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Chapter Zero
THE LENS
If you look at zero you see nothing; but look through it and you will see the world. For zero brings into focus the great, organic sprawl of mathematics, and mathematics in turn the complex nature of things. From counting to calculating, from estimating the odds to knowing exactly when the tides in our affairs will crest, the shining tools of mathematics let us follow the tacking course everything takes through everything else — and all of their parts swing on the smallest of pivots, zero.
With these mental devices we make visible the hidden laws controlling the objects around us in their cycles and swerves. Even the mind itself is mirrored in mathematics, its endless reflections now confusing, now clarifying insight.
Zero's path through time and thought has been as full of intrigue, disguise and mistaken identity as were the careers of the travellers who first brought it to the West. In this book you will see it appear in Sumerian days almost as an afterthought, then in the coming centuries casually alter and almost as casually disappear, to rise again transformed. Its power will seem divine to some, diabolic to others. It will just tease and flit away from the Greeks, live at careless ease in India, suffer our Western crises of identity and emerge this side of Newton with all the subtlety and complexity of our times.
My approach to these adventures will in part be that of a naturalist, collecting the wonderful variety of forms zero takes on — not only as a number but as a metaphor of despair ordelight; as a nothing that is an actual something; as the progenitor of us all and as the riddle of riddles. But we, who are more than magpies, feather our nests with bits of time. I will therefore join the naturalist to the historian at the outset, to relish the stories of those who juggled with gigantic numbers as if they were tennis balls; of people who saw their lives hanging on the thread of a calculation; of events sweeping inexorably from East to West and bearing zero along with them — and the way those events were deflected by powerful personalities, such as a brilliant Italian called Blockhead or eccentrics like the Scotsman who fancied himself a warlock.
As we follow the meanderings of zero's symbols and meanings we'll see along with it the making and doing of mathematics — by humans, for humans. No god gave it to us. Its muse speaks only to those who ardently pursue her. And what is that pursuit? A mixture of tinkering and inspiration; an idea that someone strikes on, which then might lie dormant for centuries, only to sprout all at once, here and there, in changed climates of thought; an ongoing conversation between guessing and justifying, between imagination and logic.
Why should zero, that O without a figure, as Shakespeare called it, play so crucial a role in shaping the gigantic fabric of expressions which is mathematics? Why do most mathematicians give it pride of place in any list of the most important numbers? How could anyone have claimed that since 0 x 0 = 0, therefore numbers are real? We will see the answers develop as zero evolves.
And as we watch this maturing of zero and mathematics together, deeper motions in our minds will come into focus. Our curious need, for example, to give names to what we create — and then to wonder whether creatures exist apart from their names. Our equally compelling, opposite need to distance ourselves ever further from individuals and instances, lunging always toward generalities and abbreviating the singularity of things to an Escher array, an orchard seen from the air rather than this gnarled tree and that.
Below these currents of thought we will glimpse in successive chapters the yet deeper, slower swells that bear us now toward looking at the world, now toward looking beyond it. The disquieting question of whether zero is out there or a fiction will call up the perennial puzzle of whether we invent or discover the way of things, hence the yet deeper issue of where we are in the hierarchy. Are we creatures or creators, less than — or only a little less than — the angels in our power to appraise?
Mathematics is an activity about activity. It hasn't ended — has hardly in fact begun, although the polish of its works might give them the look of monuments, and a history of zero mark it as complete. But zero stands not for the closing of a ring: it is rather a gateway. One of the most visionary mathematicians of our time, Alexander Grothendieck, whose results have changed our very way of looking at mathematics, worked for years on his magnum opus, revising, extending — and with it the preface and overview, his Chapter Zero. But neither now will ever be finished. Always beckoning, approached but never achieved: perhaps this comes closest to the nature of zero.
Chapter Three
TRAVELERS' TALES
What happened in that autumnal world, long ago, as the thought of Athens shifted to Alexandria, its power to Rome and its culture, carried eastward by invasion and trade, changed in new surroundings while those surroundings changed to absorb it? We are past the days when geometry snubbed arithmetic, and so would expect to find zero coming into its own. Here was this symbol with immense power to describe, explain and control locked up in its little ring, being passed from language to language, from one mathematician or astronomer to another, with none realizing what he had in his pocket.
As in all the best adventure stories, it didn't turn up where it should have: in Sicily, for example, when during the third century BC a passion for huge numbers blossomed. You would think that botanizing among such growths would lead inevitably to a full positional notation and zero, its genius loci: not different kinds of heaps present or absent, but numbers as such, written positionally and abstracted from what they counted. You've seen how hard it is to think up and manipulate new names for ever larger ensembles — but how easy to add another zero to a row stretching out after a harmless '1'. This is certainly how we picture the zillions invoked to express awe or desire. In my neck of the woods, where we tried to outdo each other as kids with bazillions and kazillions, it always came down to who could squeeze one last zero on to the page — like the barmen of Dublin who always manage to fit yet one more drop of Guinness into a brimming pint (what a history could be written of our reaching toward the infinite, and the fitful evolution of fantasy to imagination, by looking at our changing ways of naming vast multitudes). Yet an inventor who gloried in mindboggling numbers did so with never a zero in sight.
