Pindar’s claim remained unassailable for some two centuries, which of course is not bad at all as far as a line of poetry goes. The eventual refutation, composed in the middle of the third century BCE, can be fairly listed among the finest achievements of the mathematician Archimedes.
Introducing his academic paper – the first in recorded history – to the king of his day, Archimedes made a spectacularly audacious argument.
Some people believe, King Gelon, that the grains of sand are infinite in number. I mean not only the sand in Syracuse and the rest of Sicily, but also the sand in the whole inhabited land as well as the uninhabited. There are some who do not suppose that they are infinite, but that there is no number that has been named which is so large as to exceed its multitude . . . I will attempt to prove to you through geometrical demonstrations, which you will follow, that some of the numbers named by us . . . exceed . . . the number of grains of sand having a magnitude equal to the earth filled up.
Archimedes’s estimations for the measurements of the Earth, moon, sun, and the other stars were generous: for example, making the Earth’s perimeter ten times larger than the calculations of earlier astronomers. Similarly, Archimedes went to great lengths to provide a capacious margin for error concerning the estimated size of a grain of sand. He compared ten thousand grains to the scale of a poppy-seed, and then patiently lined the seeds end to end on a smooth ruler. In this way he measured the number of poppy-seeds required to reach an inch as being twenty-five. This figure he adjusted still further, changing it to forty seeds per inch-length, so as to ‘prove indisputably what is proposed’. Thus he calculated as sixteen million (10,000 x 40 x 40) the maximum number of grains of sand that could fill one square inch.
Archimedes assumed that the universe was spherical. He estimated a value for the diameter of the universe using calculations for the diameter of the Earth’s orbit around the sun. The universe, according to his reckoning, had a diameter no greater than 100,000,000,000,000 stadia (about two light years). 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 grains of sand would more than saturate the whole of space.
Next, Archimedes showed that the Greek term ‘myriad’ (ten thousand or a hundred hundreds) more than sufficed for the purpose of counting even the largest worldly quantities. The phrase ‘myriad myriads’, he pointed out, allowed the counter to reach the equivalent of one hundred million – the largest named number in his time. But, he continued, if it is possible to count in myriads, it should be equally possible to count in ‘myriad myriads’ so that multiplying the latter by itself the counter could attain ‘myriad myriad myriad myriads’ or 10,000,000,000,000,000. And considering this new figure likewise as a unit, as respectable as a ‘myriad’ or ‘myriad myriads’, the counter could multiply ‘myriad myriad myriad myriads’ by itself and proceed to: ‘myriad myriad myriad myriad myriad myriad myriad myriads’ or 100,000,000,000,000,000,000,000,000,000,000.
Up till now we have multiplied a myriad by itself a total of eight times. Archimedes’s next step possessed all the elegance of simple logic: multiply a myriad myriads by itself myriad myriads times over. The ‘1’ that starts the resulting number is tailed by eight hundred million zeroes.
Doggedly pursuing his logic, Archimedes proposed multiplying this new number by itself up to as many as a myriad myriads times over: a number requiring the insertion of eighty quadrillion (80,000,000,000,000,000) zeroes after the one.
Archimedes concluded his paper in confident, if understated tones.
King Gelon, to the many who have not also had a share of mathematics I suppose that these will not appear readily believable, but to those who have partaken of them and have thought deeply about the distances and sizes of the Earth and sun and moon and the universe this will be believable on the basis of demonstration. Hence, I thought that it is not inappropriate for you too to contemplate these things.
We find the same comparison between immensity and grains of sand in the sutras of India, many of which were set down on paper in Archimedes’s time. In the Lalitavistara sutra, a hagiographical account of the Buddha’s life, we read of a meeting between the young Siddhartha and the ‘great mathematician Arjuna’. Arjuna asks the boy to multiply numbers a hundredfold beginning with one koti (generally considered the equivalent of ten million). Without the slightest hesitation, Siddhartha correctly replies that one hundred kotis equals an ayuta (which would equate to one billion), and then proceeds to multiply this number by one hundred, and the new number by one hundred, and so on, until – after twenty-three successive multiplications – he reaches the number called tallaksana (the equivalent of 1 followed by 53 zeroes).
