Thinking in Numbers: How Maths Illuminates Our Lives

by Daniel Tammet

  Not that I wish to imply that playing chess is all instinct. Alacrity has its virtues (the absence of indecision for one), but they are not the virtues of the longer form. At its best, chess – like language – privileges reflection and careful thought. Looked at one way, a game of chess amounts to a long series of shifting problems, the most intriguing among them capable of making unique demands upon our imagination. As with a beautiful passage in a novel, we feel that we could happily spend almost any amount of time in its company.

  Occasionally an amateur player will scour the back pages of certain newspapers (the kind whose pages end up wrapping glass vases, but never greasy chips) to savour and solve these chess problems. Illustrations of the board, its pieces frozen in print, compete for space with the week’s crossword puzzle. ‘White to play and mate in three’ or ‘Black to play and draw’ its title will announce. Quite often the position displayed is at least half cleared of its starting pieces, the game has attained its final moments. The amateur stares at the inky pieces and the smudged squares, waiting for a surge of inspiration. It is more or less the same experience as when we read some striking lines of a poem.

  Many of these published chess positions have never known an actual board; they are the figment of some inventor’s imagination. Among these inventors figures one ‘Vladimir Sirin’, better known as the multilingual novelist and poet Vladimir Nabokov, for whom such compositions equated to the ‘poetry of chess’ (his 1969 anthology, Poems and Problems, was illustrated by eighteen of his own efforts).

  Those sixty-four squares, their peculiar geometry, fascinated the great wordsmith to the point of obsession. The right-angled triangle, for instance, traced by a king’s vertical (or horizontal), and then diagonal, retreat across the board disobeys Pythagoras’s famous theorem. Off the board, in the ‘real world’, moving say three paces across and as many paces up (or down) would produce a greater diagonal distance (four and a quarter paces long) between the start and end points. The king’s ‘triangle’, on the other hand, has identical length (traversing the exact same number of squares) on every side. This optical illusion (expecting a diagonal route to require a greater number of moves than either a vertical or horizontal path) plays an essential role in the chess problem composer’s craft.

  Like his words, Nabokov’s chess pieces relied on precise positions and combinations for their meaning. He saw to it that the pieces’ values differed sharply from the amateur’s expectations. The ‘value’ of a position’s queen, for example, – while generally considered to be twice that of a rook, thrice the worth of a knight or bishop, and nine times as important as a pawn – might decrease to next to nothing if immobilised to a corner or back row by the careful set up of lesser pieces.

  Numerous mathematicians also double up as inventors of chess problems. They create positions to solve questions like, what is the maximum number of possible checkmates in one move? Answer, forty-seven. Or what is the smallest number of bishops needed to occupy or attack every square? Answer, eight (the same as for the smallest number of rooks). Or else they compose whole games of a specific number of moves to reach a given position.

  There is a final analogy between chess and language that I should mention. I began these pages reflecting on mistakes. I said that grandmasters make magisterial mistakes, founded on great creative intuition. Young children do, too. From ambient talk they get wind of words, but their minds then make of them what they will. As a matter of fact, the toddler’s speech far exceeds the mimicry of adults; its features are distinct and demonstrate invention. For example, we have all heard a young child say something like ‘three mouses’ or ‘I goed’, instead of ‘mice’ and ‘went’, though no parent would utter such a thing. More inventive still were the kids who reportedly suggested that a ‘myth’ was a female moth, or that benign is ‘what you will be after you be eight’.

  Perhaps they grew up to become grandmasters.

  Selves and Statistics

  For each one of us, nothing is so personal, so intimate and so selfish as our death. From time immemorial, men have sought to predict its hour. They scattered bird entrails, or elaborated bad dreams, or consulted the office of an oracle. When these prophecies were not plainly wrong, deluded or hysterical, they often proved unhelpfully vague and misleading. According to one legend, a Scythian prince consulted a Greek oracle to discover how he would die. A mus (mouse), he was told, would be the agent of his demise. The prince took the forewarning to heart. He had his houses cleared of mice and even snubbed the company of any man called Mus. All the same, shortly thereafter, death came for him. The cause? He died from an infected arm muscle (the word muscle, in Greek, means ‘mouse’).

