The Art of the Infinite: The Pleasures of Mathematics

For Michael, Jane, and Felix

Contents

Frontispiece

An Invitation

Chapter One

Time and the Mind

Chapter Two

How Do We Hold These Truths?

Chapter Three

Designs on a Locked Chest

Interlude

The Infinite and the Indefinite

Chapter Four

Skipping Stones

Chapter Five

Euclid Alone

Interlude

Longing and the Infinite

Chapter Six

The Eagle of Algebra

Chapter Seven

Into the Highlands

Interlude

The Infinite and the Unknown

Chapter Eight

Back of Beyond

Interlude

The Infinite There—But the Finite Here

Chapter Nine

The Abyss

Acknowledgments

Appendix

Annex

Bibliography

Footnotes

Notes

A Note on the Authors

Frontispiece

The Tower of Mathematics is the Tower of Babel inverted: its voices grow more coherent as it rises. The image of it is based on Pieter Brueghel’s Little Tower of Babel (1554).

An Invitation

Less than All cannot satisfy Man.

—William Blake

We commonly think of ourselves as little and lost in the infinite stretches of time and space, so that it comes as a shock when the French poet Baudelaire speaks of cradling our infinite on the finite seas. Really? Is it ourself, our mind or spirit, that is infinity’s proper home? Or might the infinite be neither out there nor in here but only in language, a pretty conceit of poetry?

We are the language makers, and what we express always refers to something—though not, perhaps, to what we first thought it did. Talk of the infinite naturally belongs to that old, young, ageless conversation about number and shape which is mathematics: a conversation most of us overhear rather than partake in, put off by its haughty abstraction. Mathematics promises certainty—but at the cost, it seems, of passion. Its initiates speak of playfulness and freedom, but all we come up against in school are boredom and fear, wedged between iron rules memorized without reason.

Why hasn’t mathematics the gentle touches a novelist uses to lure the reader into his imagination? Why do we no longer find problems like this, concocted by Mahāvīrā in ninth-century India:

One night, in a month of the spring season, a certain young lady was lovingly happy with her husband in a big mansion, white as the moon, set in a pleasure garden with trees bent down with flowers and fruits, and resonant with the sweet sounds of parrots, cuckoos and bees which were all intoxicated with the honey of the flowers. Then, on a love-quarrel arising between husband and wife, her pearl necklace was broken. One third of the pearls were collected by the maidservant, one sixth fell on the bed—then half of what remained and half of what remained thereafter and again one half of what remained thereafter and so on, six times in all, fell scattered everywhere. 1,161 pearls were still left on the string; how many pearls had there been in the necklace?

Talking mostly to each other or themselves, mathematicians have developed a code that is hard to crack. Its symbols store worlds of meaning for them, its sleek equations leap continents and centuries. But these sparks can jump to everyone, because each of us has a mind built to grasp the structure of things. Anyone who can read and speak (which are awesomely abstract undertakings) can come to delight in the works of mathematical art, which are among our kind’s greatest glories.

The way in is to begin at the beginning and move conversationally along. Eccentric, lovable, laughable, base, and noble mathematicians will keep us company. Each equation in a book, Stephen Hawking once remarked, loses half the potential readership. Our aim here, however, is to let equations—those balances struck between two ways of looking—grow organically from what they look at.

Many small things estrange math from its proper audience. One is the remoteness of its machine-made diagrams. These reinforce the mistaken belief that it is all very far away, on a planet visited only by graduates of the School for Space Cadets. Diagrams printed out from computers communicate a second and subtler falsehood: they lead the reader to think he is seeing the things themselves rather than pixellated approximations to them.

We have tried to solve this problem of the too far and the too near by putting our drawings in the human middle distance, where diagrams are drawn by hand. These reach out to the ideal world we can’t see from the real world we do, as our imagination reaches in turn from the shaky circle perceived to the conception of circle itself.

Fuller explanations too will live in the middle distance: some in the appendix, others—the more distant excursions—(along with notes to the text) in an online Annex, at www.themathcircle.org.