Siddhartha proceeds to multiply this number in turn, though it is unclear whether he does so by one hundred or some other amount. In a phrase reminiscent of Archimedes, he claims that with this new number the mathematician could take every grain of sand in the river Ganges ‘as a subject of calculation and measure them’. Again and again, the bodhisattva multiplies this number, until at last he reaches sarvaniksepa, with which, he tells the mathematician, it would be possible to count every grain of sand in ten rivers the size of the Ganges. And if this were not enough, he continues, we can multiply this number to reach agrasara – a number greater than the grains of sand in one billion Ganges.
Such extreme numerical altitudes, we are told, are the preserve of the pure and enlightened mind. According to the sutra, only the bodhisattvas, beings who have arrived at their ultimate incarnation, are capable of counting so high. In the closing verses, the mathematician Arjuna concedes this point.
This supreme knowledge I do not have – he is above me.
One with such knowledge of numbers is incomparable!
The story of the enlightenment of Siddhartha Gautama, to give him his full name, begins in his father’s palace. It is said that the Nepalese king resolved to seclude his son at birth from the heartbreaking nature of the world. Shut up behind gilded doors, the boy would remain forever innocent of suffering, aging, poverty and death. We can imagine his constricted royal life: the fine meals of rich food, lessons in literacy and military arts, ritual music and dance. In his ears he wore precious stones heavy enough to make his earlobes droop. But of course he was not free: he had only walls for a horizon, only ceilings for a sky. Bangle strings and brass flutes displaced all birdsong. Cloying aromas of cooked food overlay the smell of rain.
Nearly thirty years, a marriage and even the birth of his own son all passed before Siddhartha learned of a world beyond the palace walls. Having resolved to go forth and see it, he made a trip through the countryside, accompanied only by the charioteer who drove him. The prince saw for the first time men enfeebled by ill health, old age and want of money. He was not even spared the sight of a corpse. Deeply shocked by all that he had seen, he fled his old life for the ascetic’s road.
The story of the prince’s seclusion in a palace reads like a fairytale – it may very well be such a tale – with all its peculiar and thought-provoking charm. One particular aspect of Siddhartha’s revelation of the outside world has always struck me. Quite possibly he lived his first thirty years without any knowledge of numbers.
How must he have felt, then, to see crowds of people mingling in the streets? Before that day he would not have believed that so many people existed in all the world. And what wonder it must have been to discover flocks of birds, and piles of stones, leaves on trees and blades of grass! To suddenly realise that, his whole life long, he had been kept at arm’s length from multiplicity.
Later, his followers would associate Siddhartha’s enlightened mind with a profound knowledge of numbers. Perhaps, as much as all the other surprises he witnessed from his chariot, it was this first sighting of multiplicity that set him on the path to Nirvana.
I am reminded of another story. This time the man was not a king but a mathematician. Unlike the Buddha’s father, big numbers pleased him; he enjoyed talking about them with his nine-year-old n
ephew. One day, a mid-twentieth-century day in America, the mathematician Edward Kasner invited the boy to name a number that contains a hundred zeroes. ‘Googol,’ the boy replied, after a little thought.
No explanation for the origin of this word is given in Kasner’s published account ‘Mathematics and the Imagination’. Probably it came intuitively to the boy. According to linguists, English speakers tend to associate an initial G sound with the idea of bigness, since the language employs many G- words to describe things which are ‘great’ or ‘grand’, ‘gross’ or ‘gargantuan’, and which ‘grow’ or ‘gain’. I could point out another feature: both the elongated ‘oo’ vowel and the concluding L suggest indefinite duration. We hear this difference in verbs like ‘put’ and ‘pull’, where ‘put’ – with its final T – implies a completed action, whereas an individual might ‘pull’ at something for any conceivable amount of time.
