Zero

by Charles Seife

  Before Arabic numerals came around, money counters had to make do with an abacus or a counting board. The Germans called the counting board a Rechenbank, which is why we call moneylenders banks. At that time, banking methods were primitive. Not only did they use counting boards, they used tally sticks to record loans: a money value was written along the stick’s side, and it was split in two (Figure 16). The lender kept the biggest piece, the stock. After all, he was the stockholder.*

  Figure 16: A tally stick

  Italian merchants loved the Arabic numbers. They allowed the bankers to get rid of their counting boards. However, while businessmen saw their usefulness, the local governments hated them. In 1299, Florence banned Arabic numerals. The ostensible reason was that the numbers were easily changed and falsified. (A 0 could be turned into a 6 with a simple flourish of a pen, for instance.) But the advantages of zero and the other Arabic numerals were not so easily dispensed with; Italian merchants continued to use them, and even used them to send encrypted messages—which is how the word cipher came to mean “secret code.”

  In the end the governments had to relent in the face of commercial pressure. The Arabic notation was allowed into Italy and soon spread throughout Europe. Zero had arrived—as had the void. The Aristotelian wall was crumbling, thanks to the influence of the Muslims and the Hindus, and by the 1400s even the staunchest European supporters of Aristotelianism had their doubts. Thomas Bradwardine, who was to become archbishop of Canterbury, tried to disprove atomism, Aristotle’s old nemesis. At the same time, he wondered whether his own logic was faulty, since he based his arguments on geometry, whose infinitely divisible lines automatically reject atomism. However, the battle against Aristotle was far from over. If Aristotle were to fall, the proof of God—a bulwark of the church—was no longer valid. A new proof was needed.

  Worse yet, if the universe were infinite, then there could be no center. How could Earth, then, be the center of the universe? The answer was found in zero.

  Chapter 4

  The Infinite God of Nothing

  [ THE THEOLOGY OF ZERO ]

  And new philosophy calls all in doubt,

  The element of fire is quite put out;

  The sun is lost, and th’ earth, and no man’s wit

  Can well direct him where to look for it….’

  Tis all in pieces, all coherence gone;

  All just supply, and all relation:

  Prince, subject, Father, Son, are things forgot.

  —JOHN DONNE, “AN ANATOMY OF THE WORLD”

  Zero and infinity were at the very center of the Renaissance. As Europe slowly awakened from the Dark Ages, the void and the infinite—nothing and everything—would destroy the Aristotelian foundation of the church and open the way to the scientific revolution.

  At first the papacy was blind to the danger. High-ranking clergymen experimented with the dangerous ideas of the void and the infinite, even though the ideas struck at the core of the ancient Greek philosophy that the church cherished so much. Zero appeared in the middle of every Renaissance painting, and a cardinal declared that the universe was infinite—boundless. However, the brief love affair with zero and infinity was not to last.

  When the church was threatened, it retreated into its old philosophy, turning back toward the Aristotelian doctrine that had supported it for so many years. It was too late. Zero had taken hold in the West, and despite the papacy’s objections, it was too strong to be exiled once more. Aristotle fell to the infinite and to the void, and so did the proof of God’s existence.

  Only one option remained for the church: accept zero and infinity. Indeed, to the devout, God could be found, hidden within the void and the infinite.

  The Nutshell Cracked

  O God, I could be bounded in a nutshell and count myself a king of infinite space, were it not that I have bad dreams.

  —WILLIAM SHAKESPEARE, HAMLET

  At the beginning of the Renaissance, it was not obvious that zero would pose a threat to the church. It was an artistic tool, an infinite nothing that ushered in the great Renaissance in the visual arts.

  Before the fifteenth century, paintings and drawings were largely flat and lifeless. The images in them were distorted and two-dimensional; gigantic, flat knights peered out of tiny, misshapen castles (Figure 17). Even the best artists could not draw a realistic scene. They did not know how to use the power of zero.

