The Clockwork Universe: Isaac Newton, the Royal Society, and the Birth of the Modern World - Edward Dolnick (2011)

Part II. Hope and Monsters

Chapter 21. “Shuddering Before the Beautiful”

The seventeenth century’s faith that “all things are numbers” originated in ancient Greece, like so much else. The Greek belief in mathematics as nature’s secret language began with music, which was seen not as a mere diversion but as a subject for the most intense study. Music was the great exception to the general rule that the Greeks preferred to keep mathematics untainted by any connection with the everyday world.

Pluck a taut string and it sounds a note. Pluck a second string twice as long as the first, Pythagoras found, and the two notes are an octave apart. Strings whose lengths form other simple ratios, like 3 to 2, sound other harmonious intervals.27 That insight, the physicist Werner Heisenberg would say thousands of years later, was “one of the truly momentous discoveries in the history of mankind.”

Pythagoras believed, too, that certain numbers had mystical properties. The world was composed of four elements because 4 was a special number. Such notions never lost their hold. Almost a thousand years after Pythagoras, St. Augustine explained that God had created the world in six days because 6 is a “perfect” number. (In other words, 6 can be written as the sum of the numbers that divide into it exactly: 6 = 1 + 2 + 3.)28

The Greeks felt sure that nature shared their fondness for geometry. Aim a beam of light at a mirror, for example, and it bounces off the mirror at the same angle it made on its incoming path. (Every pool player knows that a ball hit off a cushion follows the same rule.)


When light bounces off a mirror, the two marked angles are equal.

What looked like a small observation about certain angles turned out to have a big payoff—of the infinitely many paths that the light beam might take on its journey from a to a mirror to b, the path it actually does take is the shortest one possible. And there’s more. Since light travels through the air at a constant speed, the shortest of all possible paths is also the fastest.

Even if light obeyed a mathematical rule, the rule might have been messy and complicated. But it wasn’t. Light operated in the most efficient, least wasteful way possible. This was so even in less straightforward circumstances. Light travels at different speeds in different mediums, for instance, and faster in air than in water. When it passes from one medium to another, it bends.

Look at the drawing below and imagine a lifeguard at a rather than a flashlight. If a lifeguard standing on the beach at a sees a person drowning at b, where should she run into the water? It’s tricky, because she’s much slower in the water than on land. Should she run straight toward the drowning man? Straight to a point at the water’s edge directly in front of the flailing man?


Light bends as it passes from air into water.

Curiously, this riddle isn’t in the least tricky for light, which “knows” exactly the quickest path to take. “Light acts like the perfect lifeguard,” physicists say, and over the centuries they’ve formulated a number of statements about nature’s efficiency, not just to do with light but far more generally. The eighteenth-century mathematician who formulated one such principle proclaimed it, in the words of the historian Morris Kline, “the first scientific proof of the existence and wisdom of God.”

Light’s remarkable behavior was only one example of the seventeenth century’s favorite discovery, that if a mathematical idea was beautiful it was virtually guaranteed to be useful. Scientists ever since Galileo and Newton have continued to find mysterious mathematical connections in the most unlikely venues. “You must have felt this, too,” remarked the physicist Werner Heisenberg, in a conversation with Einstein: “the almost frightening simplicity and wholeness of the relationships which nature suddenly spreads out before us and for which none of us was in the least prepared.”

For the mathematically minded, the notion of glimpsing God’s plan has always exerted a hypnotic pull. The seduction is twofold. On the one hand, delving into the world’s mathematical secrets gives a feeling of having one’s hands on nature’s beating heart; on the other, in a world of chaos and disaster, mathematics provides a refuge of eternal, unchallengeable truths and perfect order.

The intellectual challenge is immense, and the difficulty of the task makes the pursuit even more obsessive. In Vladimir Nabokov’s novel The Defense, Aleksandr Luzhin is a chess grand master. He speaks of chess in just the way that mathematicians think of their field. While pondering a move and lighting a cigarette, Luzhin accidentally burns his fingers. “The pain immediately passed, but in the fiery gap he had seen something unbearably awesome—the full horror of the abysmal depths of chess. He glanced at the chessboard, and his brain wilted from unprecedented weariness. But the chessmen were pitiless; they held and absorbed him. There was horror in this, but in this also was the sole harmony, for what else exists in the world besides chess?”

Mathematicians and physicists share that passion, and unlike chess players they take for granted that they are grappling with nature’s deepest secrets. (The theoretical physicist Subrahmanyan Chandrasekhar, a pioneer in the study of black holes, spoke of “shuddering before the beautiful.”) They sustain themselves through the empty years with the unshakable belief that the answer is out there, waiting to be found. But mathematics is a cruel mistress, indifferent to the suffering of those who would woo her. Only those who themselves have wandered lost, wrote Einstein, know the misery and joy of “the years of searching in the dark for a truth that one feels but cannot express; the intense desire and the alternations of confidence and misgiving, until one breaks through to clarity and understanding.”

The abstract truths that enticed Einstein and his fellow scientists occupy a realm separate from the ordinary world. That gulf between the everyday world and the mathematical one has, many times through the centuries, served as a lure rather than a barrier. When he was a melancholy sixteen-year-old, the modern-day philosopher and mathematician Bertrand Russell recalled many years later, he used to go for solitary walks “to watch the sunset and contemplate suicide. I did not, however, commit suicide, because I wished to know more of mathematics.”

A deep dive into mathematics has special appeal, for it serves at the same time as a way to flee the world and to impose order on it. “Of all escapes from reality,” the mathematician Gian-Carlo Rota observed, “mathematics is the most successful ever. . . . All other escapes—sex, drugs, hobbies, whatever—are ephemeral by comparison.” Mathematicians have withdrawn from the dirty, dangerous world, they believe, and then, by thought alone, they have added new facts to the world’s store of knowledge. Not just new facts, moreover, but facts that will stand forever, unchallengeable. “The certainty that [a mathematician’s] creations will endure,” wrote Rota, “renews his confidence as no other pursuit.” It is heady, seductive business.

Perhaps this accounts for the eagerness of so many seventeenth-century intellectuals to look past the wars and epidemics all around them and instead to focus on the quest for perfect, abstract order. Johannes Kepler, the great astronomer, barely escaped the religious battles later dubbed the Thirty Years’ War. One close colleague was drawn and quartered and then had his tongue cut out. For a decade his head, impaled on a pike, stood on public display next to the rotting skulls of other “traitors.”

Kepler came from a village in Germany where dozens of women had been burned as witches during his lifetime. His mother was charged with witchcraft and, at age seventy-four, chained and imprisoned while awaiting trial. She had poisoned a neighbor’s drink; she had asked a grave digger for her father’s skull, to make a drinking goblet; she had bewitched a villager’s cattle. Kepler spent six years defending her while finishing work on a book called The Harmony of the World. “When the storm rages and the shipwreck of the state threatens,” he wrote, “we can do nothing more worthy than to sink the anchor of our peaceful studies into the ground of eternity.”