  The idea of death as a statistical phenomenon amenable to calculation occurs only toward the end of the seventeenth century, with the publication of the world’s first mortality table. This new conception of death had gestated so long for a good reason: it upended completely how men and women understood their individual lives. ‘Society’ – if and when this term existed at all – had always been held to be only a loose confederation of free souls. The precise business and destiny of each was an impenetrable mystery. Crowds were merely freaks, monsters with many heads and countless limbs. That a person might be in any way comparable to the crowd, that he might be deciphered by studying the behaviour of his family, his village, his fellow countrymen, seemed both impossible and absurd.

  And if the individual was beyond the pale of understanding, so too was his eventual extinction. No scholar, bitter experience told most people, could hope to get inside the Grim Reaper’s skull. Death touched rosy-cheeked infants and decrepit widows alike, with no apparent rhyme or reason. An ancient man on his last legs might somehow survive another decade, while his grandson, beaming with youth and health, not live to see the following spring.

  Stories took the place of science, their tellers repeating the same message over and over: life is full of surprises. Remember Old John, one tale would go, Old John who laughed so hard at his neighbour’s joke that his heart gave out? The farmer’s wife who was butted into her grave by the goat? The squire who caught cold sleeping in church?

  It was in this atmosphere of ambivalence that Edmond Halley, who found lasting fame calculating a comet’s orbit, published An Estimate of the Degrees of the Mortality of Mankind in 1693. Halley based his figures on the city of Breslaw, capital of the province of Silesia, ‘near the confines of Germany and Poland and very near the latitude of London’ with a total population of 34,000. For five years running, monthly figures for every birth and death in the city had been collated: 6,193 births and 5,869 burials in all. Of the newborns, Halley discovered, twenty-eight per cent perished in their first year; only a little more than half lived to celebrate their sixth birthday. Most of these however would go on to have children of their own. ‘From this age the infants being arrived at some degree of firmness grow less and less mortal.’

  Of citizens between the ages of nine and twenty-five, the number of deaths each year equated to about one per cent. This figure rose to three per cent for those aged between twenty-five and fifty, jumping to ten per cent for those having reached the ripe age of seventy. ‘From thence the number of the living being grown very small, they gradually decline till there be none left to die.’

  Halley used this combined data to calculate ‘the differing degrees of mortality, or rather vitality in all ages’. For example, to estimate the odds that a person aged twenty-five would not die within the next twelve months, he compared the city’s twenty-five-year-olds (totalling 567) to its twenty-six-year-olds (560), and thereby concluded that an ‘average’ twenty-five-year-old would have odds of 560 to 7, or 80 to 1, that he outlive the year.

  What is the probability that a man of forty live a further seven years? Halley took the number of forty-seven-year-olds (377) and subtracted it from the number of forty-year-olds (445) to find the difference between the two ages (68). The odds that a forty-year-old become a forty-seven-year-old were t
herefore 377 divided by 68, or 5.5 to 1.

  How many years might a man of thirty reasonably expect to have still ahead of him? To answer this question, Halley would first determine the number of thirty-year-olds (531), and then halve it (equivalent to considering as even odds that the person would die within the period). This halved figure (265), he then finds, equates to the number of citizens between fifty-seven and fifty-eight years of age. The ‘average’ thirty-year-old could look forward to a further twenty-seven or twenty-eight years.

  From his findings Halley drew a suitably pious conclusion.

