Thinking in Numbers: How Maths Illuminates Our Lives (27 page)

In Australia, the Aborigines believe that time, place and people are one. A glance at a tree or a face suffices to know the hour and the day. Their discrimination of the seasons is precise, depending on such factors as plant life and changes in the wind: the Eastern Gunwinggu, for example, speak of six seasons – three ‘dry’ and three ‘wet’ – where non-Aborigines see only one of each.

For these and other tribes, time is the product of our actions. It appears when we sing a song, climb a mountain or smoke a pipe, and vanishes when we sleep. They do not think of time as something pervasive, like the air. Seconds, minutes and hours – these are all things that we do. In place of these terms, they speak of a ‘time of harvesting’ or a ‘river fish time’. Ask an African herdsman how long such-and-such task might take and he replies, ‘Cow milking time’, meaning the time it takes to milk a cow. What is an hour to such a man? Perhaps the time it takes to milk ten cows.

We can put it another way: 1 hour = 10 milkings. My equivalent would be 1 hour = 10 tea-makings. Let us call it ‘tea time’. A short walk that lasts eighteen minutes equates to three milkings or makings-of-tea; a two-minute advert break amounts to one-third of a cup of tea. Between the opening and closing whistles of a football referee, time enough would pass to milk fifteen cows, or make fifteen cuppas.

I do not mean by this digression to suggest that approximations necessarily trump exactitude. It is not at all my intention to run clocks down. But the particular words and images our respective cultures deploy, shape the way in which we experience time. I said just now that time is not money; we might say instead that it is closer to the spending of money. According to the tribesmen’s way of thinking, it is what happens when, for example, we enter a marketplace. This emphasis on activity in how we think about time strikes me as being very healthy. When I hear someone complain about all the hours or weekends he has to fill, I stop and think that it is a mistake to speak of days as we would speak of holes. One hole is much the same as any other, whereas every day is different. In this, it is more like dough that we can sculpt into infinitely varying shapes.

On the journey back to my childhood home, I paused outside the train station, then made my way north toward the high street. The buildings were more or less the same as in my recollection: the same squat walls, tattooed with graffiti; the same ‘50% off’ signs in shop windows; the same boys and girls, their busy fingers unwrapping sweets. No bravado in the architecture, no colour or charm. Along the pavements, no bustle either – either too early or too late for shoppers. Few cars animated the road. I walked mechanically, turning here and there, smelling the sugar of freshly laid tarmac on Waterbeach Road.

I landed finally on my old street. I took it all in. On the left stood metal railings and distantly behind them the classroom buildings of my former primary school, factory-long. To the right, a chain of brick houses, close set. Their thin walls, I recall, made bad neighbours. Down the road, I spotted a small man in the distance. The man grew bigger with every step. He was wearing a blue and red football shirt, but he did not look like a footballer. The tightness of the shirt pronounced a sizeable paunch. His dark hair was cut penitentiarially short. His breathing rasped as he passed me by. And then he was gone.

I was surprised by how little had been altered. Painted house number signs, wooden gates, hedgerows all long forgotten, I recognise instantly. And yet it all seems so different from my kid days. Something has shifted out of sync, something I try to put my finger on. In frustration, I walk up and down the street until my legs tire. Only as I ready myself for the ride back does it hit me. What has changed here is: time.

In his 1890 classic work,
Principles of Psychology
, the American philosopher William James noted, ‘The same space of time seems shorter as we grow older – that is, the days, the months, and the years do so; whether the hours do so is doubtful, and the minutes and seconds to all appearance remain about the same.’

James goes on to cite a mathematical explanation for this phenomenon, by a contemporary French professor. According to this professor, Paul Janet, our experience of time is proportional to our age. For a ten-year-old child, one year represents one tenth of his existence; whereas for a man of fifty, the same year equates only to one fiftieth (two per cent). The older man’s year will thus seem to elapse five times faster than the child’s; the child’s, five times slower than the man’s.

