Thinking in Numbers: How Maths Illuminates Our Lives

by Daniel Tammet

  Five more years passed and Madame Calment begrudgingly moved in to a nursing home. At 110, her age exceeded the maximum figure of most mortality tables. At 113 she became the oldest person in the world: la doyenne de l’humanité.

  Even then, she did not die. Her eyes dimmed, her joints stiffened, but she remained in good spirits. Statisticians who had long estimated a maximal human life span of between 110 and 115 years were proven wrong. Madame Calment became the first person in history to celebrate her one hundred and sixteenth birthday, her one hundred and seventeenth birthday, her one hundred and eighteenth birthday, her one hundred and nineteenth birthday.

  In 1995, ‘Jeanne’ as people now called her, attained 120 years of age. The nursing home, which until then had received few visitors, suddenly swelled with reporters from around the world. Tiny, desiccated, she sat before a bank of cameras. She does not smile, complained one of the photographers. Ask her if she wants to go on living, demanded a journalist. The nursing home director cupped her hand around the old woman’s right ear, as though playing an absurd game of Chinese whispers. ‘The monsieur would like to know if you want to live a little longer?’ she shouted into the ear.

  Yes.

  Her notary was absent that day. Long retired, nearing eighty, he was not well enough to attend. Later that year, he died. He had never set foot inside Madame Calment’s house. Under the terms of the thirty-year-old contract, his widow continued to pay. By the time Madame Calment eventually died two years later, aged 122, the notary and his family had paid nearly a million francs: many times the anticipated payout and twice what the property had been worth when they took on the arrangement.

  From where, we might well ask ourselves, did this strange idea of an average cancer patient or an average nonagenarian originate? A Treatise on Man and the development of his faculties, introduced l’homme moyen (‘the average man’) to the nineteenth-century world. ‘If an individual at any given epoch of society possessed all the qualities of the average man, he would represent all that is great, good, or beautiful.’

  The author – a Belgian mathematician called Alphonse Quetelet – imagined how Nature created men. He pictured Nature as a bowman who aimed continuously for the statistical centre. The bull’s-eye would be a perfectly median man (or woman): a rational, temperate person free from any excess or deficiency. Most individuals, according to this view, were Nature’s errant arrows. They straggled, nearer or farther, around the mean. The taller-than-average flaunted the shorter man’s missing inches; the drunkard lacked his part of the teetotaller’s (excessive) self-control; the hairy man wore the bald man’s follicles.

  Quetelet therefore urged his readers to take a wide view of humanity. Every person, after all, was ‘but a fraction of mankind’. Instead of interesting themselves with men and women, scientists should study the population as a whole. ‘The greater the number of individuals observed, the more do individual peculiarities, whether physical or moral, become effaced, and allow the general facts to predominate, by which society exists and is preserved.’

  So the mathematician averages data collected from a thousand or ten thousand households and learns, for example, that the ‘normal’ male height is five feet and seven inches, that the ‘normal’ time spent reading the newspaper is twelve minutes, that the ‘normal’ diet consists of eggs, potatoes, and meat broth. A height of five feet and five inches, or six feet and two inches, is aberrant; spending five minutes, or thirty, with the newspaper is abnormal; too much fish or too few eggs on a plate is deviant.

  Drawing from this data, the mathematician observes regularities: most men are two inches too tall or too squat, most readers linger three minutes too long (or skim three minutes too short) over their newspaper, most housewives cook six potatoes per week too few or too many.

  But it was not only physical or mental traits that a mathematician might average. Morality, too, could offer itself up to calculation. An analysis of police statistics would reveal the defining characteristics of the ‘average’ criminal, ‘even for such crimes that seem to escape all human foresight like murder since they are committed in general without motive and in circumstances, apparently, the most fortuitous.’ According to Quetelet’s figures, the ‘typical’ murderer would be male, in his twenties, literate and a white-collar worker. He would have alcohol on his breath, and be wearing the light clothes of summer. Probability would put a pistol (rather than a knife, or a bat, or a vial of poison) in his hand.

