Thinking in Numbers: How Maths Illuminates Our Lives Page 2

  Second, defining a set owes more to art than it does to science. Faced with the problem of a near endless number of potential categories, we are inclined to choose from a few – those most tried and tested within our particular culture. Western descriptions of the set of all elephants privilege subsets like ‘those that are very large’, and ‘those possessing tusks’, and even ‘those possessing an excellent memory’, while excluding other equally legitimate possibilities such as Borges’s ‘those that at a distance resemble flies’, or the Hindu ‘those that are considered lucky’.

  Memory is a further example of the privileging of certain subsets (of experience) over others, in how we talk and think about a category of things. Asked about his birthday, a man might at once recall the messy slice of chocolate cake that he devoured, his wife’s enthusiastic embrace and the pair of fluorescent green socks that his mother presented to him. At the same time, many hundreds, or thousands, of other details likewise composed his special day, from the mundane (the crumbs from his morning toast that he brushed out of his lap) to the peculiar (a sudden hailstorm on the mid-July-afternoon that lasted several minutes). Most of these subsets, though, would have completely slipped his mind.

  Returning to Borges’s list of subsets of animals, several of the categories pose paradoxes. Take, for example, the subset (j): ‘innumerable ones’. How can any subset of something – even if it is imaginary, like Borges’s animals – be infinite in size? How can a part of any collection not be smaller than the whole?

  Borges’s taxonomy is clearly inspired by the work of Georg Cantor, a nineteenth-century German mathematician whose important discoveries in the study of infinity provide us with an answer to the paradox.

  Cantor showed, among other things, that parts of a collection (subsets) as great as the whole (set) really do exist. Counting, he observed, involves matching the members of one set to another. ‘Two sets A and B have the same number of members if and only if there is a perfect one-to-one correspondence between them.’ So, by matching each of my siblings and myself to a player in a baseball team, or to a month of the year not beginning with the letter J, I am able to conclude that each of these sets is equivalent, all containing precisely nine members.

  Next came Cantor’s great mental leap: in the same manner, he compared the set of all natural numbers (1, 2, 3, 4, 5 . . .) with each of its subsets such as the even numbers (2, 4, 6, 8, 10 . . .), odd numbers (1, 3, 5, 7, 9 . . .), and the primes (2, 3, 5, 7, 11 . . .). Like the perfect matches between each of the baseball team players and my siblings and me, Cantor found that for each natural number he could uniquely assign an even, an odd, and a prime number. Incredibly, he concluded, there are as ‘many’ even (or odd, or prime) numbers as all the numbers combined.

  Reading Borges invites me to consider the wealth of possible subsets into which my family ‘set’ could be classified, far beyond those that simply point to multiplicity. All grown up today, some of my siblings have children of their own. Others have moved far away, to the warmer and more interesting places from where postcards come. The opportunities for us all to get together are rare, which is a great pity. Naturally I am biased, but I love my family. There is a lot of my family to love. But size ceased long ago to be our defining characteristic. We see ourselves in other ways: those that are studious, those that prefer coffee to tea, those that have never planted a flower, those that still laugh in their sleep . . .

  Like works of literature, mathematical ideas help expand our circle of empathy, liberating us from the tyranny of a single, parochial point of view. Numbers, properly considered, make us better people.

  Eternity in an Hour

  Once upon a time I was a child, and I loved to read fairy tales. Among my favourites was ‘The Magic Porridge Pot’ by the Brothers Grimm. A poor, good-hearted girl receives from a sorceress a little pot capable of spontaneously concocting as much sweet porridge as the girl and her mother can eat. One day, after eating her fill, the mother’s mind goes blank and she forgets the magic words, ‘Stop, little pot’.

  ‘So it went on cooking and the porridge rose over the edge, and still it cooked on until the kitchen and whole house were full, and then the next house, and then the whole street, just as if it wanted to satisfy the hunger of the whole world.’

  Only the daughter’s return home, and the requisite utterance, finally brings the gooey avalanche to a belated halt.

