Thinking in Numbers: How Maths Illuminates Our Lives (19 page)

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.

Of course, many of these potential
Lolitas
would simply not have been viable. And yet among the bewildering, nonsensical or ham-fisted editions, readable alternatives must exist. How many? A hundred? A thousand? A million? More. Many more. Publishers could produce enough of them to give every reader on the planet his or her very own
Lolita
. In one, the famous opening couplet, ‘Lolita, light of my life, fire of my loins. My sin, my soul’ would appear halfway down page thirty-nine (perhaps replaced with a line that Nabokov placed in his chapter two: ‘My very photogenic mother died in a freak accident (picnic, lightning) . . .’). In another reader’s edition, the couplet shows up at the top of page 117. This
Lolita
begins instead, ‘I saw her face in the sky, strangely distinct, as if it emitted a faint radiance of its own.’ In a third version, the original couplet serves as the story’s closing lines.

For all I know, some of these incalculably many editions were actually published, each with their subtle yet striking alterations. Perhaps this would explain why the
Atlantic Monthly
’s reviewer called the book ‘one of the funniest serious novels I have ever read’, the
Los Angeles Times
declared it ‘a small masterpiece . . . an almost perfect comic novel’, and the
New York Times Book Review
announced, ‘technically it is brilliant . . . humor in a major key’, whereas Kingsley Amis read a book leading to ‘dullness, fatuity, and unreality’ and Orville Prescott, writing for the
New York Times
, found the story ‘dull, dull, dull.’  

Which
Lolita
did they read?

It is the writer and reader together who compose their infinite tale. The Argentinian writer Julio Cortázar created a novel in which he made this principle explicit.
Rayuela
(
Hopscotch
) was published fifty years ago, not long after
Lolita
. It contains 155 chapters (over some 550 pages), which can be read in two distinct ways. Either the reader starts at chapter one and continues reading linearly until the end of chapter fifty-six (the chapters and 200 pages that remain being considered ‘expendable’), or else he begins at chapter seventy-three, then turns back to chapter one, continues to chapter two before jumping forward to chapter 116, then back to chapter three, forward to chapter eighty-four, and so on back and forth between the chapters according to a ‘Table of Instructions’ at the front of the book.

In one of his ‘expendable’ chapters, Cortázar describes the book’s goal.

It would seem that the usual novel misses its mark because it limits the reader to its own ambit; the better defined it is, the better the novelist is thought to be. An unavoidable detention in the varying degrees of the dramatic, the psychological, the tragic, the satirical, or the political. To attempt on the other hand a text that would not clutch the reader but which would oblige him to become an accomplice as it whispers to him underneath the conventional exposition other more esoteric directions.

As Cortázar’s accomplice, we follow the novel’s hero – an Argentine bohemian – through the streets of Paris as he contemplates his life and its inexhaustible potential paths. We begin the book either at chapter one, like this: ‘Would I find La Maga?’ Or else at chapter seventy-three, on page 383: ‘Yes, but who will cure us of the dull fire, the colourless fire that at nightfall runs along the Rue de la Huchette . . .’

Turning the pages, reading different stories. For instance, the reader who begins at chapter one will soon reach this line in the fourth chapter: ‘[She] picked up a leaf from the edge of the sidewalk and spoke to it for a while.’ For the other reader, however, ‘chapter four’ is really the seventh chapter of the story, preceded by the sixth chapter, which is labelled ‘chapter eighty-four’. In this chapter, on page 405, he reads, ‘I keep on thinking of all the leaves I will not see, the gatherer of dry leaves, about so many things that there must be in the air and which these eyes will not see . . . there must be leaves that I will never see.’ These lines enrich the second reader’s understanding of the woman who, several pages later, on page twenty-five, will pick up a leaf from the edge of the sidewalk and speak to it.

A consequence of reading in this way is disorientation; the leapfrogging reader lacks any sense of having completed the book. He reads the final lines on the final page of the physical book long before he comes to any conclusion of the story. Arriving later at the one hundred and fifty-third chapter (labelled ‘chapter 131’), he proceeds to the following chapter (labelled ‘chapter fifty-eight’), only to discover that he should return again to chapter 131. An interminable loop between the two ‘final’ chapters appears. What is more, assuming he has kept count, the reader notices that the chapters – read in this order – total 154. One of the chapters – ‘chapter fifty-five’ – is absent from the list.

Hopscotch
’s structure demands that readers make their own sense of the story. One might decide to read the chapters consecutively, but in descending order, starting at chapter 155. Another decides to read all the even chapters before the odd: two, four, six, eight . . . one, three, five, seven . . . A third does the same, but in reverse, reading all the odd chapters before the even. A fourth reads only the prime numbered chapters: two, three, five, seven, eleven, thirteen, seventeen, nineteen, twenty-three, twenty-nine, thirty-one . . . finishing at chapter 151 (thirty-six chapters in all). A fifth begins at the first chapter, then reads the third (1 + 2), turning next to the sixth (1 + 2 + 3), followed by the tenth (1 + 2 + 3 + 4) and so on.

Just when the plucky reader has attained the end of one story, another story beckons him to pick up its pages and start again. The book of ascending chapters becomes a book of descending chapters. The book of odd number chapters becomes a book of even number chapters. Every reading differs; every reading offers something new. It is impossible to dip into the same book twice.

I am reminded of Nabokov’s view that we can never read a book: we can only reread it. ‘A good reader, a major reader, an active and creative reader,’ says Nabokov, ‘is a re-reader.’ Initial readings, he explains, are always laborious, a ‘process of learning in terms of space and time what the book is about, this stands between us and artistic appreciation.’