Thinking in Numbers: How Maths Illuminates Our Lives Read online

  Drake wrote out his reasoning in the impressive-looking shorthand of an equation.

  N = N* x fp x ne x fl x fi x fc x fL

  Where N is the number of communicating civilisations in our galaxy.

  N* is the number of stars in the Milky Way.

  Fp is the fraction of stars with planets orbiting them.

  Ne is the number of planets per star ecologically capable of life.

  Fl is the fraction of those planets where life evolves.

  Fi is the fraction of these living planets that evolves intelligent life.

  Fc is the fraction of these that communicate with others.

  fL is the fraction of a planet’s life during which the civilisation survives.

  Now to a journalist’s question, ‘Do other intelligent civilisations exist?’ came Drake’s reply: ‘With almost absolute certainty.’ A certainty identical to and inherited from the one he had heard in his father’s voice.

  What would the astronomer give to actually find another world! (His Russian colleagues reminded him that ‘world’ in their language is a synonym for ‘peace’.) Every day, in his observatory, he set to work. Through thick glasses he followed the chart recorder’s progress, watched the needle fidget, saw the ink illustrate the contours of random noise. Occasionally, growing more and more impatient, he would snatch up headphones and listen in to the reception. He sat stock still in the room, his heart in his mouth, and listened. Listened for what exactly? For a beep, a buzz, an electronic whisper. He watched and listened and waited and the hours turned. But the surprise did not come. Nothing came, except the months, the years, the hours.

  With time, the technology became refined and upgraded. More and more assistants leant their ears and patience to the expanding project. ‘Probability’ was on everyone’s lips. The numbers are on our side, they told the press, their friends, themselves. It is only a matter of time.

  Queasy silence was all that they heard.

  The radio telescopes grow, and the doubts grow. You would have to be superhuman not to doubt. Except perhaps for Drake, who had invested so heavily in his hope.

  The problem posed by the silence was striking, though. For if thousands of communicating civilisations have evolved throughout our galaxy over millions (or even billions) of years, why has not a single one colonised its environs, including the Earth? A civilisation far older than our own, out of the thousands predicted by Drake’s equation, would need only a few million years – a cosmic blink of an eye – to have colonised the Milky Way. Or, at the very least, to have flooded us with signs of its presence.

  Look longer! Listen harder! Drake’s response was emphatic. Perhaps the other civilisations are checking us out before they get in touch. Or perhaps they content themselves with colonising only their solar systems. Or perhaps the cost of interstellar travel is too steep. Or perhaps, they never invented the radio. Perhaps, perhaps, perhaps.

  Is Anyone Out There? To coincide, in 1992, with the five-hundredth anniversary of Columbus’s discovery of America, Drake published a book posing this question. The answer, he felt, was closer than ever. He wanted to ‘prepare thinking adults’ for the ‘imminent detection of signals from an extraterrestrial civilization.’ What kind of civilisation? They would resemble humans, with heads on the top of their bodies and two legs to walk on. But instead of two arms, they would have four: ‘four make for a much better design’. They would also be immortal, innocent of death. ‘This discovery, which I fully expect to witness before the year 2000, will profoundly change the world.’

  In the same year, NASA’s computers performed fifty million tests per second on the data from the largest ever and most sophisticated radio scan of the heavens. They found nothing.

  Biologists, meanwhile, were offering a reassessment of the equation’s assumptions. Drake and his colleagues applied ‘strictly deterministic thinking,’ wrote the Harvard biologist Ernst Mayr. ‘Such thinking is often quite legitimate for physical phenomena, but is quite inappropriate for evolutionary events or social processes such as the origin of civilizations.’ Another biologist, Leonard Ornstein, pointed out that ‘even if we allow that the universe may be awash with planets with flourishing “protometabolism”, and even “protocells”, it does not necessarily follow that contingent events that contributed to the next step in the origin of life have been mimicked on even one of these hypothetical worlds.’

  Ornstein suggested an analogy: imagine that we dipped our hand only once into a bag of marbles and withdrew just one marble. This marble is blue-green. Conclusion? We could as equally suppose that the bag contained only one-in-a-million other blue-green marbles, or none at all, as suppose that all or many still in the bag would possess a similar colour.

  The only thing we can know for certain is that the probability of intelligent life in our universe is above zero (for were it zero, I would not be here to write this sentence and you would not be here to read it). The rest is speculation. Since the Big Bang, there have been billions of civilisations. There have been millions of civilisations. There have been thousands of civilisations. There have been hundreds of civilisations, or tens. Or one.

  Why not? Probability is often expressed using large but finite numbers: ‘one in a thousand’, ‘one in a million’. But perhaps the probability of life, intelligent life, appearing somewhere in our universe is infinitesimal. If so, a universe would need infinitely many planets to produce even a finite number of civilisations (i.e., one).

  Such a conclusion ought to be at least as motivating to us as Drake’s, especially in an age of high-stakes international diplomacy, atomic bombs and climate change. As the astronomer Michael Papagiannis concluded, ‘Knowing we are the only ones might make us realise that we are too valuable to destroy.’

  The Calendar of Omar Khayyam

  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 ap
proximate 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.

  Exquisitely contrived clocks sprang up across the Islamic world throughout the medieval period. The most impressive of these their makers filled not with sand, but water. A Chinese traveller’s report from a visit to Antioch, three centuries after Muhammad, tells of a water clock within the royal residence that suspended twelve golden balls corresponding to the twelve hours of the day. At each hour one of the balls fell, splashing two o’clock, three o’clock, four o’clock, ‘the sound of which makes known the divisions of the day without the slightest error’. Another water clock, described by Al-Jazari in his 1206 Book of Knowledge of Ingenious Mechanical Devices, stood as high as two men and featured robotic drummers, trumpeters and cymbalists who played in turns according to the time of day.

