Thinking in Numbers: How Maths Illuminates Our Lives

  Naturally, not all these ‘causes’ can be right, and some are even mutually contradictory. Either Napoleon’s decision to invade Russia was impetuous and instinctive, or else it was carefully calculated (against Russian weakness) and deliberate (to keep his forces busy). Either Russian weakness impelled the attention of France’s army, or else Napoleon’s mania invented such weakness for its own purpose. Either the war resulted from French initiative, or from British interference.

  A Briton who lives in France, I see how each nation selects its own causes, and edits them convincingly into its own version of history. In Britain, Napoleon’s name is synonymous with tyranny and a small comic man’s delusions of grandeur. In France, au contraire, he is a revolutionary who stood up for the new Republic against the hostile monarchies of Europe. The puffed-up Napoleon with ‘small white hands’, as depicted by Tolstoy, is of course a Napoleon from the Russian perspective.

  This third Napoleon, as conceived by Tolstoy, had at least one cardinal virtue: he knew to keep out of the way of his soldiers, not to tread on anyone’s boots, to give a fair impersonation of someone who is in command. It is the soldiers who shoot, and stab, and cough, and groan and bleed. They constituted the vast majority of the French emperor’s army, but they issued not a single command. The commands came from the officers above the soldiers, who took them in turn from the generals above the officers, who took them from the commander-in-chief above the generals. The most important commands always come from those who participate least in the physical action. Consequently, most of these commands, thousands of them, not corresponding to conditions ‘on the ground’ at the moment and in the place that they finally filtered down to the troops, were never executed. They did not coincide with the reality of circumstances that remained beyond the chief’s control. As far as Tolstoy is concerned, then, to say that Napoleon invaded Russia is only to say that a few of his commands, out of the thousands that came to nothing, coincided with certain broader events between the peoples of France and Russia in the year of 1812.

  What were these broader events ‘on the ground’? As the novel’s calculus analogy suggests, they were innumerable, infinitesimal. At a given moment, in a given place, the wishes and desires and intentions of hundreds and thousands of people temporarily coalesced. Tolstoy illustrates such a moment in the life of a backwoods Russian region.

  In the vicinity of Bogucharovo were large villages belonging to the crown or to owners whose serfs . . . could work where they pleased . . . In the lives of the peasantry of those parts the mysterious undercurrents in the life of the Russian people, the causes and meaning of which are so baffling to contemporaries, were more clearly and strongly noticeable than among others. One instance, which had occurred some twenty years before, was a movement among the peasants to emigrate to some unknown ‘warm rivers.’ Hundreds of peasants . . . suddenly began selling their cattle and moving in whole families toward the southeast. As birds migrate to somewhere beyond the sea, so these men with their wives and children streamed to the southeast, to parts where none of them had ever been. They set off in caravans, bought their freedom one by one or ran away, and drove or walked toward the ‘warm rivers.’ Many of them were punished, some sent to Siberia, many died of cold and hunger on the road, many returned of their own accord, and the movement died down of itself just as it had sprung up, without apparent reason. But such undercurrents still existed among the people and gathered new forces ready to manifest themselves just as strangely, unexpectedly, and at the same time simply, naturally, and forcibly. Now in 1812, to anyone living in close touch with these people it was apparent that these undercurrents were acting strongly and nearing an eruption.

  Contemporary historians, to believe Tolstoy, took no notice of these ‘undercurrents’ in the life of a people. They failed to see the ocean of history for the waves. Aware only of the tides they called ‘causes’, they ignored the vast depths from which these ripples emerge. A man called Napoleon has an impetuous character; six months later Moscow is under siege. The historian looks at these two situations and asserts a link: namely, that hundreds of thousands of Muscovites fled their homes, and whole battalions of soldiers lost their lives, because of the impetuousness of this single man called Napoleon. Or, the historian notices that in, say, Liverpool and London there occurred local riots caused by a shortage of bread and that within a year Russian troop masses were fending off the French. Entire theories, each more elaborate and ingenious than the next, are spun in order to thread the rowdy fisticuffs in these English cities with the subsequent slashing and burning and murdering at Borodino.