Gradually, then, the music of mathematics will grow more distinct. We will hear in it the endless tug between freedom and necessity as playful inventions turn into the only way things can be, and timeless laws are drafted—in a place, at a time, by a fallible fellow human. Just as in listening to music, our sense of self will widen out toward a more than personal vista, vivid and profound.

Whether we focus on the numbers we count with and their offspring or the shapes that evolve from triangles, ever richer structures will slide into view like beads on the wire of the infinite. And it is this wire, running throughout, that draws us on, until we stand at the edge of the universe and stretch out a hand.

Chapter One

Time and the Mind

Things occupy space—but how many of them there are (or could be) belongs to time, as we tick them off to a walking rhythm that projects ongoing numbering into the future. Yet if you take off the face of a clock you won’t find time there, only human contrivance. Those numbers, circling round, make time almost palpable—as if they aroused a sixth sense attuned to its presence, since it slips by the usual five (although aromas often do call up time past). Can we get behind numbers to find what it is they measure? Can we come to grips with the numbers themselves to know what they are and where they came from? Did we discover or invent them—or do they somehow lie in a profound crevice between the world and the mind?

Humans aren’t the exclusive owners of the smaller numbers, at least. A monkey named Rosencrantz counts happily up to eight. Dolphins and ferrets, parrots and pigeons can tell three from five, if asked politely. Certainly our kind delights in counting from a very early age:

One potato, two potato, three potato, four;

Five potato, six potato, seven potato, more!

Not that the children who play these counting-out games always get it right:

Wunnery tooery tickery seven

Alibi crackaby ten eleven

Pin pan musky Dan

Tweedle-um twoddle-um twenty-wan

Eerie orie ourie

You are out!

This is as fascinating as it is wild, because whatever the misconceptions about the sequence of counting numbers (alibi and crackaby may be eight and nine, but you’ll never get seven to come right after tickery), the words work perfectly well in counting around in a circle—and it’s always the twenty-first person from the start of the count who is out, if you and are still act as numerals as they did in our childhood. We can count significantly better than rats and raccoons because we not only recognize different magnitudes but

know how to match up separate things with the successive numbers of a sequence:

a little step, it seems, but one which will take us beyond the moon.

The first few counting numbers have puzzlingly many names from language to language. Two, zwei, dva, and deux is bad enough, even without invoking the burla of Queensland Aboriginal or the Mixtec ùù. If you consult just English-speaking children, you also get twa, dicotty, teentie, osie, meeny, oarie, ottie, and who knows how many others. Why is this playful speciation puzzling? Because it gives very local embodiments to what we think of as universal and abstract.

A friend of ours, whose art is the garden, has since childhood always imagined the numbers as lying on a zigzag path:

What happens, however, if we follow Isobel’s route past 60? It continues into the blue on a straight line. Almost everyone lets the idiosyncrasies go somewhere before a hundred, as not numbers but Number recedes into the distance. 3 and 7, 11 and 30 will have distinct characters and magical properties, perhaps, for many—but is 65,537 anyone’s lucky number? What makes mathematics so daunting from the very start is how its atoms accelerate away. A faceless milling crowd has elbowed out the kindly nursery figures. Its sheer extent and anonymity alienate our humanity, and carry us off (as Robert Louis Stevenson once put it) to where there is no habitable city for the mind of man.

. The notion of one—one partridge, a pear tree, the whole—feels too comfortable to be anything but a sofa in the living-room of the mind.

Almost as familiar, like a tool whose handle has worn to the fit of a hand, is the action of adding. We take in 1 + 1, as a new whole needing a new name, so easily and quickly that we feel foolish in trying to define what addition is. Housman wrote:

To think that two plus two are four

And neither five nor three

The heart of man has long been sore

And long ’tis like to be.

Perhaps. But the head has long been grateful for this small blessing.

