Letters to a Young Mathematician (5 page)

Dear Meg,

By now, you’ve surely noticed that the quality of teaching in a college setting varies widely. This is because, for the most part, your professors and their teaching assistants are not hired, kept on, or promoted based on their ability to teach. They are there to do research, whereas teaching, while necessary and important for any number of reasons, is decidedly secondary for many of them. Many of your professors will be thrilling lecturers and devoted mentors; others, you’ll find, will be considerably less thrilling and devoted. You’ll have to find a way to succeed even with teachers whose greatest talents are not necessarily on display in the classroom.

I once had a lecturer who, I was convinced, had discovered a way to make time stand still. My classmates disagreed with this thesis but felt that his sleep-inducing powers must surely have military uses.

The vast amounts that have been written about teaching math might give the impression that all of the difficulties encountered by math students are caused by teachers, and it is always the teacher’s responsibility to sort out the student’s problems. This is, of course, one of the things teachers are paid to do, but there is some onus on the student as well. You need to understand how to learn.

Like all teaching, math instruction is rather artificial. The world is complicated and messy, with lots of loose ends, and the teacher’s job is to impose order on the confusion, to convert a chaotic set of episodes into a coherent narrative. So your learning will be divided into specific modules, or courses, and each course will have a carefully specified syllabus and a text. In some settings, such as some American public schools, the syllabus will specify exactly which pages of the text, and which problems, are to be tackled on a given day. In other countries and at more advanced levels, the lecturer has more of a free hand to pick his or her own path through the material, and the lecture notes take the place of a textbook.

Because the lectures progress through set topics, one step at a time, it is easy for students to think that this is how to learn the material. It is not a bad idea to work systematically through the book, but there are other tactics you can use if you get stuck.

Many students believe that if you get stuck, you should stop. Go back, read the offending passage again;
repeat until light dawns—either in your mind or outside the library window.

This is almost always fatal. I always tell my students that the first thing to do is read on. Remember that you encountered a difficulty, don’t try to pretend that all is sweetness and light, but continue. Often the next sentence, or the next paragraph, will resolve your problem.

Here is an example, from my book
The Foundations of
Mathematics
, written with David Tall. On page 16, introducing the topic of real numbers, we remark that “the Greeks discovered that there exist lines whose lengths, in theory, cannot be measured exactly by a
rational
number.”

One might easily grind to a halt here—what does “measured by” mean? It hasn’t been defined yet, and— oh help—it’s not in the index. And how did the Greeks discover this fact anyway? Am I supposed to know it from a previous course? From this course? Did I miss a lecture? The previous pages of the book offer no assistance, however many times you reread them. You could spend hours getting nowhere.

So don’t. Read on. The next few sentences explain how Pythagoras’s theorem leads to a line whose length is the square root of two, and state that there is no rational number
m
/
n
such that (
m
/
n
)
²
= 2. This is then proved, cleverly using the fact that every whole number can be expressed as a product of primes in only one way. The result is summarized as “no rational number can have
square 2, and hence that the hypotenuse of the given triangle does not have rational length.”

By now, everything has probably slotted into place. “Measured by” presumably means “has a length equal to.” The Greek reasoning alluded to in such an offhand manner is no doubt the argument using Pythagoras’s theorem; it helps to know that Pythagoras was Greek. And you should be able to spot that “the square root of two is not rational” is equivalent to “no rational number can have square 2.”

Mystery solved.

If you are
still
stuck after plowing valiantly ahead in search of enlightenment, now is the time to go to your class tutor or the lecturer and ask for assistance. By trying to sort the problem out for yourself, you will have set your mind in action, and thus you are much more likely to understand the teacher’s explanation. It’s much like Poincaré’s “incubation” stage of research. Which, with fair weather and a following wind, leads to illumination.