  By employing water – a precious substance in a peninsula largely composed of desert – the clockmakers demonstrated the reverence they accorded to time. Indeed, the Prophet had told his faithful to pay minute attention to its reckoning. Mosques employed muwaqqits (time-keepers) to calculate the official hours for each prayer. Generations of scholars scrupulously debated the world’s age. The historian Al-Tabari, for example, figured that the world would endure in total seven thousand years, of which his generation had before them only two hundred. He based his calculation on a saying of the Prophet, in which Muhammad likens the time remaining until the Last Day to the fraction of time between the Asr (afternoon) prayer and sunset (about one-fourteenth of a day).

  Writing in the fourth and fifth century of hegira (the eleventh century according to our way of reckoning), a near contemporary of Al-Tabari, Al-Biruni, compiled his Chronology of Ancient Nations. In it, he compares the calendars of the other great civilisations. The Greeks, the Syrians and the Egyptians, he noted, all use a calendar of 365 and one one-quarter days, summing the quarters to make one complete extra day every four years.

  ‘The ancient Egyptians followed the same practice, but with this difference, that they neglected the quarters of a day till they had summed up to the number of days in one complete year, which took place every 1,460 years; then they added one extra year.’

  The ancient Persians, he continued, also neglected these quarters of a day, though for a period of 120 years, after which they added one extra month.

  As fate would have it (though Muslims believe no such thing exists, and that every moment is the conscious creation of God), 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’.

  One story relates how a student visited Khayyám daily before dawn to learn from him, before badmouthing him to the other townspeople. Hearing of this, Khayyám secretly invited the town’s musicians to call on him at dawn the next day. When the unsuspecting student arrived as usual for his lesson, Khayyám ordered the musicians to beat their drums and blow their trumpets; the commotion awakened people from every quarter. ‘Men of Nishapur,’ Khayyám said, addressing the crowd, ‘He comes every day at this hour to my house, and studies with me, but to you he speaks of me in the manner you know. If I am really as he says, then why does he come and study with me? And if not, why does he abuse his teacher?’

  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. In one of several treatises on algebraic problems, Khayyám complained about the trials of the mathematician’s life.

  I was unable to devote myself to the learning of this algebra and the continued concentration upon it, because of obstacles . . . which hindered me; for we have been deprived of all the people of knowledge save for a group, small in number, with many troubles, whose concern in life is to snatch the opportunity, when time is asleep, to devote themselves meanwhile to the investigation and perfection of a science; for the majority of people who imitate philosophers confuse the true with the false, and they do nothing but deceive and pretend knowledge, and they do not use what they know of the sciences except for base and material purposes.

  In 452 (1074 CE by the Julian calendar), Sultan Jalal Al-Din invited Omar to the capital. His long Farsi texts, packed with numbers and equations, had preceded him. Even at the zenith of the golden age of Islamic mathematics, Khayyám’s talent marked him out. Anxiously, expectantly, he must have followed his guide through the halls of the turquoise-studded palace. Tessellating tiles ran the length of the floor; mirrors hanging on the walls returned to him every one of his features, down to the wrinkles of laughter around his eyes.

  The Sultan that Khayyám met hardly looked the part: he was very young, not yet twenty years of age. He was keen to make his mark. His illustrious guest, the Sultan’s Vizier reports, was promptly showered with the prince’s praises, and offered an annual pension of 1,200 mithkáls of gold. For this, Khayyám agreed to accept an important commission: he was to create – provided that God were willing – a new civil calendar in the young Sultan’s name.

  Persia, geographically vast and intricately governed, had long lived on two times: while the imams practised the faith using the hegira calendar, the bureaucrats counted their days by looking to the sun. The old civil calendar consisted of twelve months each of thirty days, excepting the eighth month, which contained thirty-five days. Its total of 365 days (eleven days more than in a lunar year) helped fix administrative dates far more closely to the seasons: an essential requirement when the nation’s annual tax revenues relied heavily on the autumn harvests. But even these eleven extra days could not keep up precisely with the seasons: each year, as Al-Biruni had recorded, accumulated a lag of a quarter day. This was the problem that Khayyám set out to solve.

  Day and night, Omar pondered how best to reform the old civil calendar. Astronomy had never be
fore found itself so flattered and moneyed. A large observatory duly arose, from which Khayyám and his colleagues stalked the sky. He studied the sun’s pathway through the twelve constellations of stars, and compiled detailed statistical tables. With this data he mapped a calendar based on the seasons, beginning the year (which the Persians called Nowruz) on the spring equinox (March 20 or 21), with the fourth month falling on the summer solstice (June 21), the seventh month on the autumn equinox (September 21), and the tenth on the winter solstice (December 20 or 21).

  To resolve the lagging of quarter days, Khayyám devised an ingenious formula. His calendar interleaved eight extra days over each thirty-three-year span. The calculation: 365 + 8/33 = 365.2424 days, aligned almost perfectly with the actual year length (365.2423 days), and proved even more accurate than that of the later Gregorian calendar: 365 + ¼ – 1/100 + 1/400 = 365 + 97/400 = 365.2425 days.

  The Sultan officially adopted Khayyám’s reform on Friday March 15, 1079 (Farvardin 1, 458 according to the new calendar). Drums and canon blasts across the nation proclaimed the first New Year.

  Concerning Khayyám’s later years, we can say very little. The Sultan’s premature death, just ten years after the calendar’s adoption, brought an end to his generous patronage. Khayyám quit the royal court, only the ritual pilgrimage to Mecca delaying a return to his hometown. He continued to write poems.

  Ah, but my computations, People say,

  Reduced the year to better reckoning? Nay,