Let us pause to make one more observation, which also seems utterly obvious. If we think of a group—everyone who lives in Italy, say—and then we think of a smaller group contained within it—everyone who lives in Rome—then it seems beyond question that the original group is bigger than the subgroup.
In a moment, we will see why these points are worth belaboring. Suppose, said Galileo, we take not simply a big group, like the citizens of Italy, but an infinite group, like the counting numbers. Galileo wrote them in a line like this:
1 2 3 4 . . .
Next, said Galileo, suppose we think of a smaller group contained within the large one. Take, for instance, the numbers 12, 22, 32, 42, and so on. (In other words, the numbers 1, 4, 9, 16 . . . ). Galileo wrote them in a line of their own:
12 22 32 42 . . .
Then he sprung his trap. Since the list 1, 4, 9, 16 . . . plainly leaves out a great many numbers, it is beyond question smaller than the collection of all numbers. But Galileo arranged the two lines of numbers one below the other and paired them up tidily.
Each number in the top line had exactly one partner in the bottom line, and vice versa. Every number had a partner; no number had more than one partner; and no number in either line was left out. (The reason Galileo chose the numbers 12, 22, 32 . . . in the first place was that they could so readily be paired up with 1, 2, 3 . . .) That left only one conclusion. Since the two collections matched exactly, they were the same size. Galileo had found a working definition of infinity—a collection is infinitely big if part of it is the same size as the whole thing!
Few thinkers in history have been as bold as Galileo, but even for him this was too much. For the sake of an idea, he would one day challenge the Inquisition. But faced with the paradoxes of infinity, he blinked and hurried away.
Galileo, brilliant in so many domains, had pinpointed infinity’s strangest property. In a sense, big numbers are all alike. One million is bigger than one thousand, but you can get from one to the other. All it takes is patience. Add one. Add one more. Add another. Eventually you get there. But infinity sits on the far side of a chasm that you can never bridge. When it comes to infinity, it’s not only that another (and another and another) doesn’t bring you to the goal; worse than that, it doesn’t bring you any nearer to the goal.
That idea, so remote from anything in the everyday world, continues to baffle even the deepest thinkers. In Portrait of the Artist as a Young Man, James Joyce took a stab at conveying the notion of infinity. The damned suffer eternally in hell. “For ever! For all eternity!” Joyce wrote. “Not for a year or for an age but for ever. Try to imagine the awful meaning of this. You have often seen the sand on the seashore. How fine are its tiny grains! And how many of those tiny little grains go to make up the small handful which a child grasps in its play. Now imagine a mountain of that sand, a million miles high, reaching from the earth to the farthest heavens, and a million miles broad, extending to remotest space, and a million miles in thickness. . . .”
On and on Joyce went, the limitlessly talented writer multiplying grains of sand by drops of water in the ocean by stars in the sky. And still he came up short, still he failed to narrow the gap between the finite and the infinite. Because the essential point is that infinity is not just a big number but something much, much more bizarre than that.
Photographic Insert
Isaac Newton was one of the greatest of all geniuses and one of the strangest men who ever lived. “The most fearful, cautious, and suspicious Temper that I ever knew,” in one contemporary’s words, he was born sickly and premature but lived to 84 (and died a virgin). Born on Christmas Day, Newton believed with all his heart that he had been selected by God to decode His secrets. This portrait shows him at the peak of his powers, at age 46, just after he had unveiled the theory of gravitation.
Charles II, whose father had been beheaded in 1649, ruled England from 1660 to 1685. Witty and restless, the “Merrie Monarch” presided over a self-indulgent court in which everyone, from the king on down, was “engaged in an endless game of sexual musical chairs.” Science, too, fascinated Charles. He founded the Royal Society and poses here with a telescope and other scientific instruments.
Robert Boyle, an aristocrat whose father was one of Britain’s richest men, was the most-respected member of the Royal Society in its early days. Boyle was a brilliant scientist who also believed, among other things, that the best cure for cataracts was to blow powdered, dried, human excrement into the patient’s eyes.
Gottfried Leibniz was Newton’s great rival and himself a man of astonishing genius. As energetic as a puppy, Leibniz was the temperamental opposite of the austere Newton. Leibniz was a philosopher, a mathematician, an inventor, and a man so pleased with himself that his favorite wedding gift to new brides was a collection of his own maxims.
The 1600s saw the birth of science and the modern age, but old beliefs and fears still held prominent places in men’s hearts. Comets were dreaded, as they had been for ages. This scene from the Bayeux tapestry (stitched in the 11th century) shows men cowering in fear as Halley’s Comet appears overhead, in 1066.
When our forebears looked at comets, they saw not merely glowing lights but fiery, death-wielding swords, as in these illustrations from 1668. Comets had appeared overhead in 1664 and 1665, and England braced for “a MORTALITY which will bring MANY to their Graves.”