With nothing more than the number one and the notion of adding, we are on the brink of a revelation and a mystery. Rubbing those two sticks together will strike the spark of a truth no doubting can ever extinguish, and put our finite minds in actual touch with the infinite. Ask yourself how many numbers there are; past Isobel’s 60, do they come to a halt at 65,537 or somewhere out there, at the end of time and space? Say they do; then there is a last number of all—call it n for short. But isn’t n + 1 a number too, and even larger? So n can’t have been the last—there can’t be a last number.

There you are: a proof as profound, as elegant, as imperturbable as anything in mathematics. You needn’t take it on faith; you need neither hope for nor fear it, but know with all the certainty of reason that the counting numbers can’t end. If you are willing to put this positively and say: there are infinitely many counting numbers—then all those differences between the small numbers you know, and the large numbers you don’t, shrink to insignificance beside this overwhelming insight into their totality.

This entente between 1 and addition also tells you something important about each point in the array that stretches, like Banquo’s descendants, even to the crack of doom. Every one of these counting numbers is just a sum of 1 with itself a finite number of times: 1 + 1 + 1 + 1 + 1 = 5, and with paper and patience enough, we could say that the same is true of 65,537.

These two truths—one about all the counting numbers, one about each of them—are very different in spirit, and taken together say something about how peculiar the art of mathematics is. The same technique of merely going on adding 1 to itself shows you, on the one hand, how each of the counting numbers is built—hence where and what each one is; on the other, it tells you a dazzling truth about their totality that overrides the variety among them. We slip from the immensely concrete to the mind-bogglingly abstract with the slightest shift in point of view.

Armies of Unalterable Law

Does number measure time, or does time measure number? And in one case or both, have we just proven that ongoing time is infinite? Like those shifts from the concrete to the abstract, mathematics also alternates minute steps with gigantic leaps, and to make this one we would have to go back to what seemed no more than a mere form of words. We asked if you were willing to recast our negative result (the counting numbers never end) positively: there are infinitely many counting numbers. To put it so seems to summon up an infinite time through which they are iterated. But are we justified in taking this step?

To speak with a lawyerly caution, we showed only that if someone claimed there was a last number we could prove him wrong by generating—in time—a next. Were we to turn our positive expression into a spatial image we might conjure up something like a place where all the counting numbers, already generated, lived—but this is an image only, and a spatial image, for a temporal process at that. Might it not be that our proof shows rather that our imaging is always firmly anchored to present time, on whose moving margin our thought is able to make (in time) a next counting number—but with neither the right, ability, nor need to conjure up their totality all at once? The tension between these two points of view—the potentially infinite of motion and the actual infinity of totality—continues today, unresolved, opening up fresh approaches to the nature of mathematics. The uneasy status of the infinite will accompany us throughout this book as we explore, return with our trophies, and set out again.

Here is the next truth. We can see that the sizable army of counting numbers needs to be put in some sort of order if we are to deploy it. We could of course go on inventing new names and new symbols for the numbers as they spill out: why not follow one, two, three, four, five, six, seven, eight, and nine with kata, gwer, nata, kina, aruma (as the Oksapmin of Papua New Guinea do, after their first nine numerals, which begin: tipna, tipnarip, bumrip …)? And surely the human mind is sufficiently fertile and memory flexible enough to avoid recycling old symbols and follow 7, 8, and 9 with @, ¤, β—dare we say and so on?

The problem isn’t a lack of imagination but the need to calculate with these numbers. We might want to add 8 and 9 and not have to remember a fanciful squiggle for their sum. The great invention, some five thousand years ago, of positional notation brought the straggling line of counting numbers into squadrons and regiments and battalions. After a conveniently short run of new symbols from 1 (for us this run stops at 9), use 1 again for the next number, but put it in a new column to the left of where those first digits stood. Here we will keep track of how many tens we have. Then put a new symbol, 0, in the digits’ column to show we have no units. You can follow 10 with 11, 12 and so on, meaning (to its initiates) a ten and one more, two more, … Continue these columns on, ever leftward, after 99 exhausts the use of two columns and 999 the use of three. Our lawyer from two paragraphs ago would remind us that those columns weren’t already there but constructed when needed: 65,537, for example, abbreviates

As always in mathematics, great changes begin off-handedly, the way important figures in Proust often first appear in asides. Zero was only a notational convenience, but this nothing, which yet somehow is, gave a new depth to our sense of number, a new dimension—as the invention of a vanishing point suddenly deepened the pictorial plane of Renaissance art (a subject to which we shall return in Chapter Eight).

