When we hear the word “mathematics,” the first thing that springs to mind is numbers. Numbers are the heart of mathematics, an all-pervading influence, the raw materials out of which a great deal of mathematics is forged. But numbers on their own form only a tiny part of mathematics. I said earlier that we live in an intensely mathematical world, but that whenever possible the mathematics is sensibly tucked under the rug to make our world “user-friendly.” However, some mathematical ideas are so basic to our world that they cannot stay hidden, and numbers are an especially prominent example. Without the ability to count eggs and subtract change, for instance, we could not even buy food. And so we teach arithmetic. To everybody. Like reading and writing, its absence is a major handicap. And that creates the overwhelming impression that mathematics is mostly a matter of numbers—which isn’t really true. The numerical tricks we learn in arithmetic are only the tip of an iceberg. We can run our everyday lives without much more, but our culture cannot run our society by using such limited ingredients. Numbers are just one type of object that mathematicians think about. In this chapter, I will try to show you some of the others and explain why they, too, are important.
Inevitably my starting point has to be numbers. A large part of the early prehistory of mathematics can be summed up as the discovery, by various civilizations, of a wider and wider range of things that deserved to be called numbers. The simplest are the numbers we use for counting. In fact, counting began long before there were symbols like 1, 2, 3, because it is possible to count without using numbers at all—say, by counting on your fingers. You can work out that “I have two hands and a thumb of camels” by folding down fingers as your eye glances over the camels. You don’t actually have to have the concept of the number “eleven” to keep track of whether anybody is stealing your camels. You just have to notice that next time you seem to have only two hands of camels—so a thumb of camels is missing.
You can also record the count as scratches on pieces of wood or bone. Or you can make tokens to use as counters—clay disks with pictures of sheep on them for counting sheep, or disks with pictures of camels on them for counting camels. As the animals parade past you, you drop tokens into a bag—one token for each animal. The use of symbols for numbers probably developed about five thousand years ago, when such counters were wrapped in a clay envelope. It was a nuisance to break open the clay covering every time the accountants wanted to check the contents, and to make another one when they had finished. So people put special marks on the outside of the envelope summarizing what was inside. Then they realized that they didn’t actually need any counters inside at all: they could just make the same marks on clay tablets.
It’s amazing how long it can take to see the obvious. But of course it’s only obvious now.
The next invention beyond counting numbers was fractions—the kind of number we now symbolize as 2/3 (two thirds) or 22/7 (twenty-two sevenths—or, equivalently, three and one-seventh). You can’t count with fractions—although two-thirds of a camel might be edible, it’s not countable—but you can do much more interesting things instead. In particular, if three brothers inherit two camels between them, you can think of each as owning two-thirds of a camel—a convenient legal fiction, one with which we are so comfortable that we forget how curious it is if taken literally.
Much later, between 400 and 1200 AD, the concept of zero was invented and accepted as denoting a number. If you think that the late acceptance of zero as a number is strange, bear in mind that for a long time “one” was not considered a number because it was thought that a number of things ought to be several of them. Many history books say that the key idea here was the invention of a symbol for “nothing.” That may have been the key to making arithmetic practical; but for mathematics the important idea was the concept of a new kind of number, one that represented the concrete idea “nothing.” Mathematics uses symbols, but it no more is those symbols than music is musical notation or language is strings of letters from an alphabet. Carl Friedrich Gauss, thought by many to be the greatest mathematician ever to have lived, once said (in Latin) that what matters in mathematics is “not notations, but notions.” The pun “non notationes, sed notiones” worked in Latin, too.
The next extension of the number concept was the invention of negative numbers. Again, it makes little sense to think of minus two camels in a literal sense; but if you owe somebody two camels, the number you own is effectively diminished by two. So a negative number can be thought of as representing a debt. There are many different ways to interpret these more esoteric kinds of number; for instance, a negative temperature (in degrees Celsius) is one that is colder than freezing, and an object with negative velocity is one that is moving backward. So the same abstract mathematical object may represent more than one aspect of nature.
Fractions are all you need for most commercial transactions, but they’re not enough for mathematics. For example, as the ancient Greeks discovered to their chagrin, the square root of two is not exactly representable as a fraction. That is, if you multiply any fraction by itself, you won’t get two exactly. You can get very close—for example, the square of 17/12 is 289/144, and if only it were 288/144 you would get two. But it isn’t, and you don’t—and whatever fraction you try, you never will. The square root of two, usually denoted √2, is therefore said to be “irrational.” The simplest way to enlarge the number system to include the irrationals is to use the so-called real numbers—a breathtakingly inappropriate name, inasmuch as they are represented by decimals that go on forever, like 3.14159. . . , where the dots indicate an infinite number of digits. How can things be real if you can’t even write them down fully? But the name stuck, probably because real numbers formalize many of our natural visual intuitions about lengths and distances.