  This idea spread quickly. It proved popular with scientists and the general public alike. Most frequently it was reinforced neither by logic nor analysis, but by simple prejudice. People laughed, and sneered, and denounced, and ridiculed variously identified ‘types’ of the average man. But worst of all, they believed that such beings really did exist.

  Images propagated the idea with particular efficacy; as the proverb says: a picture is worth ten thousand words. One newspaper caricature was enough to wipe out all trace of Quetelet’s wordy disclaimers (his original book had run to several hundred pages). If the caricature depicted an Irishman – all Irishmen – with a misshapen jaw, feathered cap, protruding teeth, it was plain enough that the ‘average Irishman’ bore more than a passing resemblance. If it showed a filthy, vulgar, gin-swilling beggar, it confirmed how many readers imagined the ‘average poor’.

  Photography – then still a young and innovative technique – was similarly enlisted. Mugshots of eight different individuals were superimposed to reveal the blurry face of the ‘average criminal’. Nine photos of consumptive patients were merged to produce a portrait of tuberculosis. Images from six medals depicting Alexander the Great were morphed to reveal the ancient king’s probable features. ‘It is now proposed,’ announced one magazine, ‘to get a clear idea of Nebuchadnezzar from the various stone and brick slabs upon which his face is graven.’

  But if photography served to popularise the idea of ‘average men’, it also produced an alternative way of looking at ourselves. Identikits, created at the end of the nineteenth century, helped to divert some of the focus back from the typical towards the individual. Photos that magnified and emphasised all manner of different facial features now replaced the phantom faces typifying this kind of man, or that sort of person. Instead of a single abstract representative of this or that contrived category, the images highlighted a wealth of actual noses, foreheads, wrinkles, ears, chins, eyelids, lips and mouths.

  Take the nose, for example. Plainly, nobody has a ‘nose’ – a person has a low nose or a high nose, a wide nose or a narrow nose, a hooked nose or a straight nose, a long nose or a snub nose. Perhaps the tip is bulbous, the nostrils dilated. And what about the chin? Big or small? Flat or bumpy? Does it retreat towards the throat, or jut out proudly? Does it form a square, or slope down into a point?

  A new picture of the person emerges. Yes, of course, commonalities remain. His name equally belongs to other people, his nose to other faces. We are all made of the same blood and bones. But take a closer look. See the proportions, the interplay between all the various parts? Every combination, like a mosaic, is unique. He has his father’s eyes, and his mother’s curls, and his uncle’s lop-sided smile. Together they create something, and someone, new. Someone who will look through those eyes in his own way, who will wear that hair according to his own style, who will deploy that smile for his own reasons. Talk to that person. Watch the skein of laughter lines that diagram his face. And how his eyes glisten, or darken, at the sound of certain words. He is simply being himself.

  Quetelet (and many others after him) believed that the essence of human nature could be found in the average, but he was mistaken. The essence of human nature is its endless variety. As Stephen Jay Gould would later remark, ‘All evolutionary biologists know that variation itself is nature’s only irreducible essence. Variation is the hard reality, not a set of imperfect measures for a central tendency. Means and medians are the abstractions.’

  The Cataract of Time

  If, as
is often said, lifetimes flow like a river, they begin with a trickle and culminate in a cataract. Heraclitus, the ancient Greek philosopher, put it well when he said, ‘Time is a game played beautifully by children.’ Perhaps this is the root of nostalgia: less the desire to return to our early years, than to the more capacious experience of time that we inhabited as children.

  Time. You know how it goes. After the age of thirty, I found, the days begin to run away from us. We struggle to keep up. That is when the nostalgic impulse awakens, burgeons, pesters. Last year I moved to Paris. Something about being back in a big city after so many years meant I could not help thinking, more and more, about the old London neighbourhood of my youth. I had reached that age when the past becomes so big and so deep that your mind finds itself increasingly drawn there. It is like living on a fragile coast, by an imposing sea whose smells and sounds gradually overwhelm your senses.