  The Brothers Grimm introduced me to the mystery of infinity. How could so much porridge emerge from so small a pot? It got me thinking. A single flake of porridge was awfully slight. Tip it inside a bowl and one would probably not even spot it for the spoon. The same held for a drop of milk, or a grain of sugar.

  What if, I wondered, a magical pot distributed these tiny flakes of porridge and drops of milk and grains of sugar in its own special way? In such a way that each flake and each drop and each grain had its own position in the pot, released from the necessity of touching. I imagined five, ten, fifty, one hundred, one thousand flakes and drops and grains, each indifferent to the next, suspended here and there throughout the curved space like stars. More porridge flakes, more drops of milk, more grains of sugar are added one after another to this evolving constellation, forming microscopic Big Dippers and minuscule Great Bears. Say we reach the ten thousand four hundred and seventy-third flake of porridge. Where do we include it? And here my child’s mind imagined all the tiny gaps – thousands of them – between every flake of porridge and drop of milk and grain of sugar. For every minute addition, further tiny gaps would continue to be made. So long as the pot magically prevented any contact between them, every new flake (and drop and granule) would be sure to find its place.

  Hans Christian Andersen’s ‘The Princess and the Pea’ similarly sent my mind spinning towards the infinite, but this time, an infinity of fractions. One night, a young woman claiming to be a princess knocks at the door of a castle. Outside, a storm is blowing and the pelting rain musses her clothes and turns her golden hair to black. So sorry a sight is she that the queen of the castle doubts her story of high birth. To test the young woman’s claim, the queen decides to place a pea beneath the bedding on which the woman will sleep for the night. Her bed is piled to a height of twenty mattresses. But in the morning the woman admits to having hardly slept a wink.

  The thought of all those tottering mattresses kept me up long past my own bedtime. By my calculation, a second mattress would double the distance between the princess’s back and the offending pea. The tough little legume would therefore be only half as prominent as before. Another mattress reduces the pea’s prominence to one third. But if the young princess’s body is sensitive enough to detect one-half of a pea (under two mattresses) or one third of a pea (under three mattresses), why would it not also be sensitive enough to detect one-twentieth? In fact, possessing limitless sensitivity (this is a fairy tale after all), not even one-hundredth, or one-thousandth or one-millionth of a pea could be tolerably borne.

  Which thought brought me back to the Brothers Grimm and their tale of porridge. For the princess, even a pea felt infinitely big; for the poor daughter and her mother, even an avalanche of porridge tasted infinitesimally small.

  ‘You have too much imagination,’ my father said when I shared these thoughts with him. ‘You always have your nose in some book.’ My father kept a pile of paperbacks and regularly bought the weekend papers, but he was never a particularly enthusiastic reader. ‘Get outdoors more – there’s no good being cooped up in here.’

  Hide-and-seek in the park with my brothers and sisters lasted all of ten minutes. The swings only held my attention as long. We walked the perimeter of the lake and threw breadcrumbs out on to the grimy water. Even the ducks looked bored.

  Games in the garden offered greater entertainment. We fought wars, cast spells and travelled back in time. In a cardboard box we sailed along the Nile; with a bed sheet we pitched a tent in the hills of Rome. At other times, I would simply stroll the local streets to my heart
’s content, dreaming up all manner of new adventures and imaginary expeditions.

  Returning one day from China, I heard the low grumble of an approaching storm and fled for cover inside the municipal library. Everyone knew me there. I was one of their regulars. The staff and I always exchanged half-nods. Corridors of books pullulated around them. Centuries of learning tiled the walls, and I brushed my fingertips along the seemingly endless shelves as I walked.

  My favourite section brimmed with dictionaries and encyclopaedias: the building blocks of books. These seemed to promise (though of course they could not deliver) the sum of human knowledge: every fact, and idea and word. This vast panoply of information was tamed by divisions – A-C, D-F, G-I – and every division subdivided in turn – Aa-Ad . . . Di-Do . . . Il-In. Many of these subdivisions also subdivided – Hai-Han . . . Una-Unf – and some among them subdivided yet further still – Inte-Intr. Where does a person start? And, perhaps more importantly, where should he stop? I usually allowed chance the choice. At random I tugged an encyclopaedia from the shelf and let its pages open where they may, and for the next hour I sat and read about Bora Bora and borborygmi and the Borg scale.

