Letters to a Young Mathematician

by Ian Stewart

  A second reason why few students ever realize that there is mathematics outside the textbook is that no one ever tells them that.

  I don’t blame the teachers. Math is actually very important, but because it genuinely is difficult, nearly all of the teaching slots are occupied with making sure that students learn how to solve certain types of problem and get the answers right. There isn’t time to tell them about the history of the subject, about its connections with our culture and society, about the huge quantity of new mathematics that is created every year, or about the unsolved questions, big and little, that litter the mathematical landscape.

  Meg, the World Directory of Mathematicians contains fifty-five thousand names and addresses. These people don’t just sit on their hands. They teach, and most of them do research. The journal Mathematical Reviews appears twelve times per year, and the 2004 issues totaled 10,586 pages. But this journal does not consist of research papers; it consists of brief summaries of research papers. Each page summarizes, on average, about five papers, so for that year the summaries covered about fifty thousand actual papers. The average size of a paper is perhaps twenty pages—roughly a million pages of new mathematics every year!

  Kathleen’s friend would have been horrified.

  Many teachers are aware of this, but they have a good reason not to say much about it. If your students are having problems remembering how to solve quadratic equations, the wise teacher will stay well clear of cubic equations, which are even more difficult. When the issue in class is finding solutions of simultaneous equations that possess solutions, it would be demoralizing and confusing to inform the students that many sets of simultaneous equations have no solutions at all, and others have infinitely many. A process of self-censorship sets in. In order to avoid damaging the students’ confidence, the texts do not ask questions that the methods being taught cannot answer. So, insidiously, we absorb the lesson that every mathematical question has an answer.

  It’s not true.

  Our teaching of mathematics revolves around a fundamental conflict. Rightly or wrongly, students are required to master a series of mathematical concepts and techniques, and anything that might divert them from doing so is deemed unnecessary. Putting mathematics into its cultural context, explaining what it has done for humanity, telling the story of its historical development, or pointing out the wealth of unsolved problems or even the existence of topics that do not make it into school textbooks leaves less time to prepare for the exam. So mostly these things aren’t discussed. Some teachers—my Mr. Radford was an example—find time to fit them in anyway. Ellen and Robert Kaplan, an American husband-and-wife team with a refreshing approach to mathematics education, have started a series of “math circles,” where young children are encouraged to think about mathematics in an atmosphere that could not be more different from that of a classroom.

  Their success shows that we need to set aside more time in the syllabus for such activities. But since math already occupies a substantial part of teaching time, people who teach other subjects might object. So the conflict may well remain unresolved.

  Now let me explain a wonderful thing: the more mathematics you learn, the more opportunities you will find for asking new questions. As our knowledge of mathematics grows, so do the opportunities for fresh discoveries. This may sound unlikely, but it is a natural consequence of how new mathematical ideas build on older ones.

  When you study any subject, the rate at which you can understand new material tends to accelerate the more you already know. You’ve learned the rules of the game, you’ve gotten good at playing it, so learning the next level is easier. At least it would be, except that at higher levels you set yourself higher standards. Math is like that. To perhaps an extreme degree, it builds new concepts on top of old ones. If math were a building, it would resemble a pyramid erected upside down. Built on a narrow base, the structure would tower into the clouds, each floor larger than the one below.

  The taller the building becomes, the more space there is to build more.

  That’s perhaps a little too simple a description. There would be funny little excrescences protruding all over the place, twisting and turning; decorations like minarets and domes and gargoyles; stairways and secret passageways unexpectedly connecting distant rooms; diving boards suspended over dizzying voids. But the inverted pyramid would dominate.

  All subjects are like that to some extent, but their pyramids do not widen so rapidly, and new buildings are often put up beside existing ones. These subjects resemble cities, and if you don’t like the building you are in, you can always move to another one and start afresh.

  Mathematics is all one thing, and moving house is not an option.

  Because school math is heavily biased toward numbers, many people think that math comprises only numbers, that mathematical research must consist in inventing new numbers. But of course there aren’t any, are there? If there were, someone would already have invented them. But this belief is a failure of imagination, even when it comes to numbers.

  Most schoolwork on numbers is arithmetical. Add 473 to 982. Divide 16 by 4. A lot is about notation: fractions like 7/5, decimals like 1.4, recurring decimals like 0.3333. . ., or more obstreperous numbers like π, whose decimal digits go on forever without any repetitive pattern.

  How do we know that about π? Not by listing every digit, or by listing lots of them and failing to observe any repetition. By proving it, indirectly. The first such proof was published in 1770 by Johann Lambert, and it is based not on geometry but on calculus. It occupies about a page and is mostly a calculation. The trick is not the calculation but figuring out which calculation to do.

  A few more inventive topics also appear at school level, such as prime numbers, which cannot be obtained by multiplying two smaller (whole) numbers together. But pretty much everything students are exposed to boils down to buttons you could push on a pocket calculator.

  The higher floors of the mathematical anti-pyramid do not look like this at all. They support concepts, ideas, and processes. They address questions very different from “add these two numbers,” such as, “Why do the digits of π not repeat?” The floors that do deal with numbers rapidly get to extremely difficult questions, which often appear deceptively straightforward.