What matters, then, is the relationship between one sequence of years and another sequence. The interval spanning the ages of thirty-two and sixty-four will seem to the individual of similar duration to that experienced between the ages of sixteen and thirty-two, and to the interval between the ages of eight and sixteen, and to that from the age of four to eight, each having the same ratio. For the same reason, all the years from the age of sixty-four to one hundred and twenty-eight (assuming such an age were ever attainable) would seem to us to occupy no more of our feeling, thought, pain, fear, joy and wonder than that big bang epoch between our second and fourth year.

More recently, from an American called T.L. Freeman, we have a formula using Janet’s insight that yields the individual’s ‘effective age’. Freeman’s calculations suggest that we experience a quarter of our entire lifetime by age two, over half by age ten, and more than three-quarters by our thirtieth birthday. At only about the chronological midway point, a forty-year-old will experience his remaining time as seemingly but one-sixth of what has gone before. For a sixty-year-old, the future will seem to last merely one-sixteenth the duration of his past.

Are all our attempts to look back, to relive some bygone period, in vain? We can never walk down the same street twice. Those streets of my youth belong to another time, which is no longer my own. Except, that is, when I dream.

Fast asleep, I become a visitor there. I see a schoolgirl at the edge of the hopscotch grid, contemplating her throw. A man, atop a ladder, is washing his windows. His free hand glides rhythmically upon the glass. On the pavement, a neighbour’s tabby squirms in the sunshine: stretching, and stretching his paws. The grunts and sighs of passing traffic fill my ears. I see my grandfather, alive, standing with his cane at the gate, as though keeping guard over my father’s vegetable patch. I stop and watch my father. Sleeves hoisted to his elbows, he picks beans, sows herbs and counts cucumbers. I watch without hurry, without a care in the world. Time is dilated; there is no time.

Our body keeps time a great deal better than our brain. Hair and nails grow at a predictable rate. An intake of breath is never wasted; appetite hardly ever comes late or early. Think about animals. Ducks and geese need only follow their instinct for when to up sticks and migrate. I have read of oxen that carried their burden for precisely the same duration every day. No whip could persuade them to continue beyond it.

We wear the tally of our years on our brow and cheeks. I doubt our body could ever lose its count. Like the ox, each knows intimately the moment when to stop.

Higher than Heaven

On 22 January 1886 Georg Cantor, who had discovered the existence of an infinite number of infinities, wrote a letter to Cardinal Johannes Franzen of the Vatican Council, defending his ideas against the possible charge of blasphemy. A devout believer, the mathematician considered himself a friend of the Church. God, he believed, had used his preoccupation with numbers to reveal a further aspect of His infinite nature. Fellow logicians had mostly sidestepped the young man’s thinking; hardly anyone yet took seriously the outstanding insights that would make his name.

Before Cantor it had been impossible to speak mathematically about different sorts of infinity. All collections without a final object (the sequence of odd or even numbers, for example, or the primes) were simply conceived as being of equal size. Cantor proved that this was false. His papers were the first to demonstrate uncountable sets of numbers, that is to say, numerical sequences that even an infinitely long recitation could not exhaust. What is more, each uncountable set of numbers spawned another set of numbers that was even ‘bigger’ than the last. Of the making of such sets, Cantor realised, there was no end.

The mathematician Leopold Kronecker, for whom ‘God created the integers [whole numbers], all else is the work of man,’ had no truck with Cantor’s (infinite) tower of ‘smaller’ and ‘bigger’ infinities. He hounded his rival with violent words, called him a charlatan, a corrupter of youth. In the absence of his peers’ understanding, Cantor turned at last for support to the Holy See.

The dialogue between theology and mathematics – varied, fitful, and singular – has a long history. Above all, infinity became the favourite topic. God is infinite, therefore mathematics is religion: a pathway to knowledge of the divine. This is what the Church fathers reasoned, and this is why the monks long ago proceeded where the mathematicians had feared to tread.

A thousand years before Cantor, in an Irish monastery, a man sat day after day at a table smelling of wicks and manuscripts. He spent years almost immobile, in deep and sustained contemplation, meditating on a perfect sphere that exists beyond space, universal and without limit. Of course, it is contradictory to think about a shape that has no border. The monk knew this. He knew that to think about infinity is to think in contradictions.