  Figure 17: Flat knights and misshapen castles

  It was an Italian architect, Filippo Brunelleschi, who first demonstrated the power of an infinite zero: he created a realistic painting by using a vanishing point.

  By definition, a point is a zero—thanks to the concept of dimension. In everyday life you deal with three-dimensional objects. (Actually, Einstein revealed that our world is four-dimensional, as we shall see in a later chapter.) The clock on your dresser, the cup of coffee you drink in the morning, the book you’re reading right now—all these are three-dimensional objects. Now imagine that a giant hand comes down and squashes the book flat. Instead of being a three-dimensional object, the book is now a flat, floppy rectangle. It has lost a dimension; it has length and width, but no height. It is now two-dimensional. Now imagine that the book, turned sideways, is crushed once again by the giant hand. The book is no longer a rectangle. It is a line. Again, it has lost a dimension; it has neither height nor width, but it has length. It is a one-dimensional object. You can take away even this single dimension. Squashed along its length, the line becomes a point, an infinitesimal nothing with no length, no width, and no height. A point is a zero-dimensional object.

  In 1425, Brunelleschi placed just such a point in the center of a drawing of a famous Florentine building, the Baptistery. This zero-dimensional object, the vanishing point, is an infinitesimal dot on the canvas that represents a spot infinitely far away from the viewer (Figure 18). As objects recede into the distance in the painting, they get closer and closer to the vanishing point, getting more compressed as they get farther away from the viewer. Everything sufficiently distant—people, trees, buildings—is squashed into a zero-dimensional point and disappears. The zero in the center of the painting contains an infinity of space.

  This apparently contradictory object turned Brunelleschi’s drawing, almost magically, into such a good likeness of the three-dimensional Baptistery building that it was indistinguishable from the real thing. Indeed, when Brunelleschi used a mirror to compare the painting and the building, the reflected image matched the building’s geometry exactly. The vanishing point turned a two-dimensional drawing into a perfect simulation of a three-dimensional building.

  Figure 18: The vanishing point

  It is no coincidence that zero and infinity are linked in the vanishing point. Just as multiplying by zero causes the number line to collapse into a point, the vanishing point has caused most of the universe to sit in a tiny dot. This is a singularity, a concept that became very important later in the history of science—but at this early stage, mathematicians knew little more than the artists about the properties of zero. In fact, in the fifteenth century, artists were amateur mathematicians. Leonardo da Vinci wrote a guide to drawing in perspective. Another of his books, about painting, warns, “Let no one who is not a mathematician read my works.” These mathematician-artists perfected the technique of perspective and could soon depict arbitrary objects in three dimensions. No longer would artists be restricted to flat likenesses. Zero had transformed the art world.

  Zero was, quite literally, at the center of Brunelleschi’s painting. The church, too, dabbled with zero and the infinite, though church doctrine was still dependent on Aristotelian ideas. A contemporary of Brunelleschi, a German cardinal named Nicholas of Cusa, looked at infinity and promptly declared, “Terra non est centra mundi”: the earth is not the center of the universe. The church didn’t yet realize how dangerous, how revolutionary, that idea was.

  One of the old declarations of the medieval Aristotelian doctrine—as strong as the ban o
n the vacuum—was the statement that Earth was unique. It was at the universe’s very center. Earth’s special position at the center of the universe made it the only world capable of containing life, as Aristotle held that all objects sought out their proper place. Heavy objects, like rocks or people, belonged on the ground; light objects, like air, belonged in the heavens. Not only did this imply that the planets—in the heavens—were made of light, airy stuff, but it also meant that any people in the heavens would naturally fall to Earth. Thus creatures could only inhabit the nutmeat in the center of the nutshell cosmos. Having other planets with life on them was as silly as having a sphere with two centers.

  When Tempier declared that the omnipotent God could create a vacuum if he so desired, Tempier insisted that God could break any Aristotelian law. God could create life on other worlds if he wished. There could be thousands of other Earths, each teeming with creatures; it was certainly within God’s power, whether Aristotle agrees or not.