  For instance, you will be aware that a triangle with sides 3, 4, and 5 units long has a right angle; allegedly the ancient Egyptians used a string divided into such lengths by knots to survey the building site for the pyramids. I am skeptical about the practical use of the 3–4–5 triangle, because string can stretch and I doubt that the measurements can be carried out to the required accuracy, but the Egyptians probably knew the triangle’s properties. Certainly the ancient Babylonians did.

  Pythagoras’s theorem—one of the few theorems mentioned at school that bears the name of its (traditional) discoverer—tells us that the squares of the two shorter sides add up to the square of the longer one: 32 + 42 = 52. There are infinitely many such “Pythagorean triangles,” and ancient Greek mathematicians already knew how to find them all. Pierre de Fermat, a seventeenth-century French lawyer whose hobby was mathematics, asked the kind of imaginative question (not very imaginative; you don’t have to go far beyond what is already known to encounter yawning gaps in human knowledge) that creates new mathematics. We know about sums of two squares making squares, but can you do it with cubes? Can two cubes add up to a cube? Or two fourth powers to a fourth power? Fermat could not discover any solutions. He found an elegant proof that it can’t be done with fourth powers. In his copy of an ancient Greek number theory text, he stated that he had a proof that it can’t be done generally—that there are no solutions in whole numbers to the equation xn+ yn= zn, where n is greater than 2—but “this margin is too small to contain it.”

  Leave aside the question of the utility of such mathematics; applications are important too, but right now we’re talking about creativity and imagination. Take too “practically minded” an attitude and you
stifle true creativity, to everyone’s detriment. Fermat’s last theorem, as the problem came to be known, turned out to be very deep and very hard. It is unlikely that Fermat’s proof, if it existed, was correct. If it was, no one else has ever thought of it, not even now, when we know Fermat was right. Generations of mathematicians attacked the problem and came away with nothing. A few chipped the odd corner off it; they proved that it couldn’t be done with fifth powers, say, or seventh powers. Only in 1994, after a hiatus of 350 years, was the theorem proved, by Andrew Wiles; his proof was published the following year. You probably remember a TV documentary about it.

  Wiles’s methods were revolutionary, and much too difficult even for a university course at undergraduate or introductory graduate level. His proof is very clever and very beautiful, incorporating results and ideas from dozens of other experts. A breakthrough of the highest order.

  The TV program was very moving. Many viewers burst into tears.

  The proof of Fermat’s last theorem leaps right over the undergraduate syllabus. It is too advanced for the courses you will take. But you will certainly take more elementary courses in number theory, proving theorems like “every positive integer is a sum of at most four squares.” You may elect to study algebraic number theory, where you will see how the great mathematicians of past eras chipped pieces off Fermat’s last theorem, and understand how the whole of abstract algebra emerged from that process. This is a new world that goes almost totally unnoticed by the great majority of humanity.

  Nearly everyone makes use of number theory every day, if only because it forms the basis of Internet security codes and the data-compression methods employed by cable and satellite television. We don’t need to be able to do number theory to watch TV (otherwise ratings of many shows would be way down), but if nobody knew any number theory, crooks would be helping themselves to our bank accounts, and we’d be stuck with three channels. So the general area of math in which Fermat’s last theorem lives is undoubtedly useful.

  The theorem itself, though, is unlikely to be of much use. Very few practical problems rest on adding two big powers together to get another such power. (Though I am told that at least one problem in physics does depend on this.) Wiles’s new methods, on the other hand, have opened up significant new connections between hitherto separate areas of our subject. Those methods will surely turn out to be important one day, very likely in fundamental physics, which is today’s biggest consumer of deep, abstract mathematical concepts and techniques.

  Questions like Fermat’s last theorem are not important because we need to know the answer. In the end it probably doesn’t matter that the theorem was proved true rather than false. They are important because our efforts to find the answer reveal major gaps in our understanding of mathematics. What counts is not the answer itself but knowing how to get it. It can only go in the back of the book when someone has worked out what it is.

  The further we push out the boundaries of mathematics, the bigger the boundary itself becomes. There is no danger that we will ever run out of new problems to solve.

  5

  Surrounded by Math

  Dear Meg,

  I’m not surprised that you’re “both excited and a little bit intimidated,” as you put it, by your imminent move to university. Let me commend your good intuition on both counts. You’ll find the competition tougher, the pace faster, the work harder, and the content far more interesting. You’ll be thrilled by your teachers (some of them) and the ideas they lead you to discover, and daunted that so many of your classmates seem to get there ahead of you. For the first six months you’ll wonder why the school ever let you in. (After that you’ll wonder how some of the others were let in.)

  You asked me to tell you something inspirational. Nothing technical, just something to hold on to when the going gets tough.

  Very well.

  Like many mathematicians, I get my inspiration from nature. Nature may not look very mathematical; you don’t see sums written on the trees. But math is not about sums, not really. It’s about patterns and why they occur. Nature’s patterns are both beautiful and inexhaustible.

  I’m in Houston, Texas, on a research visit, and I’m surrounded by math.

