Thinking in Numbers: How Maths Illuminates Our Lives Read online

  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.

  So, Icelanders think of the smallest numbers with the nuance that we reserve for colour. We can only speculate as to the reason why they stop at the number five (for which, like every number thereafter, a single word exists). According to psychologists, humans can count in flashes only up to quantities of four. We see three buttons on a shirt and say ‘three’; we glance at four books on a table and say ‘four’. No conscious thought attends this process – it seems to us as effortless as the speech with which we pronounce the words. The same psychologists tell us that the smallest numbers loom largest in our minds. Asked to pick a number between one and fifty, we tend toward the shallow end of the scale (far fewer say ‘forty’ than ‘fourteen’). It is one possible explanation for why only the commonest quantities feel real to us, why most numbers we accept only on the word of a teacher or textbook. Forty, to us, is but a vague notion; fourteen, on the other hand, is a sensation within our reach. Four, we recognise as something solid and definite. In Icelandic, you can give your baby the name ‘Four’.

  I do not know Chinese, but I have read that counting in this language rivals in its sophistication even that of the Icelanders. A shepherd in rural China says sì zhî when his flock numbers four, whereas a horseman – possessing the same quantity of horses – counts them as sì pi.This is because mounts are counted differently in Chinese to other animals. Domesticated animals, too. Asked how many cows he had milked that morning, a farmer would reply sì tóu (four). Fish are a further exception. Sì tiáo is how an angler would count his fourth catch of the day.

  Unlike Icelandic, in Chinese these fine distinctions apply to all quantities. What saves its speakers from endless trouble of recall is generalisation. Sì tiáo means ‘four’ when counting fish, but also trousers, roads, rivers (and other long, slender and flexible objects). A locksmith might enumerate his keys as qî ba (seven), but so too would a housewife apropos her seven knives, or a tailor when adding up his seven pairs of scissors (or other handy items). Imagine that, with his scissors, the tailor snips a sheet of fabric in two. He would say he has liãng zhâng (two) sheets of fabric, using the same number word as he would for paper, paintings, tickets, blankets and bed sheets. Now picture the tailor as he rolls the fabric into long inflexible tubes. This pair he counts as liãng juan (two scrolls, or rolls of film would be counted in the same way). Scrunching the sheets into balls, the tailor counts them as liãng tuán, insofar as they resemble other pairs of round things.

  When counting people, the Chinese start from yî ge (one), though for villagers and family members they begin yî kou, and yî míng for lawyers, politicians and royals. Numbering a crowd thus depends on its composition. A hundred marchers would be counted as yîbãi ge if they consist, for example, of students, but as yîbãi kou if they hail from the villages.

  So complex is this method of counting that, in some regions of China, the words for certain numbers have even taken on the varying properties of a dialect. Wushí li, for example, a standard Mandarin word meaning ‘fifty’ (when counting small, round objects like grains of rice), sounds truly enormous to the speakers of southern Min, for whom it is used to count watermelons.

  This profusion of Icelandic and Chinese words for the purpose of counting appears to be an exception to the rule. Many of the world’s tribal languages, in contrast, make do with only a handful of names for numbers. The Veddas, an indigenous people of Sri Lanka, are reported to have only words for the numbers one (ekkamai) and two (dekkamai). For larger quantities, they continue: otameekai, otameekai, otameekai . . . (‘and one more, and one more, and one more . . .’). Another example is the Caquintes of Peru, who count one (aparo) and two (mavite). Three they call ‘it is another one’; four is ‘the one that follows it’.

  In Brazil, the Munduruku imitate quantity by according an extra syllable to each new number: one is pug, two is xep xep, three is ebapug, and four is edadipdip. They count, understandably, no higher than five. The imitative method, while transparent, has clear limitations. Just imagine a number word as many syllables long as the quantity of trees leading to a food source! The drawling, seemingly endless, chain of syllables would prove far too expensive to the tongue (not to say the listener’s powers of concentration). It pains the head even to think about what it would be like to have to learn to recite the ten times tables in this way.

  All this may sound almost incomprehensible for those brought up speaking languages that count to thousands, millions and beyond, but it does at least make the relationship between a quantity and its appointed word sound straightforward and conventional. Quite often though, it is not. In many tribal languages, we find that the names for numbers are perfectly interchangeable, so that a word for ‘three’ will also sometimes mean ‘two’, and at other times ‘four’ or ‘five’. A word meaning ‘four’ will have ‘three’ and ‘five’ – occasionally ‘six’ – as synonyms.

