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?
Before this potentially unsettling line of reasoning, Aquinas remained unmoved and unimpressed. Half hearted were his remonstrations. Any past event, like the present moment, is finite: therefore the duration between them is also finite, ‘for the present marks the end of the past’.
And what about the succession of past events? Aquinas says the arguments for them can go either way. Perhaps God, in all His power, has created a world without end. If so, nothing obliged Him to populate it before Mankind.
A contemporary, Bonaventure, disagreed with Aquinas’s equity. His blood thumped at the thought of an interminable past. ‘To posit that the world is eternal or eternally produced, while positing likewise that all things have been produced from nothing, is altogether opposed to the truth and reason.’ And what about the contradictions? For instance, if the world were eternal, tomorrow would be a day longer than infinity. But how can something be greater than the infinite?
In the fourteenth century, Henry of Harclay also faulted Aquinas for saying that an eternal world was possible, but from an entirely opposed point of view to Bonaventure’s. For Harclay it was in fact probable, and every supposed contradiction dissolved on careful scrutiny. How can something be greater than the infinite? Look, said Harclay, at the infinite number of numbers: we can count from two, or from one hundred, and in both instances never reach a final number, though there are more numbers to count in the first infinity than there are in the second. He invoked Aquinas’s proportions to defend the thesis of an infinite universe in which the infinitely many months occur twelve times more frequently than the infinitely many years.
To those who point out that an infinite past would have produced an infinite number of souls with infinite power like God, Harclay refutes the argument as follows: infinitely many souls would not constitute an infinite power. They would be not ‘any species of number, but a multitude of infinitely many numbers.’ Within this endless multitude, every possible number (59, 1,043,962, 999,999,999,999,999,999,999,999,999,999 . . .) could be found, distinct and finite, each corresponding to a soul; save, that is, for an infinitieth number/soul since this would produce a contradiction: ‘there is not a number of infinite numbers, for then it would contain itself, which is impossible.’
We trace to the same period, in the monk Gregory of Rimini’s hand, the first definition of an infinite number as that which has parts equally great as the whole: an infinite sequence can be part of another infinite sequence and is equal to the infinite of which it is a part. Every twenty-third number for example (we might just as well have taken every ninety-ninth number, or every third, or every five billionth) in the infinite succession of counting numbers (1, 2, 3, 4, 5, 6 . . .) produces a sequence as long – infinitely long – as all the counting numbers combined: match one with twenty-three, two with forty-six, three with sixty-nine, four with ninety-two, five with one hundred and fifteen, and so on, ad infinitum.
Gregory articulated his defining idea fully five centuries before Cantor. He taught for many years in Paris, at the Sorbonne, where his pupils called him Lucerna splendens. Perhaps in him they sensed, as future scholars would claim, the last great scholastic theologian to wrestle with the infinite.
John Murdoch, a historian of mathematics at Harvard University, remarked that Gregory’s insight received hardly any notice from his peers or successors.
Since the ‘equality’ of an infinite whole with one or more of its parts is one of the most challenging, and as we now realise, most crucial aspects of the infinite, the failure to absorb and refine Gregory’s contentions stopped other medieval thinkers short of the hitherto unprecedented comprehension of the mathematics of infinity which easily could have been theirs.
In his writings, Cantor described himself as a servant of God and the Church. His ideas had struck him with the force of revelation. It had been with God’s help, he said, that he had worked day after day, alone, at his mathematics. But the mathematician was far from angelic; his humility sometimes slipped. To a friend, in 1896, Cantor confided in an excess of pride. ‘From me, Christian Philosophy will be offered for the first time the true theory of the infinite.’
The Art of Maths
I met a mathematician at a ‘conference of ideas’ in Mexico at which we had both been invited to speak. He was from the United States, and like all the mathematicians that I have ever crossed in my travels he fell immediately to talking shop. Moving to a corner of the conference green room, he talked to me about the history of numbers in Cambodia. The concept of zero, he believed fervently, the familiar symbol of nothingness, hailed from there. He dreamed of trekking the kingdom’s dirt tracks, in pursuit of any surviving trace. More than a millennia separated him from the decimal system’s creation; the odds of turning up any new evidence were slim. But he did not mind.
