Letters to a Young Mathematician Read online

  The answer lies in the geometry of the rainbow. When the light bounces around inside a raindrop, the spherical shape of the drop causes the light to emerge with a very strong focus along a particular direction. Each drop in effect emits a bright cone of light, or, rather, each color of light forms its own cone, and the angle of the cone is slightly different for each color. When we look at a rainbow, our eyes detect only the cones that come from raindrops lying in particular directions, and for each color, those directions form a circle in the sky. So we see lots of concentric circles, one for each color.

  The rainbow that you see and the rainbow that I see are created by different raindrops. Our eyes are in different places, so we detect different cones, produced by different drops.

  Rainbows are personal.

  Some people think that this kind of understanding “spoils” the emotional experience. I think this is rubbish. It demonstrates a depressing sort of aesthetic complacency. People who make such statements often like to pretend they are poetic types, wide open to the world’s wonders, but in fact they suffer from a serious lack of curiosity: they refuse to believe the world is more wonderful than their own limited imaginations. Nature is always deeper, richer, and more interesting than you thought, and mathematics gives you a very powerful way to appreciate this. The ability to understand is one of the most important differences between human beings and other animals, and we should value it. Lots of animals emote, but as far as we know, only humans think rationally. I’d say that my understanding of the geometry of the rainbow adds a new dimension to its beauty. It doesn’t take anything away from the emotional experience.

  The rainbow is just one example. I also look at animals differently, because I’m aware of the mathematical patterns that underlie their movements. When I look at a crystal, I am aware of the beauties of its atomic lattice as well as the charm of its colors. I see mathematics in waves and sand dunes, in the rising and the setting of the sun, in raindrops splashing in a puddle, even in birds sitting on telephone cables. And I’m aware—dimly, as if looking out over a foggy ocean—of the infinity of things we don’t know about these everyday wonders.

  Then there’s the inner beauty of mathematics, which should not be underrated. Math done “for its own sake” can be exquisitely beautiful and elegant. Not the “sums” we all do at school; as individuals those are mostly ugly and formless, although the general principles that govern them have their own kind of beauty. It’s the ideas, the generalities, the sudden flashes of insight, the realization that trying to trisect an angle with straightedge and compass is like trying to prove that 3 is an even number, that it makes perfect sense that you can’t construct a regular seven-sided polygon but you can construct one with seventeen sides, that there is no way to untie an overhand knot, and why some infinities are bigger than others whereas some that ought to be bigger are actually equal, that the only square number (other than 1, if you want to be picky) that is the sum of consecutive squares, 1 + 4 + 9 + . . ., is the number 4900.

  You, Meg, have the potential to become an accomplished mathematician. You have a logical mind and also an inquiring one. You’re not convinced by vague arguments; you want to see the details and check them out for yourself. You don’t just want to know how to make things work, you want to know why they work. And your letter made me hope that you’ll come to see mathematics as I see it, as something fascinating and beautiful, a way of seeing the world that is like no other.

  I hope this sets the scene for you.

  Yours, Ian

  2

  How I Almost Became a Lawyer

  Dear Meg,

  You ask how I got into mathematics. As with anyone, it was a combination of talent (there’s no point in being modest), encouragement, and the right sort of accident, or more accurately, being rescued from the wrong sort of accident.

  I was good at math from the start, but when I was seven, I very nearly got put off the subject for life. There was a math test, and we were supposed to subtract the numbers, but I did the same as the previous week and added them. So I got a zero and was put in the lower section of the class. Because the other kids in that section were hopeless at math, we didn’t do anything interesting. I wasn’t being challenged, and I got bored.

  I was saved by two things: a broken bone and my mom.

  One of the other kids pushed me over in the playground as part of a game, and I broke my collarbone. I was out of school for five weeks, so Mom decided to make good use of the time. She borrowed the arithmetic book from the school, and we did some remedial work. Because I couldn’t write—my right hand was in a sling—I dictated the numbers, and she wrote them in the exercise book.

  My mother was rather sensitive about schooling. Her own education had been pretty much ruined by the mistaken good intentions of a well-meaning but unimaginative school inspector. Because she was quick on the uptake, she was advanced rapidly through the grades until, by the age of eight, she was in with a class of ten-year-olds. The school inspector came by one day, observed the class, and asked the intelligent little girl who was answering all the questions, “How old are you, my dear?” On being told “eight,” he informed the school principal that the bright little girl must stay in the same class for three years in a row, until the other kids her age caught up with her. He wasn’t trying to hold her back academically; he was worried that she was out of her depth socially. But repeating the same lessons three years in a row killed off my mother’s interest in school; all she learned was how to goof off.

  Later she worked out what had happened, but by then it was too late. She wanted to be an English teacher, but she failed her chemistry exam. In those days, in the United Kingdom, failing just one subject, even one that was totally irrelevant to the subject you wanted to teach, meant that you could not train as a teacher.