Archimedes was born around 287 BC, the son of an astronomer. Among his amazing works is one he sent to Gelon, King of Syracuse, in which he shows how to name quantities greater than the number of grains of sand not only on all the beaches around Syracuse, but on all the beaches of Sicily; and in all the lands of the world, known or unknown; and in the world itself, were it made wholly of sand; and, he says, 'I will try to show you by means of geometrical proofs which you will be able to follow, that, of the numbers named by me ... some exceed the number of the grains of sand ... in a mass equal in magnitude to the universe.'
This monstrous vision, which puts in the shade such fairytale excesses as the mill grinding out salt forever on the floor of the sea, Archimedes makes precise by an ingenious sequence of multiplications.
Take it, he says, that there are at most 10,000 grains of sand in a heap the size of a poppyseed; and that a row of 40 poppyseeds will be as wide as a finger. To keep things simple, picture each seed as a sphere. Since the volumes of spheres are to each other as the cubes of their diameters, this line of 40 seeds becomes the diameter of a sphere with a volume (40)³ = 64,000 times the volume of one seed; and since that one holds 10,000 grains of sand, we're already talking about 64,000 x 10,000, that is, 640,000,000 grains. In our modern notation, that's 4³ x 10^{7} grains. Round 64 up to 100 for convenience, and we'll have 10^{9} grains in a sphere whose diameter is a fingerbreadth. Don't worry that all these estimates may be too large: exaggeration, as you'll see, is part of Archimedes' game.
Now 10,000 (10^{4}) fingerbreadths make a Greek unit of length called a stade (roughly a tenth of our mile). A sphere whose diameter is 10^{4} fingerbreadths will have a volume (10^{4})³ = 10^{12} times the volume of one with a diameter of one fingerbreadth, which we know contains 10^{9} grains of sand; so a sphere with a onestade diameter will contain 10^{12} x 10^{9} = 10^{21} grains of sand.
Archimedes next draws on the work of a great astronomer some 25 years his senior, Aristarchus of Samos, to estimate the diameter of the universe (which for the Greeks stretched out to the sphere of the fixed stars). Aristarchus — so long before Copernicus! — held that the earth circled the sun. Archimedes juggled with Aristarchus' observations and calculations to come up with an imaginary sphere (call it S) whose radius is the distance from the earth to the sun; then he assumes that
the diameter of the earth   the diameter of S 
 = 

the diameter of S   the diameter of the universe. 
This gives him (after modifying Aristarchus' figures) 100,000,000,000,000 or 10^{14} stadia for the diameter of the universe. Its volume is therefore (10^{14})³ = 10^{42} times the volume of the sphere whose diameter is one stade, which held 10^{21} grains of sand. Hence there would be 10^{21} x 10^{42} = 10^{63} grains of sand in a universe compacted wholly of sand.
'I suppose, King Gelon,' says Archimedes, 'that all this will seem incredible to those who haven't studied mathematics, but to a mathematician the proof will be convincing. And it was for this reason that I thought it worth your while to learn it.'
When you consider that in the 1940s two persistent New Yorkers estimated that the number of grains of sand on Coney Island came to about 10^{20}; and that present estimates for the total number of much smaller particles in our much larger universe weigh in at between 10^{72} and 10^{87}, you have to say that Archimedes' estimate wasn't all that bad.
This is a spectacular application of the Greek insight that the world afar can be grasped by analogy to the world at hand. But it is made much more spectacular when you realize that Archimedes hadn't our convenient notation for powers of ten, all built on the use of zero.
'How could he have missed it?' asked one of the greatest of all mathematicians, Karl Friedrich Gauss, who very much admired Archimedes. 'To what heights science would have risen by now,' he wrote in the nineteenth century, 'if only he had made that discovery!' But the fact remains that Archimedes worked with number names rather than digits, and the largest of the Greek names was 'myriad', for 10,000. This let him speak of a myriad myriads (10^{8}), and he then invented a new term, calling any number up to 10^{8} a number of the first order.
He next took a myriad myriads as his unit for numbers of the second order, which therefore go up to 10^{16} (as we would say — but he didn't); and 10^{16} as the unit for numbers of the third order (up to 10^{24}), and so on; so that the unimaginably gigantic 10^{63} is a number somewhere in the eighth order.
But Archimedes doesn't stop there. In fact, he has hardly begun. In a spirit personified for us by Paul Bunyan, he leaves the sandfilled universe behind, diminished itself to a grain of sand, as he piles up order on order, even unto the 10^{8} order, which frighteningly enough contains all the numbers from [(100,000,000).sup.99,999,999 to ([100,000,000).sup.100,000,000].