In a universe teeming with numbers, no physical quantity exists that coincides with a googol. A googol dwarfs the number of grains of sand in all the world. Collecting every letter of every word of every book ever published gets us nowhere near. The total number of elementary particles in all of known space falls some twenty zeroes short.
The boy could never hope to count every grain of sand, or read every page of every published book, but, like Archimedes and the Siddhartha of the sutras, he understood that no cosmos would ever contain all the numbers. He understood that with numbers he might imagine all that existed, all that had once existed or might one day exist, and all that existed too in the realms of speculation, fantasy and dreams.
His uncle, the mathematician, liked his nephew’s word. He immediately encouraged the boy to count higher still and watched as his small brow furrowed. Now came a second word, a variation of the first: ‘googolplex’. The suffix -plex (duplex) parallels the English -fold, as in ‘tenfold’ or ‘hundredfold’. This number the boy defined as containing all the zeroes that a hand could write down before tiring. His uncle demurred. Endurance, he remarked, varied a great deal from person to person. In the end they agreed on the following definition: a googolplex is a 1 followed by a googol number of zeroes.
Let us pause a brief moment to contemplate this number’s size. It is not, for instance, a googol times a googol: such a number would ‘only’ consist of a 1 with 200 zeroes. A googolplex, on the other hand, contains far more than a thousand zeroes, or a myriad zeroes, or a million or billion zeroes. It contains far more than the eighty quadrillion zeroes at which even the painstaking and persistent Archimedes ceased to count. There are so many zeroes in this number that we could never finish writing them all down, even if every human lifetime devoted itself exclusively to the task.
The googolplex is so vast a number that it encompasses virtually every conceivable probability. Physicist Richard Crandall gives the example of a can of beer that spontaneously tips over, ‘an event made possible by fundamental quantum fluctuations’, as having vastly greater than 1-in-a googolplex odds. A further illustration, by the English mathematician John Littlewood, asks us to imagine the plight of a mouse in outer space. Littlewood calculated as being well within a googolplex the likelihood that this mouse – helped by sufficient random fluctuations in its environment – might survive a whole week on the surface of the sun.
But of course a googolplex is not infinite. We can, as perhaps the boy did, continue to count by simply adding one. Modern computers, impervious to zero vertigo, have calculated that this number, googolplex + 1, is not prime. Its smallest known factor is: 316,912,650,057,057,350,374,175,801,344,000,001.
What did the mathematician make of his nephew’s ‘googolplex’ as the biggest number he could conceive of? His reply is not recorded, but he might have told him about some of the infinitely many numbers that exceed the googolplex’s scope. He might, for example, have mentioned ‘googol factorial’, being the product of multiplying every whole number between 1 and a googol (1 × 2 × 3 x . . . 950,345 x . . . 1,000,000,000,000,008,761 x . . . googol). This number, which computers tell us begins 16294 . . . easily surpasses every other number that we have encountered in these pages.
For a universe of such limited dimensions, these monstrous numbers seem quite useless. Worse, they can appear to us excessive, disproportionate. Every number, after a certain point, feels gratuitous as a joke. Who knows? It is possible they are not intended for our attention. The Flower Adornment sutra speaks of immense aeons, kalpas, in which the universe is continuously destroyed and reborn. At the kalpa’s peak, men live for an average of eighty-four thousand years. In other realms, so the Heart sutra reports, a single life spans eighty-four thousand kalpas – that is to say, eighty-four thousand epochs, each one many zeroes long. For such beings, a googol or its factorial would belong merely to the tangible and the convenient.
Mathematicians aspire to these heavenly realms. Vast numbers that split our senses, enrich their work. But they also produce paradoxes. For instance, which is greater: 10 or 27, when each is multiplied by itself exactly a googolplex number of times? The latter, of course, although even the most powerful calculators – plunging one hundred digits deep – struggle to tell the two apart. This difficulty confounds our expectations: intuitively, we feel that the ordering of a number should remain straightforward, even when the number’s precise value cannot be known. And yet, there exist numbers so large that we cannot easily distinguish them from their double, or triple, or quadruple or any other amount. There exist magnitudes so immense that they escape all our words, and all our numbers.