The real numbers are one of the most audacious idealizations made by the human mind, but they were used happily for centuries before anybody worried about the logic behind them. Paradoxically, people worried a great deal about the next enlargement of the number system, even though it was entirely harmless. That was the introduction of square roots for negative numbers, and it led to the “imaginary” and “complex” numbers. A professional mathematican should never leave home without them, but fortunately nothing in this book will require a knowledge of complex numbers, so I’m going to tuck them under the mathematical carpet and hope you don’t notice. However, I should point out that it is easy to interpret an infinite decimal as a sequence of ever-finer approximations to some measurement—say, of a length or a weight—whereas a comfortable interpretation of the square root of minus one is more elusive.
In current terminology, the whole numbers 0, 1, 2, 3, . . . are known as the natural numbers. If negative whole numbers are included, we have the integers. Positive and negative fractions are called rational numbers. Real numbers are more general; complex numbers more general still. So here we have five number systems, each more inclusive than the previous: natural numbers, integers, rationals, real numbers, and complex numbers. In this book, the important number systems will be the integers and the reals. We’ll need to talk about rational numbers every so often; and as I’ve just said, we can ignore the complex numbers altogether. But I hope you understand by now that the word “number” does not have any immutable god-given meaning. More than once the scope of that word was extended, a process that in principle might occur again at any time.
However, mathematics is not just about numbers. We’ve already had a passing encounter with a different kind of object of mathematical thought, an operation; examples are addition, subtraction, multiplication, and division. In general, an operation is something you apply to two (sometimes more) mathematical objects to get a third object. I also alluded to a third type of mathematical object when I mentioned square roots. If you start with a number and form its square root, you get another number. The term for such an “object” is function. You can think of a function as a mathematical rule that starts with a mathematical object—usually a number—and associates to it another object in a specific manner. Functions are often defined using algebraic formulas, which are just shorthand ways to explain what the rule is, but they can be defined by any convenient method. Another term with the same meaning as “function” is transformation: the rule transforms the first object into the second. This term tends to be used when the rules are geometric, and in chapter 6 we will use transformations to capture the mathematical essence of symmetry.
Operations and functions are very similar concepts. Indeed, on a suitable level of generality there is not much to distinguish them. Both of them are processes rather than things. And now is a good moment to open up Pandora’s box and explain one of the most powerful general weapons in the mathematician’s armory, which we might call the “thingification of processes.” (There is a dictionary term, reification, but it sounds pretentious.) Mathematical “things” have no existence in the real world: they are abstractions. But mathematical processes are also abstractions, so processes are no less “things” than the “things” to which they are applied. The thingification of processes is commonplace. In fact, I can make out a very good case that the number “two” is not actually a thing but a process—the process you carry out when you associate two camels or two sheep with the symbols “1, 2” chanted in turn. A number is a process that has long ago been thingified so thoroughly that everybody thinks of it as a thing. It is just as feasible—though less familiar to most of us—to think of an operation or a function as a thing. For example, we might talk of “square root” as if it were a thing—and I mean here not the square root of any particular number, but the function itself. In this image, the square-root function is a kind of sausage machine: you stuff a number in at one end and its square root pops out at the other.
In chapter 6, we will treat motions of the plane or space as if they are things. I’m warning you now because you may find it disturbing when it happens. However, mathematicians aren’t the only people who play the thingification game. The legal profession talks of “theft” as if it were a thing; it even knows what kind of thing it is—a crime. In phrases such as “two major evils in Western society are drugs and theft” we find one genuine thing and one thingified thing, both treated as if they were on exactly the same level. For theft is a process, one whereby my property is transferred without my agreement to somebody else, but drugs have a real physical existence.
Computer scientists have a useful term for things that can be built up from numbers by thingifying processes: they call them data structures. Common examples in computer science are lists (sets of numbers written in sequence) and arrays (tables of numbers with several rows and columns). I’ve already said that a picture on a computer screen can be represented as a list of pairs of numbers; that’s a more complicated but entirely sensible data structure. You can imagine much more complicated possibilities—arrays that are tables of lists, not tables of numbers; lists of arrays; arrays of arrays; lists of lists of arrays of lists. . . . Mathematics builds its basic objects of thought in a similar manner. Back in the days when the logical foundations of mathematics were still being sorted out, Bertrand Russell and Alfred North Whitehead wrote an enormous three-volume work, Principia Mathematica, which began with the simplest possible logical ingredient—the idea of a set, a collection of things. They then showed how to build up the rest of mathematics. Their main objective was to analyze the logical structure of mathematics, but a major part of their effort went into devising appropriate data structures for the important objects of mathematical thought.
