Thinking in Numbers: How Maths Illuminates Our Lives Read online

  What is this strange building to which every street in the city addresses itself? Filarete called it the ‘House of Vice and Virtue’. Every floor housed a different class of activity. A brothel, on the ground floor, would entertain the majority of the building’s callers. Alcoholic drinks and games could be had on the floors immediately above. Ascending further, a university and lecture halls offered its few visitors instruction. An observatory topped them all.

  The homes to which the citizens would return after their day’s work or play had been just as intricately imagined. Filarete planned his houses according to the resident’s social rank: the artisans’ quarters taking up far less space than the houses of the city’s merchants or gentlemen. In comparison with his artist and painter neighbours the architect’s own house would be twice as spacious.

  Filarete’s plans are long, his writing spidery. Spread across its twenty-five volumes his treatise contains an entire city in waiting. But not long after its completion, in 1466, the Duke Sforza died and Filarete’s vision survived only on paper.

  Plato’s ideal limits and Filarete’s volumes sought out new dreamers. They passed from mind to mind, and from century to century. It should perhaps come as no surprise that they would go on to inspire the grandest and most ambitious city plans, those drawn up in the United States of America.

  King Camp Gillette, who would later invent the safety razor, once dreamed of an immense city he called Metropolis. In 1894, he published a short illustrated book aimed at its promotion. The city, Gillette wrote, would be ‘situated in the vicinity of Niagara Falls, extending east into New York State and west into Ontario’. It would take the shape of a rectangle, sixty miles long and thirty miles wide. Gillette considered its construction, ‘In the light of a machine, or rather a part of the machine of production and distribution; and, as such, the objects to be attained must be known and understood. It must have no unnecessary parts to cause friction or demand unnecessary labour, and yet it must combine within itself all the necessary parts which will contribute to the happiness and comfort of all.’

  Sixty million inhabitants would live in circular skyscrapers, six hundred feet in diameter, ‘upon a scale of magnificence such as no civilisation has ever known.’ A beehive-like distribution of apartments throughout the city would lend space enough between the buildings for wide avenues and parks. No citizen, it also ensured, would reside any nearer or farther away from a school, shop, or theatre than any other.

  Elevators, a still recent invention in Gillette’s day, had gradually shifted city design from the horizontal toward the vertical. But Metropolis took the idea of a vertical city to a whole other level. Its skyscrapers would be truly colossal, attaining a height of twenty-five storeys. Habitable monoliths, lots and lots of them, shining with glass and progress, smooth and steel-coloured, a monotony of monuments.

  The city’s highly regular order appears again in miniature in Gillette’s plan for every home. For Gillette, as for Filarete, the home is a tiny city. The interior would be completely symmetrical, with parallel sitting rooms, bedrooms, and bathrooms on either side. And in each room, the windows would be so arranged as to make it impossible to look out from one apartment into a neighbour’s.

  Conscious of the artificiality of his vision (even the hexagonal lawns that surrounded each building would be composed of artificial grass), Gillette proposed distributing thousands of public gardens filled with trees and ‘urns of flowers’ at regular intervals throughout the city. Being constructed with complete regularity did not mean that it would suffer from sameness, he insisted. Looking out from his window, the citizen’s roving eye would encounter ‘a continuous and perfectly finished facade from every point of view, each building and avenue surrounded and bordered by an ever changing beauty in flowers and foliage.’

  Gillette summed up his utopian vision.

  Imagine for a moment these thirty-odd thousand buildings of Metropolis, each standing alone, a majestic world of art . . . a never-ending city of beauty and cleanliness, and then compare it with our cities of filth, crime, and misery, with their ill-paved and dirty thoroughfares, crowded with the struggling masses of humanity and the system of necessary traffic. And then compare the machinery of both systems, and take your choice; for I believe the only obstacle that lies in the way of the building of this great city is man.