  Lost in thought, I did not immediately notice the insistent tap tap of approaching footsteps on the polished floor. They belonged to one of the senior librarians, a neighbour; his wife and my mother were on friendly terms. He was tall (but then, to a child is not everyone tall?) and thin with a long head finished off by a few random sprigs of greying hair.

  ‘I have a book for you,’ said the librarian. I craned upward a moment before taking the recommendation from his big hands. The front cover wore a ‘Bookworms Club Monthly Selection’ sticker. The Borrowers was the name of the book. I thanked him, less out of gratitude than the desire to end the sudden eclipse around my table. But when I finally left the desk an hour later, the book left with me, checked out and tucked firmly beneath my arm.

  It told the story of a tiny family that lived under the floorboards of a house. To furnish their humble home, the father would scamper out from time to time and ‘borrow’ the household’s odds and ends.

  My siblings and I tried to imagine what it would be like to live so small. In my mind’s eye, I pictured the world as it continued to contract. The smaller I became, the bigger my surroundings grew. The familiar now became strange; the strange became familiar. All at once, a face of ears and eyes and hair becomes a pink expanse of shrub and grooves and heat. Even the tiniest fish becomes a whale. Specks of dust take flight like birds, swooping and wheeling above my head. I shrank until all that was familiar disappeared completely, until I could no longer tell a mound of laundry or a rocky mountain apart.

  At my next visit to the library, I duly joined the Bookworms Club. The months were each twinned with a classic story, and some of the selections enchanted me more than others, but it was December’s tale that truly seized my senses: The Lion, the Witch, and the Wardrobe by C. S. Lewis. Opening its pages I followed Lucy, as she was sent with her siblings ‘away from London during the war because of the air raids . . . to the house of an old professor who lived in the heart of the country.’ It was ‘the sort of house that you never seem to come to the end of, and it was full of unexpected places.’

  With Lucy I stepped into the large wardrobe in one of the otherwise bare spare rooms, tussled with its rows of dense and dust-fringed clothes as we fumbled our way with outstretched fingers toward the back. I, too, suddenly heard the crunch of snow beneath my shoes, and saw the fur coats give way all at once to the fir trees of this magical land, a wardrobe’s depth away.

  Narnia became one of my favourite places, and I visited it many times that winter. Repeated readings of the story would keep me in bright thoughts and images for many months.

  One day, on the short walk home from school, it so happened that these images came to the front of my mind. The lamp posts that lined the street reminded me of the lamp post I had read about in the story, the point in the landscape from which the children return to the warmth and mothballs of the professor’s wardrobe.

  It was mid-afternoon, but the electric lights were already shining. Fluorescent haloes stood out in the darkening sky at equidistant points. I counted the time it took me to step with even paces from one lamp post to another. Eight seconds. Then I retreated, counting backwards, and arrived at the same result. A few doors down, the lights now came on in my parents’ house; yellow rectangles glowed dimly between the red bricks. I watched them with only half a mind.

  I was contemplating those eight seconds. To reach the next lamp post I had only to take so many steps. Before I reached there I would first have to arrive at the midway point. Four seconds that would take me. But this observation implied that the remaining four seconds also contained a midway point. Six seconds from the start, I would land upon it. Two seconds now would separate me from my destination. Yet before I made it, another halfway point – a second later – would intervene. And here I felt my brain seethe hot under my woollen hat. For after seven seconds, the eighth and final second would likewise contain a halfway point of its own. Seven and one half seconds after starting off, the remaining half a second would also not elapse before I first passed its midway point in turn. After seven seconds and three-quarters, a stubborn quarter of a second of my journey would still await me. Going halfway through it would leave me an eighth of a second still to go. One sixteenth of a second would keep me from my lamp post, then 1/32 of a second, then 1/64, then 1/128, and so on. Fractions of fractions of fractions of a second would always distance me from the end.