The image of mathematics raised by this description of its basic objects is something like a tree, rooted in numbers and branching into ever more esoteric data structures as you proceed from trunk to bough, bough to limb, limb to twig. . . . But this image lacks an essential ingredient. It fails to describe how mathematical concepts interact. Mathematics is not just a collection of isolated facts: it is more like a landscape; it has an inherent geography that its users and creators employ to navigate through what would otherwise be an impenetrable jungle. For instance, there is a metaphorical feeling of distance. Near any particular mathematical fact we find other, related facts. For example, the fact that the circumference of a circle is π (pi) times its diameter is very close to the fact that the circumference of a circle is 2π times its radius. The connection between these two facts is immediate: the diameter is twice the radius. In contrast, unrelated ideas are more distant from each other; for example, the fact that there are exactly six different ways to arrange three objects in order is a long way away from facts about circles. There is also a metaphorical feeling of prominence. Soaring peaks pierce the sky—important ideas that can be used widely and seen from far away, such as Pythagoras’s theorem about right triangles, or the basic techniques of calculus. At every turn, new vistas arise—an unexpected river that must be crossed using stepping stones, a vast, tranquil lake, an impassable crevasse. The user of mathematics walks only the well-trod parts of this mathematical territory. The creator of mathematics explores its unknown mysteries, maps them, and builds roads through them to make them more easily accessible to everybody else.
The ingredient that knits this landscape together is proof. Proof determines the route from one fact to another. To professional mathematicians, no statement is considered valid unless it is proved beyond any possibility of logical error. But there are limits to what can be proved, and how it can be proved. A great deal of work in philosophy and the foundations of mathematics has established that you can’t prove everything, because you have to start somewhere; and even when you’ve decided where to start, some statements may be neither provable nor disprovable. I don’t want to explore those issues here; instead, I want to take a pragmatic look at what proofs are and why they are needed.
Textbooks of mathematical logic say that a proof is a sequence of statements, each of which either follows from previous statements in the sequence or from agreed axioms—unproved but explicitly stated assumptions that in effect define the area of mathematics being studied. This is about as informative as describing a novel as a sequence of sentences, each of which either sets up an agreed context or follows credibly from previous sentences. Both definitions miss the essential point: that both a proof and a novel must tell an interesting story. They do capture a secondary point, that the story must be convincing, and they also describe the overall format to be used, but a good story line is the most important feature of all.
Most of us are irritated by a movie riddled with holes, however polished its technical production may be. I saw one recently in which an airport is taken over by guerrillas who shut down the electronic equipment used by the control tower and substitute their own. The airport authorities and the hero then spend half an hour or more of movie time—several hours of story time—agonizing about their inability to communicate with approaching aircraft, which are stacking up in the sky overhead and running out of fuel. It occurs to no one that there is a second, fully functioning airport no more than thirty miles away, nor do they think to telephone the nearest Air Force base. The story was brilliantly and expensively filmed—and silly.
That didn’t stop a lot of people from enjoying it: their critical standards must have been lower than mine. But we all have limits to what we are prepared to accept as credible. If in an otherwise realistic film a child saved the day by picking up a house and carrying it away, most of us would lose interest. Similarly, a mathematical proof is a story about mathematics that works. It does not have to dot every i and cross every t; readers are expected to fill in routine steps for themselves—just as movie characters may suddenly appear in new surroundings without it being necessary to show how they got there. But the story must not have gaps, and it certainly must not have an unbelievable plot line. The rules are stringent: in mathematics, a single flaw is fatal. Moreover, a subtle flaw can be just as fatal as an obvious one.
Let’s take a look at an example. I have chosen a simple one, to avoid technical background; in consequence, the proof tells a simple and not very significant story. I stole it from a colleague, who calls it the SHIP/DOCK Theorem. You probably know the type of puzzle in which you are given one word (SHIP) and asked to turn it into another word (DOCK) by changing one letter at a time and getting a valid word at every stage. You might like to try to solve this one before reading on: if you do, you will probably understand the theorem, and its proof, more easily.
Here’s one solution:
SHIP
SLIP
SLOP
SLOT
SOOT
LOOT
LOOK
LOCK
DOCK
There are plenty of alternatives, and some involve fewer words. But if you play around with this problem, you will eventually notice that all solutions have one thing in common: at least one of the intermediate words must contain two vowels.
O.K., so prove it.
I’m not willing to accept experimental evidence. I don’t care if you have a hundred solutions and every single one of them includes a word with two vowels. You won’t be happy with such evidence, either, because you will have a sneaky feeling that you may just have missed some really clever sequence that doesn’t include such a word. On the other hand, you will probably also have a distinct feeling that somehow “it’s obvious.” I agree; but why is it obvious?
