Thinking in Numbers: How Maths Illuminates Our Lives

  One possible (if partial) explanation for this structure is the marked Japanese preference for odd numbers. In the annual Shichigosan (Seven-Five-Three) festival, three-year-old children of both sexes, five-year-old boys and seven-year-old girls visit shrines to celebrate their growth. Cheer groups at a sports match clap in three-three-seven beats. Even numbers, meanwhile, are virtual bogeymen. The number two represents parting and separation, while four is associated with death. In several phrases the number six translates roughly as ‘good-for-nothing’.

  Prime numbers contribute to the haiku form’s elemental simplicity. Each word and image calls out for our undivided attention. The result is an impression of sudden, striking insight, as if the poem’s objects had been put into words for the very first time. A sample from the form’s most celebrated practitioner, the seventeenth-century poet Matsuo Basho, illustrates this.

  Michinobe no (The mallow flower)

  Mukuge wa uma ni (Against the side of the road)

  Kuwarekeri (Eaten by my horse)

  There exists as well a slightly longer version of the haiku, the tanka form, which compliments the haiku’s trio of lines with two further lines of seven syllables each (known as the shimo-no-ku or ‘lower phrase’). The tanka’s total number of syllables – thirty-one – is, once again, a prime.

  Basho, whose name today is synonymous with the haiku form, counted various influences on his work, but chief among them was a wandering monk who lived at the time of Arnaut Daniel and wrote some of the finest tanka verse. His name was Saigyo. Something of the master’s evocative simplicity is suggested in the following tanka.

  Michi no be ni (On that roadside lea)

  Shimizu nagaruru (Where pure, crystal waters flowed)

  Yanagi kage (Grew a willow tree)

  Shibashi to te koso (For a little while I stayed)

  Tachidomaritsure (There and rested in its shade)

  It seems that the image of Saigyo’s willow tree took root in the imagination of many generations of poets. Some five centuries after the poem’s composition, Basho underwent a pilgrimage to its site in the north. In his travel diary he noted that ‘[t]he willow about which Saigyo wrote the famous poem still stood by a rice field in Ashino village. As Mr Koho, who governed this county, always wanted to show the tree to me, I had been anxious to discover its location. I was happy to stop by the tree today.’

  Basho’s homage culminated in a haiku dedicated to Saigyo’s tree.

  Ta ichimai (Over a whole field)

  Uete tachisaru (They have planted rice, before)

  Yanagi kana (I leave the willow)

  As I think of the complicity between poems and primes, perhaps the only surprise is that we should even find it surprising. Viewed one way, the relationship makes a perfect kind of sense. Poetry and prime numbers have this in common: both are as unpredictable, difficult to define and multiple-meaning as a life.

  This life-like quality connecting them is too often overlooked. Many poems, it is true, lie mothballed in slim anthologies; many primes languish in mathematicians’ sums. Picked and pored over by their experts, they lose the public’s attention (and affection) for the academic company they keep.

  And yet we see her so clearly, Dante’s woman, as she wanders through our memory, and Basho’s horse, munching his flower, looks and sounds all too real. Free (as a prime number) from a rhyme’s reassurance or a storybook’s rules, the images swerve and forestall our expectations – and keep all clichés at bay.

  Poems and primes are tricky things to recognise. A glance will usually not suffice to tell us if such-and-such a number has factors, or whether a given text contains much meaning. Even old hands can find it hard to tell genuinely felt verse from a trite list of nice-sounding words, or spot a composite number as being merely a pastiche of lesser primes.

  Dante’s sestina, haiku verses and the hither and thither of the primes, for each we ask ourselves, what does it mean? Are we, in the end, any closer to the woman whose beauty ‘has more virtue than rare stone’? Her face changes with the stanzas, offering us a multiplicity of perspectives. And what of Saigyo’s willow tree, which provides at once shade and reflection?

  The same is true of the prime numbers – an ancient mathematical mystery. Thirty-one, the number of syllables in the tanka, is a twin prime (being two apart from its neighbour, twenty-nine) and a Mersenne prime (being one less than a power of two: (2 × 2 × 2 × 2 × 2) – 1), but such labels fall far short of explanation. That is because we do not really know why the prime numbers appear where they do. Many unproven conjectures remain. The reader of poetry and the mathematician are left finally only with hints and fragments, minus any big picture – as in life.

