The Clockwork Universe Read online

  Newton made two purchases. They seemed innocuous, but they would revolutionize the intellectual world. “In ’63 [Newton] being at Stourbridge fair bought a book of astrology to see what there was in it,” according to a young admirer who had the story from Newton himself. Perhaps in the same year—scholars have not settled the matter—he bought a trinket, a glass prism. Children liked to play with prisms because it was pretty to see how they caught the light.

  The astrology book had no significance in itself, but it helped change history. Newton “read it ’til he came to a figure of the heavens which he could not understand for want of being acquainted with trigonometry,” he recalled many years later. “Bought a book of trigonometry, but was not able to understand the demonstrations. Got Euclid to fit himself for understanding the ground of trigonometry.”

  At that point Newton’s backtracking came to an end. To his relief, he found that Euclid was no challenge. “Read only the titles of the propositions,” he would recall, “which he found so easy to understand that he wondered how anybody would amuse themselves to write any demonstrations of them.”

  Newton turned from Euclid’s classical geometry to Descartes’ recent recasting of the entire subject. This was not so easy. He made it through two or three pages of Descartes but then lost his way. He started over and this time managed to understand three or four pages. He slogged along in this fashion, inching his way forward until he lost his bearings and then doubling back to the beginning “& continued so doing till he made himself Master of the whole without having the least light or instruction from any body.” Every aspiring mathematician knows the frustration of spending entire days staring at a single page in a textbook, or even a single line, waiting for insight to dawn. It is heartening to see one of the greatest of all mathematicians in almost the same plight.

  Newton’s pride in finally mastering Descartes’ Geometry had two aspects, and both were typical of him. He had accomplished a great deal, and he had done it without a word of guidance “from any body.” And he had only begun. To this point he had studied work that others had already done. From here on, he would be advancing into unexplored territory. In early 1665, less than two years from the day he had picked up the astrology booklet, he recorded his first mathematical discovery. He proved what is now called the binomial theorem, to this day one of the essential results in all of mathematics.45 This was the opening salvo of the “miracle years.”

  Newton’s summary of what came next remains startling three and a half centuries later. Even those unfamiliar with the vocabulary cannot miss the rat-tat-tat pacing of discoveries that spilled out almost too quickly to list. “The same year in May I found the method of Tangents . . . & in November had the direct method of fluxions & the next year in January had the Theory of Colours & in May following I had entrance into ye inverse method of fluxions. And the same year I began to think of gravity extending to ye orb of the Moon. . . .”

  Over the course of eighteen months, that is, Newton first invented a great chunk of calculus, everything to do with what is now called differentiation. Then he briefly put mathematics aside and turned to physics. Taking up his Stourbridge Fair prisms (he had bought a second one) and shutting up his room except for a pinhole that admitted a shaft of sunlight, he discovered the nature of light. Then he turned back to calculus. The subject falls naturally into two halves, although that is by no means evident early on. In early 1665 Newton had invented and then investigated the first half; now he knocked off the other half, this time inventing the techniques now known as integration. Then he proved that the two halves, which looked completely different, were in fact intimately related and could be used in tandem in hugely powerful ways. Then he began thinking about the nature of gravity. “All this,” he wrote, “was in the two plague years of 1665–1666. For in those days I was in the prime of my age for invention & minded Mathematicks & Philosophy more than at any time since.”

  Newton was indeed in his prime at twenty-three, for mathematics and physics are games for the young. Einstein was twenty-six when he came up with the special theory of relativity, Heisenberg twenty-five when he formulated the uncertainty principle, Niels Bohr twenty-eight when he proposed a revolutionary model of the atom. “If you haven’t done outstanding work in mathematics by 30, you never will,” says Ronald Graham, one of today’s best-regarded mathematicians.

  The greats flare up early, like athletes, and they burn out just as quickly. Paul Dirac, a physicist who won his Nobel Prize for work he did at twenty-six, made the point with wry bleakness, in verse. (He wrote his poem while still in his twenties.)

  Age is, of course, a fever chill

  that every physicist must fear.

  He’s better dead than living still

  when once he’s past his thirtieth year.

  In the most abstract fields—music, mathematics, physics, even chess—the young thrive. Child prodigies are not quite common, but they turn up regularly. Perhaps it makes sense that if a Mozart or a Bobby Fischer were to appear anywhere, it would be in a self-contained field that does not require insight into the quirks of human psychology. We are unlikely ever to meet a twelve-year-old Tolstoy.

  But that is only part of the story. Penetrating to the heart of abstract fields seems to demand a degree of intellectual firepower, an intensity of focus and stamina, that only the young can muster. For the greats, these truly are miracle years. “I know that when I was in my late teens and early twenties the world was just a Roman candle—rockets all the time,” recalled I. I. Rabi, another Nobel Prize–winning physicist. “You lose that sort of thing as time goes on. . . . Physics is an otherworld thing. It requires a taste for things unseen, even unheard of—a high degree of abstraction. . . . These faculties die off somehow when you grow up.”