You have now entered a phase of existence in which most mathematicians spend most of their time: frustration. You know what you want to prove, you believe it, but you don’t see a convincing story line for a proof. What this means is that you are lacking some key idea that will blow the whole problem wide open. In a moment I’ll give you a hint. Think about it for a few minutes, and you will probably experience a much more satisfying phase of the mathematician’s existence: illumination.
Here’s the hint. Every valid word in English must contain a vowel.
It’s a very simple hint. First, convince yourself that it’s true. (A dictionary search is acceptable, provided it’s a big dictionary.) Then consider its implications. . . .
O.K., either you got it or you’ve given up. Whichever of these you did, all professional mathematicians have done the same on a lot of their problems. Here’s the trick. You have to concentrate on what happens to the vowels. Vowels are the peaks in the SHIP/DOCK landscape, the landmarks between which the paths of proof wind.
In the initial word SHIP there is only one vowel, in the third position. In the final word DOCK there is also only one vowel, but in the second position. How does the vowel change position? There are three possibilities. It may hop from one location to the other; it may disappear altogether and reappear later on; or an extra vowel or vowels may be created and subsequently eliminated.
The third possibility leads pretty directly to the theorem. Since only one letter at a time changes, at some stage the word must change from having one vowel to having two. It can’t leap from having one vowel to having three, for example. But what about the other possibilities? The hint that I mentioned earlier tells us that the single vowel in SHIP cannot disappear altogether. That leaves only the first possibility: that there is always one vowel, but it hops from position 3 to position 2. However, that can’t be done by changing only one letter! You have to move, in one step, from a vowel at position 3 and a consonant at position 2 to a consonant at position 3 and a vowel at position 2. That implies that two letters must change, which is illegal. Q.E.D., as Euclid used to say.
A mathematician would write the proof out in a much more formal style, something like the textbook model, but the important thing is to tell a convincing story. Like any good story, it has a beginning and an end, and a story line that gets you from one to the other without any logical holes appearing. Even though this is a very simple example, and it isn’t standard mathematics at all, it illustrates the essentials: in particular, the dramatic difference between an argument that is genuinely convincing and a hand-waving argument that sounds plausible but doesn’t really gel. I hope it also put you through some of the emotional experiences of the creative mathematician: frustration at the intractability of what ought to be an easy question, elation when light dawned, suspicion as you checked whether there were any holes in the argument, aesthetic satisfaction when you decided the idea really was O.K. and realized how neatly it cut through all the apparent complications. Creative mathematics is just like this—but with more serious subject matter.
Proofs must be convincing to be accepted by mathematicians. There have been many cases where extensive numerical evidence suggested a completely wrong answer. One notorious example concerns prime numbers—numbers that have no divisors except themselves and 1. The sequence of primes begins 2, 3, 5, 7, 11, 13, 17, 19 and goes on forever. Apart from 2, all primes are odd; and the odd primes fall into two classes: those that are one less than a multiple of four (such as 3, 7, 11, 19) and those that are one more than a multiple of four (such as 5, 13, 17). If you run along the sequence of primes and count how many of them fall into each class, you will observe that there always seem to be more primes in the “one less” class than in the “one more” class. For example, in the list of the seven pertinent primes above, there are four primes in the first class but only three in the second. This pattern persists for numbers up to at least a trillion, and it seems entirely reasonable to conjecture that it is always true.
However, it isn’t.
By indirect methods, number theorists have shown that when the primes get sufficiently big, the pattern changes and the “one more than a multiple of four” class goes into the lead. The first proof of this fact worked only when the numbers got bigger than 10’10’10’10’46, where to avoid giving the printer kittens I’ve used the ’ sign to indicate forming a power. This number is utterly gigantic. Written out in full, it would go 10000. . . 000, with a very large number of 0s. If all the matter in the universe were turned into paper, and a zero could be inscribed on every electron, there wouldn’t be enough of them to hold even a tiny fraction of the necessary zeros.
No amount of experimental evidence can account for the possibility of exceptions so rare that you need numbers that big to locate them. Unfortunately, even rare exceptions matter in mathematics. In ordinary life, we seldom worry about things that might occur on one occasion out of a trillion. Do you worry about being hit by a meteorite? The odds are about one in a trillion. But mathematics piles logical deductions on top of each other, and if any step is wrong the whole edifice may tumble. If you have stated as a fact that all numbers behave in some manner, and there is just one that does not, then you are wrong, and everything you have built on the basis of that incorrect fact is thrown into doubt.
Even the very best mathematicians have on occasion claimed to have proved something that later turned out not to be so—their proof had a subtle gap, or there was a simple error in a calculation, or they inadvertently assumed something that was not as rock-solid as they had imagined. So, over the centuries, mathematicians have learned to be extremely critical of proofs. Proofs knit the fabric of mathematics together, and if a single thread is weak, the entire fabric may unravel.