  All Things Are Created Unequal

  Unlike diabetes or curly hair, poverty rarely skips a generation. A parent’s bank balance will often dictate his child’s destiny to a far greater extent than his blood. Blonde mothers sometimes produce brunette babies; tall men may not always spawn basketball pros; but more than ninety times out of every hundred the poor beget more poor.

  I am the son of poor parents, poor grandparents, poor great-grandparents, and so on. Suffice it to say that more than a few grisly tales of hard times have come down to me. One of these was told to me by my father, back in the early 2000s, not long after I had flown the family nest to serve my apprenticeship in the adult world. I was living in a house, south of London, that belonged to someone else. It was the first time I had shared with a housemate. The place had little to commend it. Small, out of the way, it was very modestly decorated. In my box of a bedroom I slept on a sofa bed, pale green, the colour of a plant in shade. I subsisted on students’ fare: small plates of pasta, sandwiches and beans on toast.

  Some evenings I took calls from home. On one occasion my father rang, and we fell into a long conversation. His chattiness, I must admit, took me rather by surprise. He had never been one for self-disclosure. Why then was he opening up to me? For an eldest son’s compassion, for a walk accompanied down memory lane? I do not know. We had been talking about nothing in particular when all of a sudden he said, ‘We moved around a lot when I was a kid.’

  ‘Sorry?’

  ‘My parents, well, my mum and her man. It is a long story.’

  And, just like that, in a matter-of-fact voice, he told me his tale. He told it with such simplicity and such precision, not a word or image out of place, that it dawned on me only later that he had certainly long rehearsed this moment in his mind. Even as I listened I knew better than to interrupt him with my questions or comments. He spoke, as we say, ‘from the heart’; this man, my father, telling me, his son, such things as he considered important and valuable for me to hear.

  One scene in particular struck me. As a boy of ten or thereabouts, he and his parents had been returning home one summer evening from a trip to the town fair. The front lawn, they noticed as they approached the house, had been eerily altered. Running ahead, my father was stunned by what he found. Chairs and tables and pots and pans and beds and lamps had been evacuated. The furniture huddled destitute in a big pile. A padlock on the front door barred their entry.

  I might have asked, ‘Why had the landlord not given my grandparents more time to clear their debt?’ But I did not ask. The thought of the furniture left out in the garden made a deep impression on my mind. It was the image of a home turned inside out, its intimacy smashed, its innards spewed. How terrible they must have looked, those useless lamps and long bare table legs and the embarrassing tea-coloured letters in a half-open desk drawer. The scene felt so vivid that it made my eyes smart.

  My father was born in 1954, the year that followed Queen Elizabeth’s coronation. My grandparents’ expulsion from their home happened ten years later – in the midst of the Swinging Sixties, the decade of ‘peace and love’. In their book The Poor and the Poorest, published in 1965, the sociologists Peter Townsend and Brian Abel-Smith estimated that this ten-year period, the first ten years of my father’s life, had produced
a near doubling in the percentage of Britons living beneath the breadline, rising from eight per cent to fourteen.

  In 1979, the year of my birth, Peter Townsend published a further study, Poverty in the UK, showing that relative hardship stifled the lives of twenty-one per cent of the population. This figure appears to have remained more or less stable ever since.

  A final statistic, this one from 2008, describing the generation after mine: according to the London School of Economics, the household wealth of the top ten per cent of the population now towers one hundred times above that of the lowest ten per cent.

  Inequality is invidious. It is also universal. No country, to judge from the comparative data, has been spared it. Every land has its share both of hovels and five-star hotels. Talk of a ‘classless society’ has, time and time again, proved to be nothing but hot air. Western sympathisers who visited the Soviet Union in the 1930s were disappointed to find that its October Revolution had hardly ‘abolished’ the gulf that divided rich and poor. Zero in imperial units, they learned, was still zero under the metric system. Meanwhile beneath their mud-brown uniforms, stiff with starch, the Kremlin’s rulers continued to wear the slaughtered Emperor’s clothes.