  Unjustly we repine at the shortness of our lives, and think our selves wronged if we attain not old age; whereas it appears hereby, that the one half of those that are born are dead in seventeen years time . . . so that instead of murmuring at what we call an untimely death, we ought with patience and unconcern to submit to that dissolution which is the necessary condition of our perishable materials, and of our nice and frail structure and composition: And to account it as a blessing that we have survived, perhaps by many years, that period of life, whereat the one half of the whole race of mankind does not arrive.

  It was a mortality table just like the one first drawn up by Halley that the American palaeontologist Stephen Jay Gould would scrutinise at length, three centuries later, in the summer of 1982. It was the summer when the United States’ Equal Rights Amendment was voted down, when Italy beat West Germany in the final of the football World Cup, when a debt crisis broke loose across South America, and when Gould, sat in his doctor’s office, learned that he would shortly die. He was forty and had just been diagnosed with a rare and incurable form of cancer. Poring over books at Harvard’s Medical Library, thick as hand-spans, subsequently taught him all that there was then to know about the condition and its survival rates. In summary: he had a median of eight months left to live.

  Gould could not take seriously Halley’s well-meaning words of advice. He would not submit readily to his body’s dissolution; he wanted to survive. He thought of his wife, his two young sons, his high-flying career. A million other things went through his mind: the museum’s Tyrannosaurus rex, all gigantic teeth and bulky bones, that he had seen as a small boy; his father, in his evening slippers, reading Das Kapital; the opening bars of Gilbert and Sullivan’s Mikado; his Yankees season ticket; the Pepperidge Farm cookies in his office drawer. His office. What would his microscopes do, and his favourite rattan chair? Sit on the wrong shelf; stand in the unlit corner. Gather dust.

  What did Gould do in such dire circumstances? What could he do? He did what virtually every recipient of bad news does: he hunted feverishly after even the slightest, even the faintest positive angle. He would not abandon hope. A median of eight months; that is what the statistics said. If half of all the patients with his cancer would die within eight months of their diagnosis, it meant that half would live beyond eight months. Some would live on for years. This thought comforted him. His mind tightened around it. Age? He was still young. Class? His family lived on the better side of town. Health? A little on the heavy side, but he had no other baggage. Attitude? He identified a strong will, even temper and a clear purpose for living. His chances of landing in the latter half of the patients’ prospects seemed to him great.

  He would have only one future death, not thousands, and the median said next to nothing about it. This became his mantra. Friends and family asked him to explain. Averages talk about populations, not persons, he would reply. If I died a thousand times, approximately half of those deaths would occur within eight months. The other half would follow one by one: days, weeks, months or years later. Who can tell where my single death, out of the thousand possible deaths, will fall?

  The following months were tough and turbulent for Gould, full of boredom, pain and exhaustion. Doctors radiated his body, pumped it with drugs and put it under the knife. He haemorrhaged weight – in all, he lost a third of his 180 pounds. His hair embarrassed him by falling out. The lonely and tedious hours of treatment piled one upon the other, oppressing and enfeebling him. And yet he survived. His cancer went into remission. Two years later he was well enough to write a long article about his experience, ‘The Median is Not the Message’. A decade after its publication he was still going strong. ‘I am a member of a very small, very fortunate, and very select group – the first survivors of the previously incurable cancer,’ he wrote.

  In March 2002, Gould – aged sixty – published his magnum opus, The Structure of Evolutionary Theory, 1,342 pages long. It was the seventeenth book to bear his name since his diagnosis twenty years before.

  Two months later, his personal death finally arrived, the result of a second, unrelated cancer.

  Knowing how to read the mortality table’s numbers probably extended Gould’s life by many years (he affirmed a likely link between an individual’s state of mind and his immune system). Not knowing how to read them, on the other hand, cost one man and his family very dear.

  André-François Raffray’s story is an extreme illustration, perhaps the most extreme, of confusing persons with percentages. Raffray was a long-serving and successful public notary, living and working in the southern French city of Arles. Among his clients was a ninety-year-old widow, without heirs, whose name was Jeanne Calment. One day, in 1965, Raffray agreed to purchase the widow’s house under a scheme the French call rente viagère: in return for paying Calment a monthly sum of 2,500 francs, he would own the property upon her death.

