Khayyam

Al-Biruni died in the same year that a boy was born to a Persian tentmaker. The Farsi word for tentmaker is khayyám; the tentmaker named his son, Omar.

It is probable that as a child he studied the Koran. He would have learned to recite its verses aloud, for tradition holds that the scripture is akin to a chant, which is why the angel Gabriel chose to speak its words to the illiterate Muhammad. Perhaps the boy recited such a verse as, ‘Most surely in the creation of the heavens and the earth, and the alteration of the night and the day, there are signs for men who understand.’

Many other books, on many subjects, must also have passed through his hands: books on geometry and the movement of the stars, books on arithmetic and music. He learned many of the pages by heart. It is likely that he also read or heard of Al-Biruni’s compendium of calendars, and smiled at Al-Tabari’s apocalyptic prediction. From long years of cloistered study, indifferent to the company of other people, he earned the bookish reputation of a ‘bad character’.

When he was not reading books, he wrote them. A gifted poet, he was better known in his day as a talented mathematician. ‘The notion that one could use geometric constructions for certain types of algebraic problems was certainly recognised by Euclid and Archimedes,’ writes the mathematician Ramesh Gangolli, ‘but before Omar Khayyam’s construction, only simple types of equations . . . were thought to be amenable to the geometric method . . . Khayyám opened the door to the study of the more general question: What kind of algebraic problems can be represented and solved successfully in this manner?’

The young Persian’s receptivity to inspiration must have been immense. When the sunlight shone through the latticework windows of his study, it danced upon the walls in geometrical shapes. Khayyám’s pen traced rubai (poems) of four short rhyming lines, tight as theorems, writing the words from right to left. Some say he composed only sixty such poems; others, six hundred. He also wrote a commentary on Euclid’s Elements that Gangolli tells us explained ‘in more detail many aspects that were left implicit and clarified many misconceptions about the structure of axiomatic systems.’

Polyvalent talent like his is rare in any age. It likely led to jealousies, snide comments, upturned noses from certain quarters among his fellow countrymen.

Khayyam

Al-Biruni died in the same year that a boy was born to a Persian tentmaker. The Farsi word for tentmaker is khayyám; the tentmaker named his son, Omar.

It is probable that as a child he studied the Koran. He would have learned to recite its verses aloud, for tradition holds that the scripture is akin to a chant, which is why the angel Gabriel chose to speak its words to the illiterate Muhammad. Perhaps the boy recited such a verse as, ‘Most surely in the creation of the heavens and the earth, and the alteration of the night and the day, there are signs for men who understand.’

Many other books, on many subjects, must also have passed through his hands: books on geometry and the movement of the stars, books on arithmetic and music. He learned many of the pages by heart. It is likely that he also read or heard of Al-Biruni’s compendium of calendars, and smiled at Al-Tabari’s apocalyptic prediction. From long years of cloistered study, indifferent to the company of other people, he earned the bookish reputation of a ‘bad character’.

When he was not reading books, he wrote them. A gifted poet, he was better known in his day as a talented mathematician. ‘The notion that one could use geometric constructions for certain types of algebraic problems was certainly recognised by Euclid and Archimedes,’ writes the mathematician Ramesh Gangolli, ‘but before Omar Khayyam’s construction, only simple types of equations . . . were thought to be amenable to the geometric method . . . Khayyám opened the door to the study of the more general question: What kind of algebraic problems can be represented and solved successfully in this manner?’

The young Persian’s receptivity to inspiration must have been immense. When the sunlight shone through the latticework windows of his study, it danced upon the walls in geometrical shapes. Khayyám’s pen traced rubai (poems) of four short rhyming lines, tight as theorems, writing the words from right to left. Some say he composed only sixty such poems; others, six hundred. He also wrote a commentary on Euclid’s Elements that Gangolli tells us explained ‘in more detail many aspects that were left implicit and clarified many misconceptions about the structure of axiomatic systems.’

Polyvalent talent like his is rare in any age. It likely led to jealousies, snide comments, upturned noses from certain quarters among his fellow countrymen.

Numbers and Birth of Islam

For the Bedouins living before the era of Muhammad, time did not exist. Or rather, they thought of it as an all-enveloping and all-enfeebling mist, without clear shape or pattern. Only the bright stars overhead pricked the pervasive gloom, helping the nomads to anticipate rain and decide when to take their animals to pasture. To make sense of their lives, the men sang songs, and the songs they sang told of distant earthquakes and battles. It was the sole history that they knew.

The Prophet’s birth, to believe tradition, coincided with one of these battles: the so-called ‘Event of the Elephant’ (occurring in the latter half of the sixth century of the Christian Era), when Mecca fell under the siege of a foreign king’s army with a white elephant at its helm. According to a tale later told in the Koran, God sent a cloud of birds to pelt the attackers with stones until they fled.

Alongside Muhammad’s revelation of a new religion came his revelation of time. Gone now was the idea of life as constituted by a flux of vague, discontinuous and casual moments. Five compulsory prayers – Fajr (at dawn), Dhuhr (after the sun’s zenith), Asr (during late afternoon), Maghrib (at sunset) and Isha (at twilight) – regulated each day. All our days, said the Prophet, are numbered. Each follows the other in meaningful succession.

‘God wraps night around day, and He wraps day around night.’

Seven of them together make a week (beginning on what we call Saturday), the span in which, it was said, God had progressively created the world: the earth on the first day, the hills on the second, the trees on the third, all unpleasant things on the fourth, the light on the fifth, the beasts on the sixth, and Adam, who was the last of creation, about the time of the Asr prayer on the seventh.