There is another possibility, but it’s one where help from the teacher is probably essential. Even so, you can try to prepare the ground. Whenever you get stuck on a piece of mathematics, it usually happens because you do not properly understand some
other
piece of mathematics, which is being used without explicit mention on the assumption that you can handle it easily. Remember the upside-down pyramid of mathematical knowledge? You may have forgotten what a rational number is, or what
Pythagoras proved, or how square roots relate to squares. Or you may be wondering why the uniqueness property of prime factorization is relevant. If so, you don’t need help to understand the proof that the square root of two is irrational; you need help rehearsing rational numbers, prime factors, or basic geometry.

It takes a certain insight into your own thought processes as well as a certain discipline to pinpoint exactly what you don’t understand and relate it to your immediate difficulty. Your tutors know about such things and will be on the lookout for them. It is, however, a very useful trick to master for
yourself
, if you can.

To sum up: If you think you are stuck, begin by plowing ahead regardless, in the hope of gaining enlightenment, but remember where you got stuck, in case this doesn’t work. If it doesn’t, return to the sticking point and backtrack until you reach something you are confident you understand. Then try moving ahead again.

This process is very similar to a general method for solving a maze, which computer scientists call “depth-first search.” If possible, move deeper into the maze. If you get stuck, backtrack to the first point where there is an alternative path, and follow that. Never go over the exact same path twice. This algorithm will get you safely through any maze. Its learning analogue does not come with such a strong guarantee, but it’s still a very good tactic.

As a student I took this method to extremes. My usual method for reading a mathematics text was to thumb through it until I spotted something interesting, then work backward until I had tracked down everything I needed to read the interesting bit. I don’t really recommend this to everyone, but it does show that there are alternatives to starting at page 1 and continuing in sequence until you get to page 250.

Let me urge upon you another useful trick. It may sound like a huge amount of extra work, but I assure you it will pay dividends.

Read around your subject
.

Do not read only the assigned text. Books are expensive, but universities have libraries. Find some books on the same topic or similar topics. Read them, but in a fairly casual way. Skip anything that looks too hard or too boring. Concentrate on whatever catches your attention. It’s amazing how often you will read something that turns out to be helpful next week, or next year.

The summer before I went off to Cambridge to study math, I read dozens of books in this easygoing way. One of them, I remember, was about “vectors,” which the author defined as “quantities that have both magnitude and direction.” At the time this made very little sense to me, but I liked the elegant formulas and simple diagrams with lots of arrows, and I skimmed through it more than once. I then forgot it. In the opening lecture on vectors,
suddenly it all clicked into place. I understood exactly what the author had been trying to tell me,
before
the lecturer got that far. All those formulas seemed obvious: I knew why they were true.

I can only assume that my subconscious had been stirred up, just as Poincaré claimed, and during the intervening period, it had created some order out of my desultory wanderings through that book on vectors. It was just waiting for a few simple clues before it could form a coherent picture.

When I say “read around your subject,” I don’t mean just the technical material. Read Eric Temple Bell’s
Men
of Mathematics
, still a cracking read even if some of the stories are invented and women are almost invisible. Sample the great works of the past; James Newman’s
The
World of Mathematics
is a four-volume set of fascinating writings about math from ancient Egypt through to relativity. There has been a spate of popular math books in recent years, on the Riemann hypothesis, the four color theorem, π, infinity, mathematical crackpots, how the human brain thinks mathematical thoughts, fuzzy logic, Fibonacci numbers. There are even books on the applications of mathematics, such as D’Arcy Thompson’s classic
On Growth and Form
, about mathematical patterns in living creatures. It may be outmoded in biological terms—it was written long before the structure of DNA was found—but its overall message remains as valid as ever.

Such books will broaden your appreciation of what math is, what it can be used for, and how its sits in human culture. There will likely be no questions about any of these topics on your exams. But awareness of these issues will make you a better mathematician, able to grasp the essentials of any new topic more confidently.

There are also some specific techniques that will improve your learning skills. The great American mathematical educator George Pólya put a lot of them into his classic
How to Solve It
. He took the view that the only way to understand math properly is the hands-on method: tackling problems and solving them. He was right. But you can’t learn this way if you get stuck on every problem you try. So your teachers will set you a carefully chosen sequence of problems, starting with routine calculations and leading up to more challenging questions.