  My mother was determined that nothing similar should happen to me. She knew I was clever; she’d taught me to read when I was three. After we had done 400 math problems and I’d gotten 396 of them right, she took the exercise book into school, showed it to the head teacher, and demanded that I be moved into the top section of the math class.

  When my collarbone healed and I went back to school, I was ten weeks ahead of the rest of the class in math. We’d overdone it a bit. Fortunately, I didn’t suffer too much while the class caught up.

  My teacher wasn’t a bad teacher. In fact, he was a very kind man. But he lacked the imagination to realize that he’d put me in the wrong section, and that his mistake was going to damage my education. I’d gotten a zero on the test because I was careless, not because I didn’t understand the material. If he’d simply told me to read the questions carefully, I’d have gotten the point.

  I was lucky then, thanks mostly to my mother’s good sense and willingness to fight for me. But I also owe a debt to my schoolmate for putting me in the hospital. He’d done it quite unintentionally—we were all shoving each other around—but it saved my mathematical bacon.

  After that I had several really brilliant math teachers. And those, let me tell you, are rare. There was one named W. E. Beck (we nicknamed him “spider”) whose Friday math test was a long-standing institution. Those were not easy tests. They were graded out of twenty points, and as the weeks passed, each kid’s grades were added up. The kids who were good at math were desperate to come in first for the year; the others were just desperate. I’m not sure it was acceptable educational practice—in fact I’m sure it wasn’t—but the competitive element was good for me and a few buddies.

  One of Beck’s rules was that if you missed a test, even if you were sick, you got zero. No excuses. So those of us who were in the running needed to make every point count. We knew we needed a cushion, since you weren’t safe unless you were ahead by more than twenty points. So you absolutely did not lose points by making silly mistakes. You read every question, made sure you’d done what was asked, checked everything, and then you checked it again.

  Later, when I was sixteen, I had a math
teacher named Gordon Radford. Normally he was lucky to get one boy who was really talented at math, but in my class there were six of us. So he spent all of his free periods teaching us extra math, outside the syllabus. During the regular math lessons he told us to sit at the back and do our homework; not just math, any homework. And to shut up. Those lessons weren’t for us; we had to give the others a chance.

  Mr. Radford opened my eyes to what math was really like: diverse, creative, full of novelty and originality. And he did one more crucial thing for me.

  In those days, there was a public entrance exam called a State Scholarship that provided funding to go to college. You still needed to be offered a place, but a State Scholarship was a big step in the right direction. In the last year that State Scholarships were to be offered, I and two friends were a year too young to take the exam. Mr. Radford had to persuade the headmaster to put us in for the exam one year early, something the headmaster never did.

  One morning when my two friends and I arrived at school, Mr. Radford told us we would be joining the class one year ahead of us to take a “mock exam” for the State Scholarship in math. A practice run. The older kids had done a year’s more math and had been practicing for weeks; we had five minutes’ warning. I came in first, and my friends were second and third.

  So the headmaster had no choice but to let us take the exam for the State Scholarship. After all, he was letting the older kids take it, and we had proved we were better prepared for it than they were.

  All three of us were awarded State Scholarships.

  At that point Mr. Radford got in touch with David Epstein, whom he had taught some years before and who had become a mathematician at Cambridge University, along with Oxford, the United Kingdom’s leading university, especially renowned for its math.

  “What do I do with this boy?” Gordon asked.

  “Send him to us,” said David.

  So I went to study math at Cambridge, the home of Isaac Newton, Bertrand Russell, and Ludwig Wittgenstein (along with many lesser lights), and I never looked back.

  Some careers seem to accumulate people who might easily have preferred to do something else. You will run into people who tell you that they practice law as a day job but they are really novelists or playwrights or jazz trombonists. Other people can’t settle on something, or they see their careers in more purely practical terms, and they drift into human resources management or advertising sales. Which is not to say that these people are not dedicated or fulfilled in what they do, but few of them consider their work a calling.

  No one drifts into being a mathematician. On the contrary, it’s a pursuit from which even the talented are too easily turned away. If I hadn’t broken my collarbone, if Mr. Beck hadn’t fostered all-out competition among his students, if there hadn’t been an unusually large group of strong students for Mr. Radford to promote— and if he hadn’t done it so aggressively—instead of writing you today I might be telling your parents how to save more on their tax return. And perhaps no one, least of all me, would suspect that things could have turned out differently.

  In short, Meg, you should not expect your teachers to look at you once and simply see, in a brief glance, how bright you are. You should not expect them to unerringly spot your talents and know where they might lead you. Some will, and you will be grateful to them for the rest of your life. But others, sadly, can’t tell, or don’t much care, or are caught up in their own worries and resentments. Then again, the ones who stand in awe of your gifts are not the ones from whom you will ultimately learn the most. The best teachers will occasionally, perhaps more than occasionally, make you feel a bit stupid.