  Fifty years after Gillette’s book was published, the World’s Fair in New York exhibited its own ‘City of the Future’. It was 1939.

  Millions queued for hours to see the model Democracity. (This fact alone is truly remarkable: New Yorkers hate to queue. They hate the involuntary intimacy, and the excruciating foot shuffles, and the boredom of being so long in their own company. And yet wait in line they did.)

  How long must it have taken the model makers to build? It was housed inside a sphere eighteen storeys tall. An escalator, the world’s longest, ascended visitors fifty feet above the ground to the exhibition. Music blared triumphant over loudspeakers as they entered, and then a resounding voice spoke. ‘The city of man in the world of tomorrow. Here are grass and tree as well as stone and steel. Not a dream city but a symbol of life as lived by the man of tomorrow. As man helps man so nation leans on nation, united by a thousand roads of commerce . . . Here brain and brawn, faith and courage, are interlinked in high endeavour.’

  Standing on slowly rotating balconies, visitors peered down at the city as though at an altitude of seven thousand feet. They saw a vast ring, brightly lit and painted, representing eleven thousand square miles of terrain. In its centre loomed a single block: an awesome office complex to which a quarter of a million inhabitants (one citizen in six) would commute every day.

  Five satellite suburbs surrounded this central district in concentric circles. Even the most distant Pleasantville (as the residential areas were called) lay within sixty miles of the hub. Larger Millvilles, home to the city’s factories, exiled their noise and pollution to the outskirts. Greenbelts were interspersed with suburbs; wide highways communicated between the centre and the various sectors.

  Mobility, an American obsession, had never been so well catered for. Traffic lights would be a thing of the past. The city’s highways would always flow freely, in a straight line, planned in such a way as to avoid all jams and pedestrian crossings. All other roads would be built a safe distance from any school.

  Two minutes into the performance, all of a sudden the lights were dimmed; the concave ceiling sparkled as if with stars. A chorus started up, and a film showed marching men: artisans, farmers, mechanics, everyone who would work together to help build tomorrow. The voices rose; the marching figures grew; the fairgoers held their breath.

  And then, just as quickly, the music died away and the men in the film disappeared behind thick plumes of smoke. The show was over.

  Are We Alone?

  Democritus, a contemporary of Plato and Aristotle, imagined all matter as composed of indivisible elements he called ‘atomos’, and was also the first thinker to propose a cosmos of many worlds. Every world was different. Some had neither a sun nor a moon, while others had larger moons or smaller moons or moons more numerous than our own.

  The Pythagoreans also believed that there was nothing unique about our world. For them the moon was Earth-like and inhabited, containing larger beings and more beautiful plants. The lunar residents, they believed, stood fifty times our height, lived on air and for this reason produced no excrement.

  Refutations of these ideas by both Plato (to call the number of worlds indefinite requires definite knowledge) and Aristotle did not prevent later thinkers from endorsing them.

  Since space lies empty and infinite in all directions and since atoms in countless numbers fly every which way through its furthest reaches . . . it is utterly unrealistic to think that ours are the only world and sky to have been born and that so many atoms outside our world are doing nothing . . . there are other worlds in other parts of the universe, different races of humans and species of animals
.

  These lines come from the epic poem On the Nature of Things, written by the Roman poet Lucretius in the first century before the Christian era. The poet’s ideas would later fill the fathers of the early Church with consternation. If other worlds existed, wrote St Augustine, they would each require their own Saviour, which would contradict the unique role of Christ.

  But by the medieval period, not everyone in the Church shared Augustine’s view. In 1277, the Bishop of Paris denounced the proposition that God could not create more than one world. Three centuries later, the friar Giordano Bruno put forth an elaborate argument in favour of an infinite number of worlds: if man can imagine so many worlds, then so can God, who creates what He thinks. The friar pictured Gardens of Eden, infinite in number: in half, Adam and Eve eat the fruit of Knowledge; in half, they do not. An infinite number of worlds will fall from grace and require an infinite number of Saviours to redeem them. Unlike Augustine, Bruno had no difficulty imagining an infinite number of Christs. For this and other ‘theological errors’, the authorities denounced the heretic and burned him at the stake.