Pólya offers many tricks for boosting your problem-solving abilities. He describes them far better than I can, but here is a sample. If the problem seems baffling, try to recast it in a simpler form. Look for a good example and try your ideas out on the example; later, you can generalize to the original setting. For instance, if the problem is about prime numbers, try it on 7, 13, or 47. Try working backward from the conclusion: what steps must we take to get there? Try several examples and look for common patterns; if you find one, try to prove that it must
always
happen.

As you remarked in your letter, Meg, one of the main differences between high school and college is that in college the students are treated much more like adults.
This means that to a much greater extent, it’s sink or swim: pass, fail, or find another major. There is plenty of help available for the asking, but that too takes more initiative than it did in high school. No one is likely to take you by the hand and say, “It looks like you’re having trouble.”

On the other hand, the rewards for self-sufficiency are much greater. Your high school was mainly grateful if you were not a problem requiring some sort of extra attention, and unless you were extremely lucky, the most it could offer an exceptional student (beyond the grades certifying him or her to move on) was an extracurricular club and perhaps an award or two. In a university you will encounter real scholars who are on the lookout for young people capable of doing real mathematics, and they are just waiting for you to stand out, if you can.

Dear Meg,

You’re quite correct: One of the biggest differences between school math and university math is proof. At school we learn
how
to solve equations or find the area of a triangle; at university we learn
why
those methods work, and prove that they do. Mathematicians are obsessed with the idea of “proof.” And, yes, it does put a lot of people off. I call them proofophobes. Mathematicians, in contrast, are proofophiles: no matter how much circumstantial evidence there may be in favor of some mathematical statement, the true mathematician is not satisfied until the statement is
proved
. In full logical rigor, with everything made precise and unambiguous.

There’s a good reason for this. A proof provides a cast-iron guarantee that some idea is correct. No amount of experimental evidence can substitute for that.

Let’s take a look at a proof and see how it differs from other forms of evidence. I don’t want to use anything
that involves technical math, because that will obscure the underlying ideas. My favorite nontechnical proof is the SHIP–DOCK theorem, which is about those word games in which you have to change one word into another by a sequence of moves: CAT, COT, COG, DOG.
At each step, you are allowed to change (but not move) exactly one letter, and the result must be a valid word (as determined by, say, Webster’s).

Solving this word puzzle isn’t particularly hard: for instance,

SHIP

SHOP

SHOT

SLOT

SOOT

LOOT

LOOK

LOCK

DOCK

There are plenty of other solutions. But I’m not after a solution as such, or even several: I’m interested in something that applies to
every
solution. Namely, at some stage, there must be a word that contains two vowels. Like SOOT (and LOOT and LOOK) in this particular answer. Here I mean
exactly
two vowels, no more and no less.

To avoid objections, let me make it clear what “vowel” means here. One thorny problem is the letter Y. In YARD the letter Y is a consonant, but in WILY it is a vowel. Similarly, the W in CWMS acts as a vowel: “cwm” is Welsh, and refers to a geological formation for which there seems to be no English word, although “corrie” (Scottish) and “cirque” (French) are alternatives. We need to be very careful about letters that sometimes act as vowels but on other occasions are consonants. In fact, the safest way to avoid the kinds of words that all Scrabble players love is to throw away Webster’s and redefine “vowel” and “word” in a more limited sense. For the purposes of this discussion, a “vowel” will mean one of the letters A, E, I, O, U, and a “word” will be required to contain
at least one
of those five letters. Alternatively, we can require Y and W
always
to count as vowels, even when they are being used as consonants. What we can’t do, in this context, is allow letters to be sometimes vowels, sometimes consonants.
I’ll come back to that later.

It’s not a question of what the correct convention is in linguistics; I’m setting up a temporary convention for a specific mathematical purpose. Sometimes in math the best way to make progress is to introduce simplifications, and that’s what I’m doing here. The simplifications are not assertions about the outside world: they are ways to restrict the domain of discourse, to keep it manageable. A more complicated analysis could probably
handle the exceptional letters like Y too, but that would complicate the story too much for my present purpose.