He began to explain his current research in number theory, talking quickly with the compression of passion, and I listened intently and tried to understand. When I understood, I nodded, and when I did not understand, I nodded twice, as if to encourage him to move on. His words came fast and enthusiastic, opening up vistas that I could not quite see and mental regions into which I could not follow, but still I listened and nodded and enjoyed the experience very much. Occasionally I supplemented his ideas and observations with some of mine, which he received with the utmost hospitality. It always feels exciting to me, the camaraderie of conversation: no matter whether it involves words or numbers.
He had none of the strange tics or quirks of the mathematicians that we find in books or see in movies. From experience, I was not the least surprised. Middle-aged, he looked fit and slender, though with skin as pasty as a writer’s. His shirt was open at the neck. His face wore many laughter lines. When our time was up, too soon, he patted his pockets and withdrew from one of them a small notebook in which he habitually jotted down his random thoughts and sudden illuminations. As he wrote out his contact information for me, I noticed the smallness and smoothness of his hands.
‘Great meeting you.’ We promised to stay in touch.
It was still a pleasant surprise, coming down next morning to the hotel restaurant for an early breakfast, to hear the mathematician’s voice call me over to his family’s table. I passed the assorted reporters munching their bowls of cereal, and various conference ‘stars’, dodging coffee-flecked waiters and pushing empty chairs out of my way, until I reached them. The mathematician smiled at his wife (also a mathematician, I learnt) and the surprisingly placid teenage girl sat in between who looked a lot like her mother. Their flight out was still a few hours away: over tea and toast, we talked.
We talked about the Four Colour Theorem, which states that all possible maps can be coloured in such a way that no district or country touches another of the same colour – using only (for instance) red, blue, green, and yellow. ‘At first sight it seems likely that the more complicated the map, the more colours will be required,’ writes Robin Wilson in his popular account of the puzzle’s history, Four Colours Suffice, ‘but surprisingly this is not so.’ Redrawing a country’s boundary lines, or imagining wholly alternative continent shapes, makes no difference whatsoever.
One aspect of the problem, in particular, had long intrigued me. After more than a century of fruitless endeavours to demonstrate the theorem conclusively, in 1976 a pair of mathematicians in the United States finally came up with a proof. Their solution, however, proved controversial because it relied in
part on the calculations of a computer. Quite a few mathematicians refused to accept it: computers cannot do maths!
‘I actually met one of those guys who came up with the proof,’ my new friend recalled, ‘and we discussed how they had found just the right way to feed the data into the machine and get an answer back. It really was a smart result.’
What did he and his wife think of the computer’s role in mathematics? In answer to this general question they were more circumspect. The Four Colour Theorem’s proof, they admitted, was inelegant. No new ideas had been stimulated by its publication. Worse, its pages were almost unreadable. It lacked the intuitive unity, and beauty, of a great proof.
Beauty. How often have I heard mathematicians employ this word! The best proofs, they tell me, possess ‘style’. One can often surmise who authored the pages simply from the distinctive way that they were put together: the selection, organisation and interplay of ideas are as personal, and as particular, as a signature. And how much time might they spend on polishing their proofs. Superfluous expressions, out! Ambiguous terms, out! Yes, but it was worth all the trouble: well-written proofs could become ‘classics’ – to be read and enjoyed by future generations of mathematicians.
‘What time is it?’ None of us was wearing a watch. We stopped a waiter and asked. ‘Already?’ said the mathematician’s wife when she heard his answer. They drained their cups, and dispersed their crumbs, and made shuffling sounds with their feet.
‘Oh,’ said the mathematician, turning back to me, ‘I forgot: where did you say you were based again?’ What with the history of the decimals, and the winding numerical vistas, and the painting of the entire globe with the colours of a single flag, the accidental features of our lives – where we lived, with whom, under what roof and colour of sky – had been completely absent from our conversations.
Thinking in Numbers: How Maths Illuminates Our Lives Page 21