  Nerve and brashness are as vital as brainpower. A novice sets out to change the world, confident that he can find what has eluded every other seeker. The expert knows all the reasons why the quest is impossible. The result is that the young make the breakthroughs. The pattern is different in the arts. “Look at a composer or a writer—one can divide his work into early, middle, and late, and the late work is always better, more mature,” observed Subrahmanyan Chandrasekhar, the astrophysicist who won a Nobel Prize for his work on black holes (and worked into his eighties). Even so, he declared in his old age, “For scientists, the early work is always better.”

  At age thirty-five or forty, when a politician would still count as a fresh face, when doctors in certain specialties might only recently have completed their training, mathematicians and physicists know they have probably passed their peak. In the arts, talent often crests at around forty. Michelangelo completed the ceiling of the Sistine Chapel at thirty-seven; Beethoven finished his Fifth Symphony at thirty-seven; Tolstoy published War and Peace at forty-one; Shakespeare wrote King Lear at forty-two. But the list of artists who continued to produce masterpieces decades later than that—Monet, Cervantes, Titian, Picasso, Verdi—is long.

  Science and mathematics have no such roster. In the end, the work simply becomes too difficult. Newton would make great advances in mathematics after his miracle years, but he would never again match the creative fervor of that first outburst. Looking back at his career in his old age, he remarked that “no old Men (excepting Dr. Wallis)”—this was Newton’s eminent contemporary John Wallis—“love Mathematicks.”

  From his earliest youth, Newton had seen himself as different from others, set apart and meant for special things. He read great significance into his birth on Christmas Day, his lack of a father, and his seemingly miraculous survival in infancy. The depth and sincerity of his religious faith are beyond question, and so was his belief that God had set him apart and whispered His secrets into his ear. Others had studied the prophecies in the Bible just as he had, Newton noted, but they had met only “difficulty & ill success.” He was unsurprised. Understanding was reserved for “a remnant, a few scattered persons which God hath chosen.” Guess who.


  He took the Latin form of his name, Isaacus Nevtonus, and found in it an anagram, Ieova sanctus unus, or the one holy Jehovah. He drew attention to the passage in Isaiah where God promises the righteous that “I will give thee the treasures of darkness, and hidden riches of secret places.”

  By the end of the miracle years, Newton found himself awash in hidden riches. He knew more mathematics than anyone else in the world (and therefore more than anyone who had ever lived). No one even suspected. “The fact that he was unknown does not alter the other fact that the young man not yet twenty-four, without benefit of formal instruction, had become the leading mathematician of Europe,” wrote Richard Westfall, Newton’s preeminent biographer. “And the only one who really mattered, Newton himself, understood his position clearly enough. He had studied the acknowledged masters. He knew the limits they could not surpass. He had outstripped them all, and by far.”

  Newton had always felt himself isolated from others. Now at twenty-three, wrote Westfall, he finally had objective proof that he was not like other men. “In 1665, as he realized the full extent of his achievement in mathematics, Newton must have felt the burden of genius settle upon him, the terrible burden which he would have to carry in the isolation it imposed for more than sixty years.”

  Chapter Thirty-Nine

  All Mystery Banished

  Isaac Newton believed that he had been tapped by God to decipher the workings of the universe. Gottfried Leibniz thought that Newton had set his sights too low. Leibniz shared Newton’s yearning to find nature’s mathematical structure, which in their era meant almost inevitably that both men would mount an assault on calculus, but in Leibniz’s view mathematics was only one piece in a much larger puzzle.

  Leibniz was perhaps the last man who thought it was possible to know everything. The universe was perfectly rational, he believed, and its every feature had a purpose. With enough attention you could explain it all, just as you could deduce the function of every spoke and spring in a carriage.

  For Leibniz, one of the greatest philosophers of the age, this was more than a demonstration of almost pathological optimism (though it was that, too). More important, Leibniz’s faith was a matter of philosophical conviction. The universe had to make perfect sense because it had been created by an infinitely wise, infinitely rational God. To a powerful enough intellect, every true observation about the world would be self-evident, just as every true statement in geometry would immediately be obvious. In all such cases, the conclusion was built in from the start, as in the statement “all bachelors are unmarried.” We humans might not be clever enough to see through the undergrowth that obscures the world, but to God every truth shines bright and clear.

  In fact, though, Leibniz felt certain that God had designed the world so that we can understand it. Newton took a more cautious stand. Humans could read the mind of God, he believed, but perhaps not all of it. “I don’t know what I may seem to the world,” Newton famously declared in his old age, though he knew perfectly well, “but, as to myself, I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.”

  Newton’s point was not simply that some questions had yet to be answered. Some questions might not have answers, or at least not answers we can grasp. Why had God chosen to create something rather than nothing? Why had He made the sun just the size it is? Newton believed that such mysteries might lie beyond human comprehension. Certainly they were outside the range of scientific inquiry. “As a blind man has no idea of colors,” Newton wrote, “so have we no idea of the manner by which the all-wise God perceives and understands all things.”

  Leibniz accepted no such bounds. God, he famously declared, had created the best of all possible worlds. This was not an assumption, in Leibniz’s view, but a deduction. God was by definition all-powerful and all-knowing, so it followed at once that the world could not have been better designed. (Even for one of the ablest of all philosophers, this made for an impossible tangle. If logic compelled God to create the very world we find ourselves in, didn’t that mean that He had no choice in the matter? But surely to be God meant to have infinite choice?)