But is zero a number at all? It took centuries to free it from sweeping the hearth, a humble punctuation mark, and find that the glass slipper fitted it perfectly. For no matter how convenient a notion or notation is, you can’t just declare it to be a number among numbers. The deep principle at work here—which we will encounter again and again—is that something must not only act like a number but interact companionably with other numbers in their republic, if you are to extend the franchise to it.

This was difficult in the case of zero, for it behaved badly in company. The sum of two numbers must be greater than either, but 3 + 0 is just 3 again. Things got no better when multiplication was in the air: 3 · 17 is different from 4 · 17, yet 3 · 0 is the same as 4 · 0—in fact, anything times 0 is 0. This makes sense, of course, since no matter how many times you add nothing to itself (and multiplication is just sophisticated addition, isn’t it?), you still have nothing. What do you do when someone’s services are vital to your cause, for all his unconventionality? You do what the French did with Tom Paine and make him an honorary citizen. So zero joined the republic of numbers, where it has stirred up trouble ever since.

Our primary mathematical experience, individually as well as collectively, is counting—in which zero plays no part, since counting always starts with one. The counting numbers (take 17 as a random example), parthenogenetic offspring of that solitary Adam, 1, came in time to be called the natural as their symbol. Think of them strolling there in that boundless garden, innocent under the trees. For all that we have now found a way to organize them by tens and hundreds, they seem at first sight as much like one another as such offspring would have to be. Yet look closer, as the Greeks once did, to see the beginnings of startling patterns among them. Are they patterns we playfully make in the ductile material of numbers, as a sculptor prods and pinches shapes from clay? Or patterns only laid bare by such probing, as Michelangelo thought of the statue which waited in the stone? Of all the arts, mathematics most puts into question the distinction between creation and discovery.

, as if it were a triangular number by default (extending the franchise again).

Here are the first six triangular numbers:

Each is bigger than the previous one by its bottom row, which is the next natural number. This pattern clearly undulates endlessly on.

and the next would be

. The first six square numbers, each gotten by adding a right angle of dots to the last,

are 1, 4, 9, 16, 25, 36. Another endless rhythm in this landscape.

But isn’t all this messing about indeed idle? What light does it shed on the nature of things, what use could it possibly be?

Light precedes use, as Sir Francis Bacon once pointed out. Think yourself into the mind of that nameless mathematician who long ago made triangular and square patterns of dots in the sand and felt the stirrings of an artist’s certainty that there must be a connection between them:

If there was, it was probably well hidden. Perhaps he recalled what the Greek philosopher Heraclitus had said: A hidden connection is stronger than one we can see. Hidden how? Poking his holes again in the sand, looking at them from one angle and another, he suddenly saw:

each of these square numbers was the sum of two triangular ones! Then the leap from seeing with the outer to the inner eye, which is the leap of mathematics to the infinite: this must always be so.

Our insight sharpens: the second square number is the sum of the first two triangular numbers; the third square of the second and third triangulars, and so on. You might feel the need now for a more graceful vessel in which to carry this insight—the need for symbols—and make up these:

where that always is stored in the letter n for any number.

By itself this is a dazzling sliver of the universal light, and its discovery a model of how mathematics happens: a faith in pattern, a taste for experiment, an easiness with delay, and a readiness to see askew. How many directions now this insight may carry you off in: toward other polygonal shapes such as pentagons and hexagons, toward solid structures of pyramids and cubes, or to new ways of dividing up the arrays.

