Thinking in Numbers: How Maths Illuminates Our Lives

by Daniel Tammet

  When Keynes spoke of the value of inequality he did not mean unbridled inequality. He meant an inequality that was consensual and that served a collective purpose. The selfish motive of making money, he admitted, helped to produce goods and services that benefit many. The same motive could also turn certain ‘dangerous human proclivities’ – cruelty, self-aggrandisement and the tyrannical pursuit of power – to more harmless pursuits. But, he was not at all complacent.

  It is not necessary for the stimulation of these activities and the satisfaction of these proclivities that the game should be played for such high stakes as at present. Much lower stakes will serve the purpose equally well, as soon as the players are accustomed to them. The task of transmuting human nature must not be confused with the task of managing it.

  How we lower those stakes is a question I leave to our politicians. I will not hold my breath. There is no easy solution – none analogous to a politician’s promise. Money’s abstractness is complex, evasive. It turns our world upside down. For instance, a farmer’s cow calves more prodigiously than other cattle – this is normal, part of nature. What, though, should we think of houses that beget more houses? Like many of those born poor my father never had the chance to own the roof above his head, whereas the proprietor of four houses will likely wind up with six, or twelve, or twenty.

  All of which brings us to the question that Tolstoy posed his readers in the short story ‘How much land does a man need?’ Pahom, the greedy peasant, kills himself in the endless pursuit of more and more acres.

  ‘They are poor in the midst of riches,’ Seneca observed of some of ancient Greece’s wealthiest – and greediest – men, ‘which is the worst kind of poverty.’

  A Model Mother

  Not long ago, my mother’s age reached double mine. Two lives when compared with my young man’s span – half of her that I cannot see.

  My mother has always been a mystery to me. We have had my lifetime to get to know one another, but it still feels nowhere near long enough. Her behaviour eludes me; it outpaces my powers of comprehension. Try as I might, I cannot figure her out.

  Her face has not changed much with the years. She often wears an expression that flits between smugness and fright. The same stubborn creases, made by steady clenching, around the mouth; the same defiant twinkle in her eyes. She smiles fitfully, unexpectedly, as if bestowing a favour. Beneath the fine greying hair and wrinkles, I can still find myself in her gaze.

  Memories. Take the kitchen of my childhood for example, where my mother would spend the lion’s share of her day. I remember her prowling the linoleum, biro and notepaper in hand, alive to every sound and suspicious of the slightest incursion. She was compiling a list of groceries to buy. She poked her nose into cupboards and the fridge, looking for the tins of beans and the bottles of milk and the packets of cheese and sliced bread bought only a day or two before. Thin air had taken their place.

  ‘The children eat us out of house and home,’ she complained to my father.

  My father assumed a posture of resignation.

  We kids knew which of our parents held command. Or, at least, we thought we did. At other times my mother would suddenly come over all shy and prone to blushing. My father could hardly draw a word out of her.

  Then there were the Christmas presents. Year-round she would ferret out bargains at local car-boot sales, sequestering the toys and games in closets or under beds until Santa’s sleigh arrived. Of course, we always knew where to find them, but in the spirit of the season we turned a blind eye. It was never easy: buried treasure seemed to lie in every nook and cranny of every room. Why then, many Decembers later, under piles of old clothes, did we bring unopened presents to the surface? Had she simply mislaid them, forgotten their location? Could it really be that the buying of them had meant more to her than the offering?

  A mathematician would say, ‘graph the data.’ That is how mathematicians speak. And it is true enough: puzzling occurrences usually require the long view – and a firm grasp of context. Early on in my childhood, I decided that if only I could assemble enough of my recollections, and settle on some parameters for their analysis, it might be possible to make a predictive model of my mother’s behaviour.

  It was around the time that I began to look like her, once the primary school blackboard became a blur. Our myopia, it could be said, brought us closer together. ‘A mummy’s boy,’ my father sometimes called me. None of my brothers earned this epithet. Spending more and more time in her company, I felt the puzzle of her presence intensely.

  Back then, a thigh thinner, she was always on the go. I took to mapping her movements. Saturday mornings my mother would return from the local library, carrying a couple of romantic paperbacks in her arms, smelling faintly of must. In the space in the living room before the television, I would sit for what seemed like hours half-listening to the yellow rustling of the pages that originated from the settee. Every other Sunday, robed in her best dress, she would take us – my brother, sister, and I – to the neighbour’s on the corner to drink tea and trade gossip. Midweek, she would venture out to do the rounds of second-hand shops, returning with bags fattened on goods that nobody seemed to need.

  Perhaps she noticed me tracking her and wished to catch me out, or perhaps she simply grew bored with her own repetition, but for whatever reason she would sometimes decide to mix it all up. Come Saturday, the living room would contain the same atmosphere. But now, dog-eared biographies exhaled the musty smell. Without warning, the front door would stay shut for the Lord’s Day of rest; we took tea with the neighbour an evening after school instead. Even her favourite shops suddenly became good only for returning things.