  Raffray must have imagined that he had struck an excellent deal. The widow’s house was worth close to half a million francs. Assuming that she lived another three years – the current life expectancy for an average ninety-year-old French woman – he would shell out fewer than 100,000 francs in all. Over twenty per cent of nonagenarians died before their next birthday. The statistics, he thought, were on his side. ‘Even if she makes it to ninety-four, or ninety-five, or ninety-six, I will still have the house in the end for only a fraction of its value. And what if she goes on and on, to ninety-seven, or ninety-eight, or – God forbid – to one hundred? But how many people ever live to one hundred? Not even one person in a thousand! To think that she might go on another ten years! I cannot imagine it. But what do I care? Let her keel over at one hundred: I will still make a tidy profit.’ Such was probably his line of thought.

  It was a mistaken assumption that the very old are all more or less the same. The notary’s acquaintance with Madame Calment was slight. Her snow-white hair, her birdlike frame, her papery skin, he mistook for frailty. He saw these features and thought at once of every elderly person he had ever met. Faces, bodies, lives blended together in his mind. What did these people, averaged, have in common? Sickness, sadness, shortness of breath.

  But before Madame Calment was old she had been young, and she had ridden a bicycle through the cobbled streets of Paris, and stretched after fluffy tennis balls, and eaten tins of fruit and salads slick with olive oil. Early marriage to a wealthy merchant had freed her hands for tinkling piano keys and theatre applause. No illness had ever troubled her.

  Little had changed for her since moving to the warm and sunny South. She had taken all her favourite things with her, except for the buried husband. But she had long grown comfortable with her solitude. She was not afraid of silence, of hearing the rhythm of her heart. Nor did she worry about her looks: make-up, she had learned, could not resist her frequent tears of laughter. At age eighty-five, she wore silly plump padding for her first fencing lessons without giving it a second’s thought. She still loved walking in the open air. Regular sips of port wine and snacks of chocolate brightened her day.

  Careful study of the mortality table had taught the notary with what frequency previous nonagenarians had died and after how many years, but he had not thought about the future. It was a fact, for example, that France counted no more than a few hundred centenarians in 1965. But that was then, and the widow would reach her hundredth birthday, if she reached it, in ten years’ time. How many ce
ntenarians would France have in 1975? In 1980? In 1990? That is the kind of question that the notary forgot to ask. Around the world, medicine and technology was rapidly improving. Once significant causes of death – from flu, vitamin deficiency or high blood pressure – were dwindling. Within a generation, the number of centenarians in France would increase twenty fold.

  And what of the table’s statistics? They deserve a closer look. For one thing, the data available for the very old was necessarily sparse and unreliable. Too few, before the generation of Madame Calment, had lived long enough to give being ninety years old a proper try. Statisticians knew next to nothing of a nonagenarian’s medical needs, eating habits, daily routines and much else besides. Guesswork had to fill in the gaps.

  Life expectancy: three years, announced the table. But let us see what this actually means. If there were ten thousand ninety-year-olds in 1965, there would be around five thousand still living in 1968. The life expectancy of these ninety-three year olds would not be zero. What would it be? (This was another question that the notary did not think to ask.) Almost three years. Live three years in your nineties and the chances are fair you will live three years more. And if there were five thousand ninety-three-year olds in 1968, there would be around two thousand surviving in 1971. These ninety-six-year olds could expect, on average, to live a further two years. In 1973, about a thousand – ten per cent of the original number of ninety-year-olds – would still be alive. Close to half of these in turn would live long enough to reach their hundredth birthday.

  Madame Calment, in February 1975, numbered among them. One hundred years old, but still in fine shape, on her feet every day. She left the dying to others. At the age of 105, her notary’s monthly cheques amounted to the full value of her house. But still he had to keep on paying.