The most famous paradox concerning big numbers takes us back once more to the ancient Greeks. Tradition attributes it to the philosopher Eubulides. It has been suggested that Eubulides’s inspiration owes something to his fellow sceptic Zeno, who argued that every falling grain of wheat makes a noise proportionate to the noise made by a falling bushel. Eubulides’s formulation does not feature wheat, however. As would Archimedes a century later, Eubulides built his argument on sand.
It goes as follows: first, we agree that one grain of sand does not make a heap. Adding a second grain does not make a heap either. Nor do we produce a heap with the third grain. From this it follows that adding one to any small number generates another number we call ‘small’. But if this is true, a billion is a small number. So, too, is a googolplex.
Understandably wary of this conclusion, the reader might propose that a heap of sand, like a big number, begins at a certain point: say, ten thousand. As a resolution of the paradox, this answer is unsatisfactory. It is unclear why nine thousand nine hundred and ninety nine should be considered small, but not nine thousand nine hundred and ninety nine plus one.
Of course, in at least one sense all numbers are small. Given any number ‘n’, there will be only n-1 numbers less than n, but an infinite group of numbers greater than it.
Addressing the unjustly forgotten mathematician Archytas, the Roman poet Horace, who was regarded as the finest lyric poet of the era of Emperor Augustus, expressed in his verse perhaps the greatest paradox of them all: that of finite men who spend their lives attempting to scale the infinite.
You who measured the sea and the earth and the numberless sands,
you, Archytas, are now confined in a small mound of dirt
near the Matine shore, and what good does it do you that you
attempted the mansions of the skies and that you traversed
the round celestial vault – you with a soul born to die?
Snowman
Outside it is cold, cold. Ten degrees below, give or take. I step out with my coat zipped up to my chin and my feet encased in heavy rubber boots. The glittering street is empty; the wool-grey sky is low. Under my scarf and gloves and thermals I can feel my pulse begin to make a racket. I do not care. I observe my breath. I wait.
A week before, not even a whole week, the roads showed black tyre tracks and the trees’ bare branches stood clean against blue sky. Now, Ottawa is buried in snow. My friends’ house is buried in snow. Chi
lling winds strafe the town.
The sight of falling flakes makes me shiver; it fills the space in my head that is devoted to wonder. How beautiful they are, I think. How beautiful are all these sticky and shiny fragments. When will they stop? In an hour? A day? A week? A month? There is no telling. Nobody can second-guess the snow.
The neighbours have not seen its like in a generation, they tell me. Shovels in hand, they dig paths from their garage doors out to the road. The older men affect expressions both of nonchalance and annoyance, but their expressions soon come undone. Faint smiles form at the corners of their wind-chapped mouths.
Granted, it is exhausting to trudge the snowy streets to the shops. Every leg muscle slips and tightens; every step forward seems to take an age. When I return, my friends ask me to help them clear the roof. I wobble up a leaning ladder and lend a hand. A strangely cheerful sense of futility lightens our labour: in the morning, we know, the roof will shine bright white again.
Hot under my onion layers of clothing, I carry a shirtful of perspiration back into the house. Wet socks unpeel like plasters from my feet; the warm air smarts my skin. I wash and change my clothes.
Later, round a table, in the dusk of a candlelit supper, my friends and I exchange favourite recollections of winters past. We talk sleds, and toboggans and fierce snowball fights. I recall a childhood memory, a memory from London: the first time I heard the sound of falling snow.
‘What did it sound like?’ the evening’s host asks me.
‘It sounded like someone slowly rubbing his hands together.’
Frowns describe my friends’ concentration. Yes, they say, laughing. Yes, we can hear what you mean.
One man laughs louder than the others. Above his mouth he sports a grey moustache. I do not catch his name; he is not a regular guest. I gather he is some kind of scientist, of indeterminate discipline.
Thinking in Numbers: How Maths Illuminates Our Lives Page 7
