 3.141592653589793238462643…
whose digits display no discernible pattern. π is believed to be a “normal number” with respect to base ten: that is, each of the 10 decimal digits appears to occur equally often, each of the 100 two-digit combinations appears to occur equally often, each of the 1000 three-digit combinations appears to occur equally often, and so on.
But r is different. The base ten expansion of r doesn’t repeat (the blocks of 0’s and blocks of 9’s get longer and longer) but the 10 decimal digits don’t appear equally often; in fact, only the digits 0, 1, 8 and 9 occur, no matter how far out you go! So r is an example of an irrational number that has a highly patterned decimal expansion.
A pretty way to see the structure in the decimal expansion of r is to arrange the digits in a square spiral. For instance, if we list the numbers from 1 to 25 in the pattern
and then replace each number n by the nth digit of r after the decimal point, we get
If we do this for a 49-by-49 array instead of a 5-by-5 array, and replace the digits 0, 1, 8, and 9 by pink, yellow, orange, and red respectively, we get the image shown below.
What’s going on is that r can be written not just as the infinite product .9 × .99 × .999 × .9999 × … but also as the infinite sum 1 − (.1)1 − (.1)2 + (.1)5 + (.1)7 − (.1)12 − (.1)15 + … This should remind you a lot of the formula p(n) = p(n−1) + p(n−2) − p(n−5) − p(n−7) + p(n−12) + p(n−15) − …, and it’s no coincidence, though explaining why would take us into technicalities that most of my readers won’t enjoy and that I’ll therefore skip. (I say a bit more in Endnote #2. Interested readers may turn to chapter 19 of Hardy and Wright’s classic An Introduction to the Theory of Numbers, from which I learned about this sort of thing.)
6. In mathematics, asking to see a proof is not inherently an act of skepticism or aggression.
A high point of the film is Hardy’s impassioned advocacy of Ramanujan to an audience of skeptical academics, culminating in the fiery exclamation “Who are we to question him?”
But I want to exercise the mathematician’s prerogative of interpreting a rhetorical question as if it weren’t rhetorical (and if you know a mathematician, chances are you’ve seen this prerogative in action). And I’ll shift the meaning of the question while I’m at it.
Who are the people who will typically ask to see the proof of some mathematician’s claims? Usually they are the people who want to build on what that mathematician has done! That is, the people asking to see the proof are the loyalists, who think that such results are worth understanding, as opposed to the people who couldn’t care less. When I tell a colleague about a result, I’d much rather hear “How did you prove that?” than “That’s a nice result, and what else are you working on?”
If “proof” were just about validation, mathematicians would be satisfied with knowing just one proof of any given proposition. But for us, a proof isn’t just about knowing what’s true, but also about knowing why it’s true, and what new truths might follow. It’s very hard to breed flowers from the petals alone; you need the seeds.
But also, when we ask for a proof, we’re often asking “What’s the real story here?” A formula is a kind of poem, or perhaps a short story, but some short stories cry out for a larger context. Right now I’m reading Kelly Link’s “Pretty Monsters”, a collection of short works of fantasy, and I’m frustrated because each story seems to me like a fragment of an unwritten novel; she builds all these richly-imagined worlds that leave me hungry for more details, nonchalantly refusing to tie up all the loose ends (such as the one in Endnote #1). Seeing a theorem without a proof can induce the same sort of unpleasant tension.
Let me extend this point by inviting you to imagine you’re on the receiving end of an elevator-pitch for the very movie we’ve been talking about. It might go something like this: “There are these two mathematicians from really different backgrounds, a nobody and a bigshot, but it turns out that, deep down, they’re equals.” You might respond by asking what “different backgrounds” means. The reply might be: “One is a self-taught genius from India, and the other is a witty Bloomsbury-type.” Then you might want to know what “equals” means. The answer might be: “Together they achieve some deep mathematical insights that neither of them could have arrived at on his own.” And then you’d ask more questions, until you learned the whole story (or more probably until the elevator reached its destination).
The capsule summary of the movie is in no sense a replacement for the movie itself. In the same way, knowing that a certain formula is true isn’t really satisfying for mathematicians; at the end of the ride, we want to know why it’s true.
Thank to George Andrews, Sandi Gubin, David Jacobi, Evelyn Lamb, Ken Ono, “Shecky Riemann”, and Tom Roby.
Next month (August 17): Ballots and Boltzmann.