  I have, admittedly, offered only gross approximations of these historical theories, and of course they are often far more complex, discerning many separate causes – one cause after another – of a war. The temperament of a man called Napoleon is only one cause, they say, and the bread shortage in a city like Liverpool another. Often they will find a third, and perhaps a fourth or fifth cause to supplement the first and second. All the same, Tolstoy’s chief objection remains. Historians tend by their very nature to adopt a flawed approach, he argued, because a mass conflict can no more be reduced to a handful of causes than can a ship’s course be reduced to a few waves. Between a French port and a Russian port lie innumerable points in the sea: why label the fifteen thousand four hundred and third point, say, or the seventy-one thousand nine hundred and sixty-eighth point, as being ultimately responsible for the ship’s arrival?

  An equivalent mistake would be to inquire of an age-beaten man, in which hour of your life was the blow delivered? Which blow? Why, the blow that loosened your teeth, and broke your bones, and thrashed your skin. Of course, there is no sense in such a question. The flow of time erodes patiently, continuously. What then could our elderly man say by way of a reply? He might recall that during a particularly hot summer night in 1968 he rolled out of bed and broke a shin. Perhaps he would smell once more the harsh carbolic soap with which, as a child growing up in the 1940s, he scrubbed his face. A game with his grandson in 1997 might come to his mind, in which a hard rubber ball accidentally struck his jaw. But none of these events, not less in combination, could truly help us to understand the elderly man’s present condition.

  Change appears to us mysterious because it is invisible. It is impossible to see a tree grow tall or a man grow old, except with the precarious imagination of hindsight. A tree is small, and later it is tall. A man is young, and later he is old. A people are at peace, and later they are at war. In each case, the intermediate states are at once infinitely many and infinitely complex, which is why they exceed our finite perceptions.

  Even a dramatic change can thus be accomplished without our knowledge. A friend once related to me the following illustrative tale. An American friend of my friend inherited a house in southern Europe. This house contained many fine pieces of furniture and works of art. Every summer, the American flew to Europe and lived for several months in the house among these objects. She sat on the same cushions, walked past the same paintings and heard the same ticking of the grandfather clock. The house’s upkeep she entrusted to a small and loyal staff, so that its rooms were always cleaned and polished and in good repair whenever she came through the door. And then one day, several summers following her inheritance, the American’s kid sister came to stay. The sister felt excited; she had heard many good things about the house and was longing to see it. But this feeling quickly gave way first to curiosity, then to confusion and finally to astonishment. A distinguished-looking chair in the hall, upon closer inspection, revealed itself to be cheap and rickety. Removed from its frame, the painting that hung above the fireplace flapped poster-thin. The marble-coloured statuette in the guest bedroom gave off the unmistakeable whiff of plastic. Fakes! Frantically, the sisters raced from room to room, until the whole place was left upside down. Every chair, every vase, every painting, virtually everything in the house – over one hundred items – unbeknownst to the American, had been meticulously replaced. Li
ttle by little, piece by piece, a wily member of staff had stolen the house away from under her nose.

  Sometimes, revolutions turn over a country in the same way that the American’s staff member turned over her house. Imperceptibly, dissidence grows across a land long before the dictator calls out his tanks. And as recent events in the Arab world remind us, nobody predicts a revolution before it happens and nobody controls it once it is under way. ‘Why war and revolution occur we do not know,’ affirms Tolstoy. ‘We only know that to produce the one or the other action, people combine in a certain formation in which they all take part.’ Hearing the sudden drumbeat of shoes, the roar of voices, the upturned faces – flushed with anger – the almighty, most wise and beneficent ruler does not comprehend. In his incomprehension, he asks the same question that many of his fellow dictators end up asking themselves: from where did they come? All these people with their fists and their voices. He shakes his head in disbelief. And yet, the answer is simple because only one answer is possible. Simply put, the people were always there: on the streets, in the mosques, in the bazaars. Only now, this distributed mass of people suddenly and noisily combines. Only now does the man without work, and the woman without dignity, and the teenager without anything to eat make themselves seen and heard.