  Suddenly I could no longer depend on those eight seconds to deliver me to my destination. Worse, I could no longer be sure that they would let me move one inch. Those same interminable fractions of seconds that I had observed toward the end of my journey applied equally to the start. Say my opening step took one second; this second, of course, contained a halfway point. And before I could cross this half of a second, I would first have to traverse its own midway point (the initial quarter second), and so on.

  And yet my legs disposed of all these fractions of seconds as they had always done. Adjusting the heavy schoolbag on my back, I walked the length between the lamp posts and counted once again to eight. The word rang out defiantly into the cold crisp air. The silence that followed, however, was short-lived. ‘What are you doing standing outside in the cold and dark?’ shouted my father from the yellow oblong of the open front door. ‘Come inside now.’

  I did not forget the infinity of fractions that lurked between the lamp posts on my street. Day after day, I found myself slowing involuntarily to a crawl as I passed them, afraid perhaps of falling between the whole seconds into their interspersed gaps. What a sight I must have made, inching warily forward tiny step after tiny step with my round woolly head and the lumpy bag upon my back.

  Numbers within numbers, and so tiny! I was amazed. These fractions of fractions of fractions of fractions of fractions went on forever. Add any of them to zero and they hardly registered at all. Add tens, hundreds, thousands, millions, billions of them to zero and the result is still almost exactly zero. Only infinitely many of these fractions could lead from zero to one, from nothing to something:

  ½ + ¼ + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256 + 1/512 + 1/1024 . . . = 1.

  One evening in the New Year, my mother, very flustered, asked me to be on my best behaviour. Guests – a rarity – were due any moment now, for dinner. My mother, it seemed, intended to repay some favour to the librarian’s wife. ‘No funny questions,’ she said, ‘and no elbows on the table. And after the first hour, bed!’

  The librarian and his wife arrived right on time with a bottle of wine that my parents never opened. With their backs to one another, they thrashed themselves out of their coats before sitting at the dining room table, side by side. The wife offered my mother a compliment about the chequered tablecloth. ‘Where did you buy it?’ she asked, over her husband’s sigh.

  We ate my father’s roast chicke
n and potatoes with peas and carrots, and as we ate the librarian talked. All eyes were on him. There were words on the weather, local politics and all the nonsense that was interminably broadcast on TV. Beside him, his wife ate slowly, one-handed, while the other hand worried her thin black hair. At one point in her husband’s monologue, she tried gently tapping his tightly bunched hand with her free fingers.

  ‘What? What?’

  ‘Nothing.’ Her fork promptly retired to her plate. She looked to be on the verge of tears.

  Very much novices in the art of hospitality, my mother and father exchanged helpless glances. Plates were hurriedly collected, and bowls of ice cream served. A frosty atmosphere presently filled the room.

  I thought of the infinitely many points that can divide the space between two human hearts.

  Counting to Four in Icelandic

  Ask an Icelander what comes after three and he will answer, ‘Three of what?’ Ignore the warm blood of annoyance as it fills your cheeks, and suggest something, or better still, point. ‘Ah,’ our Icelander replies. Ruffled by the wind, the four sheep stare blankly at your index finger. ‘Fjórar,’ he says at last.

  There is a further reason to be annoyed. When you take the phrasebook – presumably one of those handy, rain-resistant brands – from your pocket and turn to the numbers page, you find, marked beside the numeral 4, fjórir. This is not a printing error, nor did you hear the Icelander wrong. Both words are correct; both words mean ‘four’. This should give you your first inkling of the sophistication with which these people count.

  I first heard Icelandic several years ago during a trip to Reykjavík. No phrasebook in my pocket, thank God. I came with nothing more useful than a vague awareness of the shape and sounds of Old English, some secondary-school German, and plenty of curiosity. The curiosity had already seen service in France. Here in the North, too, I favoured conversation over textbooks.