With that caveat, am I right? Is it true that every solution of the SHIP–DOCK puzzle includes a word (in the new, restricted sense) with exactly
two
vowels (in the new, restricted sense)?

One way to investigate this is to look for other solutions, such as

SHIP

CHIP

CHOP

COOP

COOT

ROOT

ROOK

ROCK

DOCK

Here we find two vowels in COOP, COOT, ROOT, and ROOK. But even if a lot of individual solutions have two vowels somewhere, that doesn’t prove that they all have to. A proof is a logical argument that leaves no room for doubt.

After a certain amount of experiment and thought, the “theorem” that I am proposing here starts to seem obvious. The more you think about how vowels can change their positions, the more obvious it becomes that
somewhere along the way there must be exactly two vowels. But a feeling that something is “obvious” does not constitute a proof, and there’s some subtlety in the theorem because some four-letter words contain
three
vowels, for instance, OOZE.

Yes, but . . . on the way to a three-vowel word, we surely have to pass through a two-vowel word? I agree, but that’s not a proof either, though it may help us find one.
Why
must we pass through a two-vowel word?

A good way to find a proof here is to pay more attention to details. Keep your eye on where the vowels go. Initially, there is one vowel in the third position. At the end, we want one vowel in the second position. But—a simple but crucial insight—a vowel cannot change position in one step, because that would involve changing two letters. Let’s pin that particular thought down, logically, so that we can rely on it. Here’s one way to prove it. At some stage, a consonant in the second position has to change to a vowel, leaving all the other letters unchanged; at some other stage, the vowel in the third position has to change to a consonant. Maybe other vowels and consonants wander in and out, too, but whatever else happens, we can now be certain that a vowel cannot change position in one step.

How does the number of vowels in the word change? Well, it can stay the same; it can increase by 1 (when a consonant changes to a vowel), or it can decrease by 1 (when a vowel changes to a consonant). There are no
other possibilities. The number of vowels starts at 1 with SHIP and ends at 1 with DOCK, but it can’t be 1 at every step, because then the unique vowel would have to stay in the same place, position three, and we know that it has to end up in position two.

Idea: think about the
earliest
step at which the number of vowels changes. The number of vowels must have been 1 at all times before that step. Therefore it changes from 1 to something else. The only possibilities are 0 and 2, because the number either increases or decreases by 1.

Could it be 0? No, because that means the word would have no vowels at all, and by definition no “word”
in our restricted sense can be like that. Therefore the word contains two vowels; end of proof. We’ve barely started analyzing the problem, and a proof has popped out of its own accord. This often happens when you follow the line of least resistance. Mind you, things really start to get interesting when the line of least resistance leads precisely nowhere.

It’s always a good idea to check a proof on examples, because that way you often spot logical mistakes. Let’s count the vowels, then:

SHIP      1 vowel

SHOP      1 vowel

SHOT      1 vowel

SLOT      1 vowel

SOOT      2 vowels

LOOT      2 vowels

LOOK      2 vowels

LOCK      1 vowel

DOCK      1 vowel

The proof says to find the first word where this number is not 1, and that’s the word SOOT, which has two vowels. So the proof checks out in this example. Moreover, the number of vowels does indeed change by at most 1 at each step. Those facts alone do not mean that the proof is correct, however; to be sure of its correctness you have to check the chain of logic and make sure that each link is unbroken. I’ll leave you to convince yourself that this is the case.

Notice the difference here between intuition and proof. Intuition tells us that the single vowel in SHIP can’t hop around to a different position unless a new vowel appears somewhere. But this intuition doesn’t constitute a proof. The proof emerges only when we try to pin the intuition down: yes, the number of vowels changes, but
when
? What must the change look like?

Not only do we become certain that two vowels must appear, we understand why this is inevitable. And we get additional information free of charge.

