Thinking in Numbers: How Maths Illuminates Our Lives (28 page)

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.

I told him. ‘Paris,’ he echoed. ‘Why, we love Paris!’

France’s capital has something of a one-sided reputation as the consummate city of artists. We know it as the city of Manet, of Rodin, of Berlioz; as the city of street singers and can-can dancers; as the city of Victor Hugo and of young Hemingway in
A Moveable Feast
: scribbling in a café corner, turning coffee and rum and the strictures of Gertrude Stein into stories. But Paris is also the city of mathematicians.

Its researchers, a thousand strong, make the
Fondation Sciences Mathématiques de Paris
(FSMP)
the largest group of mathematicians in the world. About one hundred of the city’s streets, squares and boulevards are named after their predecessors. In the twentieth
arrondissement
, for example, one can walk the length of
rue
Evariste Galois, named after a nineteenth-century algebraist felled at the age of twenty by a dueller’s bullet. On the opposite side of the Seine, in the fourteenth
arrondissement
, lies
rue
Sophie Germain whose namesake introduced important ideas in the fields of prime numbers, acoustics and elasticity before her death in 1831. According to her biographer Louis Bucciarelli, ‘She did not wish to meet others in the streets or houses of the day, but in the purer realm of ideas outside time, where person was indistinguishable from mind and distinctions depended only on qualities of intellect.’ A few minutes’ walk away is Fermat’s little road. There are also streets called Euler, and Leibniz, and Newton.

Among the letters waiting for me on my return to my adopted home was one from the city’s
Fondation
Cartier. A museum for contemporary art, it had sent me a preview invitation to its upcoming exhibition ‘Mathematics: A Beautiful Elsewhere’: the first in Europe to showcase the work of major living mathematicians in collaboration with world-class artists. The timing seemed doubly auspicious: October 2011 happened to be the two-hundredth anniversary of Galois’s birth.

The museum stands in the fourteenth
arrondissement
at the lower end of one of the long boulevards that diagram the city. It is an ostentatiously modern building, all shiny glass and geometric steel, bright and spacious, an example of ‘dematerialised’ architecture. Reflected in the glass, scraggly trees denuded of their summer foliage appeared twice. I looked up at the symmetrical branches as I passed and entered.

Mathematics and contemporary art may seem to make an odd pair. Many people think of mathematics as something akin to pure logic, cold reckoning, soulless computation. But as the mathematician and educator Paul Lockhart has put it, ‘There is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics.’ The chilly analogies win out, Lockhart argues, because mathematics is misrepresented in our schools, with curricula that often favour dry, technical and repetitive tasks over any emphasis on the ‘private, personal experience of being a struggling artist.’

It was the mathematicians’ artistic impulse, and inner struggle, that the exhibition’s organisers intended both to communicate and celebrate. A white interior, zero-shaped, was the work of the American filmmaker David Lynch. Walls usually reserved for frames and canvases lent their space to equations, light effects and number displays. I walked through the rooms, now bare and silent, now colourful and stimulating, stopping here and there to take a closer look. I watched the other guests stand back and point and converse in low voices. Before a bright collage of sunrays and leopard spots, waves and peacock tails, and the underlying equations for each, fingers swayed and eyes widened. Another hall arrested visitors’ feet around a lean aluminium sculpture, its curves reaching toward infinity.

But, for me, the highlight of the exhibition took place in a darkened room downstairs. Here the visitors melted into twilight, rendered homogeneous in the darkness, sitting or standing in silence, all eyes, observing a large screen where a film shot in black and white was playing. A youngish face, screen big, was talking about his life as a mathematician. I pressed my back against the far wall and listened as he spoke of ‘fat triangles’ and ‘lazy gases’. Three or four minutes old, the film suddenly altered: the face gave way to another, wearing glasses. Four minutes after this, the face changed again: this time, a woman’s began to speak about chance. In total, the film lasted thirty-two minutes – eight faces long. The men and women featured came from a wide range of mathematical sub-disciplines – number theory, algebraic geometry, topology, probability – and spoke either in French, or English, or Russian (with subtitles), but their passion and wonder linked each personal testimony into a fascinating and involving whole.