  But enough with statistics: I wonder if mathematics can do more about the phenomenon of disparity than simply measure it. I wonder if it can tell us something about what kind of thing disparity is: where does it come from? What makes it grow or shrink? Can mathematical thinking address these kinds of questions?

  It can. Mathematics and money both originated in abstraction. As with mathematics, we owe the concept of money to the ancient Greeks. They who first abstracted ‘five’ from the fingers on one hand, first stamped ‘five drachmas’ on a metal coin. And just as the concept of ‘five’ slipped from the fingers that had described it, becoming applicable to anything – men, crumbs, daydreams – with identical quantity, so a coin’s ‘value’ exceeded the metal of its composition, capable of transforming into anything agreed as having equal worth.

  Substituting numbers for objects changed the world, for better or worse. At once everything became quantifiable, even the light of the moon, which Aristophanes in one of his plays describes as saving people a drachma’s worth of torches per month. Where previously, bartering and the exchange of gifts had settled all Athenian transactions, now most social dealings came down to sums. Reciprocity between citizens gave way to the potentially unlimited accumulation of individual ‘wealth’. We read with a feeling of familiarity Aristotle’s lament (in his Politics) that some doctors turn their skills into the art of making money. Sophocles goes much further, placing into one of his character’s mouths a blistering denunciation of money as that which ‘lays waste even cities, expels men from their homes,’ and ‘thoroughly teaches and transforms good minds . . . to know every act of impiety.’

  In its abstraction, money acquired the impersonal neutrality of numbers. Goods no longer embodied the donor’s generosity or personality; calculation replaced feelings. Individual autonomy grew, but so too did an egoism that made money the measure of all things. And like abstract numbers, money became invisible. Coins might be concealed far more easily than cows. Lycurgus, ruler of Sparta, found that he could only fight the ‘injustice’ of the rich hiding away large sums by making his iron coins so large and so heavy that ten of them alone required transportation upon a wagon.

  Because numbers can go on forever, money has no limit. Of wealth, Aristophanes tells us, a person can never have enough. Bread and sex and music and courage all sate the preceding appetite, but wealth does not. It is impossible to put a check on moneymaking. If a man receives thirteen coins, he will hanker after sixteen, and possessing them he considers life unbearable unless he now earns forty. Nature, it might be observed, imposes strict boundaries on a person’s height and age span, so that in even the most extreme instances no one can rise or fall too far or too short from the rest, whereas no such boundary inhibits money. Think of King Croesus who had so much gold that he gave it away. It was Croesus whom Solon the legislator memorably warned when he said that he who has much, has much to lose.

  I am reminded here of the story I first heard on the morning radio, two years ago, of an elderly Parisian heiress who had been flattered into great feats of generosity by a much younger man. Of course, the story immediately spread to print: I read the fine details across consecutive pages of my daily paper. The billionaire had allegedly been pampered out of paintings by Munch, Picasso, Matisse, seduced into gifting precious tomes and manuscripts and, all in all, fussed over to the tune of scores of millions of euros in handouts over several years.

  Poor thing!

  Solon, I should mention, became the first man in history to make laws addressing inequality. We read in Plutarch how the mass of ancient Athenians in Solon’s day found themselves in hock to the city’s wealthy aristocrats. Some had been sold into slavery or else had handed over their children as collateral, while others had fled with their family into exile. When he was elected Chief Magistrate, Solon promptly divided the city’s populace into categories, granting to each category proportional responsibilities and rights. The highest class consisted of those earning an income in excess of five hundred bushels. In the second class were those citizens able to afford a horse (and therefore to pay a ‘horse tax’ upon the purchase). Yoke-of-oxen men comprised the third class, with an annual income between two hundred and three hundred bushels. Freed from the fear of enslavement, the remainder of landless citizens could attend public assemblies for the first time, and sit on juries.

  We can express the distribution of wealth in any society as a formula, using a number x between 50 and 100 such that x per cent of the society’s wealth belongs to (100 – x) per cent of the population. In a highly egalitarian society (where x equals, say, 55 or 60), forty-five per cent (or forty per cent) of the people would hold fifty-five per cent (or sixty per cent) of the assets. Most Western societies, however, show a far more skewed distribution. Economists have found that x in most developed countries equates to a number in the region of 80, meaning that eighty per cent of the society’s fortune pads the pockets of only twenty per cent of its members.