  Nicholas of Cusa was bold enough to say that God must have done so. “The regions of the other stars are similar to this,” he said, “for we believe that none of them is deprived of inhabitants.” The sky was littered with an infinite number of stars. The planets glowed in the heavens; the moon and the sun each glowed with light. Why couldn’t the stars in the sky be planets or moons or suns on their own? Maybe Earth glows brightly in their heavens, just as they glow in ours. Nicholas was sure that God had, indeed, created an infinite number of other worlds. Earth was no longer at the center of the universe. Yet Nicholas was not declared a heretic, and the church didn’t react to the new idea.

  In the meantime another Nicholas turned Cusa’s philosophy into a scientific theory. Nicolaus Copernicus showed that Earth is not the center of the universe. It revolves around the sun.

  A Polish monk and a physician, Copernicus learned mathematics so he could cast astrological tables, the better to cure his patients with. Along the way, Copernicus’s dabblings with the planets and stars showed him how complicated the old Greek system of tracking the planets was. Ptolemy’s clockwork heavens—with Earth at the center—were extremely accurate. However, they were terribly complex. Planets course around the sky throughout the year, but every so often they stop, move backward, and then shoot ahead once more. To account for the planets’ bizarre behavior, Ptolemy added epicycles to his planetary clockwork: little circles within circles could explain the backward, or retrograde, motion of the planets (Figure 19).

  The power of Copernicus’s idea was in its simplicity. Instead of placing Earth at the center of the universe filled with epicycle-filled clockworks, Copernicus imagined that the sun was at the center instead, and the planets moved in simple circles. Planets would seem to zoom backward as Earth overtook them; no epicycles were needed. Though Copernicus’s system didn’t agree with the data completely—the circular orbits were wrong, though the heliocentric idea was correct—it was much simpler than the Ptolemaic system. The earth revolved around the sun. Terra non est centra mundi.

  Nicholas of Cusa and Nicolaus Copernicus cracked open the nutshell universe of Aristotle and Ptolemy. No longer was the earth comfortably ensconced in the center of the universe; there was no shell containing the cosmos. The universe went on into infinity, dotted with innumerable worlds, each inhabited by mysterious creatures. But how could Rome claim to be the seat of the one true Church if its authority could not extend to other solar systems? Were there other popes on other planets? It was a grim prospect for the Catholic Church, especially since it was beginning to have trouble with its subjects on even its own world.

  Copernicus published his magnum opus on his deathbed—in 1543, just before the church started clamping down on new ideas. Copernicus’s book, De Revolutionibus, was even dedicated to Pope Paul III. However, the church was under attack. As a result, the new ideas—the questioning of Aristotle—could no longer be tolerated.

  The attack on the church began in earnest in 1517, when a constipated German monk nailed a list of complaints to the door of the church in Wittenberg. (Luther’s constipation was legendary. Some scholars believe that his great revelation about faith came to him when he was sitting on the privy. “Luther’s release from the constricting bondage of fear corresponded to the release of his bowels,” notes one text, commenting on this theory.) This was the beginning of the Reformation; intellectuals everywhere began to reject the authority of the pope. By the 1530s, in a quest to ensure an orderly succession to the throne, Henry VIII had spurned the authority of the pope, declaring himself the head of the English clergy.

  Figure 19: Epicycles, retrograde motion, and heliocentrism

  The Catholic Church had to strike back. Though it had been experimenting with other philosophies for several centuries, when threatened with schism it turned orthodox once again. It fell back upon its orthodox teachings—the Aristotelian-based philosophies of scholars like Saint Augustine and Boethius, as well as Aristotle’s proof of God. No longer could cardinals and clerics question the ancient doctrines. Zero was a heretic. The nutshell universe had to be accepted; the void and the infinite must be rejected. One of the key groups that spread these teachings was founded in the 1530s: the Jesuit order, a collection of highly trained intellectuals well suited to attack Protestantism. The church had other tools to fight heresy as well; the Spanish Inquisition started burning Protestants in 1543, the same year Copernicus died and the same year that Pope Paul III issued the Index of forbidden books. The Counter-Reformation was the church’s attempt to rebuild the old order by crushing the new ideas. An idea embraced by Bishop Étienne Tempier in the thirteenth century and Cardinal Nicholas of Cusa in the fifteenth century could mean a death sentence in the sixteenth century.