  Voltaire would later take endless delight, in Candide, in pummeling Leibniz. On Candide’s very first page, we meet Leibniz’s stand-in, Dr. Pangloss, the greatest philosopher in the world. Pangloss’s specialty is “metaphysico-theologo-cosmolonigology.” The world, Pangloss explains contentedly, has been made expressly for our benefit. “The nose is formed for spectacles, therefore we wear spectacles. . . . Pigs were made to be eaten, therefore we eat pork all the year round.”

  Pangloss and the hero of the novel, a naïve young man named Candide, spend the book beset by calamity—Voltaire cheerily throws in an earthquake, a bout of syphilis, a stint as a galley slave, for starters. Bloodied and battered though both men may be, Pangloss pops up from every crisis as undaunted as a jack-in-the-box, pointing out once more that this is the best of all possible worlds.

  This was great fun—Voltaire was an immensely popular writer, and Candide was his most popular work—but it was a bit misleading. Leibniz knew perfectly well that the world abounded in horrors. (He had been born during the Thirty Years’ War.) His point was not that all was sunshine, but that no better alternative was possible. God had considered every conceivable universe before settling on this one. Other universes might have been good, but ours is better. God could, for instance, have made humans only as intelligent as dogs. That might have made for a happier world, but happiness is not the only virtue. In a world of poodles and Great Danes, who would paint pictures and write symphonies?

  Or God might have built us so that we always chose to do good rather than evil. In such a world, we would all be kind, but we would all be automatons. In His wisdom, God had decided against it. A world with sin was better than a world without choice. Not perfect, in other words, but better than any possible alternative. It was this complacency that infuriated Voltaire. He raged against Leibniz not because Leibniz was blind to the world’s miseries but because he so easily reconciled himself to them.

  But Leibniz’s God was as rational as he was. For every conceivable world, He totted up the pros and cons and then subtracted the one from the other to compute a final grade. (It is perhaps no surprise that Leibniz invented calculus; in searching for the world that would receive the highest possible score, God was essentially solving a calculus problem.) Since God had necessarily created the best of all possible worlds, Leibniz went on, we can deduce its properties by pure thought. The best possible world was the one that placed the highest value on the pursuit of intellectual pleasure—here the philosopher showed his hand—and the greatest of all intellectual pleasures was finding order in apparent disorder. It was certain, therefore, that God meant for us to solve all the world’s riddles. Leibniz was “perhaps the most resolute champion of rationalism who ever appeared in the history of philosophy,” in the words of the philosopher Ernst Cassirer. “For Leibniz there . . . is nothing in heaven or on earth, no mystery in religion, no secret in nature, which can defy the power and effort of reason.”

  Surely, then, Leibniz could solve the problem of describing the natural world in the language of mathematics.

  Chapter Forty

  Talking Dogs and Unsuspected Powers

  Leibniz gave the impression that he intended to pursue every one of nature’s secrets himself. “In the century of Kepler, Galileo, Descartes, Pascal, and Newton,” one historian wrote, “the most versatile genius of all was Gottfried Wilhelm Leibniz.” The grandest topics intrigued him, and so did the humblest. Leibniz invented a new kind of nail, with ridged sides to keep it from working free. He traveled to see a talking dog and reported to the French Academy that it had “an aptitude that was hard to find in another dog.” (The wondrous beast could pronounce the French words for tea, coffee, and chocolate, and some two dozen more.)


  He drew up detailed plans for “a museum of everything that could be imagined,” roughly a cross between a science exhibition and a Ripley’s Believe It or Not museum. It would feature clowns and fireworks, races between mechanical horses, rope dancers, fire eaters, musical instruments that played by themselves, gambling halls (to bring in money), inventions, an anatomical theater, transfusions, telescopes, demonstrations of how the human voice could shatter a drinking glass or how light reflected from a mirror could ignite a fire.

  Leibniz’s energy and curiosity never flagged, but he could scarcely keep up with all the ideas careening around his head. “I have so much that is new in mathematics, so many thoughts in philosophy, so numerous literary observations of other kinds, which I do not wish to lose, that I am often at a loss what to do first,” he lamented.

  Many of these ventures consumed years, partly because they were so ambitious, partly because Leibniz tackled everything at once. He continued to work on his calculating machine, for example, and on devising a symbolic language that would allow disputes in ethics and philosophy to be solved like problems in algebra. “If controversies were to arise, there would be no more need of disputing between two philosophers than between two accountants. For it would suffice to take their pencils in their hands, to sit down to their slates, and to say to each other (with a friend as witness, if they liked): ‘Let us calculate.’ ”

  Leibniz wrote endlessly, at high speed, often while bumping along the road in a coach. Today a diligent team of editors is laboring to turn well over one hundred thousand manuscript pages into a Collected Works, but they do not expect to complete the project in their lifetimes. Volume 4, to choose an example at random, comes under the heading of “Philosophical Writings,” and consists of three “books.” Each book contains over a thousand pages. The editors envision sixty such volumes.