  One afternoon she took me with her to return a pair of shoes. Approaching the shop, I compared the imaginary mother in my head with the real thing. The imaginary mother would choose a male shop assistant (my mother, I knew, hated to haggle with other women). She would complain that the shoes pinched her young son’s toes, and when the man lifted the offending footwear from its container, she would add that the leather scuffed immediately. To the request for a receipt (which she always lost), her voice would rise in volume, detailing all the children’s feet that she had to keep dry and warm. The man would then nod patiently before offering an exchange.

  Unfortunately, on this occasion my real mother bore no resemblance to her model.

  A young woman with tight bound hair assisted all the customers. My mother’s voice as she handed over the shoebox sounded weak, her words unacquainted with one another. ‘I’m sorry to hear it,’ the woman interrupted, as though offering her condolences, ‘I’m sorry to hear it, but there really is nothing that I can do.’

  I confidently expected my mother to put up a fight. But she promptly slumped into one of the seats intended for the trying-on of shoes, with only a long sigh for a rejoinder. The shop assistant reiterated that a refund was quite out of the question. My mother looked down at the floor and merely sighed again. Eventually, when my mother showed no signs of relenting, the young woman said, ‘Please leave’ and then, ‘Please leave, or I will have to call the police.’ My mother sagged even deeper into her seat and crossed her legs.

  My ten-year-old self was filled with apprehension. My imaginary mother would never have behaved like that! It took a long time for me to understand my real mother’s sit-in. Of course, she knew all too well what she was doing. She knew that the young lady had conceived her own version of my imaginary mother. On that day, in that shop, my mother in her flesh and blood defied them both.

  At last, driven to exasperation, the woman pulled something small and shiny from her pocket. ‘Do not breathe a word of this to anyone,’ she said, before sketching a long gash along one of the shoes with the blade of her penknife. ‘This way we get a refund from the manufacturer for damaged goods.’

  In fact, it did not bother me as much as you might think when the actions of my mother and her imaginary doppelganger failed to coincide. I slowly came to understand
how limited and clumsy an approximation was my model of her, how many variables I had not accounted for (whose existence I had not even guessed), and how large and liberating a role chance played in all our affairs. Besides, each variation between the imaginary and the real provided me with new clues. This variation, I hoped, growing larger or smaller by turns, would act as a sort of compass toward a greater understanding of my mother’s true nature.

  On those rare occasions when my mother became her model’s mimic, the eerie sense of déjà vu nauseated me. I worried that it suggested some dark cunning within me, or, worse, deterioration in the liberty of my mother’s will. Besides, how could I even be sure that my success showed real merit? Perhaps it could be put down to nothing more than luck: even a stopped watch gives the correct time twice a day.

  Possibly, my inability to understand my mother is the result of an anterior uncertainty, that is: what should a mother’s behaviour look like? I am not talking about an ideal mother – I do not believe, alas, in such creatures – but rather the most common and distinguishing maternal qualities. A baseline, if you will.

  This is more difficult than it sounds. For one thing, the category of ‘mother’ admits all manner of members. You can be a mother at sixteen or at sixty (with the help and expertise of scientists); with a single child, or my mother’s nine. According to the dictionary definition, a mother is any ‘female who has given birth to offspring’. This is about as roomy and heterogeneous a group of people as we can find. It is, in the words of statisticians, too large a sample.

  How, then, should I go about assembling a more workable sample of my mother’s peers, one that provides a realistic context for analysis? By looking at mothers of nine? I am not sure that Britain can boast many nine-child families, not since the days when Queen Victoria had her own royal brood of nine. Newspaper articles turn up only a couple of examples: one is a philosophy graduate and boardroom executive who thought she and her Buddhist husband ‘would stop at five’; the second, a former anorexic who says she feared her chances of falling pregnant were ‘very, very slim’. Naturally, this tiny cohort of women is no more representative of maternity than the first.

  I might try a slightly different, related, question: what does my mother’s behaviour tell us about her? But here we run into similar difficulties. For each of her acts, we can imagine a hundred more or less plausible reasons. A hundred imaginary mothers would slug it out for vindication. But each act is the offspring of another: each imaginary mother would be capable of producing one hundred more. Clearly, this approach will take us nowhere closer to any answer. For even if we could somehow locate the ‘right’ reason for each of my mother’s actions, and thus identify the ‘right’ imaginary mother within our galaxy of imaginary mothers, we would be left in the end only with an identical twin – as complex, mysterious and baffling a woman as the one who raised me.

  More empirical observation and less abstract reasoning seems necessary, if I am to arrive at any conclusions about who my mother really is. I fully admit that in this I am saying nothing new. When the psychiatrist Édouard Toulouse resolved to objectively measure Emile Zola’s genius, for example, he was entirely typical of his times. He took the novelist’s height, wrapped a tape around his shoulders, skull and pelvis, evaluated the strength of his grip, the acuity of his nose, ears and vision, grilled his powers of recollection and noted the hours when he ate, slept and wrote. Zola’s pulse, the doctor found, was sixty-one before he first put pen to paper, dropping to fifty-three when he called it a day.