#1. Here’s a typical passage from Kelly Link, taken from her story “The Wrong Grave”: “There was nothing of light or enlightenment about Bethany’s hair. It knew nothing of hope, but it had desires and ambitions. It’s best not to speak of those ambitions. As for the tattoo, it wanted to be left alone. And to be allowed to eat people, just every once in a while.” That’s all we’re ever told about the tattoo, but I sure wanted to know more about it! If fantasy worlds behaved the way mathematical worlds do, I’d be able to find out more about the tattoo by entering the world of the story, with or without Kelly Link’s help. Of course I might get eaten, so maybe I should stick to math.
#2. Notice that Euler’s formula, starting from p(0) = p(1) = 1, implies that
p(2) = p(1) + p(0) = 1 + 1 = 2,
p(3) = p(2) + p(1) = 2 + 1 = 3, and
p(4) = p(3) + p(2) = 3 + 2 = 5,
so if you thought that the sequence 1,1,2,3,5,7,11,15,… started out in a Fibonacci-ish mode, you were onto something!
The Fibonacci pattern isn’t just a coincidence. It’s a symptom of the even deeper pattern that Euler discovered. Both Fibonacci’s equation f(n) = f(n−1) + f(n-2) and Euler’s equation p(n) = p(n−1) + p(n−2) − p(n−5) − p(n−7) + p(n−12) + p(n−15) − … are what mathematicians call recurrence relations. Ramanujan’s equation was different: it was what mathematicians call a closed-form expression (albeit an infinite one). Hardy initially doubted that such a formula could be found, though the film imputes the skepticism to MacMahon, and ratchets up the skepticism to outright disbelief for dramatic effect.
#3. Your brain didn’t come with an instruction manual, but psychologists are trying to write one, and to compose an FAQ about your brain’s bugs and features and the ways in which they can adversely affect your perceptions and judgments. To see what sort of implicit ideas your brain has about various groups of people, try the IAT (Implicit Assocation Test). As is often the case in the social sciences, proving a link between “in vitro” behavior (how well you do on the IAT) and “in situ” behavior (which postdoc you choose to hire) can be a subtle matter. But researchers have found some correlations between implicit associations and real-world behavior, and in any case common sense strongly suggests that there’s a link. Depressingly, there’s also evidence suggesting that willpower alone does not suffice to defeat the power of implicit associations; we can know that they’re there and want to combat them but still be swayed by them.
In some fields of endeavor, simple counter-measures can help a lot (the way the practice of blind auditioning has boosted the participation of women musicians in symphony orchestras). It’s less clear how to do this in academia, since at some point, candidates will be interviewed and all sorts of decisionally irrelevant information will leak through. Still, Human Resources department could help make hiring less biased by “blinding” application materials, thereby deferring and lessening the impact of evaluators’ latent baises. My university doesn’t do this; do any of you know of a university that does?
I’m focusing here on sexism, but of course there are axes other that gender along which people differ, all of which can feed into bias: race, sexual orientation, etc.
#4. When you see a movie about John Nash or Srinivasa Ramanujan, there are an infinite number of movies you’re not seeing; one of them is a movie about Emmy Noether. Since you can’t see a movie about her, at least not yet, I invite you to do your own small part to combat your brain’s cinematically-induced implicit associations by imagining your own version of The Emmy Noether Movie, based on her life in Germany and the U.S. She worked in many fields, but one of her great contributions was Noether’s Theorem, which taught physicists that there was a deep mathematical link between invariance principles (going back to Galileo’s precursor to Einstein’s special theory of relativity) and conservation principles (conservation of energy, momentum, etc.). Einstein was impressed, and thought her the greatest mathematician of the age. In my movie I’d focus on that, since the theory of Noetherian rings doesn’t go down well with popcorn.
Just as Ramanujan had Hardy as his champion, Noether had David Hilbert — who in 1915 advocated for her being allowed to join the Göttingen faculty despite her being a woman by famously declaring “This is a university, not a bathhouse.” In my version of the movie, when we see Hilbert proclaiming this, it’s the second time we’ve heard the line; the first time, it was Noether saying it to Hilbert as they strolled the university grounds, and Hilbert gently reproved her, saying “You know it’s not that simple.” Now, I have no evidence that things went down that way — that Hilbert got the idea for his quip from Noether — but, hey, see the fine print on the movie poster: “based on the life of Emmy Noether.” Plus, it’s my movie. If you don’t like it, imagine your own!
In addition to the references from last month, see:
George Andrews, The Man Who Knew Infinity: a report on the movie.
Jordan Ellenberg, The Wrong Way to Treat Child Geniuses.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers.
Robert Kanigel, The Man Who Knew Infinity (the book on which the movie is based).
Evelyn Lamb, Math and the Genius Myth.
George Polya, Mathematics and Plausible Reasoning; a great discussion of the Pentagonal Number Theorem can be found on pages 91-98.
Wikipedia, The Pentagonal Number Theorem.