Thinking in Numbers: How Maths Illuminates Our Lives (23 page)

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.

I think back to the menu. For each course there are just a few options. A good opportunity to put my imaginary mother to the test. But will the model and the mother agree? Recalling the trios of starters, mains, and desserts, I assign to each a probability. I assess not only each of the individual courses but also the potential combinations across them. For example, two-thirds of the mains have meat: for this reason I downgrade the odds of my mother selecting pâté as a starter (unless, that is, she were to opt for a second course of fish). Assuming she begins with the good intentions of a salad, I plump for the caramel cake as her choice of dessert.

My imaginary mother decides to steer a middle route between the pâté and the salad. When the waiter looms, his voice announces the roast vegetable soup. I inhale the satisfying sweetness as it passes my chair.

Now the beef rises even higher up my mental shortlist. But when the table has been cleared, and the waiter next comes, I find my stare returned by the glassy eye of a baked cod. Bits of the fluffy flesh scatter around her plate and down her top as she eats.

Finally, we arrive at dessert. There is no room for doubt in my mind. My model mother’s fondness for chocolate has been corroborated many times before. But not today. My actual mother finishes her meal with a bowl of exotic fruit.

Outside the restaurant, she takes my arm in hers. She wants to show me the street on which she grew up. She grips my arm tight as we take the short walk side by side. Of her life before me I know next to nothing, only one or two titbits, gathered here and there. I ask whether it is true that before starting a family, she worked as a secretary. ‘Leave off,’ she laughs. ‘I typed addresses onto envelopes.’ She still has postcodes on the brain.

‘Name a town,’ she says suddenly.

Bethlehem, I think. ‘St Ives,’ I say.

‘St Ives,’ she repeats and it sounds twice as long when she says it. ‘TR26.’

We turn onto the street, a stone’s throw from the Palace of Westminster. My grandfather worked for the local brewery here, delivering the beer by horse and cart. The block of flats was home to several of the workers’ families. A single washroom, with enamel bath, did for everyone. We cannot see inside, however. The building has become a base for the homeless; some of the windows are boarded up. Before we leave, I pull out my camera and take a photo.

I think of the girl who would grow up to become my mother. Who was the future woman that she imagined for herself? Did she dream of having a loving partner, a big house and children who always smiled at her? In her mind’s eye, would she be well-educated and well-travelled, always generous, patient, and kind? Did she imagine that every cherished moment would be remembered forever; every grievance instantly forgotten?

And thinking of this girl I feel at once immensely happy and immensely sad. I feel as I feel when I think of myself.

Talking Chess

To win at chess is simple: victory belongs to the player who makes the last-but-one mistake.

Whoever it was that first came up with this line spoke much truth. The strongest players operate neither like machines nor angels; their superiority lies in accomplishing a better class of error.

A winning mistake would presumably owe nothing to sloppiness, incuriosity or a yellow belly. It would be far closer to the lucky slip of an artist’s brush or writer’s pen, one that suddenly infuses a picture or page with unforeseen possibility. I am thinking of the (perhaps apocryphal) story of the painter who, hot-tempered at his many failed attempts to perfect a detail of his portrait, tossed his sponge with disgust at the easel and thereby achieved just the desired effect. Or that of the book printer who inadvertently won Herman Melville lavish accolades for the phrase ‘soiled fish’ (in place of the intended ‘coiled fish’ in reference to an eel).

I will not risk stretching this argument any further. Creativity is obviously so much more than an unexpected gesture here and there. But talent in chess bears this similarity with other creative pursuits, in tolerating error, as if the grandmaster – like the great artist – is the one who truly explores possibility’s outer limits. Or, as a character in Joseph Conrad’s
Lord Jim
intones, ‘To the destructive element submit yourself and with the exertions of your hands and feet in the water make the deep, deep sea keep you up.’

Chess is a perfect arena for just such an exerted exploration of the possible. Its chequered sea is very deep indeed. The mathematics behind the game’s complexity are staggering. An initial move by each player will create one of four hundred legal positions. A second move by each: seventy-two thousand. There are some nine million possible set-ups after the players’ third move; 288 billion after their fourth. Back in 1950, the mathematician Claude Shannon calculated the possible number of forty-move games, a figure henceforth known as the ‘Shannon Number’. He estimated as about thirty the potential number of viable moves at every turn. In this way he arrived at a total number (1 followed by 120 zeroes) that easily exceeds the number of atoms in the observable universe.

For all its immensity, chess is a finite game. It is therefore at least conceivable that a machine might one day be programmed with the knowledge, deep down in its nodes, of every possible sequence of moves for every possible game. No combination, however ingenious, would ever surprise it; every board position would be as familiar as a face. Like checkers, which was solved by computer scientists in Canada in 2007, we would finally discover how chess, played perfectly by both sides, ends.

This perfect game of chess – the immaculate order and configuration of its moves, the exquisite ballet of its pieces in their precisely timed roles – is imprinted on the imagination of every player. In his innermost being, every player carries some notion of the divine game. For one, it begins with the two-square forward march of the white king’s pawn, to which the L-shaped spring of black’s queenside knight replies. Six moves later, the white queen arrives on a4 – a side square – only to be promptly rebuffed by black’s bishop. No, no, says a second player: the game begins with a white knight – either one – to which black responds in kind. The central pawns advance in pairs. But another player disagrees, citing the fact that a piece would need to be sacrificed by white after eleven moves – the exchange of his queen for a rook. For still others the white pawns creep up the far sides of the board like ivy, or the black king crouches incessantly behind his queen, or all four bishops dance the diagonals till precisely half of all the original pieces remain.