Thinking in Numbers: How Maths Illuminates Our Lives (9 page)

Definitions and axioms would shape President Lincoln’s most famous addresses. His powers of rhetoric, persuasion, deduction and logic were all subjected to the severest test. The nation was in crisis. Civil war would shortly come. The President spoke to the entire country in defence of the Union.

I hold, that in contemplation of universal law, and of the Constitution, the Union of these States is perpetual. Perpetuity is implied, if not expressed, in the fundamental law of all national governments. It is safe to assert that no government proper ever had a provision in its organic law for its own termination. Continue to execute all the express provisions of our national Constitution and the Union will endure forever, it being impossible to destroy it except by some action not provided for in the instrument itself.

Proposition: The Union of these States is perpetual.

Clarification: Perpetuity is implied in the fundamental law of all national governments.

Axiom: No government ever had a legal provision for its own termination.

Conclusion: Therefore continue to execute the Constitution and the Union will endure forever.

Throughout Lincoln’s four years in office, intense fighting saw approximately 750,000 men killed, and the nation all but tear itself apart, but the president’s proof would ultimately be vindicated.

‘We are not enemies,’ the President had said in the same national address, ‘but friends.’ Perhaps he was thinking of a proverb attributed to Pythagoras, one that he took as an axiom: ‘Friendship is equality.’

On Big Numbers

In the second of his
Olympian Odes
, the ancient lyric poet Pindar wrote, ‘the sand escapes numbering’. He was expressing the same idea that would lead his fellow Greeks to coin the term ‘sand hundred’ for an inconceivably great quantity.

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 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.

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.