  Naturally,the share will vary from person to person. Money is fickle, always changing owners. That different people have dissimilar wealth is not surprising. What surprises is the scale and constancy of the divide. The economist and mathematician Vilfredo Pareto, who first observed (at the end of the nineteenth century) that twenty per cent of Italians owned eighty per cent of the nation’s wealth, found nearly identical results when he studied the historical data from many other parts of Europe. The distribution of wealth in Paris since 1292, he discovered, had hardly moved at all. Later researchers confirmed these findings.

  Because most men and women, for want of resources, remain at the bottom, the elite, for want of competition, remain at the top. The poorest expend all their energies simply to keep body and soul together. I think of Degas’s painting of the two peasant women ironing: one is anonymous, hunched over her iron; the other yawns candidly, her mouth the shape of an ‘O’. The yawn distorts her features, undoing the individuality of her face.

  The Spartan Lycurgus, we saw, made money as big as men; imagine, an instant, men as big as their money. Picture the difference between a miller and a millionaire. The man who mills grain owns perhaps no more than one-thousandth of the rich man’s fortune: the millionaire, his advantage converted into height, would be a thousand times as tall. To him, the miller would appear no bigger than an ant. With whom will the giant do business? Only with someone big and strong enough to carry his employer’s burden. Likewise this someone, smaller than the millionaire on whom he is dependent, but still far greater in size than the grain-running ant, to whom will he entrust his dealings? To his peers. They, in turn, will do the same. Good manners and compromise characterise the bulk of these transactions, but hardly anyone thinks to condescend to those with whom they have next to nothing in common. Our lowly ant friend is simp
ly nowhere in sight.

  Every man and woman, at whichever point along the scale, prefers to look up, not down. Even the miller gives a hand to his equal or his superior, rather than to someone far below him, for fear of conceding his rank to the worse-off man. With the wealthier, he is generous; with the poorer, he is mean. He has little, but the little he has goes purely to keeping up his minor station.

  Comparisons are somewhat facile, of course. Beside a billionaire, even the millionaire is poor. The world’s hundredth-richest person has but one dollar for every eight in the pocket of the world’s richest man.

  Whether the economy expands or whether it shrinks, the obsession with ‘keeping up one’s station’ remains. The inequality this obsession feeds off is a precocious learner: the more of it there is, the faster it will grow. Take the hypothetical egalitarian society, for example, where forty-five per cent of the population own fifty-five per cent of its wealth. In such a society, around twenty per cent (forty-five per cent of the forty-five per cent) of citizens own about thirty per cent (fifty-five per cent of the fifty-five per cent) of the total resources. By the same logic, a sixth (fifty-five per cent of the thirty per cent) of the society’s goods belongs to one citizen in every eleven (forty-five per cent of twenty per cent).

  The contrast between this theoretical society and most of our modern cities – those that obey Pareto’s 80–20 principle – is striking. Wealth in these places can propagate far more ruthlessly, dramatically: the accounts of as few as four individuals in every hundred (twenty per cent of twenty per cent) will bloat with as much as two-thirds (eighty per cent of eighty per cent) of all the available income. And of these four men made of money, the very richest might have up to one half (eighty per cent of the two-thirds) all to himself.

  Human beings and their self-interest are inseparable, but inequality needs a society to invent it. The creation of any vast and ambitious social project demands an unequal allocation of resources in order to achieve its goals. Without substantial inequality, as John Maynard Keynes pointed out, Europe’s railways – a ‘monument to posterity’ – would never have been constructed. Tolstoy, for one, hated the railways precisely because they represented this inequality, going even so far as to throw his best-loved character under a murderous train. It is true that most of the men who laid the rails never had the opportunity to ride them. Why then did they agree to do so? The railwaymen chose to cooperate with the wealthy, Keynes argued, on the tacit understanding that what they produced together, using the money of one side, the labour of the other, would ultimately serve the nation as a whole, and the principle of ‘progress’. World war, however, would subsequently smash this fragile alliance between the classes, by shaking the faith of both in the future. The bludgeoning of bombs and the gutting of gunfire disclosed to all ‘the possibility of consumption . . . and the vanity of abstinence.’