Letters to a Young Mathematician (3 page)

That’s why it feels as if math is “out there.” Finding the answer to an open question feels like discovery, not
invention. Math is a product of human minds but not bendable to human will. Exploring it is like exploring a new tract of country; you may not know what is around the next bend in the river, but you don’t get to choose. You can only wait and find out. But the mathematical countryside does not come into existence until you explore it.

When two members of the Arts Faculty argue, they may find it impossible to reach a resolution. When two mathematicians argue—and they do, often in a highly emotional and aggressive way—suddenly one will stop, and say, “I’m sorry, you’re quite right, now I see my mistake.” And they will go off and have lunch together, the best of friends.

I agree with Hersh, pretty much. If you feel that the humanist description of math is a bit woolly, that this type of “shared social construct” is a rarity, Hersh offers some examples that might change your mind. One is money. The entire world runs on money, but what is it? It is not pieces of paper or disks of metal; those can be printed or minted anew, or handed into a bank and destroyed. It is not numbers in a computer: if the computer blew up, you would still be entitled to your money. Money is a shared social construct. It has value because we all agree it has value.

Again, there are strong constraints. If you tell your bank manager that your account contains more than his computer says it does, he does not respond, “No problem,
it’s just a social construct, here’s an extra ten million dollars. Have a nice day.”

It is tempting to think that even if we consider math to be a shared social construct, it has a kind of logical inevitability, that any intelligent mind would come up with the
same
math. When the
Pioneer
and
Voyager
spacecraft were sent off into space, they carried coded messages from humanity to any alien race that might one day encounter them.
Pioneer
bore a plaque with a diagram of the hydrogen atom, a map of nearby pulsars to show where our sun is located, line drawings of a naked man and woman standing in front of a sketch of the spacecraft, for scale, and a schematic picture of the solar system to show which planet we inhabit. The two
Voyager
craft carried records with sounds, music, and scientific images.

Would an alien recipient be able to decode those messages? Would a picture like o–o, two circles joined by a line,
really
look like the hydrogen atom to them? What if their version of atomic theory relied on quantum wave functions instead of primitive “particle” images, which even our own physicists tell us are wildly inaccurate? Would the aliens understand line drawings, given that humans from tribes that have never encountered such things fail to do so? Would they consider pulsars significant?

In most discussions about such questions, one eventually hears it argued that even if they grasped nothing else, any intelligent alien would be able to comprehend
simple mathematical patterns, and the rest can be built from there. The unstated assumption is that math is somehow
universal
. Aliens would count 1, 2, 3, . . . just as we do. They would surely see the implied pattern in diagrams like * ** *** **** .

I’m not convinced. I’ve been reading
Diamond Dogs
, by Alistair Reynolds, a novella about an alien construct, a bizarre and terrifying tower, through whose rooms you progress by solving puzzles. If you get the answer wrong, you die, horribly. Reynolds’s story is powerful, but there is an underlying assumption that aliens would set mathematical puzzles akin to those that a human would set. Indeed, the alien math is
too
close to human; it includes topology and an area of mathematical physics known as Kaluza–Klein theory. You are as likely to arrive on the fifth planet of Proxima Centauri and find a Wal-Mart. I know that narrative constraints demand that the math should
look
like math to the reader, but even so, it doesn’t work for me.

I think human math is more closely linked to our particular physiology, experiences, and psychological preferences than we imagine. It is parochial, not universal. Geometry’s points and lines may seem the natural basis for a theory of shape, but they are also the features into which our visual system happens to dissect the world. An alien visual system might find light and shade primary, or motion and stasis, or frequency of vibration. An alien brain might find smell, or embarrassment, but
not shape, to be fundamental to its perception of the world. And while discrete numbers like 1, 2, 3, seem universal to us, they trace back to our tendency to assemble similar things, such as sheep, and consider them property: has one of
my
sheep been stolen? Arithmetic seems to have originated through two things: the timing of the seasons and commerce. But what of the blimp creatures of distant Poseidon, a hypothetical gas giant like Jupiter, whose world is a constant flux of turbulent winds, and who have no sense of individual ownership? Before they could count up to three, whatever they were counting would have blown away on the ammonia breeze. They would, however, have a far better understanding than we do of the math of turbulent fluid flow.

I
think
it is still credible that where blimp math and ours made contact, they would be logically consistent with each other. They could be distant regions of the same landscape. But even that might depend on which type of logic you use.