In 1665, plague struck England. No one knew the cause, and no one knew a cure. Somehow the killer leaped from victim to victim. Many were healthy one day and dead that night. Plague doctors, shown here, could offer little but a kind word. Their costume was meant to protect the wearer; the beak contained herbs and spices, to counter the smell of the dead and dying.
Bills of Mortality recorded the weekly death toll. The first deaths in London came at a rate of one or two per month. At its peak, in September 1665, plague killed more than six thousand Londoners in a single week. The stricken city was nearly silent except for the tolling of church bells.
Samuel Pepys lived through the turmoil of London in the late 1600s and took notes in his diary, in a secret code. An administrator in the British navy, he reported on office politics, dalliances with mistresses, and fights with his wife. His accounts of the plague and the Great Fire of London are the most detailed descriptions we have.
As soon as the plague released England from its chokehold, a second calamity swooped down. In the year 1666—ominous because of the fearsome number 666—London caught fire. For four days the city burned. Iron bars in prison cells melted. One hundred thousand people were left homeless. Plainly an outraged God had lost patience with his creation.
In this harsh era, punishments were carried out in public, the better to warn and entertain. Drawing and quartering was the most gruesome. A man was hanged by the neck but not killed. In England he was then disemboweled (while alive) and cut in quarters. The head and body parts were nailed up around the city. In France, as in this picture, horses performed the quartering.
Spectators out for a day’s entertainment might attend a puppet show or a hanging or, perhaps, a bear-baiting. Dogs attacked a chained bear, who flailed at his attackers. Bull-baiting was popular, too. The sport gave rise to the English bulldog, whose flat face made it possible for him to keep breathing without releasing his hold on the bull.
The microscope astonished all those who looked through it. Even a lowly flea, like this one drawn by Robert Hooke, revealed God’s perfect artistry. But when Hooke turned his microscope on man-made objects, they looked shoddy in comparison. The point of a needle (at right, top) was rough and pitted, as was a razor’s edge (at right, bottom). A printed dot on a page (at right, middle) was “quite irregular, like a great Splatch of London dirt.”
Anything was still possible in science’s early days. Travelers told wondrous tales of such oddities as Indians in the New World who had no heads but sported eyes on their torsos. Isaac Newton, a devout believer in alchemy, drew the image at the bottom of the page. I
t refers to a magical substance called the philosopher’s stone, which could change ordinary objects to gold and convey immortality. Newton included instructions for coloring the diagram.
Tycho Brahe was the most eminent astronomer of the generation before Kepler and Galileo. Rich and eccentric, Tycho presided over a magnificent observatory on a private island. He had lost part of his nose in a duel and ever after sported a replacement made of gold and silver.
Johannes Kepler was an astronomer, an astrologer, and a mathematical genius. Kepler spent decades studying the astronomical data that Tycho had gathered, in search of patterns that he fervently believed God had hidden like secret messages in a text.
Kepler’s proudest achievement was devising an elaborate model that explained how God had laid out the solar system. God was a mathematician, Kepler believed, and He had arranged the planets’ orbits in keeping with this intricate geometric model. The sun sat at the center, inside a nested cage of cubes, pyramids, octahedrons, and so on. Beautiful as the model was, it eventually became clear that it had nothing to do with reality.
Galileo was brilliant and pugnacious. A mathematician and astronomer, he amazed Italy’s rulers by letting them look out to sea through a telescope, a brand-new invention that he had considerably improved. When Galileo turned his telescope to the skies, he revolutionized our picture of the heavens. Among other discoveries, he found that the moon was not a perfect sphere, as everyone had believed, but was rough and pockmarked. Galileo painted these watercolors himself.
The crucial belief of Isaac Newton and his fellow scientists was that God had designed the world on mathematical lines. All nature followed precise laws. The belief derived from the Greeks, who had been amazed to find that music and mathematics were deeply intertwined. Here they explore the relation between the weight of a bell, or the volume of a glass, and its pitch.
Pythagoras, shown below, is credited with being the first to find this connection, “one of the truly momentous discoveries in the history of mankind.”
Newton’s theory of gravity propelled him to instant fame. He liked to tell the story, depicted in this Japanese print, that the crucial insight came from watching an apple fall. The story is quite likely a myth. Before anyone knew of his mathematical genius, Newton had dazzled the Royal Society with this compact yet powerful telescope.
Edmond Halley (known today for Halley’s Comet) was a brilliant astronomer and, just as surprisingly, a man so congenial that he could get along with Isaac Newton. Halley took on the task of coaxing the reluctant, secretive Newton into publishing his masterpiece, Principia Mathematica. The 500-page book, in Latin and dense with mathematics, might never have appeared without Halley’s labors.
Newton’s tomb, in Westminster Abbey. From the moment he unveiled the theory of gravity Newton was hailed as almost superhuman. Voltaire observed Newton’s funeral and was stunned to see dukes and earls carrying the casket. “I have seen a professor of mathematics, simply because he was great in his vocation, buried like a king who had been good to his subjects.”