  This is what happened to the unfortunate Giordano Bruno. In the 1580s, Bruno, a former Dominican cleric, published On the Infinite Universe and Worlds, where he suggested, like Nicholas of Cusa, that the earth was not the center of the universe and that there were infinite worlds like our own. In 1600 he was burned at the stake. In 1616 the famous Galileo Galilei, another Copernican, was ordered by the church to cease his scientific investigations. The same year, Copernicus’s De Revolutionibus was placed on the Index of forbidden books. An attack on Aristotle was considered an attack upon the church.

  Despite the church’s Counter-Reformation, the new philosophy wasn’t easily destroyed. It got stronger as time went on, thanks to the investigations of Copernicus’s successors. In the beginning of the seventeenth century, another astrologer-monk, Johannes Kepler, refined Copernicus’s theory, making it even more accurate than the Ptolemaic system. Instead of moving in circles, the planets, including Earth, moved in ellipses around the sun. This explained the motion of the planets in the heavens with incredible accuracy; no longer could astronomers object that the heliocentric system was inferior to the geocentric one. Kepler’s model was simpler than Ptolemy’s, and it was more accurate. Despite the church’s objections, Kepler’s heliocentric system would prevail eventually, because Kepler was right and Aristotle was wrong.

  The church attempted to patch the holes in the old way of thought, but Aristotle, the geocentric world, and the feudal way of life were all mortally wounded. Everything that philosophers had taken for granted for millennia was called into doubt. The Aristotelian system could not be trusted, and at the same time it could not be rejected. What, then, could be taken for granted? Literally nothing.

  Zero and the Void

  I am in a sense something intermediate between God and nought.

  —RENÉ DESCARTES, DISCOURSE ON METHOD

  Zero and the infinite were at the very center of the philosophical war taking place during the sixteenth and seventeenth centuries. The void had weakened Aristotle’s philosophy, and the idea of an infinitely large cosmos helped shatter the nutshell universe. The earth could not be at the center of God’s creation. As the papacy lost its hold on its flock, the Catholic Church tried to reject zero and the void more strongly than ever, yet zero had already t
aken root. Even the most devout intellectuals—the Jesuits—were torn between the old, Aristotelian ways and the new philosophies that included zero and the void, infinity and the infinite.

  René Descartes was trained as a Jesuit, and he, too, was torn between the old and the new. He rejected the void but put it at the center of his world. Born in 1596 in the middle of France, Descartes would bring zero to the center of the number line, and he would seek a proof of God in the void and the infinite. Yet Descartes could not reject Aristotle entirely; he was so afraid of the void that he denied its existence.

  Like Pythagoras, Descartes was a mathematician-philosopher; perhaps his most lasting legacy was a mathematical invention—what we now call Cartesian coordinates. Anyone who has taken geometry in high school has seen them: they are the sets of numbers in parentheses that represent a point in space. For instance, the symbol (4, 2) represents a point four units to the right and two units upward. But to the right and upward of what? The Origin. Zero (Figure 20).

  Descartes realized that he could not start his two reference lines, or axes, with the number 1. That would lead to an error like the one Bede encountered when revamping the calendar. However, unlike Bede, he lived in a Europe where Arabic numerals were common, so he started counting with zero. At the very center of the coordinate system—where the two axes cross—sits a zero. The origin, the point (0, 0), is the foundation of the Cartesian system of coordinates. (Descartes’s notation was slightly different from what we use today. For one thing, he didn’t extend his coordinate system to the negative numbers, though his colleagues would soon do that for him.)