by Ian Stewart
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.
8
Fear of Proofs
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 subs
titute 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.