  So I decided to return. There was salt water to cross, and buses that jolted and trains that bored, but none of those things mattered. I simply had to go and see the place again after all this time.

  My younger siblings, when I spoke to them about the plan, chorused dissuasion. ‘There is nothing there,’ they said, perplexed. Evidently, they had all moved on. ‘Why go back?’

  I tried to explain. I opened my mouth, felt the breath on my tongue form plausible shapes. But my heart was not in it and each of my made-up reasons fell flat. I decided not to argue. I booked the tickets, packed my bags and left.

  That I could not explain my desire to return to my old London home did not disturb me. On the contrary, its surprising strength convinced. Reassured, even. Gazing out from the wobbly train, I tried to remember when I had last set foot there. My reflection in the window wore an expression of thought. Tall trees and green hills ran past. I looked away, cracked open a book, and stared at the pages till the words seemed to gel into a single inky mass.

  Five years. Already? Where had they gone? So many things that I had accomplished, people that I had met, places that I had seen, now looking back, seemed to have taken hardly any time at all. And yet how difficult, how exhausting, how important each event had struck me in the moment! And how impossibly distant, a lifetime away, these hours in the train out of Paris would have appeared to me back then.

  It was a fast service to London, without delays. Arriving in the capital, I felt more like a suburban commuter than an international traveller. I changed trains and rattled out from the centre toward the familiar periphery, my excitement building. Gradually the carriage emptied of its suits; a different class of passenger took their place. ‘We must be getting close,’ I thought, straining forward, oblivious to my wristwatch. Near the end of the line, I gathered my things – my bearings too – and stepped out. The platform was covered in litter and broken glass, but for an instant, at least, it felt unambiguously good to be back.

  Time is more than an attitude or a frame of mind. It is about more than seeing the hourglass as half empty or half full. More than ever in this age, let us call it the computer age, a lifetime has become a discrete and eminently measurable quality. To date, to believe the surveys in newspapers, I have spent some one hundred thousand minutes standing in a queue, and five hundred hours making tea. I have spent a year’s worth of waking days on the hunt for lost things. This year, I knew, contained my twelve thousand and twelfth day and night. That number equates to over a quarter of a million hours, seventeen and a quarter million minutes. Counting one number for every second since my birth, I had recently made it into the billionaire’s club.

  We occasionally liken time to money, as something to be spent wisely, but it is not money. No refunds are possible for days ill-spent; no bank exists to take savings. We cannot apportion our time like money, since we live always in ignorance of when the former will end. How to plan when a person can never know if he will see tomorrow, or survive to such an age that his eyes turn coal-black with blindness?

  Perhaps it would be better to talk about time in the manner of certain tribes. Strangers to clocks, they pace their days according to nature. Native Americans traditionally planted corn ‘when the leaf of the white oak was the size of a mouse’s ear’. Equinoxes and solstices scheduled their rituals. As for language, the Sioux have no word for ‘late’ or ‘waiting’.

  In Australia, the Aborigines believe that time, place and people are one. A glance at a tree or a face suffices to know the hour and the day. Their discrimination of the seasons is precise, depending on such factors as plant life and changes in the wind: the Eastern Gunwinggu, for example, speak of six seasons – three ‘dry’ and three ‘wet’ – where non-Aborigines see only one of each.

  For these and other tribes, time is the product of our actions. It appears when we sing a song, climb a mountain or smoke a pipe, and vanishes when we sleep. They do not think of time as something pervasive, like the air. Seconds, minutes and hours – these are all things that we do. In place of these terms, they speak of a ‘time of harvesting’ or a ‘river fish time’. Ask an African herdsman how long such-and-such task might take and he replies, ‘Cow milking time’, meaning the time it takes to milk a cow. What is an hour to such a man? Perhaps the time it takes to milk ten cows.