Look up at the heavens, Muhammad urged his followers. Each month, he declared, began when the moon appeared ‘like an old shrivelled palm-leaf’. Divinely ordained properties separated the months and made them distinct. During four of the months, it was forbidden to draw a sword. During certain others, believers could set out on pilgrimage. One month, called Ramadan, was set aside for fasting during the hours of daylight. Twelve lunar months composed one year.

Muhammad had been preaching for about a decade when, at the approximate age of fifty, the rulers of Mecca ousted him and his small band of followers from the city. On camels, they travelled north to the oasis town of  Yathrib, finding refuge there. The flight, known as hegira, became the founding date of the Islamic calendar; henceforth, every period of time would be precisely accounted.

Stories Around Big Numbers

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 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.

***

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 nephew. 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.

Numbers and Languages

The Veddas, an indigenous people of Sri Lanka, are reported to have only words for the numbers one (ekkamai) and two (dekkamai). For larger quantities, they continue: otameekai, otameekai, otameekai . . . (‘and one more, and one more, and one more . . .’).Another example is the Caquintes of Peru, who count one (aparo) and two (mavite). Three they call ‘it is another one’; four is ‘the one that follows it’.

In Brazil, the Munduruku imitate quantity by according an extra syllable to each new number: one is pug, two is xep xep, three is ebapug, and four is edadipdip. They count, understandably, no higher than five. The imitative method, while transparent, has clear limitations. Just imagine a number word as many syllables long as the quantity of trees leading to a food source! The drawling, seemingly endless, chain of syllables would prove far too expensive to the tongue (not to say the listener’s powers of concentration). It pains the head even to think about what it would be like to have to learn to recite the ten times tables in this way.

***

There is a tribe in the Amazon rainforest who know nothing whatsoever of numbers. Their name is the Pirahã or the Hi’aiti’ihi, meaning ‘the straight ones’. The Pirahã show little interest in the outside world. Surrounded by throngs of trees, their small clusters of huts lie on the banks of the Maici River. Tumbling grey rain breaks green on the lush foliage and long grass. Days there are continuously hot and humid, inducing a perpetual look of embarrassment on the faces of visiting missionaries and linguists. Children race naked around the village, while their mothers wear light dresses obtained by bartering with the Brazilian traders. From the same source, the men display colourful T-shirts, the flotsam of past political campaigns, exhorting the observer to vote Lula.

Manioc (a tough and bland tuber), fresh fish and roasted anteater sustain the population. The work of gathering food is divided along lines of sex. At first light, women leave the huts to tend the manioc plants and collect firewood, while the men go upriver or downriver to fish. They can spend the whole day there, bow and arrow in hand, watching water. For want of any means of storage, any catch is consumed quickly. The Pirahã apportion food in the following manner: members of the tribe haphazardly receive a generous serving until no more remains. Any who have not yet been served ask a neighbour, who has to share. This procedure only ends when everyone has eaten his fill.

The vast majority of what we know about the Pirahã is due to the work of Daniel Everett, a Californian linguist who has studied them at close quarters over a period of thirty years. With professional perseverance, his ears gradually soothed their cacophonic ejections into comprehensible words and phrases, becoming in the process the first outsider to embrace the tribe’s way of life.

To the American’s astonishment, the language he learned has no specific words for measuring time or quantity. Names for numbers like ‘one’ or ‘two’ are unheard of. Even the simplest numerical queries brought only confusion or indifference to the tribesmen’s eyes. Of their children, parents are unable to say how many they have, though they remember all their names. Plans or schedules older than a single day have no purchase on the Pirahã’s minds. Bartering with foreign traders simply consists of handing over foraged nuts as payment until the trader says that the price has been met.

Nor do the Pirahã count with their bodies. Their fingers never point or curl: when indicating some amount they simply hold their hand palm down, using the space between their hand and the ground to suggest the height of the pile that such a quantity could reach.

It seems the Pirahã make no distinction between a man and a group of men, between a bird and a flock of birds, between a grain of manioc flour and a sack of manioc flour. Everything is either small (hói) or big (ogii ). A solitary macaw is a small flock; the flock, a big macaw. In his Metaphysics, Aristotle shows that counting requires some prior understanding of what ‘one’ is. To count five, or ten or twenty-three birds, we must first identify one bird, an idea of ‘bird’ that can apply to every possible kind. But such abstractions are entirely foreign to the tribe.

With abstraction, birds become numbers. Men and maniocs, too. We can look at a scene and say, ‘There are two men, three birds and four maniocs’ but also, ‘There are nine things’ (summing two and three and four). The Pirahã do not think this way. They ask, ‘What are these things?’ ‘Where are they?’, ‘What do they do?’ A bird flies, a man breathes and a manioc plant grows. It is meaningless to try to bring them together. Man is a small world. The world is a big manioc.

It is little surprise to learn that the Pirahã perceive drawings and photos only with great difficulty. They hold a photograph sideways or upside down, not seeing what the image is meant to represent. Drawing a picture is no easier for them, not even a straight line. They cannot copy simple shapes with any fidelity. Quite possibly, they have no interest in doing so. Instead their pencils (furnished by linguists or missionaries) produce only repeating circular marks on the researcher’s sheet of paper, each mark a little different to the last.

Perhaps this also explains why the Pirahã tell no stories, possess no creation myths. Stories, at least as we understand them, have intervals: a beginning, middle and an end. When we tell a story, we recount: naming each interval is equivalent to numbering it. Yet the Pirahã talk only of the immediate present: no past impinges on their actions; no future motivates their thoughts. History, they told their American companion, is ‘where nothing happens, and everything is the same.