  What is the power that moves peoples? Not the power of rulers, says Tolstoy, or for that matter the power of ideas. It is ineffable, invisible.

  History seems to assume that this force is self-evident and known to everyone. But in spite of every desire to regard it as known, anyone reading many historical works cannot help doubting whether this force, so variously understood by the historians themselves, is really quite well known to everybody.

  This force is the Human Life in which every person – from the lowly peasant Karataev to the Emperor Napoleon – participates. It is the ‘hidden warmth’ of patriotism experienced by the Muscovites when suddenly confronted with the dramatic threat of foreign invasion. It is the ‘chemical decomposition’ of the fleeing mass of French soldiers once their goal (the head-on confrontation in Moscow with an appointed ‘enemy’) becomes unattainable. It is the ‘force of habit’, during a salon conversation, which makes Prince Vasili Kuragin say ‘things he did not even wish to be believed.’

  Rather than assign various degrees of responsibility to this cause or that, Tolstoy proposes that historians pay far greater attention to this force. Moscow’s conflagration, which the historians variously explain as a defensive tactic of the Russians (the so-called policy of ‘scorched earth’) or the wild vengeance of the invading French, becomes explicable in other terms.

  Moscow was burned because it found itself in a position in which any town built of wood was bound to burn . . . A town built of wood, where scarcely a day passes without conflagrations when the house owners are in residence and a police force is present, cannot help burning when its inhabitants have left it and it is occupied by soldiers who smoke pipes, make campfires of the Senate chairs in the Senate Square, and cook themselves meals twice a day.

  Readers were shocked by these arguments when the book first appeared (over several editions) in the 1860s. Turgenev denounced the historical reflections as ‘charlatanism’ and ‘puppet comedy’ while Flaubert found them merely repetitive. The historian A.S. Norov entitled his review of the book ‘Tolstoy’s Falsification of History’. Another historian, Kareev, complained that the novelist wanted to abolish history altogether. When, one and a half centuries later in the year 2000, a Russian publisher edited and released an early draft of the book, they boasted about the lack of all these troublesome elements. ‘Half the usual length. Less war and more peace. No philosophical digressions or incomprehensible French. A happy ending.’

  Mathematicians, on the other hand, have proved considerably more receptive. Tolstoy’s mathematician friend, Urusov, expressed his delight at the analogy with calculus. And more recently, in a 2005 article for the Mathematical Association of America, Stephen T. Ahearn praised Tolstoy’s mathematical metaphors as being both ‘rich’ and ‘deep’ and encouraged maths teachers to use them in their classrooms.

  What, then, are we to conclude? Is it we who have to conclude? After all, if Tolstoy is right, his book – like any event in time – cannot be understood with prior assumptions, rules and theories. Everything has its moment, its context. Earlier, in one state, you began this essay, and now later on you finish it in another. What do you think? I cannot tell you. In everyone and everything, the process of change always asserts its own meaning.

  Book of Books

  I have walked in my sleep, and talked in my sleep, but I have never written in my sleep. The Icelandic author Gyrðir Elíasson’s short story Næturskriftir (‘Nightwriting’) depicts a character whose writer’s block suddenly disappears once the lights go out. In a notebook lifted from his bedside table, he begins to write down words, sentences, and even whole stories while he dreams.

  The days passed by all the same; he could not write . . . but at night he wrote; nearly every night . . . his wife knew not to wake a sleepwriter, so she lay [in the bed] and watched the expression of his back, how he wrote with amazing confidence with the notebook on his knees.’ (My translation).

  Something about Elíasson’s tale touches a chord in me. I think it has to do with confronting the infinity that is every written and unwritten book, including the ‘Book of Life’: the infinitely many potential combinations that comprise our days. How does the author select the right word, the right phrase, the right image from among the countless conceivable possibilities? How does each person imagine a new existence; reconfigure the choices that make up another destiny?