  We can put it another way: 1 hour = 10 milkings. My equivalent would be 1 hour = 10 tea-makings. Let us call it ‘tea time’. A short walk that lasts eighteen minutes equates to three milkings or makings-of-tea; a two-minute advert break amounts to one-third of a cup of tea. Between the opening and closing whistles of a football referee, time enough would pass to milk fifteen cows, or make fifteen cuppas.

  I do not mean by this digression to suggest that approximations necessarily trump exactitude. It is not at all my intention to run clocks down. But the particular words and images our respective cultures deploy, shape the way in which we experience time. I said just now that time is not money; we might say instead that it is closer to the spending of money. According to the tribesmen’s way of thinking, it is what happens when, for example, we enter a marketplace. This emphasis on activity in how we think about time strikes me as being very healthy. When I hear someone complain about all the hours or weekends he has to fill, I stop and think that it is a mistake to speak of days as we would speak of holes. One hole is much the same as any other, whereas every day is different. In this, it is more like dough that we can sculpt into infinitely varying shapes.

  On the journey back to my childhood home, I paused outside the train station, then made my way north toward the high street. The buildings were more or less the same as in my recollection: the same squat walls, tattooed with graffiti; the same ‘50% off’ signs in shop windows; the same boys and girls, their busy fingers unwrapping sweets. No bravado in the architecture, no colour or charm. Along the pavements, no bustle either – either too early or too late for shoppers. Few cars animated the road. I walked mechanically, turning here and there, smelling the sugar of freshly laid tarmac on Waterbeach Road.

  I landed finally on my old street. I took it all in. On the left stood metal railings and distantly behind them the classroom buildings of my former primary school, factory-long. To the right, a chain of brick houses, close set. Their thin walls, I recall, made bad neighbours. Down the road, I spotted a small man in the distance. The man grew bigger with every step. He was wearing a blue and red football shirt, but he did not look like a footballer. The tightness of the shirt pronounced a sizeable paunch. His dark hair was cut penitentiarially short. His breathing rasped as he passed me by. And then he was gone.

  I was surprised by how little had been altered. Painted house number signs, wooden gates, hedgerows all long forgotten, I recognise instantly. And yet it all seems so different from my kid days. Something has shifted out of sync, something I try to put my finger on. In frustration, I walk up and down the street until my legs tire. Only as I ready myself for the ride back does it hit me. What has changed here is: time.

  In his 1890 classic work, Principles of Psychology, the American philosopher W
illiam James noted, ‘The same space of time seems shorter as we grow older – that is, the days, the months, and the years do so; whether the hours do so is doubtful, and the minutes and seconds to all appearance remain about the same.’

  James goes on to cite a mathematical explanation for this phenomenon, by a contemporary French professor. According to this professor, Paul Janet, our experience of time is proportional to our age. For a ten-year-old child, one year represents one tenth of his existence; whereas for a man of fifty, the same year equates only to one fiftieth (two per cent). The older man’s year will thus seem to elapse five times faster than the child’s; the child’s, five times slower than the man’s.

  What matters, then, is the relationship between one sequence of years and another sequence. The interval spanning the ages of thirty-two and sixty-four will seem to the individual of similar duration to that experienced between the ages of sixteen and thirty-two, and to the interval between the ages of eight and sixteen, and to that from the age of four to eight, each having the same ratio. For the same reason, all the years from the age of sixty-four to one hundred and twenty-eight (assuming such an age were ever attainable) would seem to us to occupy no more of our feeling, thought, pain, fear, joy and wonder than that big bang epoch between our second and fourth year.

  More recently, from an American called T.L. Freeman, we have a formula using Janet’s insight that yields the individual’s ‘effective age’. Freeman’s calculations suggest that we experience a quarter of our entire lifetime by age two, over half by age ten, and more than three-quarters by our thirtieth birthday. At only about the chronological midway point, a forty-year-old will experience his remaining time as seemingly but one-sixth of what has gone before. For a sixty-year-old, the future will seem to last merely one-sixteenth the duration of his past.