As for utility, what if you wanted to add all the natural numbers from 1 to 7, for example, without the tedium of adding up each and every one? Well, that sum you want is a triangular number:

and work our way backward—but this will get us into an ugly tangle—and if it isn’t beautiful it isn’t mathematics. Faith in pattern and easiness with delay: we want to look at it somehow differently, with our discovery of page 13 tantalizingly in mind. A taste for experiment and a readiness to see askew: well, that triangle is part of a square in having a right angle at its top—what if we tilt it over and put the right angle on the ground:

Why? Just messing about again, to make the pattern look squarelike; but this feels uncomfortable, incomplete—it wants to be filled out (perhaps another ingredient in the mix of doing mathematics is a twitchiness about asymmetries).

If we complete it to a square, we’re back to what proved useless before. Well, what about pasting its mirror image to it, this way?

, which is—28! Is this it?

next to it, upside-downyou get a 7 by 8 rectangle, half the dots in which give the desired 28.

So in general,

for any number n.

Or

. Experiments of light have yielded, as Bacon foretold, experiments of fruit.

Unnatural Numbers

brought up subtraction, which isn’t at home among the natural numbers.

Aren’t negative numbers in fact ridiculously unnatural? Five-year-olds—fresh from the Platonic heaven—will tell you confidently that such numbers don’t exist. But after a childhood of counting games, years of discretion approach with the shadows of commerce and exchange. I had three marbles, then lost two to you, and now I have one. I lose that one and am left with none, so I borrow one from a friend and proceed to lose that too, hence owing him one. How many have I? Even recognizing that I had one marble after giving up two is scaly, a snake in our garden, the presage of loss.

How are we even to picture the negative numbers—by dots that aren’t there?

Yesterday upon the stair

I met a man who wasn’t there.

He wasn’t there again today—

I wish that man would go away.

But the negative numbers won’t go away: Northerners are intimately familiar with them, thanks to thermometers, and all of us, thanks to debt.

Perhaps by their works shall you know them, through seeing the palpable effects of subtracting. Look again at our triumphant discovery of what the first n natural numbers added up to. If we subtract from these numbers all the evens, what sum are we left with—what is the sum of the odds?

It looks as if it might be the square of how many odd numbers we are adding. And here is a wonderful confirmation of this, in the same visual style as our last one—another piece of inspired invention. When we add right angles of dots to the previous ones, as we did on page 13,

what are we doing but adding up the successive odd numbers? So of course their sum is a square: the square of how many odds we have added up.

A thousand years of schoolchildren caught the scent of subtraction in problems like this, as they studied their Introductio Arithmetica. It had been written around A.D. 100 by a certain Nichomachus of Gerasa, in Judea. His vivid imagination conjured up some numbers as tongueless animals with but a single eye, and others as having nine lips and three rows of teeth and a hundred arms.

Subtracting—taking away the even numbers from the naturals—has left us with the odd numbers. To people making change, subtraction turns into what you would have to add to make the whole (98 cents and 2 makes 1, and 4 makes 5). But it is an act also of adding a negative quantity: $5 and a debt of 98 cents comes to $4.02. Does the fact that we can’t see the negative numbers themselves make them any less real than the naturals? The reality of the naturals, after all, is so vivid precisely because we can’t sense them: numbers are adjectives, answering the question how many, and we see not five but five oranges, and never actually see 65,537 of anything: large quantities are blurs whose value we take on faith. If we come to treat numbers as nouns—things in their own right—it is because of our wonderful capacity to feel at home, after a while, with the abstract. On such grounds the negatives have as much solidity as the positives, and ramble around with them, like secret sharers, in our thought.