Are we done? Hardly. All of those orders, up to the one just named, make up the first period. If you look at his own words your mind begins to go out of focus, and you feel like Alice as she fell toward Wonderland: 'Do cats eat bats? Do bats eat cats?' For he says:
And let the last number of the first period be called a unit of numbers of the first order of the second period. And again, let a myriad myriads of numbers of the first order of the second period be called a unit of numbers of the second order of the second period. Similarly —
Perhaps Gelon had stopped reading by now, and so missed Archimedes' eventual conclusion:
And let the process continue up to a myriadmyriad units of the myriadmyriadth order of the myriadmyriadth period
— or in our notation, up to 10^{80,000,000,000,000,000}. Of course there wouldn't be enough grains of sand in his universe or ours to trickle this number out, nor enough time, from the Big Bang to now, to recite its digits at one a second, since the last number of his first period is 1 followed by 800 million zeroes, and this one has 10^{8} times as many.
What was Archimedes after in all this, and — if it makes sense to shape negative history, speculating about paths not taken — why was zero absent from his contrivings? Some say that his SandReckoner was a tour de force, showing that the stylus was mightier than the xiphos. You might think of it as thoroughly Greek in its playfulness: for Plato said we are but playthings of the gods, and so should play the noblest games — and this exuberant work of his, having no conceivable practical use, must be a scherzo. Was it his intention to humble a king, or to glory in surpassing the magnitudes of his predecessors? His father, Phidias, for example, had declared that the sun's diameter was 12 times that of the moon. Archimedes took it to be 30 times (he would have been pleased to learn it is actually 400 times as great). And Aristarchus made use of the awesome 71,755,875 in one of his calculations, which Archimedes here strode past in a single step. Was there a kind of chestthumping rivalry among the mathematicians of his day, as with children of ours, in conjuring up larger and larger numbers? Archimedes' contemporary Apollonius seems to have responded to The SandReckoner with a system of his own for naming large numbers, which Archimedes then countered with a problem whose answer (could Archimedes possibly have known it?) was a number so large that to write it out in digits would take up the next 47 pages (for Archimedes, the answer would have begun: '7 units of 3 myriad 5,819th numbers, and 7,602 myriad 7,140 units of 2 myriad 5,818th numbers, and...') You will be amused — or perplexed — to learn that mathematicians still thus see and raise one another, but now with infinities.
More profoundly, wasn't he showing us how to think as concretely as we can about the very large, giving us a way of building up to it in stages rather than letting our thoughts diffuse in the face of immensity, so that we will be able to distinguish even such magnitudes as these from the infinite? As a mathematician I know said recently, 'Large numbers are actually very large.'
Or are we seeing here that play we have already watched between language and thought, this time not leading to but deliberately avoiding the convenience of zero? For at the beginning of The SandReckoner Archimedes made, you recall, a curious remark:
There are some who ... think that no number has been named which is great enough to exceed [the number of grains of sand in every region of the earth]. But I will try to show you ... that, of the numbers named by me ... some exceed ... the number [of grains of sand that would fill up the universe].
Why does the emphasis fall on naming? Think for a moment of the letter St Paul wrote to the Ephesians, where he speaks of Christ as
far above all principality, and power, and might, and dominion, and every name that is named, not only in this world but also in that which is to come.
Haven't we all an ancient sense that for something to exist it must have a name? Many a child refuses to accept the argument that the numbers go on forever (just add one to any candidate for the last) because names run out. For them a googol — 1 with 100 zeroes after it — is a large and living friend, as is a googolplex (10 to the googol power, in an Archimedean spirit). A sevenyearold of my acquaintance claimed that the last number of all was 23,000. 'What about 23,000 and one?' she was asked. After a pause: 'Well, I was close.' Under this Adam Impulse people have exerted themselves to come up with names for very large numbers, such as primovigesimocentillion for 10^{366}, and the mellifluous millimillillion for 10^{3,000,003}. What points to a fundamental trait of ours is that 10^{63}, or 1 with 63 zeroes after it, leaves the imagination cold: it might just as well have had a few dozen zeroes more, or a couple less. What facilitates thought impoverishes imagination.
By not using zero, but naming instead his myriad myriads, orders and periods, Archimedes has given a constructive vitality to this vastness — putting it just that much nearer our reach, if not our grasp. There are other ways, to be sure, of satisfying the Adam Impulse: invoking dread, for example, rather than awe. Eighteen hundred years after Archimedes, John Donne, haunted and haunting, said in a Lenten sermon:
Men have calculated how many particular graines of sand, would fill up all the vast space between the Earth and the Firmament: and we find, that a few lines of cyphers will designe and expresse that number ... But if every grain of sand were that number, and multiplied again by that number, yet all that, all that inexpressible, inconsiderable number, made up not one minute of this eternity; neither would this curse [of God on the inveterate sinner] be a minute shorter for having been endured so many generations, as there were graines of sand in that number ... How do men bear it, we know not; what passes between God and those men, upon whom the curse of God lieth, in their dark horrours at midnight, they would not have us know ... This is the Anathema Maranatha, accursed till the Lord come; and when the Lord cometh, he cometh not to reverse, nor to alleviate, but to ratifie and aggravate the curse.