If a letter can sometimes be a vowel and sometimes a consonant, then this particular proof breaks down. For instance, with three-letter words there is a sequence:

SPA

SPY

SAY

SAD

If we count Y as a vowel in SPY but as a consonant in SAY (which is defensible but also debatable), then each word has a single vowel, but the vowel position moves. I don’t think this effect can cause trouble when changing SHIP into DOCK, but that depends on a much closer analysis of the actual words in the dictionary. The real world can be messy.

Word puzzles are fun (try changing ORDER into CHAOS). This particular puzzle also teaches us something about proofs and logic. And about the idealizations that are often involved when we use math to model the real world.

There are two big issues about proof. The one that mathematicians worry about is, what
is
a proof? The rest of the world has a different concern: why do we need them?

Let me take those questions in reverse order: one now, and the other in a later letter.

I’ve begun to observe that when people ask why something is necessary, it is usually because they feel uncomfortable doing it and are hoping to be let off the hook. A student who knows how to construct proofs never asks what they’re for. In fact, a student who knows
how to do long multiplication in his head while doing a handstand also never asks why that’s worth doing.
People who enjoy performing an activity hardly ever feel the need to question its worth; the enjoyment alone is enough. So the student who asks why we need proofs is probably having trouble understanding them, or constructing his own. He is hoping you will answer, “There’s no need to worry about proofs. They’re totally useless. In fact, I’ve taken them off the syllabus and they won’t come up in the test.”

Ah, in your dreams.

It’s still a good question, and if I leave it at what I’ve just said, I’m ducking the issue just as blatantly as any proofophobic student.

Mathematicians need proofs to keep them honest. All technical areas of human activity need reality checks.
It is not enough to believe that something works, that it is a good way to proceed, or even that it is true. We need to know
why
it’s true. Otherwise, we don’t know anything at all.

Engineers test their ideas by building them and seeing whether they hold up or fall apart. Increasingly they do this in simulation rather than by building a bridge and hoping it won’t fall down, and in so doing they refer their problems back to physics and mathematics, which are the sources of the rules employed in their calculations and the algorithms that implement those rules. Even so, unexpected problems can turn up. The Millennium
Bridge, a footbridge across the Thames in London, looked fine in the computer models. When it opened and people started to use it, it suddenly started to sway alarmingly from side to side. It was still safe, it wasn’t going to fall down, but crossing it wasn’t an enjoyable experience. At that point it became clear that the simulations had modeled people as smoothly moving masses; they had ignored the vibrations induced by feet hitting the deck.

The military learned long ago that when soldiers cross a bridge, they should fall out of step. The synchronized impact of several hundred right feet can set up vibrations and do serious damage. I suspect this fact was known to the Romans. No one expected a similar kind of synchronization to arise when individuals walked across the bridge at random. But people on a bridge respond to the movement of the bridge, and they do so in a similar way and at roughly the same time. So when the bridge moved slightly—perhaps in response to a gust of wind— the people on it started to move in synchrony. The more closely the footsteps of the people became synchronized, the more the bridge moved, which in turn increased the synchronization of the footsteps. Soon the whole bridge was swaying from side to side.

Physicists use mathematics to study what they amusingly call the real world. It is real, in a sense, but much of physics addresses rather artificial aspects of reality, such as a lone electron or a solar system with only one planet.
Physicists are often scathing about proofs, partly out of fear, but also because experiment gives them a very effective way to check their assumptions and calculations.
If an intuitively plausible idea leads to results that agree with experiment, there’s not much point in holding the entire subject up for ten, fifty, or three hundred years until someone finds a rigorous proof. I agree entirely. For example, there are calculations in quantum field theory that have no rigorous logical justification, but agree with experiment to an accuracy of nine decimal places. It would be silly to suggest that this agreement is an accident, and that no physical principle is involved.

The mathematicians’ take on this, though, goes further. Given such impressive agreement, it is equally silly not to try to find out the deep logic that justifies the calculation. Such understanding should advance physics directly, but if not, it will surely advance mathematics. And mathematics often impinges indirectly on physics, probably a different branch of physics, but if so, it’s a bonus.