Thinking in Numbers: How Maths Illuminates Our Lives (3 page)

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 chicken 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.

I hate textbooks. I hate how they shoehorn even the most incongruous words – like ‘cup’ and ‘bookcase,’ or ‘pencil’ and ‘ashtray’ – onto the same page, and then call it ‘vocabulary’. In a conversation, the language is always fluid, moving, and you have to move with it. You walk and talk and see where the words come from, and where they should go. It was in this way that I learnt how to count like a Viking.

Icelanders, I learnt, have highly refined discrimination for the smallest quantities. ‘Four’ sheep differ in kind from ‘four’, the abstract counting word. No farmer in Hveragerði would ever dream of counting sheep in the abstract. Nor, for that matter, would his wife or son or priest or neighbour. To list both words together, as in a textbook, would make no sense to them whatsoever.

Do not think that this numerical diversity applies only to sheep. Naturally enough, the woolly mammals feature little in town dwellers’ talk. Like you and me, my friends in Reykjavík talk about birthdays and buses and pairs of jeans but, unlike English, in Icelandic these things each require their own set of number words.

For example, a toddler who turns two is
tveggja
years old. And yet the pocket phrasebook will inform you that ‘two’ is
tveir
. Age, abstract as counting to our way of thinking, becomes in Icelandic a tangible phenomenon. Perhaps you too sense the difference: the word
tveggja
slows the voice, suggesting duration. We hear this possibly even more clearly in the word for a four-year-old:
fjögurra
. Interestingly, these sounds apply almost exclusively to the passage of years – the same words are hardly ever used to talk about months, days, or weeks. Clock time, on the other hand, renders the Icelander almost terse as a tick: the hour after one o’clock is
tvö
.

What about buses? Here numbers refer to identity rather than quantity. In Britain or America, we say something like, ‘the number three bus’, turning the number into a name. Icelanders do something similar. Their most frequent buses are each known by a special number word. In Reykjavik, the number three bus is simply
þristur
(whereas to count to three the Icelander says
þrír
).
Fjarki
is how to say ‘four’ when talking buses in Iceland.

A third example is pairs of something – whether jeans or shorts, socks or shoes. In this case, Icelanders consider ‘one’ as being plural:
einar
pair of jeans, instead of the phrasebook
einn
.

With time and practice, I have learned all these words, more words for the numbers one to four than has an Englishman to count all the way up to fifty. Why do Icelanders have so many words for so few numbers? Of course, we might just as well wonder: why in English are so many numbers spoken of in so few words? In English, I would suggest, numbers are considered more or less ethereal – as categories, not qualities. Not so the smallest numbers in Icelandic. We might, for instance, compare their varieties of one, two, three and four, to our varieties of colour. Where the English word ‘red’ is abstract, indifferent to its object, words like ‘crimson,’ ‘scarlet’ and ‘burgundy’ each possess their own particular shade of meaning and application.