The belief that there is
one
mathematics—ours—is a Platonist belief. It’s possible that “the” ideal forms are “out there,” but also that “out there” might comprise more than one abstract realm, and that ideal forms need not be unique. Hersh’s humanism becomes Poseidonian blimpism: their math would be a social construct shared by
their
society. If they had a society. If they didn’t—if different blimps did not communicate—could they have any conception of mathematics at all? Just as we can’t
imagine a mathematics not founded on the counting numbers, we can’t imagine an “intelligent” species whose members don’t communicate with each other. But the fact that we can’t imagine something is no proof that it doesn’t exist.

But I am drifting off the topic. What is mathematics? In despair, some have proposed the definition “Mathematics is what mathematicians do.” And what are mathematicians? “People who do mathematics.” This argument is almost Platonic in its perfect circularity. But let me ask a similar question. What is a businessman? Someone who does business? Not quite. It is someone who
sees opportunities
for doing business when others might miss them.

A mathematician is someone who sees opportunities for doing mathematics.

I’m pretty sure that’s right, and it pins down an important difference between mathematicians and everyone else. What is mathematics? It is the shared social construct created by people who are aware of certain opportunities, and we call those people mathematicians. The logic is still slightly circular, but mathematicians can always recognize a fellow spirit. Find out what that fellow spirit does; it will be one more aspect of our shared social construct.

Welcome to the club.

Dear Meg,

In your last letter you asked me about the extent to which mathematics at university can go beyond what you have already done at school. No one wants to spend three or four years going over the same ideas, even if they are studied in greater depth. Now, looking ahead, you are also right to worry about the scope that exists for creating new mathematics. If others have already explored such a huge territory, how can you ever find your way to the frontier? Is there even any frontier left?

For once, my task is simple. I can set you at ease on both counts. If anything, you should worry about the exact opposite: that people are creating too much new mathematics, and that the scope for new research is so gigantic that it will be difficult to decide where to start or in which direction to proceed. Math is not a robotic way of replacing thought by rigid ritual. It is the most creative activity on the planet.

These statements will be news to many people, possibly including some of your teachers. It always astonishes me that so many people seem to believe that mathematics is limited to what they were taught at school, so that basically “it’s all been done.” Even more astonishing is the assumption that because “the answers are all in the back of the book,” there is no scope for creativity, and no questions remain unanswered. Why do so many people think that their school textbook contains every possible question?

This failure of imagination would amount to deplorable ignorance, were it not for two factors that together go a long way to explain it.

The first is that many students quickly come to dislike mathematics as they pass through the school system. They find it rigid, boring, repetitive, and, worst of all, difficult. Answers are either right or wrong, and no amount of clever verbal jousting with the teacher can convert a wrong answer into a correct one. Mathematics is a very unforgiving subject. Having developed this negative attitude, the last thing the student wants to hear is that there is more mathematics, going beyond the already daunting contents of the set text. Most people
want
all the answers to be at the back of the book, because otherwise they can’t look them up.

Dame Kathleen Ollerenshaw, one of Britain’s most distinguished mathematicians and educators, who continues to do research at the age of ninety, makes exactly
this point in her autobiography
To Talk of Many Things
.
(Do read it, Meg; it’s inspirational, and very wise.) “When I told a teenage friend that I was doing mathematical research, her reply was, ‘Why do that? We have enough mathematics to cope with already—we don’t want any more.’”

The assumptions behind that statement bear examination, but I content myself with just one. Why did Kathleen’s friend assume that any newly invented mathematics would automatically appear in school texts? Again we encounter the same belief, that the math you are taught at school is the entire universe of mathematics. But no one thinks that about physics, or chemistry, or biology, or even French or economics. We all know that what we are taught at school is just a tiny part of what is currently known.

I sometimes wish schools would go back to using words like “arithmetic” to describe the content of “math” courses. Calling them “mathematics” debases the currency of mathematical thought; it’s a bit like using the term “composing” to describe routine exercises in playing musical scales. However, I lack the power to change the name, and if in fact the name were changed, the main effect would be to
decrease
public recognition of mathematics. For most people, the only time they are aware of their life and mathematics intersecting is at school.

As I wrote in my first letter, this does not imply that mathematics has no relevance to our daily lives.
But the profound influence of our subject on human existence takes place behind the scenes and therefore passes unnoticed.

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: 3
²
+ 4
²
= 5
²
. 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
x
ⁿ
+
y
ⁿ
=
z
ⁿ
, 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.