  The Inquisitors’ heavy footsteps dissuaded Bruno’s contemporary, Galileo Galilei, from seeing any evidence for extraterrestrial life in the rugged lunar landscape revealed by his telescope. All the same, since the valleys and mountains that corrugated the moon’s surface seemed at least comparable to those on Earth, might not the moon also have people to dwell among them? His friend Johannes Kepler, the seventeenth-century mathematician and astronomer, thought so. Jupiter, too, he deduced ‘with the highest degree of probability’, though its inhabitants were undoubtedly inferior to humans.

  Probability: this word became the cornerstone of the argument for life on other planets. ‘As for mind beyond the confines of our tiny globe,’ wrote the American astronomer Percival Lowell in 1895, ‘modesty, backed by a probability little short of demonstration, forbids the thought that we are the sole thinkers in this great universe.’

  His argument, two millennia old, cited the most modern scientific observations: the conditions on Mars suggested hospitality. The planet enjoyed an atmosphere and its weather appeared extremely clement (with an average climate, he reported, that was similar to that in southern England). Water, essential for life, was also present. ‘Anyone looking through a telescope at the planet, early last summer,’ noted Lowell, ‘would at once have been struck by the fact that its surface was diversified by markings in three colours, white, blue-green, and reddish-ochre; the white lying in a great oval at the top of the disc. The white oval was the south polar icecap.’ And the blue-green? The colour of water. Or, of what still remained of Martian water, since ‘the signs [are] that its water supply is now exceedingly low’. Its inhabitants, he reasoned, had poured all their energies into irrigation. When he scanned the planet’s surface, he picked out a ‘network of fine, straight dark lines’. Canals. ‘All this,’ the astronomer conceded, ‘of course, may . . . signify nothing; but the probability seems the other way . . . that Mars seems to be inhabited is not the last, but the first word on the subject.’

  Lowell’s claims found many sympathetic ears. ‘Probability’, he knew, was a sesame word: it opened ears and minds. But its magic did not work on everyone. The biologist Alfred Russel Wallace, who independently of Darwin discovered the principle of natural selection, was among his fiercest critics. Yes, Mars appeared to have polar icecaps, and a day only half an hour longer than our own, and lengthy seasons that vanished one into the next. But according to Wallace’s calculations, the planet was in fact too cold to have rivers, seas or canals. The features observed by Lowell were natural landforms, all products of normal geological processes. ‘Mars, therefore, is not only uninhabitable by intelligent beings . . . but is absolutely uninhabitable.’

  Not only was the planet Mars lifeless, but there were in all likelihood no habitations on any other planet either. This, at the turn of the twentieth century, was Wallace’s stark assessment. The exceptional (and exceptionally complex) combination and sequence of events – physical, chemical, cosmological – permitting the origin of life on Earth made the prospect of finding other beings elsewhere in the universe immeasurably remote. The formation of intelligent life, quite simply, was a once-in-a-universe event.

  What? Alone? Many people could not believe it. It was one thing to be alone in a room, or a house. But to imagine being the only person in a town, a city, a country! The only people in a universe. They agreed with the ancient Greek Metrodorus, who had thought it absurd that in a vast field only one stalk should ever sprout. What is more, the sense of such all-encompassing solitude seemed to these people intolerably oppressive. They felt like foreigners in a lifeless void.

  Decades passed, and the only extraterrestrials were to be found in pulp stories and motion pictures. When the American astronomer Frank Drake, who has spearheaded research into extraterrestrial communication, was a boy he drew on this imagery as he listened to his father’s words. Look at this star, and this one, and that star, and that one. All these stars, innumerable stars, shining in the night sky. Around some of those stars, somewhere in space, circled other worlds like our own. The boy listened to his father and believed him. He believed with all his heart.