  Few circumstances within these communities require any greater numerical precision. Any number beyond their fingertips i
s superfluous to their traditional way of life. In many of these places there are, after all, no legal documents that require dates, no bureaucracies that levy taxes, no clocks or calendars, no lawyers or accountants, no banks or banknotes, no thermometers or weather reports, no schools, no books, no playing cards, no queues, no shoes (and hence, no shoe sizes), no shops, no bills and no debt to settle. It would make as much sense to tell them, say, that a group of men amounts exactly to eleven, as for someone to inform us that this same group has precisely one hundred and ten fingers, and as many toes.

  There is a tribe in the Amazon rainforest who know nothing whatsoever of numbers. Their name is the Pirahã or the Hi’aiti’ihi, meaning ‘the straight ones’. The Pirahã show little interest in the outside world. Surrounded by throngs of trees, their small clusters of huts lie on the banks of the Maici River. Tumbling grey rain breaks green on the lush foliage and long grass. Days there are continuously hot and humid, inducing a perpetual look of embarrassment on the faces of visiting missionaries and linguists. Children race naked around the village, while their mothers wear light dresses obtained by bartering with the Brazilian traders. From the same source, the men display colourful T-shirts, the flotsam of past political campaigns, exhorting the observer to vote Lula.

  Manioc (a tough and bland tuber), fresh fish and roasted anteater sustain the population. The work of gathering food is divided along lines of sex. At first light, women leave the huts to tend the manioc plants and collect firewood, while the men go upriver or downriver to fish. They can spend the whole day there, bow and arrow in hand, watching water. For want of any means of storage, any catch is consumed quickly. The Pirahã apportion food in the following manner: members of the tribe haphazardly receive a generous serving until no more remains. Any who have not yet been served ask a neighbour, who has to share. This procedure only ends when everyone has eaten his fill.

  The vast majority of what we know about the Pirahã is due to the work of Daniel Everett, a Californian linguist who has studied them at close quarters over a period of thirty years. With professional perseverance, his ears gradually soothed their cacophonic ejections into comprehensible words and phrases, becoming in the process the first outsider to embrace the tribe’s way of life.

  To the American’s astonishment, the language he learned has no specific words for measuring time or quantity. Names for numbers like ‘one’ or ‘two’ are unheard of. Even the simplest numerical queries brought only confusion or indifference to the tribesmen’s eyes. Of their children, parents are unable to say how many they have, though they remember all their names. Plans or schedules older than a single day have no purchase on the Pirahã’s minds. Bartering with foreign traders simply consists of handing over foraged nuts as payment until the trader says that the price has been met.

  Nor do the Pirahã count with their bodies. Their fingers never point or curl: when indicating some amount they simply hold their hand palm down, using the space between their hand and the ground to suggest the height of the pile that such a quantity could reach.

  It seems the Pirahã make no distinction between a man and a group of men, between a bird and a flock of birds, between a grain of manioc flour and a sack of manioc flour. Everything is either small (hói) or big (ogii ). A solitary macaw is a small flock; the flock, a big macaw. In his Metaphysics, Aristotle shows that counting requires some prior understanding of what ‘one’ is. To count five, or ten or twenty-three birds, we must first identify one bird, an idea of ‘bird’ that can apply to every possible kind. But such abstractions are entirely foreign to the tribe.

  With abstraction, birds become numbers. Men and maniocs, too. We can look at a scene and say, ‘There are two men, three birds and four maniocs’ but also, ‘There are nine things’ (summing two and three and four). The Pirahã do not think this way. They ask, ‘What are these things?’ ‘Where are they?’, ‘What do they do?’ A bird flies, a man breathes and a manioc plant grows. It is meaningless to try to bring them together. Man is a small world. The world is a big manioc.

  It is little surprise to learn that the Pirahã perceive drawings and photos only with great difficulty. They hold a photograph sideways or upside down, not seeing what the image is meant to represent. Drawing a picture is no easier for them, not even a straight line. They cannot copy simple shapes with any fidelity. Quite possibly, they have no interest in doing so. Instead their pencils (furnished by linguists or missionaries) produce only repeating circular marks on the researcher’s sheet of paper, each mark a little different to the last.

  Perhaps this also explains why the Pirahã tell no stories, possess no creation myths. Stories, at least as we understand them, have intervals: a beginning, a middle and an end. When we tell a story, we recount: naming each interval is equivalent to numbering it. Yet the Pirahã talk only of the immediate present: no past impinges on their actions; no future motivates their thoughts. History, they told their American companion, is ‘where nothing happens, and everything is the same.’