  3

  The Breadth of Mathematics

  Dear Meg,

  It’s not hard to see, in your question, a sense of—I don’t know—anticipated boredom, or perhaps some worry about what you’ve let yourself in for. It’s all reasonably interesting now, but, as you say, “Is this all there is?” You’re reading Shakespeare, Dickens, and T. S. Eliot in your English class, and you can reasonably assume that while this is of course only a tiny sample of the world’s great writing, there is not some higher level of English literature whose existence has not been disclosed to you. So you naturally wonder, by analogy, whether the math you’re learning in high school is what mathematics is. Does anything happen at higher levels besides bigger numbers and harder calculations?

  What you’ve seen so far is not really the main event.

  Mathematicians do not spend most of their time doing numerical calculations, even though calculations are sometimes essential to making progress. They do not occupy themselves with grinding out symbolic formulas, but formulas can nonetheless be indispensable. The school math you are learning is mainly some basic tricks of the trade, and how to use them in very simple contexts. If we were talking woodwork, it’s like learning to use a hammer to drive a nail, or a saw to cut wood to size. You never see a lathe or an electric drill, you do not learn how to build a chair, and you absolutely do not learn how to design and build an item of furniture no one has thought of before.

  Not that a hammer and saw aren’t useful. You can’t make a chair if you don’t know how to cut the wood to the correct size. But you should not assume that because that’s all you ever did at school, it’s all carpenters ever do.

  An awful lot of what is now called “mathematics” at school is really arithmetic: various notations for numbers, and methods for adding, subtracting, multiplying, and dividing them. As you get older, you are shown other bits of the toolkit: elementary algebra, trigonometry, coordinate geometry, maybe a little calculus. If your syllabus was “modernized” in the 1960s or 1970s, you may get two-by-two matrices and tiny bits of group theory. “Modern” is a strange word to use here: it means between one and two hundred years old, as opposed to the two-hundred-plus-year-old math that formed the bulk of the older syllabus.

  Unfortunately, it’s almost impossible to progress to the more interesting regions of the subject if you don’t know how to do sums and get them right, how to solve basic equations, or what an ellipse is. The highest levels of every human activity demand a solid grasp of the basics; think of tennis, or playing the violin. Mathematics happens to require rather a lot of basic knowledge and technique.

  At university you will encounter a much broader conception of mathematics. In addition to the familiar numbers, there will be complex numbers, where minus one has a square root. Things far more important than numbers will appear, such as functions: rules that assign, for any chosen number, some specific other number. “Square,” “cosine,” “cube root,” those are all functions. You won’t just solve simultaneous equations in two unknowns; you will understand the solutions of simultaneous equations in any number of unknowns, when they exist at all, which they sometimes don’t. (Try solving x + y = 1, 2x + 2y = 3.) You may learn how the great mathematicians of the Renaissance solved cubic and quartic equations (involving cubes and fourth powers of the unknown), not just quadratics; if so, you will probably find out why such methods fail for quintic equations (fifth powers). You will see why this becomes almost obvious if you ignore the numerical values of the equations’ solutions and instead think about their symmetries, and why it is arguably more important to understand the symmetries of equations than to be able to solve them.

  You will find out how to formalize the concept of symmetry in abstract terms, which is what group theory is. You will discover that Euclid’s geometry is not the only one possible, and move on to topology, where circles and triangles become indistinguishable. You will have your intuition challenged by Möbius bands, which are surfaces with only one side, and fractals, which are shapes so complex that they have a fractional number of dimensions. You will learn methods to solve differential equations, and eventually you will appreciate that most of them cannot be solved by those methods; then you will learn how they can still be understood and used, even when you cannot write down their solutions. You will find out why every number can be resolved
uniquely into prime factors, be puzzled by the apparent lack of patterns in primes despite their statistical regularities, and baffled by open questions like the Riemann hypothesis. You will meet different sizes of infinity, discover the real reasons why π is important, and prove that knots exist. You will belatedly realize just how abstract your subject has become, how far removed from mere numbers, and then numbers will bite you on the ankle, reemerging as key ideas.

  You will learn why tops wobble and how that affects ice ages; you will comprehend Newton’s proof that planets’ orbits are elliptical, and find out why they aren’t perfectly elliptical, opening up the Pandora’s box of chaotic dynamics. Your eyes will be opened to the vast range of uses of math, from the statistics of plant breeding to the orbital dynamics of space probes, from Google to GPS, from ocean waves to the stability of bridges, from the graphics in Lord of the Rings to antennas for mobile phones.

  You will come to feel just how much of our world would be impossible without math.

  And when you survey this glorious diversity, you will wonder what makes it all the same thing: why are such disparate types of ideas all called mathematics? You will have gone from asking “Is this all there is?” to being slightly amazed that there can be so much. By then, just as you can recognize a chair but can’t define one in a manner that permits no exceptions, you will find that you can recognize mathematics when you see it, but you still can’t define it.

  Which is as it should be. Definitions pin things down, they limit the prospects for creativity and diversity. A definition, implicitly, attempts to reduce all possible variations of a concept to a single pithy phrase. Math, like anything still under development, always has the potential to surprise.