Chapter Thirty-Five
Barricaded Against the Beast
The closer mathematicians looked at infinity, the stranger it seemed. Take one of the simplest drawings imaginable, a straight line one inch long. That line is made up of points, and there are infinitely many of them. Now draw a line two inches long. The longer line must have twice as many points as the shorter one (what else could make it twice as long?). But a matching technique, much like Galileo had used with numbers, shows that there are precisely the same number of points on both lines.
The proof is pictorial. Make a dot as in the drawing and draw a straight line from it through the two lines. Any such line pairs up a point on the short line with a point on the longer line. Like a perfectly orderly dance, everyone has a partner. No point on either line is left out, and no point has to share a partner with anyone else. How can that be?
Worse was to come. Exactly the same argument shows that a line ten inches long is made of precisely as many points as a line one inch long. So is a line ten miles long, or ten thousand. Could anything send a clearer message that infinity was a topic best left to philosophers and mathematicians, and completely unsuited to hardheaded scientists?
Infinity is built into mathematics from the beginning, because numbers go on forever. If someone made a claim about all the human beings on Earth—no person alive today is nine feet tall—in principle you could test it by gathering everyone into a line and working your way along from first person to last. But no such test can work for numbers, because the line never ends. For every number there is another, bigger number (and also another half as big).
But it was by no means clear that infinity had anything to do with the real world. That was fine. Seventeenth-century scientists, like all their predecessors, would happily have left the paradoxes of infinity to those who enjoyed such things. These practical men of science glanced at infinity, saw that it could not be tamed, and booted it out the door so that they could concentrate on the real-life questions that preoccupied them.
No sooner had they set to work than they heard a clawing at the window.
The most basic challenge in seventeenth-century science was to describe how objects move. To move is to change position. Infinity kept fighting its way into the picture because change comes in two forms. One is easy. The other would challenge and tantalize some of the most powerful thinkers the world had ever seen.
The easy form is steady change, as when a car rolls down the highway with the cruise control set at sixty miles an hour. The car is changing position, but one moment looks much like another. Now think of a rock falling off a cliff. The rock is changing position, like the car, but it is changing speed at every instant, too. That kind of changing change happens all around us. We see it when a population grows, or a bullet tears through the air, or an epidemic sweeps through a city. Something is changing, and the rate at which it is changing is changing, too.
Look again at the falling rock. Galileo showed that, as time passed, the rock fell faster and faster. At every instant its speed was different. But what did it mean to talk about speed at a given instant? As it turned out, that was where infinity came in. To answer even the most mundane question—how fast is a rock moving?—these seventeenth-century scientists would have to grapple with the most abstract, highfalutin question imaginable: what is the nature of infinity?
It was easy to talk about average speed, which posed no abstruse riddles. If a traveler in a hackney coach covered a distance of ten miles in an hour, then his average speed was plainly ten miles per hour. But what about speed not over a long interval but at a specific moment? That was trouble. What if the horses pulling the coach labored up a steep hill and then sped down the far side and then stumbled and slowed for a moment and then regained their footing and sped back up? With the coach’s speed varying unpredictably, how could you possibly know its speed at a precise instant, at, for instance, the moment it passed in front of the Fox and Hounds Tavern?
The point wasn’t that anyone needed to know precisely how fast coaches traveled. For any practical question about making a journey from here to there, a rough guess would do. The coach’s speed was only important as the key to a larger question: how could you devise a mathematical language that captured the ever-changing world and all its myriad moving parts? How could you see the world as God saw it?
Before you could tackle the world in general, then, it made sense to try to sort out something as familiar as a horse-drawn coach. For decades mathematicians had all tried to solve the mystery of instantaneous speed in the same way. Speed, they knew, was a measure of how much distance the coach covered in a given time. Suppose the coach happened to pass the Fox and Hounds at precisely noon. To get a rough guess of its speed at that moment, you might see how far down the road it was an hour later. If the coach had traveled eight miles between noon and one o’clock, its speed at noon was likely somewhere near eight miles per hour. But maybe not. An h
our is a long while, and anything could have happened during that time. The horses might have stopped to graze the grass. They might have been stung by hornets and broken into a sprint. It would be better to guess the coach’s speed at the stroke of noon by looking at how far it traveled in a shorter interval that included noon, such as from noon to 12:30. A shorter interval still, say from noon to 12:15, would be better yet. From noon to 12:01 would be even better, and from noon to one second after noon would be better than that.
Success seemed close enough to touch. To measure speed at the instant the clock struck noon, all you had to do was look at how much distance the coach covered in shorter and shorter intervals beginning at noon.
And then, with victory at hand, it flew out of reach. An instant, by definition, is briefer than the tiniest fraction of a second. How much distance did the coach cover in an instant? No distance at all, because it takes some amount of time to travel even the shortest distance. “Ten miles per hour” is a perfectly sensible speed. What could “zero distance in zero seconds” possibly mean?
The Clockwork Universe Page 17