  Lest anyone should think tribes such as the Pirahã somehow lacking in capacity, allow me to mention the Guugu Yimithirr of north Queensland in Australia. In common with most Aboriginal language speakers, the Guugu Yimithirr have only a handful of number words: nubuun (one), gudhirra (two) and guunduu (three or more). This same language, however, permits its speakers to navigate their landscape geometrically. A wide array of coordinate terms attune their minds intuitively to magnetic north, south, east and west, so that they develop an extraordinary sense of orientation. For instance, a Guugu Yimithirr man would not say something like, ‘There is an ant on your right leg,’ but rather ‘There is an ant on your southeast leg.’ Or, instead of saying, ‘Move the book back a bit,’ the man would say, ‘Move the book to the north northwest a bit.’

  We are tempted to say that a compass, for them, has no point. But at least one other interesting observation can be drawn from the Guugu Yimithirrs’ ability. In the West, young children often struggle to grasp the concept of a negative number. The difference between the numbers two (2) and minus two (-2) often evades their imagination. Here the Guugu Yimithirr child has a definite advantage. For two, the child thinks of ‘two steps east’, while minus two becomes ‘two steps west’. To a question like, ‘What is minus two plus one?’ the Western child might incorrectly offer, ‘Minus three’, whereas the Guugu Yimithirr simply takes a mental step eastward to arrive at the right answer of ‘one step west’ (-1).

  A final example of culture’s effect on how a person counts, from the Kpelle tribe of Liberia. The Kpelle have no word in their language corresponding to the abstract concept of ‘number’. Counting words exist, but are rarely employed above thirty or forty. One young Kpelle man, when interviewed by a linguist, could not recall his language’s term for seventy-three. A word meaning ‘one hundred’ frequently stands in for any large amount.

  Numbers, the Kpelle believe, have power over people and animals and are to be traded only lightly and with a kind of reverence; village elders therefore often guard jealously the solutions to sums. From their teachers, the children acquire only the most basic numerical facts in piecemeal fashion, without learning any of the rhythm that constitutes arithmetic. The children learn, for example, that 2 + 2 = 4, and perhaps several weeks or months afterwards that 4 + 4 = 8, but they are never required to connect the two sums and see that 2 + 2 + 4 = 8.

  Counting people directly is believed by the Kpelle to bring bad luck. In Africa, this taboo is both ancient and widespread. There exists as well the sentiment, shared by the authors of the Old Testament, that the counting of human beings is an exercise in poor taste. The simplicity of their numbers is not linguistic, or merely linguistic, but also ethical.

  I read with pleasure a book of essays published several years ago by the Nigerian novelist Chinua Achebe. In one, Achebe complained of the Westerners who asked him, ‘How many children do you have?’ Rebuking silence, he suggested, best answered such an impertinent question.

 
‘But things are changing and changing fast with us . . . and so I have learned to answer questions that my father would not have touched with a bargepole.’

  Achebe’s children number ano (four). In Iceland, they would say, fjögur.

  Proverbs and Times Tables

  I once had the pleasure of discovering a book wholly dedicated to the art of proverbs. It was in one of the municipal libraries that I frequented as a teenager. The title of the book escapes me now, the name of its author, too, but I still recall the little shiver of excitement I felt as my fingers caressed its quarto pages.

  ‘Penny wise, pound foolish’

  ‘Small fishes are better than empty dishes’

  ‘A speech without proverb is like a stew without salt’

  Now that I come to think of it, I doubt this book had any single author. Every proverb is anonymous. Each appears in a society’s mental repertoire by some process akin to immaculate composition. Like the verses in the Muslims’ Koran, proverbs seem pre-written, patiently awaiting a mouth to utter their existence. Some linguists contend that language happens independently of our reason, its origins traceable to a still mysterious and exclusive gene. Perhaps proverbial logic is like language in this respect, its existence as essential to our humanity as the power of speech.

  Whoever the author or the editor, this book proved to me that there are only so many proverbs a healthy man can take. A point of surfeit is reached, beyond which the reader can no longer follow: his brain starts to ache, his eyes to water. Taken in excess, proverbs lose all the familiar felicities of their compact structure. They start to read merely as repetitions – which many by then quite probably are. From my experience, I estimate this limit at about one hundred.