  Houston is a huge, sprawling city. Flat as a pancake. It used to be a swamp, and when there’s a heavy thunderstorm, it tries to revert to its natural condition. Close by the apartment complex where my wife and I always stay when we visit, there is a concrete-lined canal that diverts a lot of the runoff from the rain. It doesn’t always divert quite enough; a few years ago the nearby freeway was thirty feet under water, and the ground floor of the apartment complex was flooded. But it helps. It’s called Braes Bayou, and there are paths along both sides of it. Avril and I like to go for walks along the bayou; the concrete sides are not exactly pretty, but they’re prettier than the surrounding streets and parking lots, and there’s quite a lot of wildlife: catfish in the river, egrets preying on the fish, lots of birds.

  As I walk along Braes Bayou, surrounded by wildlife, I realize that I am also surrounded by math.

  For instance . . .

  Roads cross the bayou at regular intervals, and the phone lines cross there too, and birds perch on the phone lines. From a distance they look like sheet music, fat little blobs on rows of horizontal lines. There seem to be special places they like to perch, and it’s not at all clear to me why, but one thing stands out. If a lot of birds are perching on a wire, they end up evenly spaced.

  That’s a mathematical pattern, and I think there’s a mathematical explanation. I don’t think the birds “know” they ought to space themselves out evenly. But each bird has its own “personal space,” and if another birds gets too close, it will sidle along the wire to leave a bit more room, unless there’s another bird crowding it from the other side.

  When there are just a few birds, they end up randomly spaced. But when there are a lot, they get pushed close together. As each one sidles along to make itself feel more comfortable, the “population pressure” evens them out. Birds at the edge of denser regions get pushed into less densely populated regions. And since the birds are all of the same species (usually they’re pigeons), they all have much the same idea of what their personal space should be. So they space themselves evenly.

  Not exactly evenly, of course. That would be a Platonic ideal. As such, it helps us to comprehend a more messy reality.

  You could do the math on this problem if you wanted to. Write down some simple rules for how birds move when the neighbors get too close, plonk them down at random, run the rules, and watch the spacing evolve. But there’s an analogy with a common physical system, where that math has already been done, and the analogy tells you what to expect.

  It’s a bird crystal.

  The same process that makes birds space themselves regularly makes the atoms in a solid object line up to form a repetitive lattice. The atoms also have a “personal space”: they repel each other if they’re too close together. In a solid, the atoms are forced to pack fairly tightly, but as they adjust their personal spaces, they arrange themselves in an elegant crystal lattice.

  The bird lattice is one-dimensional, since they’re sitting on a wire. A one-dimensional lattice consists of equally spaced points. When there are just a few birds, arranged at random and not subject to population pressure, it’s not a crystal, it’s a gas.

  This isn’t just a vague analogy. The same mathematical process that creates a regular crystal of salt or calcite also creates my “bird crystal.”

  And that’s not the only math that you can find in Braes Bayou.

  A lot of people walk their dogs along the paths. If you watch a walking dog, you quickly notice how rhythmic its movement is. Not when it stops to sniff at a tree or another dog, mind you; it’s rhythmic only when the dog is just bumbling happily along without a thought in its head. Tail wagging, tongue lolling, feet hitting the ground in a careless doggy dance.

  What do the feet do?

  When th
e dog is walking, there’s a characteristic pattern. Left rear, left front, right rear, right front. The foot-falls are equally spaced in time, like musical notes, four beats to the bar.

  If the dog speeds up, its gait changes to a trot. Now diagonal pairs of legs—left rear and right front, then the other two—hit the ground together, in an alternating pattern of two beats to the bar. If two people walked one behind the other, exactly out of step, and you put them inside a cow costume, the cow would be trotting.

  The dog is math incarnate. The subject of which it is an unwitting example is known as gait analysis; it has important applications in medicine: humans often have problems moving their legs properly, especially in infancy or old age, and an analysis of how they move can reveal the nature of the problem and maybe help cure it. Another application is to robotics: robots with legs can move in terrain that doesn’t suit robots with wheels, such as the inside of a nuclear power station, an army firing range, or the surface of Mars. If we can understand legged locomotion well enough, we can engineer reliable robots to decommission old power stations, locate unexploded shells and mines, and explore distant planets. Right now, we’re still using wheels for Mars rovers because that design is reliable, but the rovers are limited in where they can go. We’re not decommissioning nuclear power stations at all. But the U.S. Army does use legged robots for some tidying-up tasks on firing ranges.

  If we learn to reinvent the leg, all that will change.

  Egrets standing in the shallows with that characteristic alert posture, long beaks poised, muscles tensed, are hunting catfish. Together they form a miniature ecology, a predator–prey system. Ecology’s connection with mathematics goes all the way back to Leonardo of Pisa, also known as Fibonacci, who wrote about a rather simple model of the growth of rabbits in 1202, in his Liber Abaci. To be fair, the book is really about the Hindu-Arabic number system, the forerunner of today’s ten-symbol notation for numbers, and the rabbit model is mainly there as an exercise in arithmetic. Most of the other exercises are currency transactions; it was a very practical book.