We extend the franchise to them by calling the collection of natural numbers, their negatives and zero, the Integers: upright, forthright, intact. The letter Z, from the German word for number, Zahl, is their symbol, and −17 a typical member of their kind. And once they are incorporated to make this larger state, we find not only our itch to symmetrize satisfied, but our sense of number’s relation to time widened. If the positive natural numbers march off toward a limitless future, their negative siblings recede toward the limitless past, with 0 forever in that middle we take to be the present. It takes a real act of generosity, of course, to extend the franchise as we have, because we so strongly feel the birthright of the counting numbers. God created the natural numbers, said the German mathematician Kronecker late in the nineteenth century, the rest is the work of man. And certainly zero and the negatives have all the marks of human artifice: deftness, ambiguity, understatement. If you like, you can preserve the Kroneckerian feeling of the difference between positives and negatives by picturing our present awareness as the knife-edge between endless discovery ahead and equally endless invention behind.

From Ratios to Rationals

You pretty much know where you are with the integers. There may be profound patterns woven in their fence-post-like procession over the horizon, but they mark out time and space, before and behind, with comforting regularity. Addition and multiplication act on them as they should—or almost: (–6) · (–4) = 24: a negative times a negative turns out, disconcertingly, to be positive. Why this should—why this must—be so we will prove to your utter satisfaction in Chapter Three. Otherwise, all is for the best in this best of all possible worlds.

Exhilarated by its widened conception of number, mind looks for new lands to colonize and sees an untamed multitude at hand. For from the moment that someone wanted to trade an ox for twenty-four fine loincloths, or a chicken for 240 cowrie shells, making sense of ratios became important. You want to scale up this 2 by 4 wooden beam to 6 by—what? Three of your silver shekels are worth 15 of your neighbor’s tin mina: what then should he give you for five silver shekels?

The Greeks found remarkable properties of these ratios and subtle ways of demonstrating them. If an architect wondered what length bore the same relation to a length of 12 units that 4 bears to 7, a trip with his local geometer down to the beach would have him drawing a line in the sand 4 units long; and at any angle to that, another of 7 units, from the same starting-point, A:

the urge for completion would lead them both to draw the third side, BC, of their nascent triangle. But now the geometer continues the lines AB and AC onward:

and marks a point D on AB’s extension so that AD is 12 units long:

ingenuity and an intimacy with similar triangles now leads him to draw from D a line parallel to BC, meeting AC at E:

AE will be in the same ratio to 12 as 4 is to 7.

couldn’t possibly be numbers, because numbers arose from the unit, and the unit was an indivisible whole.

How nightmarish it would have been for a Pythagorean to think of that whole fractured into fractions. It would mean that how things stood to one another—their ratios—and not the things themselves were ultimately real: and they could no more believe this than we would think that adjectives and adverbs rather than nouns were primary. That would have led to a world of flickering changes, of fading accords and passing dissonances, of qualities heaped on qualities, where shadowy intimations of what had been and what would be tunneled like vortices through a watery present you never stepped in twice.

If Greek philosophers and mathematicians did not have fractions, it seems their merchants did—picked up, perhaps, in their travels among the Egyptians, for whom fractions (though only with 1 in their numerators) dwelt under the hawklike eye of Horus.

Against this background of daily practice, insights into how ratios behaved kept growing, until inevitably they too became embodied in numbers. How could properties accumulate without our concluding that what has them must be a thing—especially since we are zealous to make objects out of whatever we experience? So they came to live among the rest as pets do among us, each with its cargo of domestic insects:

Great fleas have little fleas

Upon their backs to bite ’em,

And little fleas have lesser fleas,

And so ad infinitum.

. The average of the two ends falls between them, along with how many other splinters of the whole, so that infinity not only glimmers at the extremities of thought but is here in our very midst, an infinity of fractions in each least cleft of the number line.

So the franchise was hesitantly extended to ratios in the guise of fractions, although uneasiness at splitting the atomic unit remained. The fractions, preserving traces of their origin in their official name of Rational Numbers, were symbolized by the letter Q, and so on). And notice how this new flood of intermediate numbers makes number itself suddenly much more time-like: flowing with never a break, it seems, invisibly past or through us.

, where a and b can be any integer. Or almost any: a pinprick of the old discomfort remains