Minutes passed, hours passed. But what is a minute or an hour when compared to eternity? No time at all. A minute, an hour, a year, a thousand years are all equally long or short in comparison. The light in the monk’s cell would gradually disperse at the end of each long day; his mind might stutter, ‘I, I, I, I, I . . .’, but try as he would, Johannes Scottus Eriugena – John of Ireland – could not escape his senses and grasp the infinite.

According to Eriugena, God is not good, since He is beyond goodness; not great since He is beyond greatness; not wise since He is beyond wisdom. God, he writes, is more than God, more than time,
infinitas omnium infinitatum
(the infinity of all infinities), the beginning and end of all things, though He Himself had no beginning and will meet no end. Eriugena recalls the words of Job.

Can you search out the deep things of God? Can you find out the limits of the Almighty? They are higher than heaven — what can you do? Deeper than Sheol — what can you know? Their measure is longer than the earth and broader than the sea. If He passes by, imprisons, and gathers to judgment, then who can hinder Him?

If God is infinite, Holy Scripture, being inspired by God, is held to exist outside the bonds of conventional time. Eriugena cites St Augustine to affirm that the Bible often employs the past tense to express the future. Adam’s life in Paradise ‘only began,’ occupying no real time at all, so that its depiction in Genesis ‘must refer rather to the life that would have been his if he had remained obedient’.

Augustine’s teachings contributed greatly to the Irish monk’s thought, and that of the theologians who followed. In
The City of God
, Augustine insists that God knows every number to infinity and can count them all instantaneously. ‘If everything which is comprehended is defined or made finite by the comprehension of him who knows it, then all infinity is in some ineffable way made finite to God, for it is comprehensible by his knowledge.’

Two centuries after Eriugena, in 1070, Anselm provided his famous ‘ontological proof’ that God is that-than-which-nothing-greater-can-be-thought. If every number has its object, the object of infinity is God. Anselm became Archbishop of Canterbury; one of his successors, Thomas Bradwardine, in the fourteenth century, identifies the divine being with an infinite vacuum. The finite world is compared to a sponge in a boundless sea of space.

Infinity begets finitude, and thus cannot be grasped in finite terms. But how then to understand infinity in infinite terms? Alexander Neckham, a twelfth-century reviver of interest in Anselm’s work, offered this problem a vivid image. For Neckham, God’s immensity is such that even if one were to double the world in the next hour, and then triple it in the hour after that, then quadruple it in the following hour, and so on, still the world would be but a ‘quasi point’ in comparison.

Such immensity inspires in the monks at once admiration and consternation: consternation, because an infinitely remote divine being would rule out the Incarnation. For the same reason, the believer would never see God in the Beatific Vision, and neither could he ever conform his will to the divine will. The vacuum is in fact a chasm, forever separating Mankind from its Creator.

The
De Veritate
of Thomas Aquinas, written between 1256 and 1259, offers a solution: ‘as the ruler is related to the city, so is the pilot to the ship’. An infinitely powerful ruler bears no direct comparison to a humble captain, yet both possess a ‘likeness of proportions’: a finite quantity equates to another finite quantity, in the same way that the infinite is equal to the infinite. In other words, ‘three is to six as five million is to ten million’ bears a likeness to the proportion ‘God is to the angels as the infinite vacuum is to an eternal creation’. Aquinas deploys the analogy throughout his work: as our finite understanding grasps finite things so does God’s infinite understanding grasp infinite things; as our finite intellect is to what it knows, so is God’s infinite intellect to the infinitely many things He knows; just as men distribute finite goods so does God distribute all the goods of the universe. Aquinas writes that the similarity between the infinite God and His finite creation constitutes a ‘community of analogy . . . The creature possesses no being except insofar as it descends from the first being, nor is it named a being except insofar as it imitates the first being.’

Exasperated by critics he called ‘murmurers’, Aquinas sought to settle a further point of contention. The Church taught that the world had a beginning in time. ‘The question still arises whether the world could have always existed.’ He penned these words in 1270, entitling them
De Aeternitate Mundi
(On the Eternal World). His argument was that if the world has always existed, the past regresses infinitely. The world’s history must comprise an infinite sequence of past events. If there exists an infinite number of yesterdays, then an infinite number of tomorrows must also succeed. Time is infinitely past, and infinitely future, but never present. For how can any present moment arrive after infinitely many days?