  Sleep on it. Why not? Our dreams contain the infinite. Uninhibited by wakefulness, words and pictures and emotions circulate and combine freely inside our head. Across the centuries, the Unconscious mind has authored some of the greatest works in literature: Goethe and Coleridge are only two of its pseudonyms.

  Dreams defy our finite scrutiny; too often they evaporate in the narrow light of day. We are left upon waking only with sweet hints of rain and distant echoes of a song, a nose here, a smile there, some tremor of sadness or flicker of joy, a suggestive and beguiling void. Like a book, like a life, where does the explanation start? A dream has no beginning, and therefore no middle and no end.

  I dreamt I entered a house and found its inhabitants all lying on the floor. Lying, but talking and laughing and eating together. Lying instead of sitting. It was like a scene from a book that I had not read and that had not been written. How many such scenes are there to occupy our dreams, our lives, the pages of a book? Infinitely many.

  Like Elíasson’s sleepwriter, Anton Chekhov faithfully nurtured a little notebook throughout his remarkable career, though we can suppose that he used his mostly during waking hours. Filled with his day-to-day observations of existence’s minutiae, the pages preserve glimpses of ‘ordinary’ life’s infinite permutations.

  ‘Instead of sheets – dirty tablecloths.’

  ‘In the bill preserved by the hotelkeeper was, among other things: “Bugs – fifteen kopecks.” ’

  ‘If you wish women to love you, be original; I know a man who used to wear felt boots summer and winter, and women fell in love with him.’

  This endless variety inspired many of Chekhov’s tales. In The Lottery Ticket, a middle-class couple envisage the potential lives that would follow a jackpot.

  The possibility of winning bewildered them . . . ‘And if we have won,’ he said, ‘why it will be a new life, it will be a transformation!’ . . . Pictures came crowding on his imagination, each more gracious and poetical than the last. And in all these pictures he saw himself well fed, serene, healthy, felt warm, even hot! . . . ‘Yes, it would be nice to buy an estate,’ said his wife, also dreaming . . . Ivan Dmitritch stopped and looked at his wife. ‘I should go abroad you know, Masha,’ he said. And he began thinking how nice it would be in late autumn to go abroad somewhere to the south of France, to Italy, to India!


  Writing half a century after his fellow countryman, another precocious note-taker, Vladimir Nabokov, composed his novels in two alphabets and three languages (Russian, French and English). From a blank vivid room, anagrams, puns and neologisms spilled out. He compared composing a story with fitting together the pieces of a jigsaw puzzle.

  Reality is a very subjective affair . . . You can get nearer and nearer, so to speak, to reality; but you never get near enough because reality is an infinite succession of steps, levels of perception, false bottoms, and hence unquenchable, unattainable. You can know more and more about one thing but you can never know everything about one thing: it’s hopeless.

  Like a jigsaw puzzle, like a dream, Nabokov’s novels emerged non-linearly; he often wrote the middle of a story last. Chapter eight of a draft manuscript might appear long before chapter seven or chapter three. A new story he would frequently write backwards, starting out from its final lines.

  Lolita, Nabokov’s most famous (and infamous) novel, began life on a long series of three-by-five-inch index cards. He sketched out the story’s closing scenes first. On subsequent cards Nabokov jotted down not only paragraphs of text but also plot ideas and other bits of information; on one, a chart of statistics on the average height and weight of young girls; on another, a list of jukebox songs; on a third, an illustration of a revolver.

  Every so often Nabokov would rearrange his index cards, searching for the most promising combination of scenes. The number of possible permutations would have been immense. Three of Nabokov’s cards can be rearranged in a total of six different ways: (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1), while ten cards (equivalent to between two and three printed pages in a book) would be capable of permuting into more than three and a half million sequences. To compose only four or five pages (equivalent to the contents of about fifteen index cards) would require a choice from among some 1.3 trillion variations. Lolita runs to sixty-nine chapters and over three hundred and fifty pages, which means that the number of its potential versions exceeds (by an almost unimaginable margin) the number of atoms that make up our universe.