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.
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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.