  Frank Drake, the son of an engineer, was at home with big numbers. His formative years were spent in Chicago, a city so populous that it could sustain the livelihoods of more than a hundred full-time piano tuners. His thoughts ran often to the many, many worlds far above his head. He wondered about their cities, about their cars, about whether they knew war or cancer.

  After graduating from Harvard with a doctorate in radio astronomy, Drake conducted the first ever search for interstellar communication. On 8 April 1960, he aimed the radio at two stars much like our sun, twelve light years from Earth. Over the next two months he and his colleagues listened sedulously for a signal, but heard nothing. Not a peep.

  But the numbers! Drake believed the numbers were on his side. The number of stars in our galaxy amount to at least one hundred billion. One hundred billion! And how many of these stars were suns to planets? No hard facts could lend a hand. His imagination groped, sifted, hesitated. He closed his eyes and made a guess: one in two. Planets would orbit half of all the stars in the Milky Way, which equals fifty billion solar systems.

  Not every solar system will produce life, however. A shot at life requires a certain kind of solar system (a sun that is neither too cool nor too dim, nor so massive as to burn out before life appears), which would host planets in some way comparable to the Earth. Drake thought of the only solar system with which we are familiar, of its (then) nine planets, and the sole planet that had become a world. This figure – one – troubled him: it smacked of uniqueness. No, no, there had to be solar systems with multiple worlds. Some system will host a world, then it will host another, and perhaps another after that. Why not? Look at Mars, it had not been so very far off from being a second Earth. And so he made the guess of two (two being greater than one) for the number of possible Earths in each solar system.

  So far, so good. But Drake’s next estimations demanded all his powers of invention. First, he had to estimate the number of these planets on which life has emerged. This is how he reasoned about it. Four and a half billion years ago, not long after our Earth was formed, it struck out as a cold and barren rock. From this unpromising start, a few hundred million years later, the first living cells arose. What are a few hundred million years in a universe of ten or twenty billion years? It is as though life will get going just as soon as it is given the chance. Life had come quite easily to Earth, he concluded, therefore it should come easily for all the other possible worlds.

  Second, Drake pondered the question of intelligence. On how many of the hundred billion Earth-like planets might the living cells contrive intelligent forms? The Earth’s diversity presented itself to his mind. Billions of animal species had wriggled, and buzzed, and hissed, and flitted, and swum through the ages. Yet only one had ever posed itself
questions, and dreamed of life on other worlds. Furthermore, Homo sapiens were latecomers to a planet that had done without their brains for billions of years. It suggested to Drake that questioning minds were anything but universal. He finally settled upon a figure of one in a hundred, leaving him a billion potential civilisations amid the stars.

  How many of these civilisations had developed the technology (not to mention, had the desire) to communicate with others? In ways that we could understand? Drake was a radio astronomer. He knew that transmissions from Earth had already seeped far into space. Out there somewhere, twenty or thirty light years away, hands and ears with the know-how were already perhaps tuning in to the same episodes of Flash Gordon and The Lone Ranger that he had listened to as a boy. And some of these planets would certainly be broadcasting signals of their own: say, one hundred thousand in all.

  These planets would exude music, news broadcasts, coded messages, provided of course that they were still around, that this same technology had not yet blown them to pieces. Civilisation, after all, is a tricky business. Hazardous, too. The civilised human dates back only ten thousand years, and already Washington had nuclear arms aimed at Moscow (and vice versa). Mutual destruction, the threat of it, Drake knew, was no delusion. But neither was such a trajectory inevitable. Mindful of his final proviso, he set his expectations low: of the potential hundred thousand communicating civilisations, the survivors in our galaxy would number maybe ten. Signals, centuries and millennia old, from thousands of their predecessors would likewise be filtering through space, ready to brush some awaiting antenna.