  Scientists in the Soviet Union also went in for this sort of thing, counting the subject’s words rather than his pulse. In their experiments, they tried predicting the next word in a sentence from those that had come before. Young girls in conversation, they discovered, were easiest to anticipate; newspaper columnists followed close behind, while poets proved hardest to second-guess.

  Did this result surprise the scientists? They do not tell us. Perhaps the poets took the same liberties with their speech as with their pen. The best poems, the mathematicians determined, combined in equal parts the predictability of metre with the novelty of unusual words. Too much metre made a poem banal; too much freewheeling, on the other hand, rendered it hard to follow. Convention and invention, their delicate balance, give meaning to what we say.

  The lesson we can learn from these experiments is small but valuable. Mutual understanding depends on our powers of prediction, though they frequently operate beyond our control. With his microscope the psychiatrist got no nearer to what moved Zola’s pen, yet he knew intuitively how to talk his old friend into his tests. The Soviet mathematicians could not accurately preview the poets’ inspiration, but their conversations once outside the laboratory ranged as far and wide as with anyone else.

  In Edgar Allan Poe’s story The Purloined Letter, we see a boy observing and then outguessing each of his schoolmates at a game of marbles. The game consists in determining whether the number of marbles concealed in the opponent’s hand is odd or even. For every correct guess, a marble is won; for every wrong guess, one marble is yielded. Thanks to his ‘astuteness’, the boy finishes by winning all the marbles in the school. The boy, Poe explains, makes an intuitive assessment of his rival.

  For example, an arrant simpleton is his opponent, and, holding up his closed hand, asks, ‘are they even or odd?’ Our schoolboy replies, ‘odd,’ and loses; but upon the second trial he wins, for he then says to himself, ‘the simpleton had them even upon the first trial, and his amount of cunning is just sufficient to make him have them odd upon the second; I will therefore guess odd’; – he guesses odd, and wins.

  When the boy considers his opponent ‘a simpleton a degree above the first’, he reasons, ‘This fellow finds that in the first instance I guessed odd, and, in the second, he will propose to himself upon the first impulse, a simple variation from even to odd, as did the first simpleton; but then a second thought will suggest that this is too simple a variation, and finally he will decide upon putting it even as before. I will therefore guess even.’ He guesses even, and wins.

  Poe goes on to tell us how the marble winner is able to intuit the thoughts and feelings of the boy opposite him: he closely watches and mirrors the opponent’s facial expression, so that the other boy’s gaze fleetingly becomes his gaze, the boy’s smile becomes his smile, the boy’s frown becomes his own. In this posture, the winner finds himself thinking and feeling in the same way as his rival. His success hinges entirely on the precision of his mime.

  In a sense, we are always evaluating and predicting the other, though we may not heed the act. Often the people we scrutinise hardest are those we cherish the most. In love there is constant contemplation and the most intense desire to understand the object of our affections. There exists melancholy, too, as we grow to appreciate just how little we can ever truly know for sure. Our ignorance is painful. And yet we persevere. Humbly, patiently, we assiduously observe till at last we identify ourselves in some way with the other. Anticipation becomes an act of love.

  I have spent years learning how to evaluate my mother’s various tics and gestures. These days, I can read her body English with fair fluency. But still the same questions return over and over again. What, I often wonder, does that smile of hers say?

  We meet in a posh restaurant in central London. A woman’s face, old enough to be thought young-looking, smiles from a table at the back when I come through the door. I kiss my mother on the cheek. Her ecstatic eyes follow the stiff-backed young waiters carrying platters of food and wine. What shall we eat? I know the place well and have already made up my mind. I confide my choices to my mother before heading to the bathroom. When I return, the menus are gone. My mother is fiddling with her napkin. The skin on her hands is worn tight, I notice, like the peel on overripe fruit. Her fingers twist the napkin while we talk.

  Her rent has gone up again. Arabesques of graffiti continue to shout from wall to wall. Last week, an Albanian down the road set fire to his
mattress; the wailing sirens made ‘a right old racket’. And yet, my mother will not hear of moving away. She insists on sticking close to where her children were all born and raised. I am aware that any new plea will fall once more on deaf ears, and have no choice but to let the matter go.

  ‘Midday,’ I answer when she asks about my flight tomorrow out of Heathrow. Tokyo, I tell her, is nine hours ahead of her time. The trip will include my first lecture in the Far East. My mother feigns curiosity. She has never held a passport, knows nothing of the world beyond her shores. All of a sudden, she shakes with silent laughter. Some word, its sound or the mental image elicited by the word, has tickled her. Like the boy in Poe’s tale, I reciprocate and try to laugh along. But I do not understand. And then, just as quickly as it came, the laughter leaves her. With the napkin, she dabs the wet corners of her eyes.